MATHEMATICS - Badhan Educationbadhaneducation.in/assignment/class 10th/math/class... · Web viewFind the value of k for which the G.C.D. of MATHEMATICS CLASS X RATIONAL EXPRESSIONS

MATHEMATICS

CLASS XCIRCLE

MISCELLANEOUS EXERCISE

1. In fig 1.1, AB is a diameter of a circle C(O, r) and radius OD AB.If C is a point on arc DB. Find

2. In fig 1.2, O is the centre of the circle. Calculate(i)

3. In fig. 1.3, O is the centre of the circle and AOB = 70°, Find 4. In fig 1.4, O is the centre of the circle. Determine.

(i) 5. In fig . 1.5, given that AB = AC and Determine 6. In fig. 1.6, O is the centre of the circle. Determine 7. Two chords AB and CD intersect inside a circle at point O, Prove that

8. Two lines OAB and OCD drawn from an exterior point O to a circle intersect the circle at A and B, and at C and D respectively. Prove that

9. AB and XY are parallel chords of a circle. AY intersects BX at O. Prove OX = OY.10. AB is a chord of a circle whose centre is O. If p is any point on the minor arc AB, prove that

11. In a quadrilateral ABCD, in which AB = AC = AD, show that See fig. 1.8.

12. In fig. 1.9, ABC, AEG and HEC are straight lines. Prove that are

Graphics By:- Pradeep Tokas 1 Written By:- Raj Kumar Badhan

A BO

DC

32°

47°

C

A B

O

E

O

A

C

B 68°

Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 1.4

68°

A P

B C

PO

QRT

S 52° 25°

O

C B

A D

Fig. 1.5 Fig. 1.6 Fig. 1.7

supplementary.13. In a circle with centre O, chords AB and CD intersect inside the circumference at E. Prove that

.14. I is the in centre of ABC . AI when produced meets the circumcircle of ABC in D. If BAC =

66° and = 80°. Calculate [See fig 1.10](i) (ii) (iii) BID

15. Fill up the blanks with appropriate word.(i) The set of all those points in a plane that ate at a given constant distance from a given fixed

point is called a ……….(ii) Circles having the same centre are called ……….(iii) The line drawn from the centre of a circle, perpendicular to a chord … the chord.(iv) The perpendicular bisector of a chord of a circle passes through the ………(v) A line-segment joining two points on the circle passes through the ………(vi) Two circles are said to be congruent, if and only if their …… are equal.(vii) If two arcs of a circle are congruent, then the corresponding chords are ………(viii) Equal chords cut off …… arcs.(ix) Perpendicular bisectors of two non-parallel chords of a circle intersect each other at the

………… of the circle.(x) If two circles intersect in two distinct points, the line joining their centres is ………… of their

common chord.(xi) In a right triangle, the circumcentre is the ………… of the hypotenuse.(xii) For an equilateral triangle, at the circumcentre coincides with …………(xiii) If the semicircle drawn with one side of a triangle as diameter passes through the opposite

vertex, then the measure of the angle at the vertex is ……….(xiv) If opