XMaths (ENG)300__399.pmd

(300) (Maths__Xth class)

Real Numbers

Key Points

1. Euclid's division lemma :-

For given positive integers ‘a’ and ‘b’ there exist unique whole numbers ‘q’ and ‘r’ satisfyingthe relation a = bq + r, 0 ≤ r < b.

2. Euclid’s division algorithms :

HCF of any two positive integers a and b. With a>b is obtained as follows :

Step : Apply Euclid’s division lemma to a and b to find q and r such that

a = bq + r. 0 ≤ r > b

Step 2 : If r = 0, HCF (a, b), = b if r ≠ 0, apply Euclid’s lemma to b & r

3. The Fundamental Theorem of Arithmetic :

Every composite number can be expressed (factorized) as a product of primes and thisfactorizationi is unique, apart from the order in which the prime factors occur.

4. Let x = p

q , q ≠ 0 to be a rational number, such that the prime factorization of ‘q’ is of the

form 2m5n, where m, n are non-negative integers. Then x has a decimal expansion which isterminating.

5. Let x = p

q , q ≠ 0 be a rational number, such that the prime factorization of q is not of the

form 2m5n, where m, n are non-negative integers. Then x has a decimal expansion which isnon-terminating repeating.

6. √p is irrational, which p is a prime. A number is called irrational if it cannot be written in

the form p

q where p and q are integers and q ≠ 0

Real Numbers

Questions

1 Mark questions :

1. A number, which we can not write in the form of p

q where p and q are integers and q ≠ 0

then write, what we call this number.

2. How many prime numbers are there between 1 and 10.

3. Write whether rational number 29343

has a terminating decimal expansion or non-

terminating repeating decimal.

4. Write the L.C.M. of the numbers 60 and 72.


5. Find HCF × LCM for the numbers 100 and 190.

6. Which of the following rational numbers have terminating decimal expansion —

37900

432500

28720

, ,

7. Express 0.375 in the form of p

q to is lowest term.

8. Write the H.C.F. of the numbers 3.2252 and 32235.

9. Express 0.3 in the form of p

q

10. Find the L.C.M. of the numbers 2332 and 2233.

11. Fill in teh blank of the following

945 = 33 × × 7

12. Write decimal representation of the rational number 46

2 53 2

13. If x = 5 + √3 and y = 5 – √3. Write whether sum of two irrational number x and y is arational or irrational number.

14. Whether the product of two irrational numbers (2+√5) and (2–√5) is rational or irrationalnumber.

15. The L.C.M. and H.C.F. of two numbers are 180 and 6 respectively. If one of the number is30. Write the other number.

2 Marks questions –

16. State fundamental theorem of the Arithmetic.

17. Write two irrational numbers between 1 and 2.

18. Write two rational numbers between 12

and 23

.

19. Decimal expansion of two real numbers is given as (i) 0.202002000 .... (ii) 3.333..... Statewhether they are rational or irrational numbers.

20. Using Euclid’s division alogrithm. Find HCF of 135 and 225.

21. Find H.C.F. and L.C.M. of numbers 40, 70 and 90.

22. State Euclid’s division Lemma.

23. Find x and y in the following diagram.

X

Y2

3 5


24. Explain why 7 × 11 × 13 + 13 is a composite.

25. Find the largest number whether divides 245 and 1029 leaving remainder 5 in each case.

26. An army group of 308 members is to march behind an army band of 24 numbers in aparade. The two groups are to march in the same number of columns. What is the maximumnumber of column is which they can march.

3 Marks Questions –

27. Prove that √5 is an irrational number.

28. Prove that 2–5√3 is an irrational number.

29. Find the L.C.M. and H.C.F. of the numbers 306 and 657 and veryfy that “L.C.M. × H.C.F.= Pruduct of two numbers.”

30. Find the HCF of 867 and 255; by using Euclid’s division alogrithm.

31. Show that 8n cannot end with the digit ‘O’ for any natural number n.

32. Divide x4 – 3x2 + 4x + 5 by x2 – x + 1 and verify the division alogrithm.

33. The length, breath and height of a room are 8m 25 cm, 6m 75 and 4 m 50 cm respectively.Determine the longest rod wich can measure the three dimensions of the room exactly.

34. Find the largest number that will divide 398, 436 and 540 leaving remainder 7, 11 and 13respectively.

35. Find two rational and two irrational numbers between √2 and √3.

Answers

1. Irrational number

2. 4

3. Non-terminating

4. 360

5. 19000

6.43

2500

7.38

8. 24

9.79

10. 23.33

11. 5

12. 0.230

13. Rational


14. Rational

16. 36

17. Non terminating recurring decimal

(1> and <2)

18.12

pq

23

, , ; but q ≠ 0

19. (a) Irrational (b) Rational

20. 45

21. 2520

23. x = 30; y = 15

24. Hint : 13 (7×11+ 1)

13 is a factor except + 1

26. 16

26. 4

29. 22338, 9

30. 51

32. Q = x2 + x – 3

Remainder = 8

33. 75

34. 17

35. Hint √2 = 1.414..... and √3 = 1.732

We can take two rational number between √2 and √3

e.g. 1.5 = 32

, 1.6 = 85

For irrational number 2<2.1<2.2<3

so √2 < √2.1< √3

Polynomials


Key Poins

1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomialsrespectively.

2. A quadratic polynomial in x with real coefficient is of the form ax2 + bx + c, where a, b, care real number with a ≠ o.

3. The zeroes of a polynomial p(x) are precisely the x - coordinates of the points where thegraph of y = p(x) intersectes of the x-axis i.e. x = a is a zero of polynomial p(x) if p (a) = 0.

4. A polynomial can have at most the same number zeros as the degree of polynomial.

5. For quadriatic polynomial ax2 + bx + c (a ≠ 0)

Sum of roots = – ba

Product of roots = ca

6. The division alogrithm states that given any polynomial p(x) and polynomial g(x), thereare polynomials q(x) and r(x) such that :-

p(x) = g(x).q (x) + r(x), g(x) ≠ 0

wether r(x) = 0 or degree of r(x) < degree of g(x)

Polynomials

1 Mark Questions - (Q. No. 10 under HOTS)

1. The graph of y = p(x) is shown in figure write the number of zeroes of p(x)

y

y1

xx1

2. If x = 2 is zero of polynomial, x2 – 3x + k write the value of k.

3. If α, β are zeroes of quadratic polynomial 2x2 + 5x + 10. Write the value of α + β.

4. Write the degree of polynomial 2x4x3 + 5x7 + 2

5. Write the product of zeroes of the quadratic polynomial 2x2 – 2√2x + 1


6. Write the polynomial p(x) whose zeroes are – 1 and 2.

7. Write the quadratic polynomial the product and sum of zeroes are 3 and –5.

8. How many maximum zeroes can be polynomial of degree three have.

9. For what value of k, (–4) is zero of polynomial x2 –x – (2k+2)

10*. Write the zeroes of the polynomial 15x2 –x–6.

2 Marks question (Q. No. 18 under HOTS)

11. Find the zeroes of following quadratic polynomials and verify the relation between thezeroes and the coefficient of the polynomials

(a) p(x) = x2+7x + 10

(b) q(x) = 2x2+5x+3

(c) p(x) = 6x2 – 3 – 7x

(d) q(s) = s2 – 3

12. Find the quadratic polynomial whose zeroes are 3+√2 and 3–√2

13. Find the zeroes of quadric polynomial x2+4√2x+6

14. Find the quadratic polynomial whose zeroes are 2 and –35

15. Find the zeroes of polynomeal p(x) = x2–2.

16. Find the quadratic polynomial whose zeroes are 23

and 14

.

17. Find the quadratic polynomial whose product and sum of zeroes are –7, and –3.

18*. If α, β are the zeroes of the polynomial p(x) = 2x2 – 7x +3. Find the value of α2 + β2.

3 marks questions (Q. No. 19, 23 and 26 under HOTS)

19*. Find all the zeroes of 2x4 – 3x3 – 3x2 –2. If it is given that two of its zeroes are √2 and –√2.

20. Divide 4x4 + 12x3 – x2 –3x + 4 by 2x2 + 5x – 3 and veryfy the division algorithm.

21. Find the value of p for which the polynomial x3 + 4x2 – px + 8 is exactly divisible by x–2.

22. If x+a is a factor of 2x2 + 2ax 10 find the value of a.

23*. Find all zeroes of the polynomial x4 + x3 – 7x2 – 5x + 10. If its two zeroes are √5 and –√5.

24. Find the quadratic polynomial sum of whose zeroes is 8 and product is 12. Hence find thezeroes of the polynomial.

25. Using division algorithm find quotient and remainder on dividing p(x) by g(x) if

(a) p(x) = 2(x)3 + 3x2 – 5x+6, g(x) = 2x–3

(b) p(x) = x3 + 4, g (x) = x+1

26*. If two zeroes of the polynomial x4 – 6x3 – 26x2 + 138 x – 35 are 2 ± √3. Find other zeroes.


Answers

1. 3

2. k = 2

3.– 5

2

4. 7

5.12

6. x2 – x – 2

7. x2 + 3x + 5

8. Three

9. k = 9

10. 2/3, –3/5

11. (a) (–2, 5) (b) (–1, –3/2)

(e) (–13

, 32

) (d) ±√3

12. x2 – 6x + 7

13. –√2, –3√2

15. ± √2

16. 12x2 – 5x – 2

17. x2 + 3x + 7

18. 37/4

19. √2, –√.2, 12

, 1

20. (2x2 + 5x –3) (2x2 + x) + 4

21. p = 16

22. a = 2

23. 2.1, √5, –√5

24. x2 – 8x + 12, zeroes 6, 2

25. (a) x2 + 3x + 2, 0

(b) x2 – x + 1, 3

26. –5, 7


Pair of Linear Equation in two variable

Key points

1. The most general form of a pair of linear equations is :

a1x + b

1y + c

1 = 0

a2x ± b

2y + c

2 = 0

Where a1, a

2, b

1, b

2, c

1, c

2 are real numbers and a

12 + b

12 ≠ 0, a

22+b2

2 ≠ 0

2. The graph of a pair of linear equations in two variables is represented by two lines ;

(i) If the lines intersect at a point, the pair of equations is consistent. The point ofintersection gives the unique solution of the equation.

(ii) If the lines coincide, then there are infinitely many solutions. The pair of equationsis consistent. Each point on the line will be a solution.

(iii) If the lines are parallel, the pair of the linear equations has no solution. The pairof linear equations is inconsistent.

3. If a pair of linear equations is given by a1x + b

1y + c

1 = 0 and a

2x + b

2y + c

2 = 0

(i) a

a1

2

b

b1

2

≠ the pair of linear equations is consistent. (Unique solution)

(ii) a

a1

2

b

b1

2

= ≠c

c1

2

the pair of linear equations is inconsistent (No solution)

(iii) a

a1

2

b

b1

2

= =c

c1

2

the pair of linear equations is dependent & consistent (infinitely

many solutions).

Pair of linear equation in two variables

1 Mark question

1. Express y is terms of x in the equation.

3x – y = 5

2. For what value of k the pair of linear equation ?

kx – 2y = 3

3x + y = 5 has unique solutions

3. For what value of m the pair of linear equation and represent parallel lines ?

3x + my – 8 = 0

3x – 5y + 7 = 0

4. For what value of k the following pair of linear equation

2x + 3y = 7

4x + ky = 14 has infinite many solutions.


5. Write the condition for which pair of linear equations.

a1x + b

1y + c

1 = 0

a2x + b

2y + c

2 = 0 has no solution

6. The difference between two number is 36. One number is four times of other. Form pair oflinear equation of this word problem.

7. How many solution of the equation 5x – 4y + 6 = 0 are possible.

8. Write value of x if :

x + y = 5

x – y = 3

2 Marks Questions (Q. No. 17, 18 and 19 under HOTS)

Solve for x and y (qn. 9-15)

9. x – 4y = 13

3x + 2y + 3 = 0

10.x + 1

2 +

y + 12

= 0

11. 4x – 3y – 8 = 0

6x – y –293

= 0

12.2x

+ 3y = 13

5x

– 4y = – 2

13. 31x + 43y = 117

43x + 31y = 105

14. 3x + 4y = 14

2x + 1y = 1

15.1

2x +

13y = 2

13x

+ 1

2y = 136

16. For what value of p will be the following pair of linear equations have unique solutions.

3x = 4 – 2y

y = 3 – px


17*. Solve for x and y, by cross multiplication method

ax + by + a = 0

bx + ay + b = 0

Solve for x and y

18*. 3(x+y) = 7xy

3(x+3y) = 11xy

19*.xa

+ yb

= 2

ax – by = a2–b2

20.44

x + y + 30

x – y = 10

55x + y +

40x – y = 13

3 marks questions (Q. No. 28, 31, 32, 33 and 35 under HOTS)

21. Gaphically show that the system of linear equation

4x + 6y - 10 = 0

2x + 3y + 13 = 0 has no solution

22. Determine graphically whether the system of linear equation

3x + 2y = 5

3x – y = 2 has unique solution.

23. Show graphically that the following linear equations have infinite solution

2y = 4x – 6

2x = y + 3

24. Solve graphically for x and y, 2x – y = 4, x + y + 1 = 0

Find the points of x-axis where the lines intersect.

25. A number consists of two digits whose sum is 9. If 27 is added to the number the digit arereversed. Find the number.

26. The ratio of income of A and B is 9:7 and the ratiio of their expenditure is 4:3 if each ofthem saves Rs. 2000 yearly. Find their annual income.

27. A fraction become 12

when 1 is substracted from numerator and 2 is added in denominator.

It becomes 13

when 7 is substracted from numerator and 2 is substracted from denominator.

Find the fraction.

28*. A person travels 600 km partly by train and partly by car. He takes 8 hours, if he travels


120 km. by train and rest by car. He takes 20 minutes longer if he travels 200 km by trainand the rest by the car. Find the speed of the train and the car separately.

29. The taxi charges in a city comporised of a fixed charge for 1st km. together with the chargefor distance covered. For a journey of 15 km the charge paid is Rs. 115 and for a journey of27 km the charged for paid is Rs. 199. What a person has to pay for a distance of 50 km.

30. Place A and B are 80 km a part from each other on a high way. A car starts from A and otherfrom B at the same time. If they move in same direction they meat in 8 hours. If they movein opposite direction they meet in 1 hour 20 minutes. Find the speed of the cars.

31*. Solve the following pair of linear equations.

px + qx = p – q

qx – py = p + q

32*. The students of a class are made to stand in rows. If 4 students are extra in a row, therewould be 2 rows less. If 4 students are less in a row there would be 4 more rows. Find thenumber of students be in the class.

33*. Solve for x and y

axb

– bya

= a + b

ax – by = 2ab

34. A father’s age is thrice the sum of ages of two children. After five year his age wil be twicethe sum of children’s ages. How old is father at present ?

35*. Sum of two numbers is 16 and the sum of their reciprocals is 13

. Find these numbers.

36. A boat goes 16 km. upstream and 24 km. down stream in 6 hours. Also it covers 12 kmupstream and 36 km down stream in the same time. Find the speed of boat in still waterand that of the stream.

37. 8 men and 12 boys can finish a piece of work in 5 days, while 6 men and 8 boys can finishit in 7 days. Find the time taken by 1 man alone and that by 1 boy alone to finish the samework.

Answers

1. y = 3x – 5

2. k ≠ – 6

3. m = – 5

4. k = 6

5.a

a1

2

= b

b1

2

+ c

c1

2

6. x – y = 36

x – 4y = 0

7. Infinite solution


8. x = 4

y = 1

9. x = 1 y = – 3

10. x = 7, y = 13

11. x = 3/2 y = –2/3

12. x = 12

y = 13

13. x = 1, y = 2

14. x = –2 y = 1/5

15. x = 12

y 13

16. p ≠ 3/2

17. x = – 1 y = 0

18. x = 1 y = 3/2

19. x = a y = b

20. x = 8 y = 3

25. Number = 36

26. (a) Rs. 18000 (b) Rs. 1400

27.1516

28. Speed of train 60 km/h

30. Speed of car = 80 k/b

31. x = 1 y = –1

32. Number of student 96

Here let no. of rows = y

No. of students = x

Total students = xy

(x–2)(x+4) = xy

(x+4)(x–4) = xy

33. x = b y –a

34. 45 years

35. 12 and 4

36. Speed of boat = 8 km/b

Speed of stream = 4km/b

37. Man - 70 days, Boys - 140 days


Quadratic Equation

Key Points

1. The equation ax2 + bx + c = 0, a ≠ 0 is the standard form of a quadratic equation, where a,b, c are real numbers.

2. A real number α is said to be a root of the quadratic equation ax2 + bx + c = 0. If 9x2 + bx+ c = 0, the zeroes of the quadratic polynomial ax2 + bx + c and the roots of the quadraticequation ax2 + bx + c = 0 are the same.

3. If we can factorize ax2 + bx + c = 0, a ≠ 0 into a product of two linear factors, then the rootsof the quadratic equation ax2 + bx + c = 0 can be found by equating each factors to zero.

4. A quadratic equation can also be solved by the method of completing the square.

5. A quadratic formula : the roots of a quadratic equation ax2 + bx + c = 0 are given by

–b ± b – 4ac

2a

√ 2

provided that b2 – 4ac ≥ 0

6. A quadratic equation ax2 + bx + c = 0 has :-

(i) Two distinct and real roots if b2 – 4ac > 0

(ii) Two equal and real roots, if b2 – 4ac = 0

(iii) Two roots are not real, if b2 – 4ac < 0

Quadratic Equations

Questions

1 Mark questions (Q. No. 9 and 10 under HOTS)

1. The product of two consecutive odd integers is 63. Represent this in form of mathematicalequation.

2. Write the discriminant of the quadratic equation 3x2 – 5x – 11 =0

3. If x = –2 is a root of the equation 3x2 – 5x + 2k = 0. Write the value of k.

4. For what value of k quadratic equation x2 – kx + 4 = 0 has equal roots.

5. Write the nature of the roots of equation 4x2 – 12x + 8 = 0

6. Show that x = – 3 is the solution of the quadratic equation x2 + 6x + 9 = 0

7. Write the value of x in equation x2 – 4 = 0

8. Form a quadratic equation whose roots are –3 and 4.

9*. For what value of m for which x = 2/3 is a solution of mx2 – x – 2 = 0

10*. For what value of p the quadratic equation

2x2 – 6x + p = 0 has real roots.


2 Marks questions (Q, No. 19 and 20 under HOTS)

11. Solve the following quadratic equations

3y2 + (6 + 4a) y + 8a = 0

12. Find the value of a and b such that x = 1, x = – 2 are the solution of the quadratic equation.

x2 + ax + b = 0

13. Find the roots of equation, x + 1x

= 3

14. Find the value of p if equation 2x2 + px + 3 = 0 has two equal roots.

15. Find the value of k for which equation 5kx2 + 8x + 2 = 0 has two equal roots.

16. Find the roots of equation a2b2x2 + (b2 – a2) x – 1 = 0

17. Find the roots of equation √2x2+7x + 5 √2= 0

18. Divide 51 in to two parts sucvh athat their product is 378.

19*. If the roots of the equation (b–c)x2 + (c–a)x + (a–b) = 0 are equal then prove that 2b = a+c

20*. Find k so that equation [k+4]x2 + (k+1)x + 1 = 0 has equal roots.

3 marks question (Q. No. 28, 29 and 30 under HOTS)

21. Solve the following quadratic equation by the method of completing square.

(a) 2x2 –5x + 3 = 0

(b) ax2 + bx + c = 0

22. Solve the following quadratic equation by using quadratic formula.

abx2 + (b2 – ac) x – bc = 0

23. Find the discriminant of the equation 3x2 – 2 x + 13

= 0 and hence find the nature of

roots, find the roots if they are real.

Solve the following equations (24 – 30)

24. p2x2 + (p2 – q2)x –q2 = 0

25.x – 1x – 2

+ x – 3x – 4

= 313

26.x + 1x – 1

+ x – 1x + 1

= 56

, x ≠ 1

25.1x+1

+ 2x+2

= 4x+4

x ≠ –2, –4

26. 2x2 + 5√3x + 6 = 0

27.1x – 2

+ 2x – 1

= 6x

x ≠ 0, 1.2

28*. 3a2x2 + 8abx + 4b2 = 0 a ≠ 0


29*.xx + 1

+ x + 1

x = 1

112

30*.1

a + b + x =

1a

+ 1b

+ 1x

6. Marks questions (Q. No. 31, 34, 35 adn 39 under HOTS)

31*. A two digit number is such that the product of digit is 35, when 18 is added to the numberthe digits interchange their places. Find the number.

32. A train travels 360 km at uniform speed. If the speed had been 5 km/h more it would havetaken 1 hour less for the same journey. Find the speed of the train.

33. Find two numbers whose sum is 27 and product is 182.

34*. A motorboat whose speed is 9 km/h is still water goes 12 km. down stream and comes backin a total time 3 hours. Find the speed of the stream.

35*. The hypotenuse of right angled triangle is 6cm more than twice the shortest side. If thethird side is 2 cm less than the hypotenuse find the sides of the triangle.

36. Sum of two number is 15, if sum of their recipocal is 3

10. Find the numbers.

37. Rs. 9000 were divided equally among a certain number of students. Had there been 20more students, each would have got Rs. 160 less. Find the original number of students.

38. In a class test sum of Kamal’s marks in Mathematics and English is 40. Had he got 3marks more in mathematics and 4 marks less in English, the product of his marks wouldhave been 360. Find his marks in two subject separately.

39*. Solve for x

9x2 – 9 (a+b) x+ (2a2 + 5ab + 2b2) = 0

40. By a reduction of Rs. 2 per kg in the price of sugar. Ram Lal can 2 kg sugar more for Rs. 244.Find the original price of sugar per kg.

41. An aeroplane takes an hour less for a journey of 1200 km. if the speed is increased by 100km/h from its usual speed. Find the usual speed.

Answers

1. (2x – 1)(2x+1) = 63

2. D = 157

3. k = – 11

4. K = = 4

5. D ≥ 0 real number

6. – 3 is solution

7. x = ± √2


8. x2 + x – 12 = 0

9. m = 6

10. p < 92

11. –2, –4a3

12. a = 1, b = –2

13.3 + 5

2

√,

3 – 5

2

√

14. p = ± 2√6

15. k = 8/5

16.–1

a2 ,

1

b2

17.–5 2

2

√, –√2

18. 9, 42

19. – proof

20. (a) 3

2.1

(b) x = – b + b – 4ac

2a

√ 2

x = –b b – 4ac

2a

√ 2

22. c/b, –b/a

23. +13

, 13

24.q

p

2

2 , –1

25. 2+√2, 2–√2

26. –2√3, –√3/2

27. 3, 43

28.–2b

a,

–2b3a

29. –4, 3

30. x = – a, –b


31. 57

32. 40 km/h

33. 13, 14

34. 3 k/h

35. 26 cm, 24 cm, 10 cm

36. 5, 10

37. 25 students

38. Maths - 21 Eng. - 19

Maths - 12 Eng. = 28

39.2a+b

3,

a+2b3

40. Rs. 16

41. 300 km/h


Arithmetic Progression

Key Points

1. Sequence : A set of numbers arranged in some definite order and formed according tosome rules is called a sequence.

2. Progression : The sequence that follows a certain pattern is called progression.

3. Arithmetic progression : A sequence in which the difference obtained by substractingfrom any term its preceeding term is consistent throughout, is called on arithmetic sequenceor arithmetic progression (A.P.)

The general form of an A.P. is a, a+d, a+2d, ....... (a : first term, d : common difference)

4. General Term : if ‘a’ is the first term and ‘d’ is common difference in an A.P., then nth term(general term) is given by

an = a + (n–1)d

5. SUM OF n TERMS OF AN A.P. : If ‘a’ is the first term and ‘d’ is the common difference ofan A.P., then sum of first n terms is given by

Sn =

n2

{ 2a+(n–1)d}

If ‘l’ is the last term of a finite A.P., then the sum is given by

sn =

n2

{a +l}

6. (i) If an is given, then common difference d = a

n –a

n–1

(ii) If sn is given, then nth term is given by a

n = s

n – s

n–1

iii) if a, b, c are in A.P., then 2b = a + c

(iv) If a sequence has n terms, its rth term from the end = (n – r + 1)th term from thebeginning.

1 Mark Questions

1. If nth term of an A.P. is 5 – 3n, write the common difference of this A.P.

2. Which term of the A.P. –7, –3, 1, 5, ............... is 73 ?

3. If 5, 2k–3, 9 are in A.P., then write the value of ‘k’

4. Is –10, a term of the A.P. 25, 22, 19, ....... ?

5. Write 13th term of the A.P. 3, 8, 13 ......

6. Write nth term of the A.P. –5, –2, 1, ......

7. Is 717

, 727

, 737

, ...... A.P. ? If yes, write the common difference.

8. The first term of an A.P. is 3 and sixth term is 23. Write common difference o the A.P.

9. Write the first term and common difference of the A.P. 7.3, 6.9, 6.5 .......

10. Write first three terms of an A.P., whose second term is –4 and common difference is –1.


11. Write the sum of first 10 natural numbers.

12. Is √2, √8, √18, √32 ... an A.P. ? If yes, then write next two terms.

13. Write the missing terms of the A.P.

3, , –1, –3,

14. Write 9th term from the end of the A.P.

7, 11, 15, ..........., 147

15. If the sum of n terms of an A.P. is n2, write its nth term.

16. For what value of ‘m’ the numbers 1

2m,

1m

, m+12m

are in A.P. ? (m ≠ 0)

17. The sum of 6th and 7th terms of an A.P. is 39 and common difference is 3. Write its firstterm.

18. The sum of 3 numbers is A.P. is 30. If the greatest number is 13, write its common difference.

19. Write an A.P. whose third term is 16 and the difference of the 9th term from 11th term is12.

20. Write the sum of first ‘n’ even natural number.

2 Marks Questions (Q. No. 36 to 40 under HOTS)

21. If –9, –14, –19, ............ is an A.P., then find a30

– a20

.

22. Find the A.P. whose second term is 10 and the sixth term exceeds the fourth term by 12.

23. The sum of 3rd and 7th terms of an A.P. is 14 and the sum of 5th and 9th terms is 34. Find thefirst term and common difference of the A.P.

24. Which term of the A.P. 41, 38, 35, ........... is the first negative term ?

25. If the sum of first n terms of an A.P. is 2n2 + n, then find nth term and common difference ofthe A.P.

26. How many terms of A.P. 22, 20, 18......... should be taken so that their sum is zero ?

27. Find the sum of odd positive integers less than 199.

28. If 9 times of 9th term is equal to 8 times the 8th term of an A.P. Find its 17th term.

29. Which term of A.P. 5, 13, 21, 29 ........ will be 48 less than its 19th term.

30. How many two digits numbers between 4 and 102 are divisible by 6 ?

31. Find an A.P. whose 3rd term is –13 and 6th term is 2.

32. The angles of triangle are in A.P. If the smallest angle is one fifth the sum of other twoangles. Find the angles.

33. Nidhi, starts a game and scores 200 points in the first attempt and she increases the pointsby 40 in each attempt. How many points will she score in teh 30th attempt ?

34. Find ‘k’, if the given value of x is the kth term of the A.P. 3, 7, 11, ........., x = 83

35. Anurag saves Re. 1 on day 1, Rs. 2 on day 2, Rs. 3, on day 3 and so on. How much money willhe save in the month of feb. 2010.


36*. Find an A.P. of 8 terms, whose first term is 12

and last term is 176

.

37*. For an A.P. a1, a

2, a

3, ........ if

a

b4

7

= 23

, then find a

a6

8

.

38*. The fourth term of an A.P. is equal to 3 times the first term and the seventh term exceedstwice the third term by 1. Find the first term and the common difference of the A.P.

39*. If 2nd, 31st and last term of an A.P. are 314

, 12

and – 132

respectively. Find the number of

terms in the A.P.

40*. For what value of ‘n’, are the nth terms of two A.P.s 2, 10, 18, ...... and 68, 70, 72 ....... equal? Also find the term.

3 Marks Questions (Q. No. 56 to 60 under HOTS)

41. Find the sum of A.P. 4 + 9 + 14 + .......... + 249

42. If pth and qth term of an A.P. are 1q and

1p respectively, then find the sum of pq terms.

43. Find the sum of the first 40 terms of an A.P., whose nth term is 3 – 2n.

44. If nth term of an A.P. is 4, common difference is 2 and sum of n terms is –14, then find firstterm and number of terms.

45. Find the sum of all the three digits numbers each of which leaves the remainder 3 whendivided by 5.

46. The sum of first six terms of an A.P. is 42. The ratio of the 10th term to the 30th term is1:3. Find first term and 11th term of the A.P.

47. The sum of three numbers in A.P. is 24 and their product is 440. Find the numbers.

48. The sum of n terms of two A.P.’s are in the ratio 3n + 8 : 7n + 15. Find the ratio of their12th terms.

49. The sum of first 16 terms of an A.P. is 528 and sum of next 16 terms is 1552. Find the firstterm and common difference of the A.P.

50. The sum of first 8 terms of an A.P. is 140 and sum of first 24 terms is 996. Find the A.P.

51. If pth, qth and rth terms of an A.P. are l,m and n respectively. then proe that

p(m–n) + q (n–l) + r(l–m) = 0

52 Find the number of terms of the A.P. 57, 54, 51....... so that their sum is 570. Expain thedouble answer.

53. If the sum of first 20 terms of an A.P. is one third of the sum of next 20 terms. If first termis 1, then find the sum of first 30 terms.

54. A picnic group for Manali consists of students whose ages are in A.P., the common diferencebeing 3 months. If the youngest students Rohit is just 12 years old and the sum of ages ofall the students is 375 years. Find the number of students in the group.

55. The digits of a three digits positive number are in A.P. and the sum of digits is 15. On


subtracting 594 from the number the digits are inerchanged, find the number.

56*. If the roots of the equation a(b–c)x2 + b(c–a)x + c (a–b) = 0 are equal, then show that 1a

,

1b

, 1c

are in A.P.

57*. If the sum of m terms of an A.P. is n and the sum of n terms is m, then show that sum of (m+ n) terms is – (m +n).

58*. The sum of 5th and 9th terms of an A.P. is 8 and their product is 15. Find the sum of first 28terms of the A.P.

59*. If mth and nth terms of an A.P. are a & b respectively, then show that the sum of its (m+n)

terms is m + n

2 { a + b +

a – bm –n

}

60*. Anita arranged balls in rows to form an equilateral triangle. The first row consists of oneball, the second of two balls, and so on. If 669 more balls are added, then all the balls can bearranged in the shape of a square and each of its sides then contains 8 ball less than eachside of the triangle. Determine the initial number of balls, Anita has.

Answers

1. –3

2. 21

3. k = 5

4. NO

5. 63

6. 3 n = 8

7. Yes, common difference = 17

8. 4

9. 7.3, common difference = – 0.4

10. – 3, –4, –5.

11. 55

12. Yes, 5√2, 6√2

13. 1 and –5

14. 115

15. 2n–1

16. m = 2

17. 3

18. 7, 10 and 13


19. 4, 10, 16, ........

20. n2 + n

21. – 50

22. 4, 10, 16, ..........

23. First term = – 13, d = 5

24. 15th term

25. nth term = 4n–1, common difference = 4

26. 23 terms

27. 9801

28. Zero

29. 13th term

30. 15

31. – 23, –18, –13, .......

32. 300, 600 and 900

33. 1360

34. k = 21

35. Rs. 406

36.12

, 56

, 76

, ......

37.45

(Hint. a + 3da + 6d

= 23

)

38. First term = 3

Common difference = 2

39. 59.

40. n = 12, term = 90

41. 6325

42.12

(pq + 1)

43. – 1520

44. First term = – 8

total terms = 7

45. 99090

46. First term = 2

11th term = 22

47. 5, 8, 11

48. 7:16


49. First term = 3

common difference = 4

50. 7, 10, 13, .....

51. (hint : an = a + (n–1) d)

52. 19 or 20 (20th term is zero)

53. 450

54. 25 studetns

55. 852

56. (Hint. : in quadratic equation, D = 0)

(for equal roots)

57. Hint : sn =

n2

{2a + (n–1)d}

58. 115, 45, {d = ± 12

}

59. Hint : sn =

n2

{2a +(n–1)d}

60. 1540 balls.

Trignometry


Key Points

1. Trignometrical Ratios :-

In ∆ABC, ∠B = 900 for angle ‘A’

sin A = perpendicular

Hypotenuse

cosA = Base

Hypotenuse

tan A = Perpendicular

Base

cot A = Base

Perpendicular

sec. A = Hypotenuse

Basecosec A =

Hypotenuse

Perpendicular

2. Reciprocal Relations :

sinθ = 1

cosec θ, cosecθ =

1

sin 'θ

cosθ = 1

sec θ, sec θ =

1

cos 'θ

tanθ = 1

cot θ, cotθ =

1

tan 'θ

3. Quotient Relations :

tanθ = Sin

cos 'θ

θ, cot θ =

cos

sin 'θ

θ

4. Identities :

sin2 θ + cos2 θ = 1 ⇒ sin2 θ = 1 – cos2θ and cos2 θ = 1 – sin2θ

1+ tan2 θ = sec2 θ ⇒ tan2θ = sec2 θ – 1 and sec2 θ – tan2 θ = 1

1+cot2 θ = cosec2 θ ⇒ cot2θ = cosec2 θ – 1 and cosec2 θ – cot2 θ = 1

5. TRIGNOMETRIC RATIOS OF SOME SPECIFIC ANGLES :

∠A 00 300 450 600 900

sin A 01

2

1

2√

√3

21

cos A 1√3

2

1

2√1

20

A B

C

Hyp

oten

use

Base

Perp

en

dic

ula

r


tan A 01

√31 √3 Not defined

cosec A Not defined 2 √22

√31

sec A 12

√3√2 2 Not defined

cot A Not defined √3 11

√30

6. Trignometric ratios of complementary angles :

sin (900 –θ) = cosθ

cos(900–θ) = sin θ

tan (900 –θ) = cot θ

cot (900 –θ) = tan θ

sec 900 –θ) = cosec θ

cosec (900 –θ) = sec θ

7. Line of sight :- The line of sight is the line drawn from the eye of an observer to the pointin the object viewed by the observer.

8. Angle of elevation : The angle of elevation is the angle formed by the line of sight with thehorizontal when it is above the horizontal level i.e. the case when we raise our head tolook at the object.

9. Angle of depression : The angle of depression is the angle formed by the line of sight withthe horizontal when it is below the horizontal i.e. case when we lower our head to look atthe object.

1 mark questions

1. Write tan A in terms of sin A

2. In ∆PQR, ∠Q = 900 and sin R = 3

5, what is the value of cos P ?

3. If A and B are acute angles and sin A = Cos B, then write the value of A + B.

4. Write the value of 7sec2620 – 7 cot2280.

5. If sinθ = 1

2, write the value of sinθ + cosecθ.

6. Write the value of sin620sin280–cos620cos280


7. Express cosec770 + tan 620 in terms of trignometrical ratios of angles between 00 and 450.

8. If 4 cot θ = 3, then write the value of tanθ + cot θ

9. If sin θ – cos θ = 0, o0< θ< 900, then write the value of ‘θ’

10. What is the value of sin2410 –cos2490 ?

11. Write the value of cot2300 + sec2450.

12. Write the value of sin2740 + sin2160.

13. If θ = 300, then write the value of sinθ + cos2θ

14 Write the value of sin(900 –θ) cosθ + cos(900 –θ) sinθ.

15. If tan (3x –150) = 1, then write the value of ‘x’.

16. In ∆ABC, write sinA+B

2 in terms of angle ‘C’.

17. Write the value of tan (550 – θ) – cot (350 + θ)

18. If tan θ + cot θ = 5, then what is the value of tan2θ + cot2θ ?

19. If θ = 300, then write the value of 1 – tan2θ

20. If θ = 450, then what is the value of 2cosec2θ + 3sec2θ ?

2 Marks questions (Question No. 36 to 40 under HOTS)

21. If θ = 300, find the falue of 1 – tan

2 θ

1 + tan 2 θ

22. If sin (A+B) = 1 and cos (A–B) = √3

2, 00 ≤ (A+B) ≤ 900, A ≤ B, then find the values of A

and B.

23. If sin2θ = cos(θ – 360), 2θ and θ – 360 are acute angles. Find the value of ‘θ’

24. If θ = 300, then verify; sin3θ = 3sin θ – 4 sin3θ

25. If tan (320 + θ) = cotθ, θ and (320 + θ) are acute angles, find the values of ‘θ’

26. Simplify : tan2600 + 4cos2450 + 3 sec2 300 + 5cos2900

27. If tan θ = √2 – 1, then find the value of

2 tan θ

1 + tan 2 θ

28. If 4 cot θ = 13, find the value of

3 cos + 4sin

5 cos – 3 sin

θ

θ

θ

θ

29. Prove that, sec4 θ – sec2 θ = tan2θ + tan4θ

30. If sin θ + sin2θ = 1, then find the value of cos2θ + cos4θ

31. Find the value of


sin 62

cos 28

0

0 + 3. tan 73

cot 17

0

0 – 5. sin 28 . sec 62

7sec 32 – 7 cot 58

0 0

2 0 2 0


13

5 sin 65

cos25

0

0 – 4

5

cos 53 . cosec. 37

7 sec 32 – 7 cot 58

0 0

2 0 2 0

33. Find the value of sin600 geometrically


cosec ( 90 – ) – tan

4 (cos 40 + cos 50 )

2 0 2

2 0 2 0

θ θ –

2 tan 30 . sec 52 , sin 38

3(cosec 70 – tan 20 )

2 0 2 0 2 0

2 0 2 0

35. If tan (A+B) = √3 and tan(A–B) = 1

3√, 00 ≤ (A+B) ≤ 900, A>B, then find the value of cos

(2A –3B)

36*. Find the value of

sin250 + sin2 100 + sin2150 + sin2200 + ............... + sin2850

37*. In ∆MNR, ∠N = 900, MN = 8cm, RN – MN = 7 cm. Find the value of sinR, tanR and secM.

38*. If sin(A+B) = sinA cosB + cosA sinB, then find the values of sin750 and cos150.

39*. If 2sin (3x –150) = √3, then find the value of sin2 (2x +100) + tan2 (x+50).

40*. If x = m sinα. cosβ, y = m sinα sinβ and z = m cos α,then prove that x2+y2+z2 = m2

3 Marks questions (question No. 56 to 60 under HOTS)

41. Prove that cos A

1 – tanA +

cos A

1 – cotA = sin A cos A


tan (900 –θ) cot θ – sec (900–θ) cosec θ + 3(cot 27 – sec 63 )

cot 26 cot41 cot45 cot49 cot64

2 0 2 0

0 0 0 0

– 2 sec 24 × sin 66

sin 62 + sin 28

0 0

2 0 2 0 + 3 tan2300

43. Prove that

1

cosec A – cot A –

1

sinA =

1

sinA –

1

cosec A – cot A

44. If sec θ + tan θ = 4, then prove that cos θ = 8

17


45. Prove that sec – 1

sec + 1

θ

θ +

sec + 1

sec – 1

θ

θ = 2 cosec θ

46. Prove that (sinθ + cosecθ)2 + (cosθ + secθ)2 = tan2θ + cot2θ + 7

47. Prove that sec – cos + 1

sec – cos – 1

θ

θ

θ

θ =

1 + sin

cos

θ

θ

48. Pove that (1 + cot A – cosec A) (1 + tan A + Sec A) = 2

49. Prove that (1 + 1

tan2θ

) (1 + 1

cot2θ

) = 1

sin – sin2 4θ θ

50. Prove that cos8θ – sin8θ = (cos2θ – sin2θ) (1–2 sin2θ. cos2θ)

51. If a sinA = b cos A and a sin3A + b cos3A = sinAcosA, then prove that a2+b2 = 1

52. Prove that 2(sin6A + cos6A) – 3 (sin4A + cos4A) + 1 = 0

53. If cosθ – sinθ = √2 sinθ, then prove that :

cosθ + sinθ = √2 cosθ

54. If secθ = x + 1

4x, then prove that secθ + tanθ = 2x or

1

2x

55. Prove that cosec A – 1

cosec A + 1 =

cot A – cos A

cot A + cos A

56*. If cos2 α – sin2α = tan2β, then prove that

cos β = 1

cos√2 α

57*. If cosecθ - sinθ = m3 and secθ – cosθ = n3, then prove that m4n2 + m2n4 = 1

58*. If x = tanA + sinA and y = tanA – sinA, then prove that x2 – y2 = 4√xy

59*. If sinα = α sinβ and tan α = b tan β, then prove that

cos2α = a – 1

b – 1

2

2

60*. If sinθ + sin2θ = 1, then prove that

cos12θ + 3cos10 θ + 3 cos8 θ + cos6θ + 2 cos4θ + cos2 θ = 2 + sin2θ

6 MARKS QUESTIONS (Question No. 76 to 80 under HOTS)

61. The shadow of a tower standing on a level ground is found to be 60 m shorter when theSun’s altitude is 600 then when it is 300, find the height of the tower.

62. The angles of elevation of a bird from a point on the ground is 600, after 50 seconds flightthe elevation changes to 300. If the bird flying at the height 500√3m. Find the speed of thebird.

63. From a point on the ground the angles of elevation of the bottom and the top of a watertank kept at the top of 30 m. high building are 450 and 600 respectively. Find the height of


the water tank.

64. A tree breaks due to storm and the broken part bends so that the top of the tree touchesthe ground making an angle 600 with the ground. The distance from the foot of the tree tothe point where the top touches the ground is 5 m. Find total height of the tree.

65. From a window (20 m high above the ground) o a house in a street. The angles of elevationand depression of the top and the foot of an other house opposite side of street are 600 and450 respectively. Find the height of the opposite house.

66. A light pole 4 m high is fixed on the top of a building, the angle of elevation of the top of thepole observed from a point ‘p’ on the ground is 600 from the top of the building is 450. Findthe height of the building.

67. An aeroplane at an altitude of 1200 meters observes the angles of depression of oppositepoints on the two banks of a river to be 600 and 450, find the width of the river.

68. The angles of elevation of the top of a pole from two points P and Q at distances of ‘x’ and‘y’ respectively from the base and in the same straight line with it, are complementary.Prove that the height of the pole is √xy.

69. The angle of elevation of a bird from a point 12 metres above a lake is 300 and the angle ofdepression of its reflection in the lake is 600. Find the distance of the bird from the point ofobservation.

70. The angle of elevation of the top of a 10 metres tall building from a point P on the groundis 300. A flag is hoisted at the top of the building and the angle of elevation of the flag stafffrom P is 450. Find the length of flag staff and the distance of the building from P.

71. Nikita standing on a bank of a river observes that the angle subtended by a tree on theopposite bank is 600, when she retires 30 metres from the bank, she finds the angle to be300, find the breadth of the river and height of the tree.

72. A man, on a cliff, observes a boat at an angle of depression of 300, which is approaching theshore to the point ‘A’ on the immediately beneath the observer with a uniform speed, 12minutes later, the angle of depression of the boat is found to be 600. Find the time taken bythe boat to reach the shore.

73. A man on the deck of a ship, 18 metres above water level, observes that the angle of elevationand depression respectively of the top and bottom of a cliff are 600 and 300. Find the distanceof the cliff from the ship and height of the cliff.

74. The angle of depression of the top and bottom of a 10 metres tall building from the top ofa tower are 300 and 450 respectively. Find the height of the tower and distance betweenbuilding and tower.

75. An aeroplane when 3000 metres high, passes vertically above another aeroplane at aninstant when the angle of elevation of two aeroplanes from the same point on the groundare 600 and 450 respectively. Find the vertical distance between the two planes.

76*. At the foot of the mountain the elevation of its summit is 450. After ascending 1000 metrestowards the mountain at an inclination of 300, the elevation is 600. Calculate the height ofthe mountain. (√3 = 1.732)

77*. From an aeroplane vertically above a straight horizontal plane, the angle of depression oftwo consecutive kilometre stones on the opposite sides of the aeroplane are found to be θ


and α. Show that the height of the aeroplane is

tan . tan

tan + tan

θ α

θ α

78*. At a point ‘P’ on level ground, the angle of elevation of a vertical tower is found to be such

that its tangent is 3

4. On walking 192 metres away from P the tangent of the angle is

5

12.

Find the height of the tower.

79*. A round balloon of radius ‘r’ subtends on an angle ‘θ’ at the eye of the observer while theangle of elevation of its centre is α. prove that the height of the centre of the balloon is r

sinα cosecθ

2

80*. A boy standing on a horizontal plane, finds a bird flying at a distance of 100 metres fromhim at an elevation of 300. A girl, standing on the roof of 20 metres high building, finds theangle of elevation of the same bird to be 450. Both the boy and girl are on opposite sides ofthe bird. Find the distance of the bird from the girl.

Answers

1. tan A = sin A

1 – sin A2

2. cos p = 3

5

3. A + B = 900

4. 7

5.5

2

6. Zero

7. sec130 + cot280

8.25

12

9. θ = 450

10. Zero

11. 5

12. 1

13. 1

14. 1

15. x = 20

16. cosc

2


17. Zero

18. 23

19. –2

20. 10

21.1

2

22. A = 600, B = 300

23. θ = 420

24. (Hint. : 3θ = 900)

25. θ = 290

26. 9

27.1

2√

28.55

53

29. Hint : sec2θ – 1 = tan2θ

30. 1

31.23

7

32.9

5

33.√3

2

34.1

36

35.1

2√

36.17

2

37. sin R = 8

17, tan R =

8

15

sec M = 17

8

38.√

√

3+1

2 2,

√

√

3+1

2 2 (hint : A = 450, B = 300 or A = 300, B = 450)


39.13

12

42. – 5

57. (Hint : m = cos

sin

2θ

θ

13

n = sin

cos

2θ

θ

13

58. (Hint : m + n = 2 tanθ)

m – n = 2sinθ)

60. (Hint : sinθ = cos2θ

(a+b)3 = a3 + 3a2b + 3ab2 + b3)

61. 30√3 metres

62. 20 metres/sec.

63. 30(√3–1) metres

64. 5(2+√3) metres

65. 20(√3+1) metres

66. 2(√3+1) metres

67. 400 (3+√3) metres

68. (Hint : complementary angles)

69. 24√3 metres

70. Length of flag staff = 10 (√2–1) metres

Distance of the building = 10√3 metres

71. 15 metres, 15√3 metres

72. 18 minutes

73. 18√3 metres, 72 metres

74. 5(3+√3) metres, 5(3+√3) metres

75. 1000 (3–√3) metres

76. 1366 metres

78. 180 metres

80. 30 metres


Co-ordinate Geometry

Key Points

1. The length of a line segment joining A & B is the distance between two points A(x1, y

1) and

B (x2, y

2) is √{(x

2–x

1)2 + (y

2 –y)2}

2. The distance of a point (x, y) from the origin is √(x2+y2). The distance of P from x-axis is yunits and from y-axis is x-units.

3. The co-ordinates of the points p(x, y) which divides the line segment joining the pointsA(x

1, y

1) and B(x

2,y

2) in the ratio m

1 : m

2 are

(m x +m x

m +m1 2 2 1

1 2

, m y +m y

m +m1 2 2 1

1 2

)

we can take ratio as k:1, k = m

m1

2

4. The mid-points of the line segment joining the points P(x1, y

1) and Q(x

2, y

2) is (

x +x

21 2 ,

y +y

21 2 )

5. The area of the triangle formed by the points (x1, y

1), (x

2, y

2) and (x

3, y

3) is the numeric

value of the expressions 12

[x1(y

2–y

3) + x

2(y

3–y

1) + x

3 (y

1–y

2)]

6. If three points are collinear then we can not draw a triangle, so the area will be zero i.e.

x1(y

2–y

3) + x

2(y

3–y

1) + x

3(y

1 –y

2) = 0

1 Mark questions (Question 8-11 are under HOTS)

1. What is the distance between (a1o) and (o

1 b).

2. What is the midpoint of the line segment joining the points (3, 4) and (11, 6) ?

3. What is the value of a and b if (2, –3) is the mid point of the line segment joining (2, a) and(b, –1) ?

4. What is the area of the triangle joining the points (2, 4), (2, 0) and (2, –11) ?

5. AB is the diameter of a circle with centre at origin. What arethe co-ordinates of B if co-ordinates of point A are (3, –4) ?

6. What is the length of the side of the rhombus A(1, –2),B(2, 5), C(–5, 4) and D(–6, –3) ?

7. In the adjoining figure what is the length of AB ?

A(1, 0)

(–3, 3)B


8. What is the value of x if (3, 5) and (7, 1) are equidistant from T(x, o) ?

9. What is the value of y if ar (∆ABC) = 0 and co-ordinates of vertices are A(1, 2), B(y, 6), C(1,3) ?

10. Given a circle with centre at origin and radius 5√2 units. State where the point (5, –7)lies?

11. A line is drawn through p(4, 6) parallel to x-axis what is the distance of the line from x-axis?

2 marks questions - (Question 26-28 are under HOTS)

12. Find x if the distance between the points (x, 2) and (3, 4) be √8 units.

13. Find the point on y-axis which is equidistant from the points (–2, 5) and (2, –3).

14. Find the co-ordinates of the point which divides the line segment joining the points (1, 3)and (2, 7) in the ratio 3:4.

15. A and B are the points (1, 2) and (2, 3). Find the co-ordinates of a point G on the

line-segment AB such that AGGB

= 43

.

16. The mid point of the line segment joining the points (5, 7) and (3, 9) is also the mid pointof the line segment joining the points (8, 6) and (a, b). Find a, b.

17. Find the distance between the points A(a, b) and B(b, a), if a – b = 4

18. Find the ratio in which the point (11, 15) divides the line segment joining the points (15,5) and (9, 20).

19. Find the point of trisection of the line segment joining the points (–3, 4) and (1, –2).

20. NICE is a parallelogram whose three vertices taken in order are (–3, 1), (1, 1) and (3, 3).Find the co-ordinate of the fourth vertex.

21. Find the point which is 34

of the way from (3, 1) to (–2, 5).

22. Prove that we can draw the line passing through the points (0, 1), (3, 5) and (6, 9). (Showthat points are collinear).

23. Find the area of the triangle whose vertices are (1, –1), (–3, 5) and (2, 7).

24. Find the value of k if (k, 1), (5, 5) and (10, 7) are Collinear.

25. The vertex of the triangle ABC are A(–1, 3), B(1, –1) and C(5, 1). Find the length of themedian drawn from the vertex A.

26. ∆ABC is an isosceles triangle with AB = AC and vertex A is on y-axis. If the co-ordinates ofvertex B and C are (–5, –2) and (3, 2) respectively then find the co-ordinates of vertex A.

27. The point K(1, 2) lies on the perpendicular bisector of line segment joining the points E(6,8) and F(2, 4). What is the distance of the point K from the line segment EF.

28. Point P(k, 3) is the mid point of AB. If the distance AB = √52 units and co-ordinates of Aare (–3, 5) then find the value of k.

3 marks questions - (Questions 40-42 are under HOTS)


29. Find the abscissa of a point whose ordinate is 4 and which is at a distance of 5 units from(5, 0).

30. In figure, CD is a median from the vertex C on the side AB of

∆ABC, P is the point on CD such that DP = 1 unit. Find DPPC

.

31. If A(–3, 2), B(x, y) and C(1, 4) are the vertices of an isoscelestriangle with AB = BC. Find the value of (2x+y).

32. Find the ratio in which the line 3x + y = 12 divides the line segment joining the points(1, 3) and (2, 7).

33. Prove that the figure obtained on joining the mid points of parallelogram PQRS is a squarewhere P(1, 0). Q(5, 3), R(2, 7) and S(–2, 4). Also find the sum of the diagonals.

34. A point P on the x-axis divides the line segment joining the points (4, 5) and (1, –3) incertain ratio. Find the co-ordinates of point P.

35. In right angled triangle ABC, ∠B= 900 and AB = √34 unit. The co-ordinates of points Band C are (4, 2) and (–1, y) respectively. If the ar (∆ABC) = 17 unit2 then find the value ofy.

36. If the point (6, 4) divides the line segment joining L (a,b) and M (8,5) in the ratio 2:5 thenfind the value of a and b. Also find the co-ordinates of the mid point of ML.

37. The vertices of quadrilateral ABCD are A(–5, 7), B(–4, –5), C(–1, –6) and D(4, 5). Find thearea of the quadrilateral ABCD.

38. In figure D and E are the mid-points of the side BCand AB respectively. Find the length of DE.

39. Find the value of y such that ar (∆ABC) = 4 units2

and co-ordinates of the vertices are A(1, 2), B(3, y) and C(5, 2)

40. If the point P(3, 4) is equidistant from the points A(a+b, b–a) and B(a–b, a+b) then provethat 3b–4a = 0

41. If the area of the quadrilateral PQRS is zero where P(1, –2), Q(–5, 6) R(7, –4) and 5(h, –2)are the vertices then find the value of h. State are the points really making quadrilateral.

42. In figure find the radius of the circle.

B(3, –1) (7, –3)C

A (5, 3)

D

P

B (–6,–1) D C (4, –2)

A (2, 1)

E


ANSWERS

1. √(a2 + b2)

2. (7, 5)

3. a = – 5

b = 2

4. Zero

5. (–3, 4)

6. √50 unit

7. 5 unit

8. x = 2

9. y =1

10. Outside

11. 6 unit

12. x = 1, 5

13. (0, 1)

14. (107

, 337

)

15. (117

, 187

)

16. a = 0, b = 10

17. 4√2 unit

x1

y1

y

–2

–1

–1

2

1 2 3

2

1

x


18. 2:1

19. (–53

, 2), (–13

, 0)

20. (–1, 3)

21. (–34

, 4)

23. 5 Sq. units

24. k = –5

25. 5 unit

26. (0, –2)

27. 5 unit

28. k = 0, –6

29. 2, 8

30. 1:4

3.1 1

32. 6:1

33. 10√2

34. (178

, 0)

35. – 1

36. a = 255

b = 185

(6610

, 4310

)

37. 72 square unit

38.12

√13 unit

39. y - 0

41. h = 3,

NO

42.5 2

7

√ units

Triangles


Key Points

1. Similar triangles :

Two triangles are said to be similar in their corresponding angles are equal and theircorresponding sides are proportional.

2. Criteria for Similarity :-

in ∆ABC and ∆DEF

(i) AAA similarity ∆ABC ~ ∆DEF when ∠A = ∠D, ∠B = ∠E and ∠C = ∠F

(ii) SAS similarity : ∆ABC ~ ∆DEF when ABDE

= BCEF

= ACDF

AND ∠A = ∠D

(iii) SAS similarity : ∆ABC ~ ∆DEF ABDE

= ACDF

= BCEF

3. The proofs of the following theorems can be asked in the examination :-

(i) Basic proportionality Theorems : If a line is drawn parallel to one side of a triangle tointersect the other sides in distinct points, the other two sides are divided in thesame ratio.

(ii) The ratio of the area of two similar triangles is equal to the square of the ratio of theircorresponding sides.

(iii) Pythagoras theorem : In a right triangle, the square of the hypotenuse is equal to thesum of the squares of the other two sides.

(iv) Converse of Pythagoras Theorem : In a triangle, if the square of ne side is equal tothe sum of the squares of the other two sides then the angle opposite to the first sideis a right angle.

Similar Triangles

1 Mark Questions

1. In fig(1) what PSSQ if ST||QR, PT = 8 cm and PR = 10 cm.

2. In fig (2) is ∆ABC ~ ∆AQP

3. In fig (2) if QC = 10 cm is it possiblethat PQ||BC ?

P

Q R

S T

(Fig. 1)A

B C

P Q

(Fig. 2)

18cm

10 c

m 15cm

12 cm


4. In ∆XYZ, P&Q are points on XY and XZ respectively XPXY

= 23

, XQ = 6 cm, QZ = 3 cm.

What type of linesegments PQ & YZ are.

5. In ∆PQR, S&T are points on PQ and PR such that ST||QR, PS = 2 cm, SQ = 3 cm, PR =15 cm. What is the length of TR.

6. An isoscels triangle ABC is similar to ∆PQR. AC = AB = 4 cm, PQ = 10 cm and BC = 6 cm.What is the length of PR ?

7. In fig (3), l||m. what is the measurement of x ?

8. In fig (4) ∆ABC ~ ∆PQR. What is the value of x ?

9. In fig (5), if DE||BC, what is the values of x ?

10. If the ratio of the corresponding sides of two similar trianglesin 4:5. What is the ratio of their areas ?

11. In fig (6) DE||QR and DE = 14

QR. How many times is PQ of

PD. (This can be asked as write PQ : QD)

135

0

120

0

x y l

m

(Fig. 3)

A

B C P Q

R

6 7.5

x6

4 5

(Fig. 4)

A

B C

D E

(Fig. 5)

x

8

x

2

P

Q R

D E

(Fig. 6)


12. The length of median of an equilateral riangle is 3 cm. What is the length of its sides ?

13. In two triangles ABC and PQR if ∠B = ∠Q and ABPQ =

12

, what is the value of PRQR ?

14. Measurement of three sides of a triangle are a, √10 a, 3a. What is the measurement of theangle opposite to the longest side ?

15. If fig (7), DE||BC what is the value of DE.

2 Marks Questions

16. In fig (8) find SR.

17. In ∆PQR, RS⊥PQ, ∠QRS = ∠P, PS = 5 cm, SR = 8cm. Find PQ

18. Two similar triangles ABC and PBC are made on opposite sides of the same base BC.

Prove that AB = BP

19. In fig (9) ABCD is a rectangle. ∆ADE and ∆ABF are two triangles.

Such that ∠E = ∠F.

Prove that ADAE

= ABAF

20. In fig. (10) DE||BC.

If ADAB

= 38

. Find ar ( ADE)

ar ( ABC)

∆∆

21. In figure (10) DE||BC, DE = 3 cm, BC = 90 cm and ar(∆ADE) = 30 cm2. Find ar (trap BCED)

A

B C

D E

(Fig. 7)

10 cm

x 2 cm

3 cm

P

(Fig. 8)

6 cm

Q RS9 cm

F

C

BA

DE

(Fig. 9)

A

B C

D E

(Fig. 10)


22. Amit is standing at a point on the ground 8 m away from a house. A mobile network toweris fixed on the root of the house. Finds that the top and bottom of the tower are 17 m and10 m away from the point. Find the heights of the tower and house.

23. In a right angled triangle right angle at B, BCAB

= √3. Find ABAC

.

24. In a right angled triangle PRO, PR is the hypotanous and the other two sides are of length6 cm and 8 cm. Q is a point outside the triangle such that PQ = 24 cm, PQ = 26 cm. Whatis the measure of ∠RPQ ? How many such triangles PQR are possible ?

25. ∠B and ∠ACD of a quadrilateral ABCD and right angles. Prove that AD2 = AB2 + BD2 +CD2.

26. In figure (11) ∆ABC is isosceles with AB = AC.

Prove taht BMCN

= MPNP

27. Find the length of the diagonal of rectangle BCDE (fig 12 (i))if ∠BCA = ∠DCF.

28. In fig. (12 (ii) ∆BEF is a rectangle. C is the mid point of BD.

If AB = 16 cm, De = 9 cm, BD = 24 cm. AE = 25 cm.

Prove that ∠ACE = 900

29. In fig. (13) Find the value of x if PQ||BC

30. PQRS is a trapezium. SQ is a diagonal. E & F are two points on PQ and RS respectivelyinteresting SQ at G. Prove that SG × QE = QG × SF.

A

B C

M

(Fig. 11)

N

P

A F

D

B

E

C5 cm 10 cm

(Fig. 12 (i))

B D

EF

C(Fig. 12 (ii))

A

A

B

C

PQ

3x

x+2 2x+

24x+

1

(Fig. 13)


31. In figure (14) prove that DE||BC.

Also find the ratio of ar ( ADE)

or (trap BCED)

∆.

Where D is the mid point of BC.

32. In ∆ABC, EF||BC, such that EF passes through the controid G. Find , where D

is the mid point of BC.

33. In fig. (15) D||BE and ADDC

= CEBE

.

Prove that PDCE is a parallelogram.

34. In a quadrilateral ABCD, ∠A + ∠D = 900. Prove that AC2 + BC2 = AD2 + BC2.

35. In fig. 15(i) PQR and S are points on the sides of quadrilateral ABCDsuch that these points divides the sides AB, CB, CD and AD in theratio 2:1.

Prove that PQRS is a parallelogram.

36. Equiangular triangles are drawn on sides of right angled triangle inwhich perpendicular is double of the base. Show that the area of the triangle on thehypotenuse is the sum of the areas of the other two triangles.

37. In a rhombus prove that four times the square of any sides is equal to sum of squares of itsdiagonals.

38. ABCD is a rectangle in which length is double of its breadth two equilateral triangles aredrawn one each on length and breadth. Find the ratio of their areas.

39. In fig. (16) ∠AEF = ∠AFE E is the mid point of CA.

Prove that BD ×CE = BF ×CD.

40. Prove that if a line is drawn parallel to one side of a triangle, it divides the other two sidesin the same ratio. Rider’s based on above theorem

A

B C

D E

(Fig. 14)

x

4 6

1.5

A B

C

D

P(Fig. 15)

E

B

A D

C

R

S

Q

R

(Fig. 15 (i))

B

A

D

C

E(Fig. 16)


(i) in the adj. fig. (18) AB||DE, BD||EF. Find CD2.

(ii) ABCD is a parallelogram (see fig.19. )

Prove that DPPQ =

DCBQ

(iii) Find x, if DE||BC (fig. 20)

(iv) ABCD is a trapezium. Find value of x (fig. 21)

41. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squaresof their corresponding sides.

Use above result to prove the following

Riders :

ar ABC

ar (trap BCED)

∆ =

1649

, BC||DE

What is the length of the altitudes to the bigger triangles if length of altitude to smallertriangle is 8 cm (fig. 22)

C

A B

D

E

(Fig. 18)

F

A(Fig. 19) B

Q

P

CD

A

B C

D E

(Fig. 20)

8 4

x–43x–19

A (Fig. 21) B

CD

3x–19

x–5

x–3

3

A

D E

B C

(Fig. 22)


42. In a triangle, if the square of one side is equal to sum of the squares on the other two sides,the angles opposite to the first side is a right triangle.

Use above result to prove the following Rider;s

(i) Dinesh, Naresh and Ashu are standing in such a way that distance between Dineshand Naresh is p meter . Naresh and Ashu are at a distance √(q2 –2) m from ech other.Dinesh and Ashu are √(p2 + q2 –2) m apart. What type of triangle they are forming.

(ii) Three sticks of length (a–1) cm, 2√a cm and (a+1) cm are joined with their end pts. tofrom a triangle. Do they form a right triangle. Show.

43. In a right triangle, the square on the hypotenuse is equal to the sum of the squares on theother two sides.

Rider based on aboe theorm.

Amar and Ashok are two friends standing at a corner of arectangular garden. They wanted to drink wter. Amar goes duenorth at a speed of 50 m/min. and Ashok due west at a speed of60 m/min. They travel for 5 minutes. Amar reaches the tapand drink water. But somebody told Ashok to go towards pointC. As there was to other tap.

Find the min. distance Ashok has to travel to reach point C.(fig. 23)

Answers

1.41

2. NO

3. No

4. Parrallel

5. 9 cm

6. 10 cm.

7. 750

8. 9 cm

9. 4 cm

10. 16:25

11. 4

12. 2 √3

13.12

B A

C

(Fig. 23)


14. Right angle

15. 2.5

16. 4

17. 12.8 cm

18. ----

19. ----

20.9

64

21. 60 cm2

22. 9 m, 6 m

23.√3

2

24. Right angle, Two

25. ----

26. -----

27. 5√10

28. ----

29. 2

30. ----

31.169

32.49

33. ---

34. ---

35. ---

36. ---

37. ---

38. 1:4

39. ---

40. (i) CF × CA, (ii) ---, (iii) –11, (iv) 8 or 9

41. 14 cm

42. (i) right angled triangle (iv) yes

43. 50 √61 m


Statistics

Key Points

1. The mean for grouped data can be found by :

(i) The direct method = Xfixi

fi=

(ii) The assumed mean method = Xfidi

fi= a + , where d

i = x

i – a

(iii) The step deviation method = Xfiui

fi= a + × h, where u

i =

x – a

hi

2. The mode for the grouped data can be found byusing the formula :-

mode = l + f – f

2f – f – f

1 0

1 0 2

× h

l = lower limit of the model class.

f1 = frequency of the model class

f0 = frequency of the proceeding class of the model class.

f2 = frequency of the succeeding class of the model class

h = size of the class interval.

Model class - class interval with highest frequency.

3. The median for the grouped data can be found by using the formula :-

median = l + –Cf

f× h

n2

l = lower limit of the median class.

n = number of observations

Cf = cumulative frequency of class interval preceeding the median class.

f = frequency of median class.

h = class size.

4. Imperical Formula :-

Mode = 3 median - 2 mean

5. Cumulative frequency curve or an Ogive :-

(i) Ogive is the graphical representation of the cumulative frequency distribution.


(ii) Less than type Ogive :-

* Construct a cumulative frequency table

* Mark the upper class limit on the x = axis.

(iii) More than type Ogive :-

* Construct a frequency table

* Mark the lower class limit on the x-axis.

(iv) To obtain the median of frequency distribution from the graph :-

* Locate point of intersection of less than type Ogive and more than type Ogive :-

Draw a perpendicular from this point to x-axis.

* The point at which it cuts the x-axis gives us the median.

Statistics

I mark Questions (Question12-15 are under HOTS)

1 What is the median of the following distribution

2,3,6,0,1,4,8,2,5

2 What is the Mean if Median= 14 and Mode=12

3 What is the mean of x, x+1, x+2, x+3, x+4

4 The following table shows the frequenay distribution of the marks of 50 students What isthe value of y.

class interval 0-5 5-10 10-15 15-20 20-25

frequency 8 11 13 y 10

5 Write the class mark of the class interval 16.5-21.5

6 A teacher ask the student to find the average marks obtauned by most of the Students.What the student will find: Mean, Mode or Median.

7 What is the mode of the following data.

1, 0, 2, 2, 3, 1, 4, 5, 1, 0,

8 In the following distribution, Write the modal class

Class interval 10-15 15-20 20-25 25-30 30-35

Frequency 4 7 20 8 1

9 For the frequency distribution ∑fi=40 and ∑fixi=2440 What is the mean of thedistribution.

10 A teacher ask the student to find the average marks obtained by the class student inMathematics What the student will find: Mean Mode or Median.


11 The following data is arranged in the ascending order

11,12,12,13,7x,7x+1,16,16,18,20 if the median of data is 14.5,what is the value of x

12 What is the value of the median of the data using the following qraph of “less then ogive’’and “ More than ogive’’

13 The following “ More than ogive’’ Shows the weight of 40 Student of a class. What is thelower limit of the Median class.


14. From the cumulative frequency table Write the frequency of the class 20-30

Marks Number

of student

less than 10 1

less than 20 14

less than 30 36

less than 40 59

less than 50 60

15. Following is a commulative frequency curve for the marks obtained by 20 Student as shownin find the Median marks obtained by the student.

6 Marks Questions (Question No.24, 25, 26, 27 are under HOTS)

16. Find the mean of the following frequency distribution:

Class interval 18-24 24-30 30-36 36-42 42-48 48-54

Frequency 6 10 12 8 4 2

17. Find the value of p, if the mean of the following distribution is 20

x 15 17 19 20+p 23

f 2 3 4 5p 6

18. The mean of the following frequency distribution is 62.8 and the sum of all the frequenciesis 50. Find the values of p and q.

Class-Interval 0-20 20-40 40-60 60-80 80-100 100-120

Frequency 5 p 10 q 7 8


19. The following frequency distribution gives the daily wage of a worker of a factory: Find themean daily wage of a worker.

Daily wage No. of

(in Rs) Workers

More than 300 0

More than 250 12

More than 200 21

More than 150 44

More than 100 53

More than 50 59

More than 0 60

20. A retailer stocks car batteries of a particular type. The lives (in years) of 40 such batteriesare recorded in the following frequency distribution:

Life of 2.0-2.5 2.5-3.0 3.0-3.5 3.5-4.0 4.0-4.5 4.5-5.0

batteries in Yrs.

Number 2 6 14 11 4 3

find the modal life of a battery in years.

21. The following frequency distribution shows the marks obtained by 100 students in a school.Find the mode.

Marks Number of students

less than 10 10

less than 20 15

less than 30 30

less than 40 50

less than 50 72

less than 60 85

less than 70 90

less than 80 95

less than 90 100

22. The following frequency distribution gives the weights of 30 students of a class. Find themedian Weight of the students.

Weight (in kg) 40-45 45-50 50-55 55-60 60-65 65-70 70-75

No.of persons 2 3 8 6 6 3 2

23. A life insurance agent found the following data for distribution of ages of 150 policy holders.calculate the median aqe. if policies are only given to persons having age 18 years. Onwards


but less than 60 year .

Age (in years) number of policy holder

Below 20 8

Below 25 10

Below 30 32

Below 35 59

Below 40 105

Below 45 125

Below 50 138

Below 55 148

Below 60 150

24. The median of the following frequency distribution is 28.5 and sum of all the frequenciesis 60. Fund the value of x and y.

Class interval 0-10 10-20 20-30 30-40 40-50 50-60 Total

frequency 5 x 20 15 y 5 60

25. Draw “More than ogive” for the following distribution:

Class Interval 50-55 55-60 60-65 65-70 70-75 75-80

Frequency 2 8 12 24 38 16

Also find median from the graph of ogive verify that by using the formula.

26. Draw “less than” ogive for the following distribution.

Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100

Number 5 3 4 3 3 4 7 9 7 8

of Students

Find the median from the graph.

27. A survey regarding the heigh (in cm) of 50 girls of class × of a school was conducted andregarding the followig data was obtained.

Height in cm 120-130 130-140 140-150 150-160 160-170 Total

Number of 2 8 12 20 8 50

girls

Find the mean, median and mode of the above data.


ANSWERS

1. I

2. 15

3. x+2

4. y=8

5. 19

6. Mode

7. 1

8. 20-25

9. 61

10. Mean

11. x=2

12. 40

13. 147.5

14. 22

15. 20

16. 33

17. p=1

18. P=8 q=12

19. Rs 182.50

20. 3.364 years

21. 41.82

22. 56.67 Kg

23. 35.93 years

24. x=8,y=7

25. 71.58

26. 66

27. Mean=149.8 cm

Median=151.5 cm

Mode=154 cm


Probablity

Key points

1. The Theoretical probablity of an event E written as (E) is .

P(E)= Number of outcomes favourable to E

Number of all possible outcomes of the experiment

2. The sum of the probability of all the elementary events of an experiment is 1.

3. The probability of a sure event is 1 and probability of an impossible event is o.

4. if E is an event, in general, it is true that P(E) + P(E) = 1

5. From the definition of the probability, the numerator is always less than or equal to thedenominator therefore O ≤ P(E) ≤ 1

PROBABILITY

1 mark questions (* questions are under HOTS)

1. Two coins are tossed once. Write the sample space (possible out comes)

2. E is an event such that P(E) = 0.32 What is P(E)?

3. If probability of success is 52% What is the probability of failure?

4. A bag contains 4 blue and 5 green balls. What is the probability of getting a red ball if a ballis drawn randomly from the bag.

5. E is an event such that P(E) =1/5 what is P(E).

6. A bag contain 9 Red and 6 Blue marbles A marble is taken out randomly,, What is theprobability of getting red marble.

7. A game of chance of a spinning wheel has numbers 1 to 10 What is the probability ofgetting a number less than and equal to 7 when wheel comes to rest.

8. In a class of 40 students there are 25 girls and 15 boys. What is the probability of getting aboy selected as class monitor.

9. In a survey it is found that every fourth person possess a vehicle what is the probability ofperson not possessing the vehicle.

10. During IPL Cricket tournament a match is played between Delhi dare Devils and KnightRiders If probability of Dare Devils winning the match is 0.579 What is the probability ofKnight Riders Winning the match.

11. Only face cards are well shuffled A card is drawn at random What is the probability ofgetting a red queen? (there are 12 face cards)

12. Humanshu and Sachin are friends What is the probability that they both have birthdayon 13th April (ignoring leap year)


13. A dice is thrown once. What is the probability of getting an “even prime number”?

14. In the coloured water 3 coke ,5 orange and 7 Nimbuz bottles are kept. What is the probabilityof getting a bottle of coke?

15. Two dice are rolled once What is the probability of getting a doublet.

16. In a family there are one son and two daughters What is the probability of son being theyoungest?

17. In the word “success” What is the probability of getting the letter ‘S’?

18. A dice is rolled once. What is the probability of getting a prine number?

19. A garden has different plants. Every sixth plant is a rose plant. A child plucks a flower.What is the probability of getting a flower other than rose.

20. A bank ATM gas notes of denomination 100, 500, 1000 in equal numbers What is theprobability of getting a note of Rs 1000.

21. In a film fare award, SHAHRUKH KHAN is nominated in the Best Actor category If thereare five nominations then what is his probability of winning an award.

22*. What is the probability of getting a number greater than 6 in a single throw of dice.

23. A selection commiltee interviewed 50 people for the post of sales Manager. Out of which35 are males and 15 are females. What is the probability of a female candidate being selected.

2 mark questions (Questions 34-40 are under HOTS)

24. Two dice are rolled simultaneously. Find the probability that the sum is more than andequal to 10.

25. From the well shuffled pack of 52 cards, Two Black Kunqe and two red Ace ‘s are reinovedWhat is the probability of getting a face card.

26. A box contains green and red balls. The probability of drawing green ball is thrice theprobability of drawing red balls. Find the number of green balls it the total number ofballs in box is 28.

27. A box contain cards marked with the number 2to17. A card is drawn randomly. What isthe probability that the card contains the number between 8 and 14.

28. A box contain card marked with the number 2 to 101. One card is drawn at random. Whatis the probability of getting a number which is a perfect square.

29. A carton consist of 100 pens, out of which 65 are good one, 15 have minor defects andremaining have major defects what is the probability that a man will pick up a pen withmajor defects.

30. A box contains orange, mango and lemon flavoured candies A candy is drawn randomly. IfP(not lemon) = 11/15 and P(mango)=1/3 then what is P(orange)?

31. A box contain cards which are numbered from 1to 20 The cards containing prime numbersare removed from the box. A card is drawn randomly What is the probability of getting acard containing even number.

32. From the well shuffled pack of 52 cards . few cards of same colour are missing et P(Red


card)=1/3 and P(Black card) =2/3 then which colour of cards are missing and how many?

33. In a beginning of session there were 40 students in Class X out of which 19 were girls and21 were boys. 7 students got promoted after compartmental exam.out of which 4 weregirls and 3 boys who gas the more probability of getting selected as the class monitor?What is the difference between their probabilities.

34. An archery target marked with its three scoring are as fromthe centre outward as Gold, Red, Blue The diameter ofthe region representing Gold score is 21 cm and each ofthe other band as 10.5cm wide it the archer hit the goldarea he is declared winner. What is the probability ofwinning the game.

35. Four Jacks are missing in a well shuffted pack of 52 cards.A child could find only 2 red jack and reshuffled it back.what is the probability of getting a face card.

36. A meeting of officials is to last for 2 hrs. What is the probability of getling it over in 1 hr. 20min?

37. An aeroplane has to do an emergency landing inthe rectangular area as shown in the fiqure. Whatis the probability that it lands safely?

38. A child organises the game in which if you drop a coinin the circle, you win the game as shown in the figure.What is the probability of winning the game?

39. A bag contains 12 balls out of which x are black. If 6more black balls are put in the box, the probability ofdrawing a black ball is double of what it was before. Find x.

GOLD

RED

BLUE

LAND

JUNGLE

6 KM

4 KM

4.5

KM

2 K

M

A B

D C

1

3 UNIT

2 U

NIT


40. A bag contain few black and red marbles The number of red marbles are two more thanhalf the black marbles. If total number of marbles are 14, then find the probability of blackand red marble.

ANSWERS

1) HH,HT,TH.TT

2) o.68

3) 48% or 12/25

4) zero

5) 4/5

6) 3/5

7) 7/10

8) 3/8

9) 3/4

10) 0.421

11) 1/6

12) 1/365

13) 1/6

14) 1/5

15) 1/6

16) 1/3

17) 3/7

18) 1/2

19) 5/6

20) 1/3

21) 1/5

22) 0

23) 3/10

24) 1/12

25) 5/24

26) 21

27) 5/16

28) 9/100

29) 1/5

30) 2/5


31) 3/4

32) Red,13

33) Boys,1/47

34) 1/9

35) 1/5

36) 2/3

37) 5/27

38) 11/21

39) 3

40) P(B)=4/7

P(R)=3/7


Mensuration (Continued)

Surface Areas and Volumes

Key Points

1. Total Surface area of cube of side a units = 6a2 units.

2. Volume of cube of side a units = a3 cubic units.

3. Total surface area of cuboid of dimensions l, b & h = 2(l ×b+b×h+h×l) square units

4. Volume of cuboid of cylinder of dimensions l, b and h = l × b ×h cubic units.

5. Curved surface area of cylinder of radius r and height h = 2πrh square units.

6. Total surface area of cylinder of radius r and height h = 2πr(r+h) square units.

7. Volume of cylinder of radius r and height h = πr2h cubic units.

8. Curved surface area of cone of radius r height h and slant height l = πrl square unitswhere

l = √r2 + h2

9. Total surface area of cone = πr(l+r) sq. units.

10. Volume of cone = 13

πr2h

11. Total curved surface area of sphere of radius r units = 4πr2 sq. units.

12. Curved surface area of hemisphere of radius r units 2πr2 sq. units.

13. Total surface area of hemisphere of radius r units = 3πr2 sq. units

14. Volume of sphere of radius r units 43

πr3 cubic units.

15. Volume of hemisphere of radius r units 23

πr3 cubic units.

16. Curved surface of frustum = πl(r+R) sq. units, where l slant height of frustum and radii ofcircular ends are r & R.

17. Total surface area of frustum = πl(r+R) + π(r2 + R2) sq. units.

18. Volume of Frustum = 13

πh(r2 + R2 + rR) cubic units


MENSURATION

QUESTION

1 mark questions (use, π π π π π = 227

)

1. The Perimeter of a protractor (semicircle) is 66 cm. What is its radius.

2. A wire is in the form of a circle of diameter 21cm. It is cut and bent to form a square. Writethe measure of the side of square.

3. The numerical difference between circumference and diameter is 30cm. What is the radiusof the circle.

4. Write the Perimeter of the adjoining figure; Where AED is a semicircle and ABCD is arectangle.

5. What is the Perimeter of a sector of angle 450of a circle With radius 7cm.

6. A chord of a circle of diameter 28cm subtends an angle of 600. at the center. Write the areaof the major sector.

7. A Pendulam of Wall clock swings through an angle of 300. describes an arc of length 8.8cm.Write length of Pendulam.

8. From each vertex of trapezium a sector of radius 7cm has been cut off. Write the total areacut off.

9. Write the ratio of the areas of two sectors I & 111.

10. If an are forms 450 at the centre ‘��O’ of the circle. What is the ratios of its length to thecircumference of the circle?

11. How many cubes of side 4cm can be cut from a cuboid measuring (16×12×8) cm3 ?

E

A

BC

D

20 cm

14 cm

1500

1200

III


12. The diameter and heigth of a cylinder and a cone are equal. What is the ratio of theirvolume?

13. A cylindrical tank has a capacity of 6160m3. What is its depth, it the radius of the base is14cm?

14. A cylinder, a cone and a hemisphere are of equal base and have the same height. What isthe ratio in their volumes?

15. A rectangular sheet of Paper 66cm.×18cm. is rolled along its length and a cylinder isformed. Write the curved surface area of the cylinder.

16. A metallic sphere of total volume πcm3, is melted and recast in to the shape of a cylinder ofradius 0.5cm. What is the height of cylinder?

17. The length of minor are is 7-20 of the cirumference of the circle. Write the angle subtendedby the arc at the centre.

18. A bicycle Wheel make s 5000 revolutions in_ moving 10km. Write the perimeter of wheel.

19. Curved surface area of a right circular cylinder is obtained on multi plying volume by ‘K’.Write the value of K.

20. The sum of the radius of the base and the height of a solid cylinder is 15cm. If total surfacearea is 660m2, Write the radius - of the base of cylinder.

21. Find the height of largest right circular cone that can be cut out of a cube whose volume is729 cm3.

22. What is the ratio of the areas of a circle and an equilateral triangle Whose diameter and aside are respectively equal?

23. A solid sphere of radius ‘s’ is melted and cast in to the shape of a solid cone of height ‘s’What is the radius of the base of the cone.

24. If the circumference of the circle exceeds its diameter by 30cm. What is the diameter ofthe circle.

25. The length of an are of a circle of radius 12cm is 10πcm. Write the angle measure of thisare.

(3 Marks Questions ) : (Que 42-51 under HOTS)

26. Find area of the shaded region shown in the adjoiningfigure where ABCD is a square of side 10cm and semicircles are drawn with each side of the square asdiameter. (useπ=3.14)

10 cm

D

A B

C

10 cm


27. Ι n the figure given along side, Find the area of the shaded portion?

28. Find the area of the shaded portion; If side of square is 28cm?

29. Find the shaded area of the figure- along side et diagonal of square is 28cm?

30. If AOB is a quadrant of a circle of radius 14cm. Find the area of shaded region?

A

D

B

C

12 cm

A

D

B

C

12 cm

5 cm

7 cm

7 cm7 cm

7 cm

7 cm

14

cm

A

O B


31. Find the area of the shaded region when PQ=QR=RS=14cm(student can find Perimeterof this figure also)

32. Find the volume of the largest right circular cone that can be cut out of a cube whosevolume is 729 cm2?

33. A hemi spherical bowl of internal radius 15cm contains a liquid. The liquid is to be filledin to cylindrical shaped bottles of diameter 5cm and height 6cm. How many bottles arenecessary to empty the bowl?

34. The base radii of two right circular cones, of the same height are in the ratio 3:5 Find theratio of their volumes?

35. The dimensions of a metallic caboid are 100cm×80cm×64cm.If it is melted and recast into a cube; Find the surface area of the cube.

36. Three equal cubes are placed adjacently in a row. Find the ratio of the total surface area ofthe new cuboid to that of the sum of the surface zreas of the three cubes.

37. A sheet of paper is in the form of a reetangle ABCD in which AB=40cm and AD= 28cm. Asemicircular portion with BC as diameter is cut off. find the area of the remaining paper.

38. A bicycle wheel makes 500 revloutions in moving 11km. find the diameter of the wheel.

39. Two circles touch externally. The sum of their areas is 130πsq cm and the distance betweentheir centres is 14cm. find the radii of the circles.

40. AB is a chord of a circle of radius 10cm. The chord subtends a right angle at the centre ofthe circle. find the area of the major segment.(take π=3.14)

41. The inner cirumference of a circular track is 440cm. The track is 14cm wide. find thecastof leveling it at 20 paise/ sam. Also find the cast of putting up a fencing along outercircle at Rs2/ metre.

42. A hemispherical bowl of internal diameter 36 cm is full of liquid. Thus liquid is to be filledin cylindrical bottles of radius 3 cm and height 6 cm. How many bottles are required toempty the bowl?

43. The area of equilateral triangle is 49√3 cm2. Taking each angular Points as centre, a circleis described with radius equal to half the lenght of the side of the triangle. find the area oftriangle not included in the circle?

44. AB is a chord of a circle of radius 10cm. The chord subtends a right angle at the centre ofthe circle. find the area of the minor segment?

P Q

R S


45. AB and CD are two perpendicular diameters and CD= 8cm. find the shaded region?

46. If the area of four sectors having same radius is 616 sq cm. What is the radius?

47. A cylindrical pipe has inner diameter of 4cm and water flows through it at the rate of 20m.per minute. How long would it take to to fill a conical tank. whose diameter of base is 80cmand depth 72cm?

48. Sum of the areas of two squares is 468πm2. If the difference of their perimeter is 24cm.find the side of the two squares.

49. PQRS is a diameterof a circle of radius 6cm, The lengthsPQ, QR and RS are equal, semi circles are drawn on PQand QS as diameter as shown in the figure. Find theperimeter of the shaded region.

50. In the adjoining figure ABC is a right angledtriangle, right angled at A. semi circle are drawnon AB;AC and BC as diameters. Find the areaof shaded region.

A

O CD

B

1200

600

r

r

r

r

PQ R S

A

B C

4 cm3

cm


51. A small torch can spread its light over a seetor of angle 900 up to a distance of 14cm. VishalRaj put two torches of same type at opposite vertices of a square of side14 cm as shown infigure. find the area which get light from both sides.

6 Marks Question (Question, 70-74 under HOTS)

52. The rain water from a roof 22m×20m drains into a cylindrical vessel having diameter bybase 2m and height 3.5m et the vessel is just full; find the rainfall in cm.

53. A cylinder whose height is two third of its diameter has the same volume as a sphere ofradius 4cm. calculate the radius of the radius of the base of the cylinder.

54. 50 circular plates, each of radius 4 cm, and thickness 1/2 cm. are placed one above theother to form a solid right circular cylinder. find the total surface area and volume of thecylinder so formed.

55. Marbles of diameter 1.4cm are dropped in to a cylindrical beaker of diameter 7cm containingsome water find the number of Marbles dropped so that water level rises by 5.6cm.

56. A bucket is in the form of a frustum of a cone and holds 28.49 litres of milk. The radii of thetop and bottom are 28cm and 21cm respectively find the height of the bucket.

57 A student Navneet, developed a model of a hut. He made a cylinder of radius 6cm andheight 6 cm and attached a hemisphore of radius new at its top.He left an open area fordoor on the cylinder of length 4cm and breath 3cm. find the surface area of the hut.

58. The slant height of right circular cone is 10cm and its height is 8cm. It is cut by a planeparallel to its base passing through the mid point of the height find ratio of the volume oftwo parts.

59. A cylindrical piece of wax of radius 2.8cm and height 2cm is placed in a metallichemispherical bowl of diameter 7cm. If the wax melts into the bowl, will the bowl overflow. If yes, how much wax over flows. If not how much more can be accomodated.

60. The interior of a building is in the form of a right circular cylinder of radius 7cm andheight 6cm, surmounted by a right circular cone of same radius and of vertical angle 600.Find the cast of painting the building from inside at the rate of Rs30/m2.

61. A container, shaped like a right circular cylinder, having diameter 12cm. and height 15cmis full of ice-cream. This ice-cream is to be filled in to cones of height 12cm and diameter6cm, having a hemi spherical shap on the top, find the number of such cones which can befilled with ice-cream.

62. A solid spherical ball of the metal is divided in to two hemispheres and joined as shown inthe figure,This solid is placed in a cylindrical tub, full of water in such a way that the


whole solid is sub merged in water. the radius and height of cylindrical tub are 4cm and5cm respectively.the radius of spherical ball is 3cm. find the volume of water left in thecylindrical tub.

63. A gulab jamun, contains sugar syrup up to about 30% of its volume. find approximatelyhow much syrup would be found in 45 gulab jamuns, each shaped like a cylinder, with twohemispherical shaped like a cylinder, with two hemispherical ends, with lenght 5cm andits diameter 2.8cm.

64. Water flows out through a circular pipe whose internal radius is 1 cm, at the rate of 80cm/second in to an empty cylindrical tank, the radius of whose base is 40cm. By how muchwill the level of water rise in the tank in half an hour?

65. A solid is in the form of a right circular cone mounted on a hemisphere. The radius of thehemisphere is 2.1cm and height of cone is 4cm. The solid is placed in a cylindrical tab, fullof water,in such a way that the whole solid is submerged in water. If the radius of cylinderis 5cm and height 9.8 cm. find the volume of water left in the tub.

66. The height of a solid cylinder is 15 cm and diameter 7cm. Two equal conical holes of radius3 cm and height 4cm are cut and surface area of remaining solid.

67. Three cubes of metal whose edge are in the ratio 3:4:5 are melted in to a single cube whosediagonal is 12�√3cm find the edge of the three cubes.

68. A toy is in the form of a cone mounted on a cone frustum.etthe radius of the top and bottom are 14cm and 7cm. find thevolume and height of cone is 5.5 cm. find the volume of thewhere the height of toy is 10.5cm.

10.5 cm


69. A tent is in the shape of a cylinder surmounted by a conical top. it the height and diameterof the cylinder part are 2.1m and 4m respectively and the slant height of the top is 2.8m,find the area of the canvas used for making the tent find the cast of the canvas of the tentat the rate of Rs 500 per m2 Also find the volume of air enclosed in the tent.

70. In the adjoining figure, ABC is a right triangle, right anglod at A. find the area of shadedregion it AB = 6cm, BC= 10cm and o is the centre of the incircle of ∆ABC.(Take π= 3.14)

71. From solid cylinder of height 28cm and radius 12cm, a conical cavity of height 16cm, andradius 12cm, is drilled out. find (a) the volume (b) total surface area of remaining solid.

72. A conical vessel of radius 6cm and height 8cm is completely filled with water. A solid sphereis immersed in to it and its size is such that when it touches the sides, it is just immerged.Find the quantity of water over flows?

73. Two maths teacher observed six pillars to support shades of metrostation at Kashmiri Gate, which are of the shape as given in the figure.One of them said that the approximately height and radius ofcylindrical portion are 1 m and 1/2 m respectively.The total height ofthe pillar is approximately 3-1/2 m She asked the other teacher. howmuch money is required to construet one such pillar it the currentrate of construction is Rs 3000 per cubic metre.

74. An orange contains juice about 15% of its volume. Find approximately how many dozenoranges are required for a gathering of 50 people of each guest is to be served with 250 mljuice. Assuming that radius of each orange is 3.5m.

A

C

8 cm

6 cm

½ m

2½ m

1 m


MENSURATION (Answers)

1. 21cm

2. 16.5cm

3. r=cm

4. 76cm

5. 19.5 cm

6. 513 1/3sq.cm

7. 16.8cm

8. 154 cm2

9. 4:3

10. 1:8

11. 24

12. 3:1

13. 10m

14. 3:1:2

15. 1188 sq.cm

16. 4cm

17. 126

18. 2m

19. 2/r

20. 7cm

21. 27cm

22. π : √3

23. 2 r

24. 14cm

25. 150

26. 57 cm2

27. 1019/14 sq cm

28. 154 sq cm

29. 308 sq cm

30. 77sq.cm

31. 462 sq cm

32. 190 13/14 cm2

33. 60

34. 9:25


35. 38400 sq cm

36. 7:9

37. 812 sq cm

38. 70cm

39. 11cm,3cm

40 28.5 sq cm

285.5sq cm

41. Rs.1355.20

Rs. 1056

42. 72

43. (49 √3 – 77) cm2

44. 28.50 sq cm

45. 108/7 dq cm

46. 14cm

47. 4 minute 48 secend

48. 18,12

49. 12π

50. 6 sq cm

51. 112 sq cm

52. 2.5 cm

53. 4 cm

54. 1408 sq cm, 3850 cm3

55. 150

56. 15cm

57. 563.14 sq cm

[HINT: total Area= 2πR2 + 2πrh + π(R2– r2) – L×B

58. 8:7

59. No, 39.55 cm3

60. 17160

[HINT: to Find slant height of cone use sin 300]

61. 10

[HINT: volume of ice cream in one cone=volume of cone+ volume of hemi sphere]

62. 138.29 cm2

63. 338.17 cm2


[HINT. volume of Gulab jamun= volume of cylindrical postion+2× volume of hemisphere]

64. 98m

65. 732.12 cm3

66. 502.07cu.cm, 444.17 sq cm

67. 6cm,8cm,10cm

68. 29 26 cu cm

69. 44 sq m Rs 22000,34.57 cu.m

70. 11.44 sq.cm

HINT: join o to A, B and C are of ABC= ar OAB+ ar. OBC+ ar.OAC 24=1/2 AB×r +1/2 BC×r+1/2 × AC×r [r=2]

71. 10258 2/7 cu.cm, 3318 6/7 cm3

72. 113.14 cu.cm

[HINT: AB=10cm

AD× AC=AE2

73. Rs16 110

74. 39 Dozen


Circle

Key points

1. Tangent to a circle : It is a line that intersects the circle at only one point.

2. There is only one tangent at a point of the circle.

3. The proofs of the following theorems can be asked in the examination :

(i) The tangent at any point of a circle is perpendicular to the radius through the pointof contact.

(ii) The lenghts of tangents drawn from an external point to a circle are equal.

1 mark questions

1. In fig(i) PS = 8 cm, SR = 6 cm. What is the value of PQ ?

2. For fig (i) which one is true

(i) PS2 + SR2 = PR2

(ii) SR2+ RP2 = SP2

(iii) RQ2 + PQ2 = PR2

(iv) RQ2 + PS2 = SR2

3. In fig(i) if ∠RPS = y + 180 & ∠PRS = 24, what is the value of y.

4. A tangent is drawn to a circle of radius 8 cm from apoint which is 17 cm from the centre. What is thelength of the tangent.

5. If perimeter of DABC is 16 cm, what is the value ofAC + AE. (fig. 2)

6. In fig. (2) what is the perimeter of quadrilateral ACDE,if CD = 8 cm, AD = 17 cm.

7. In fig (3) PR = 7.6 cm. What is the length of QS.

8. Two tangents AB and AC are drawn to a circle fromoutside point A. If AB + AC = 14.8 cm. What is thelength of AB.

Fig. (1)

R

S

Q P

Fig. (2)

D

C

F

A

E

B

Fig. (3)

S

PQ

R


9. In fig (4) if ÐB = 300 what statement is true

(a) AC > CB, (ii) AC = CB, (iii) AC<CB

10. In fig. (5) ∠BCD = 1000, what is the measure of x

11. In fig. (5) if ∠BCA = 2x and ∠BAC = x, what is themeasure of x.

12. AB is tangent to O circle of length equalto radius BC. What is the measure of ∠BCA.

13. In fig (6) PB is diameter and Qp is tangent. What is thevalue of X ?

14. In fig (7), what is the length ofAD.

2 marks question

15. An incircle is drown touching the equal sides of an isosceles triangle at E&F. Show thatthe point D. Whee the circle touches the third side is the mid point of that side.

16. The length of tangent to a circle of radius 2.5 cm from external point P is 6cm. Find thedistance of P from the nearest point of the circle.

Fig. (4)

D

A

C B30

0

Fig. (5)

C

B

A

D

x

Fig. (6)

P

A

B

Q

x y

350

y

C

Fig. (7)

17 cm

8cm

3cm 4cm

B A D

E F


17. TP & TQ are the tangents from the external point of circle with centre O. If ∠DPQ = 300,then find the measure of ∠TQP

18. In fig. (8) AP = 4 cm, BQ = 6 cm and AC = 9 cm. Find the semiperimeter of ∆ABC

19. In fig. (9) find SR

20. In fig. (10) A semi circle is drown outside thesemicircle. Diameter BE of the smaller semi circle ishalf the radius BF of the bigger semi circle. If radiusof bigger semi circle is 4√3 cm. Find the length oftangent AC from A on a smaller semi circle.

21. If ∠P = 1200. Prove that OP = 2AP (fig. 11)

3 Marks Question

22. In fig. (12) find x, if PA and PB are tangents from P.

A

C

R

(Fig. 8)

9 cm

4cm

B6 cmQRAS

QCP

4.2

cm

BD

(Fig. 9)

6.5 cm

7.3 cm

D EB

F

C

A

(Fig. 10)

Fig. (11)

O

A

P

Fig. (12)

O

A

Px

x–1

x+1

B


23. PA & PB are two tangents to a circle from an externalpoint P. Prove that perimeter(∆PAB) = 2PA or 2 PB

24. In fig. (13) the radii of two concentric circles are 5cm and 7 cm. Lengthof tangent from P to bigger circleis 15 cm. Find length of tangent to smaller circle.

25. In fig (14) AB = 13 cm, BC = 7 cm, AD = 15 cm. FindPC.

26. An incircle in drawn touching the sides of a right angledtriangle the base and perpendicular of the triangle are 6 cm and 2.5 cm respectively. Findradius of the circle.

27. AP and AP are tangents to a circle with centre O from external point A of ∠OPQ = 300,prove that ∆APQ is equiangular triangle.

28. On the side AB as diameter of a right angled triangle ABC a circle is drawn intersectinghypotenuse AC in P. Prove that PB = PC

29. In fig. (15) AB = AC, D is the mid point of AC and BD is thediameter of the circle.

Prove that AE = 14

AC

30. In fig. (16), PQ is tangent AP = 8 cm,

length of tangent exceeds the radius by 1.

Find radius of the circle.

Fig. (13)

O

B

P

A

Fig. (14)

S

R

P

C

D

Q

4 cm

A

B

E A

D

C

B

Fig. (15)Q

BA

Fig. (16)

PO


31. A chord AB of 8 cm is drawn in a circle with centre O of radius 5 cm. Find length oftangents from external point P to A & B.

32. PQ is diameter of a circle and PR is the chord S.T. ∠RPQ = 300. Tangent at R intersects PQproduced at S. Prove that RQ = QS.

33. Find r in fig. 17.

34. PA and PB are two tangents to a circle with centre with centre O fromexernal point P such that OP = 2 radius prove that ∆ABP isequilateral triangle

35. In fig. (18) if r = 30 cm. Findperimeter (∆ABC)

6 Marks questions

36. Prove that the tangents at any point of a circle is perpendicular to the radius through thepoint of contact.

Rider

1. Prove that in two concentric circles the chord of thelarger circle which to includes the smaller circle isbisected at the point of contact.

2. If PQ is tangent from Q and PB in diameter what isthe value of x (fig. 19).

Fig. (17)

S

R

P

C

D

Q

r

A

B

23 cm

5 c

m

29 cm

Fig. (17)

35√

3 cm

35

√

A

C

B

Fig. (19)

P

A

B

Q x y

350

y


3. In fig. (20) prove that 2a + b = 900.

4. In fig. (21) PAT is tangent. Find value of x.

37. Prove that the length of tangents, drawn from an external point to a circle are equal. Usethe above result in the following :

1. In fig. (22) PA & PB are tangents from point P.

Prove that KN = AK +BN.

2. AB is diameter of a circle. P is a point on the circle through which a tangent L isdrawn. AC ⊥L and BD⊥L. Prove that AC = BD = AB

Answers

1. 4 cm

2. (i)

3. 24

4. 15

5. 8

6. 46 (15+8+8+15)

7. 3.8 cm.

8. 7.4 cm.

9. (ii)

10. 400

11. 300

12. 450

Fig. (20)D

B

O

A

yzo

f

C

x

B

A

550

P TFig. (21)

Fig. (22)

O

A

N

P

B

K


13. 350

14. 33 cm (5+3+8+17)

15. -----

16. 4 cm.

17. 600

18. 15 cm.

19. 5 cm.

20. 12 cm.

21. Hint - Use trignometry

22. 4 cm.

23. ----

24. 2√66

25. 5 cm

26. r = 1 cm (h = 6.5 cm)

27. ----

28. ----

29. ----

30. r = 3 cm

31.203

cm.

32. -----

33. 11 cm.

34. -----

35. 32 cm.

36. (2) 350, (4) 550

37. -----


Chapter

Constructions

Key Points

1. Construction should be neat and clean and as per scale given in question.

2. Steps of construction should be provided only to those questions where it is mentioned.

3. Marks Questions

1. Draw a line segment AB = 8 cm. Take a point P on AB such that AP : PB = 3:5 (This

question can be asked by giving APPB

= 35

or 5×AP = 3×PB.

2. Draw a line segment xy = 10 cm. Take a point A on xy such that XAXY

= 25

(This question

can be asked by giving XAAY

= 23

or AYXY

= 35

.

3. Draw a triangle PQR such that PQ = 5 cm, QR = 7.4 cm and ∠PQR = 740. Now construct

a ∆ P’QR’ ~ ∆ PQR with its sides 32

times of the corresponding sides of DPQR. (32

times

can be written as 1.5 times.)

4. Draw an isoceles ∆ ABC with AB = AC and base AC = 7 cm and vertical angle 1200. Construct

∆AB'C'~∆ ABC with its sides 113

times of the corresponding sides of ABC.

5. Construct ∆D'EF'~∆DEF with its sides equal to 13

rd of the corresponding sides of ∆DEF

where ∠E = 900, DE = 6 cm, EF = 8 cm.

6. Draw a ∆PQR in which RQ = 7.0cm, ∠P = 600 altitude from P on RQ is 5 cm. Draw asimilar ∆PQ'R' such that RQ = 8.5 cm.

7. Draw a right angled triangle in which base is 2 times of the perpendicular. Now constructa triangle similar to it with base 1.5 times of the original triangle.

8. Draw an equilateral triangle ABC with side 5.5 cm. Now draw ∆AB'C' such that ABA'B

= 12

.

Measure A’B. What type of triangle is AB'C'.

9. Draw circle of radius 4 cm with centre O. Take a point P such that OP = 6 cm. Draw twotangents PX and PY to the circle from P. Find perimeter of the triangle PXY.

10. Draw a circle of radius 3.5 cm. Now draw a set of tangents from external point P such thatthe angle between the two tangents is half of the central angle made by joining the point ofcontacts to the centre.

11. Draw a circle with centre P and radius 3 cm. Now draw tne tangents OA and OB fromexternal point O such that ∠AOB = 450. What is the value of ∠AOB + ∠APB.

12. Draw a line segment AB = 9 cm. Draw circles of radius 5 cm and 3 cm from A and B


respectively. Draw tangents to each circle from the centre of the other.

13. Draw a diameter AB of a circle of any radius with centre O. Now draw a radius OP⊥AB.Though P draw a tangent. Is the tangent parallel to the diameter.

14. Draw a circle of radius OP = 3 cm. Draw ∠POQ = 550. Such that OQ = 5 cm. Now drawtwo tangent from Q to the given circle. If P one of the point of contact of the tangent to thecircle.

15. Draw a circle with centre O and radius 3.5 cm. Take a horizontal diameter. Extend it toboth the sides to point P&Q such that OP = OQ = 7 cm. Draw tangents PQ and QB oneabove the diameter and the other below the diameter. Is PQ//BQ.

16. Draw a right angled isocles triangle with equal side 5 cm. Draw a circle such that the twoequal sides of the triangle are tangents to the circle and the end poins of the hypotenuseare points of contacts.

17. Draw any triangle ABC a point P outside it. Join P to vertices of the triangleby dottedlines. On these doted lines draw ∆DEF similar to ∆ABC.

18. Draw a circle of radius 3 cm with centre O. Take point P such that OP = 5 cm. PO cuts thecircle at T. Draw two tangents PQ and PR. Join Q to R through T. Draw AB parallel to QRsuch that A and B are points on PQ and PR respectively. Check whether PQ + PR =perimeter (∆PAB)

PRACTICE PAPER I


Time : 3 hours M.M. : 80

General Instructions :-

(i) All questions are compulsory.

(ii) The question paper consists of thirty questions divided into 4 section – A, B, C and D.

Section A comprises of ten questions of 01 mark each. Section B comprises of five

questins of 02 marks each, Section - C comprises of ten questions of 03 marks each

and section D comprises of five questions of 06 marks each.

(iii) All questions in Section A are to be answered in one word, one sentence or as per the

exact requirement of the question.

(iv) There is no overall choice. However, internal choice has been provided in one question

of 02 marks each, three questions of 03 marks each and two questions of 06 marks

each. You have to attempt only one of the alternatives in all such questions.

(v) In question on construction, drawings should be neat and exactly as per the given

measurements.

(vi) Use of calculators is not permitted.

1. Write 3276 as a product of prime factors.

2. For what value of ‘k’, the quadratic equation

2x2 + kx+ 2 = 0, has equal roots?

3. Write a polynomial, whose zeros are 2 and -3.

4. If nth term of an A.P. is 2–5n, write its common difference.

5. If sin(2x–150) = 12

, then write the value of 'x'.

6. From a well shuffled pack of cards, a card is drawn at random, write the probability ofgetting a face card.

7. A cone and a hemisphere have equal bases and equal volumes. Write the ratio of theirheights.

8. The perimeter of two similar triangles ∆PQR and ∆ABC are respectively 36 cm and 24 cm.If AB = 10 cm, then write the value of PQ.

9. In fig., diameter AB = 12 cm & AP = PQ = QB.

Write the area of shaded region.

P QBA


10. In fig., PQ || BC and APPB

= 13

,

write the value of area of APQ

are of ABC

∆∆

.

Section-B (Two marks each)

11. Evaluate

2sin 62 + 2sin 28

sec58 . sin32

2 0 2 0

0 0 – 5(sec 72 – cot 18 )

3 tan15 tan35 tan45 tan 55 tan 75

2 0 2 0

0 0 0 0 0

12. Find the ratio in which the point (x, 2) divides the join of p(–3, 5) and Q (5, 1). Also findthe value of 'x'

13. Find all the zeros of the polynomial x4 + x3 – 9x2 – 3x + 18, if its two zeros are √3 and –√3.

14. T is a point on the side QR of a ∆PQR, such that ∠PTR = ∠QPR, show that RP2 = RQ. RT.

or

∆ABC is right angled at B and D is mid point of BC, prove that

AC2 = 4AC2 – 3AB2

15. A die thrown once. Find the probability of getting

(i) a number less than 4

(ii) a prime number.

Section - 'C' (Three mark each)

16. Prove that 1 + cos

1 – cos

θ

θ = cosecθ + cotθ

or

Prove that

2sec2θ – sec4θ – 2 cosec2θ + cosec4θ = cot4θ – tan4θ

17. How many terms of the AP,, 47, 43, 39, ........ be taken so that their sum is 299 ?

18. Show that 3 + 2 √5 is an irrational number.

19. Draw the graphs o the equation 3x + 2y –12 = 0 and x - y + 1 = 0. Find the coordinates ofthe vertices of the triangle formed by the lines and the y-axis. Also find the area of thetriangular region.

20. Construct a ∆PQR in which PQ = 6.5 cm, ∠Q = 600 and QR = 5.6 cm. Also construct a

A

P Q

B C


triangle PQ'R' similar to ∆PQR, whose each side is 32

times the corresponding side of the

triangle PQR.

21. Show that the points A(2, –2), B(14, 10), C(11, 13) and D(–1, 1) are the vertices of arectangle.

22. Two poles of height 'a' and 'b' (a>b) are 'c' metres apart. Prove that the height of the pointof intersection of the lines joining the top of each pole to the foot of the opposite pole is

ab

a+b.

or

Two triangle ∆ABC and ∆DBC are on the same base BC and on the same side of BC inwhich ∠A = ∠D= 900. If CA and BD meet each other at E, show that

AE. EC = BE. ED

23. Find the area of the shaded part of figure. Where PQRS is a square of side 28 cm. Region Xis a semi circle on RS as diameter. Region Y and Z are quadrants of circles with centres atP and Q respectively. T is the mid point of PQ.

24. Solve the following equation by completing the perfect square.

2x2 – 3x + 1 = 0

25. Find the area of the triangle formed by joining the mid-points of the sides of the triangle,whose vertices are (0, –1), (2, 1) and (0, 3). Find the ratio of this area to the area of thegiven triangle.

Section-D (Six marks each)

26. Prove that the ratio of the areas of two smilar triangles is equal to the ratio of squares oftheir corresponding sides. Using the above result, prove the following.

In a ∆ABC, XY is parallel to BC and it divides ∆ABC in to two equal parts of equal area.

Prove that BX

AB =

√

√

2 –1

2.

Or

PT Q

X

S R

Y Z


The lenghts of tangents drawn from an external point to a circle are equal. Prove it.

Use the above theorem.

Prove that, if all sides of a parallelogram touches a circle, then it is a rhombus.

27. A straight high way leads to the foot of a tower. A man standing at the top of the towerobserves a car at an angle of depression of 300, which is approaching the foot of the towerwith a uniform speed. After 6 seconds, the angle of depression of the car is found to be 600.Find the time taken by the car to reach the foot of the tower.

or

From the top of 12 metres high building the angle of elevation of the top of a tower is 600

and angle of depression of its foot is 450. Find the height of the tower.

28. The mean of the following frequency distribution is 62.8 and the sum of all frequencies is50. Find the values of missing frequencies x and y.

Class 0-20 20-40 40-60 60-80 80-100 100-120

Frequency 5 8 x 12 y 8

29. A bucket made up of a metal is in the form of a frustum of a cone of height 16 cm withdiameters of its lower and upper ends as 16 cm and 40 cm respectively. Find the volume ofthe bucket. Also find the cost of the bucket if the cost of material sheet used is Rs. 40 per100 cm2. (Use π = 3.14)

30. A fast train takes 3 hours less a slow train for a journey of 600 km. If the speed of the slowtrain is 10km/hr. less than that of the fast train, find the speed of the two trains.

Answers

1. 22× 32 × 7 ×13

2. k = ± 4

3. x2 + x – 6

4. – 5

5. x = 22.5

6.3

13

7. 2:1

8. 15 cm

9. 12πcm2

10.1

16

11.1

3

12. 3:1, x = 3


13. √3, –√3, –3, 2

15.1

2,

1

2

17. 13

19. (–1, 0), (2, 3), (0, 6) 5 square units

23. 168 cm2

24. 1, 1

2

25. 1:4

27. 3 sec. or 12 (√3+1)

28. x = 10, y = 7

29. Rs. 766.46 (Approx.)

30. 50 km/h, 40 km/m

Practice Paper - 2 (With Solution)


Section - A

1. Write down the decimal representation of 77210

.

2. Write a quadratic equation whose sum of roots is zero and product is –15.

3. Which term of an A.P. 13, 20, 27, ......... is 35 more than 19th term.

4. Write the zero of the polynomial p(x) whose graph is given below.

5. Write the value of cotθ from the adjoining figure.

6. In the word "MISSIPPI". What is the probability of getting the letter 'I'.

7. Write the median if the mode and mean of the data is 36.8 and 35.3 respectively.

8. Write the perimeter of quadrant of a circle of radius 7 cm.

9.

In the figure, ∠Q = ∠R = 500. Express x interms of a, b and c ? Where a, b and c are thelength of QP, QR and RS respectively.

24 cm

60 cm7 cm


10. If two tangents of 7 cm each are inclined at an angle of 900. Write the radius of that circle;which have that tangents.

Section-B

11. AB is a line segment, where is A(7,1) and B(3, –4), intersected by x – axis at point P. FindAP:PB.

12. In given figure DE||BC, find the value x

13. Find the value of m if following equations have no solutions.

(m+1) x + 2y = 5

3x + y = 3

14. Find the value of tan600 geometrically.

or

If √3 tanθ = 3 sinθ

Find the value of sin2θ – cos2θ

15. A box contain some cards bearing the numer 3, 4, 5, 6 ...... 19. If a card is taken out from thebox at random. Find the brobability that the card drawn bears –

(i) an even number

(ii) a number divisible by 3 or 5

Section - C

16. Prove that √11 is an irrational number.

17. Prove that :

Secθ(1–sinθ) (secθ + tanθ) = 1

18. Solve the system of linear equation

ax + by = c

bx + ay = 1 + c

or

A

B C

E

x

x–1x–

2

D

x+

2


Draw the graph of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Calculate the areabounded by these lines and x – axis.

19. The mid point of the sides of a triangle are (1, 2), (3, –1) and (5, 0). Find the vertices oftriangle.

20. Find the zeroes of the polynomial if two of its zeroes are –√3 and √3 and the given polynomialis 2x4 –3x3 –5x2 + 9x – 3

21. How many terms of the A.P.

43, 39, 35, ..........

Must be taken to give a sum of 252 ?

22. If A(4, –8), B(3, 6) and C (5, –4) are the vertices of ∆ABC AD is the median and P is a point

on AD such that APPD

= 2. Find the co-ordinate of point P.

or

Prove that only one line passes through (7, –2), (5, 1) and (3, 4)

23. In the figure a circle is inscribed in a quadrilateral ABCD in which∠B = 900. If AD = 23 cm, B = 29 cm and DS = 5 cm, find theradius of the circle.

24. Construct a ∆ABC in which AB = 4 cm, BC = 6 cm and ∠ABC =

600. Construct ∆A'BC'~∆ABC such that sides are 34

times the

corresponding sides o given triangle.

25. ABCD is a rectangle in which AB:BC = 2:1 M isthe mid point of AB MD and MC are one fourthof the circles with centres A and B respectively.Find the ratio of the area of the rectangle ABCDand the shaded portion.

or

The given figure, OACB represent a quadrant of a circle ofradius 3.5 cm with centre at O.

Calculate the area of the shaded region if OD = 2 cm.

D

C

B

A

R Q

P

Or

r

S


SECTION - D

26. An aeroplane with flying at a height of 4000 m from the ground passes vertically aboveanother aeroplane at an instant when the angles of elevation of the two aeroplanes fromthe same point on the ground are 600 and 450 respectively. Find the vertical distancebetween the two aeroplanes at that instant.

27. A piece of cloth costs Rs. 4000. If the piece was 5 m longer and each metre of cloth costs Rs.4 less, than cost of the piece will remain unchanged. How long is the piece and what is theoriginal rate per metre ?

or

An eagle is sitting on the top of a tower, which is 9 m high. From a point 27 m away fromthe bottom of the tower, a snake is coming towards his hole which is at the base of thetower. Eagle flew from the top to catch the snake. If their speed are equal, at what distancefrom the hole is snake caught ?

28. The weight of 60 students taken at random from a school is represented in the followingtable.

Weight 0-10 10-20 20-30 30-40 40-50 50-60 Total

Frequency 3 p 20 15 q 5 60

If the medium weightis 28.5, find values of p and q.

29. In a triangle of the square on one side is equal to the sum of squares on the other twosides, prove that the angle opposite to the first side is a right angle. Using this prove thefollowing.

In a quadrilateral PQRS, if ∠Q = 900, PS2 = PQ2 + QR2 + RS2. Prove that ∠PRS = 900

30. A conical hole is drilled in a circular cylinder of height 15 m and radius 8 cm. The heightand base radius of the cone are also the same as of cylinder. Find

(i) Volume of the remaining solid figure.

(ii) Total surface area of the remaining solid figure after removing cone,.

or

A nurse fills syrup in an inverted frustum shaped flask up to the level. The radii and slantheight of the flask are 4 cm, 10cm and 10 cm respectively. She has to give this syrup incylindrical shaped cups of height and radius 1.3 cm and 1 cm respectively to the childrenin children ward of a hospital. If each child has to take two cups. How many children candrink the syrup at a time.


SECTION - A

Answer (with solution)

Note : A student can write; ony the answer for 1 mark questions to get full one mark.

1.77210

= 1130

by long division :

1130

= 0.36666.......

77210

= 0.36 Ans.

2. Quadratic equation :

x2 – (sum of roots) x + product of roots = 0

x2 – (o) x + (–15) = 0

x2 – 15 = 0

3. d = 20 – 13 = 7

7 × 5 = 35

19th term + next 5 term = 24th term.

Ans. : 24th term.

4. x = 2

5. AB = √242 + 72

AB = √625 = 25

Cot θ = 6025

125

Cot θ = 125

6. P (I) = 38

7. Mode = 3 median – 2 mean

36.8 = 3 median – 2× 35.3

107.43

= median

Median = 35.8 Ans.


8.

Perimeter of the quadrant

= 1

4 (2πr) + r + r

= 1

2 ×

22

7 × 7 + 7+ 7

= 11 + 14 = 25 cm. Ans.

9. ∠Q = ∠R, TR||PQ and So ∠P = ∠T

∆STR ~ ∆SPQ

TRPQ =

SR(QR + RS) (KM= QR + RS)

xc

= a

b +c

x = ac

b +c Ans.

10. Quadrilateral formed by two radius (from the point of contact tangents) and two tangents(inclined at an angle 900 is a square so radius = length of tangents

r - 7 cm

SECTION - B

11.A Bp (x, o)

(3, –4)1:K(7, 1)

Let P (x, 0) divides the AB in K : I

y = ky + y

k + 12 1

o = – k + 1k + 1

o = – 4k + 1

4k = 1

k = 1

4

AP : PB :: 1 : 4 Ans.

7 cm

7 c

m


12. DE || BC

According to basic proportionality theorem –

ADBD

= AEEC

x(x –2) –

(x + 2)(x –1)

x2 – x = x2 – 4

x = 4 Ans.

13. Equations are in the form of ax + by + c = o as follows :

(m +1) x + 2y – 5 = 0

3x + y – 3 = 0

condition for no solution :

a

a1

2

= b

b1

2

≠ c

c1

2

a2 = m + 1 b

1 = 2 c

1 = –5

a2 = 3 b

2 = 1 c

2 = –3

m + 13

= 2

1

m + 1 = 6

m = 5 Ans.

14. Let �∠XOP = 600 PM⊥ OX and MQ = OM = a

⇒ ∆OMP ≅ ∆QMP (SAS)

so ∠MOP = ÐMQP = 600

DPOQ is an equilateral triangle

OP = OQ = 2a

MP = √OP2 – OM2

= √4a2 – a2

= √3 a

tan 600 = MPOM

= √3a

a = √3 Ans.

or

√3 + tanθ = 3 sinθ

P

P Qa X

2a

M


√3 sin

cos

θθ

= 3 sinθ

cosθ = √3

3

Now, sin2θ – cos2θ

= 1 – cos2θ – cos2θ

= 1– 2 cos2θ

= 1– 2 (√3

3)2

= 1 – 2 × 39

13

= 1 – 23

= 13

Ans.

15. Total No. of cards = 17

(i) Even number (from 3 to 19)

= (4, 6, 8, 10, 12, 14, 16, 18)

⇒ 8 = No. of favourable outcomes.

p(even number) 817

(ii) Number divisible by 3 or 5

(3, 5, 6, 9, 10, 12, 15, 18)

⇒ 8 = No. of favourable outcomes.

p ( no divisible by 3 or 5) = 817

SECTION - C

16. Let √11 is rational

that is, we can find integers a & b (C ≠ o)

such that √11 = a

b, a and b are coprime.

or b√11 = a

Squaring both sides, we get 11 b2 = a2 .............. (i)


∴ a2 is divisible by 11 ⇒ a is also divisible by 11 so we can write a = 11x ................ (2)

⇒ 11 b2 = 121x2

⇒ b2 = 11x2

⇒ b2, is divisible by 11

⇒ b, is divisible by 11

⇒ 11, is divisible common factor of a & b.

But this contradicts the fact that a & b are coprime.

∴ √11 is not rational

so √ is irrational

17- L.H.S. = secθ(1–sinθ)(secθ+tanθ)

=1

cosθ (1–sinθ) (

1

cosθ +

sin

cos

θθ

)

=1

cosθ (1 – sinθ)

(1+ sin

cos

θ)θ

= (1– sin

cos2θ

θ θ) (1+sin )

=1– sin

cos

2

2

θθ

= cos

cos

2

2

θθ

= 1 RHS

18. By cross multiplication method.

b

a

c

1 + c

a

b

b

a

x

b(1+c) – ac = y

cb – a (1+c) = –1

a – b2 2

(i) (ii) (iii)

from (i) and (iii)x

b + bc – ac =

– 1

a – b2 2

x = – b – bc + ac

a – b2 2 =

– b

a – b2 2 +

c (a – b)

a – b2 2

= –b

a – b2 2 =

c

a + b

from (ii) and (iii) = y

cb – a (1+c) = – 1

a – b2 2


y = – bc + a + ac

a – b2 2 =

a

a – b2 2 +

c ( a–b)

a – b2 2

= a

a – b2 2 +

c

a + b

x = –b

a – b2 2 +

c

a + b , y =

a

a – b2 2 +

c

a + b

or

From x – y + 1 = 0

x –1 0 2

y 0 1 3

From 3x + 2y – 12 = 0

x 0 2 4

y 6 3 3

Area ∆ACF = 1

2 × 5 × 3 = 7.5 square units.

19 In ∆ABC

P is the mid point of AB

1= a + x

2 ⇒ a + x = 2 (i)

2= b + y

2 ⇒ b + y = 4 (ii)

Q, is the mid point of AC

a + p

2 = 3 ⇒ a + b = 6 (iii)

b + q

2 = –1 ⇒ b + q = – 2 (iv)

R, is mid point of BC

x + p

2 = 5 ⇒ x + p = 10 (v)

y + p

2 = 0 ⇒ y + q = 0 (vi)

Adding (i), (iii) and (v)


a + x + a + p + x + p = 18 ⇒ a + x + p = q (vii)

Adding (ii), (iv) and (vi)

b + y + b + q + y + q = 2 ⇒ b + y + q = 1 (viii)

from (i) and (vii) p = 7

from (iii) and (vii) x = 3

from (v) and (vii) a = –1

from (ii) and (viii) q = 3

from (iv) and (viii) y = 3

from (vi) and (viii) b = 1

∴ Vertices of ∆ABC are A(–1, 1), B(3, 3), C (7, –3)

20. –√3 and √3 are zeros

∴ (x + √3) (x – √3) = x2 – 3 is a factor of the polynomial.

∴ x2 – 3) 2x4 – 3x2 – 5x2 + 9x – 3 (2x2 – 3x + 1

2x2 – 6c2

– 3 x3 + x2 + 9x – 3

± 3 x3 ± 9x

x2 – 3

– x2 ± 3

x

2x4 – 3c3 – 5x2 + 9x – 3 = (x2 –3) (2x2 – 3x + 1)

= (x2 – 3)(2x2 – 2x – x +1)

= (x2 –3) [2x–1) (x–1)]

∴ Zeros of polynomial are √3, – √3, 1

2, 1

21. Sn

= 252, a = 43, d = – 4

Sn

= n

2 [2a + (n–1) d]

252 = n

2 [2×43 + (n–1) (–4)]

252 = n

2 [86 – 4n +4)


252 = n

2 [90 –4n)

252 = n (45 – 2n)

or 2n2 – 45n + 252 = 0

(2n –21) (n –12) = 0

n = 12, = 21

2

rejecting n = 21

2, We get n = 12

22. AD is median

⇒ D, is the mid point of BC

∴ Co-ordinate of D is (4, 1)

But AP

PD =

2

1

∴ coordinate of P = (2 × 4 + 1× 4

2 +1]

2 × 1 + 1× –8

2 +1)

= ¼8 + 4

3]

2 – 8

3½

= ¼12

3]

–6

3½

= ¼4] &2½

or

If a line passes through three points, then

area of the triangle = 0

Area of the triangle

= 1

2 [x

1 (y

2 – y

3) + x

2 (y

3 – y

1) + c

3 (Y

1 – y

2)]

= 1

2 [7, (1–4) + 5 (4– (–2) + 3 (–2–1)]

= 1

2 [7×–3 + 5 × 6 + 3 × – 3)

= 1

2 [–21 + 30 – 9] = 0


∴ Points are collinear.

⇒ A line passes through ¼7] &2½] ¼5] 1½ and ¼3] 4½

23- In Fig. OP⊥BC and OQ ⊥BA

∠B = 900 , OP = OQ = r

⇒ OPBQ is a square

BP = BQ = r

DR = DS = 5 cm

AR = AD – DR = 23 – 5 = 18 cm

AQ = AR = 18 cm

BQ = AB – AQ = 29 – 18 = 11 cm

∴ r - 11 cm.

24.

25. Let BC = x

AB = 2 x

∴ M is the mid point of AB

AM = MB - x

Area of the quadrant of the circle = 1

4 πx2

Area of two quadrants of teh circle = 1

2 πx2

Aera of sheded rigion = Area of the rectangle – Area of two quadrants

= 2x × x – 1

2πx2

YA

A'

B C

4l

6 C'60

0

B1

B2

B3

B4 X

cm

cm


= 2x2 – 1

2 ×

22

7x2

= 2x2 – 11

7x2 =

3

7x2

Ratio is = 2x2 : 3

7x2 or 14:3

or

∠AOB = 900 radious of OACB = 3.5 cm.

Area of OACB = 90

360 ×

22

7 × 3.5 × 3.5

= 9.625 cm2

∆AOD is right angle :

∴ Area of ∆AOD = 1

2 × AO × OD =

1

2 × 3.5 × 2

= 3.5 cm2

Area of shaded region = 9.625 – 3.5

= 6.125 cm2

SECTION - D

26. In ∆ADC

tan600 = AC

DC

⇒ DC = 4000 3

3

√

In ∆BDC

tan450 = BC

DC

⇒ BC = DC

BC = 2306.7 m

⇒ AB = (4000 – 2306.7) m

= 1693.3 m

27. Total cost = Rs. 400

A

B

C D60

0

450

4000


Let length of the cloth piece = x m.

Cost / m = Rs. 400

x

New Cost / m = Rs. ¼400

x & 4½

New length = (x + 5) m

∴ ¼ 400

x & 4½ (x+5) = 400

x2 + x – 500 = 0

x = – 25, 20

rejected = – 25, length of the piece = 20 m.

and cost /m = Rs. 400

20 = Rs. 20

or

A : position of eagle

B : hole

C : point, where eagle caught snake

D : Position of snake

In ∆ABC

AB2 + BC2 = AC2

⇒ (9)2 + x2 = (27 – x)2

⇒ 81 + x2 = 729 + x2 – 54x

⇒ 54 x = 729 – 81

⇒ x = 12 m

28- Class-internal Frequency Cummulative frequency

0&10 5 5

10&20 p 5 + p

20&30 20 25 + p

30&40 15 40 + p

40&50 q 40 + p + q

50&60 5 45 + p + q

–b

a – b2 2

9m

A

B x

27–x

27 m

27–x


60

45 + p + q = 60

p + q = 15

Median class is 20 – 30

l = 20, f = 20, cf = 5 + p, h = 10, N = 60

Median = l +

N

2– cf

f × h

median = 20 + [

60

20– (5 + p)

20] × 10

8.5 = 25 – p

2

17 = 25 – p

p = 8, q = 7

29. Figure, given and proof

PS2 = PQ2 + QR2 + RS2 .............. (i)

In ∆PQR

PQ2 + QR2 = PR2

from (i)

PS2 = PR2 + RS2

∴ ∠PRS = 900

30. Slant height of the cone = √82 + 152

= 17

¼1½ Volume of the cylinder = πr2h

= 960π cm3

Volume of conical hole = 1

3 πr2h

= 320 π cm3

Volume of remaining solid = 960 π – 320 π

= 640 π cm3

¼2½ Curned surface area of the cylinder =2πrh

= 240 π cm2

P

Q R

S


Curved surface area of cone = πrl

= 136 π cm2

Area of base of the cylinder = πr2 cm2

= 64πcm2

Surface area of remaining solid = Curved surface area of the cylinder + area of thebase + curved surface area of the cone

= 240π + 64π + 136π cm2

= 440 πcm2

or

Given r = 4 cm, R = 10 cm l = 10 cm

∴ l = √h2 + (R–r)2

⇒ h = 8 cm

Volume of the flask = 1

3πh (R2 + r2 + Rr)

= 416π cm3

Volume of cylinderical cup = πr2h

= 1-3 π cm3

∴ No. of children = Volume of flask

2 × volume of cylinderical cup

= 416

2 × 1.3

π

π

= 160

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X__Maths (ENG)__300__399.pmd

XMaths (ENG)300__399.pmd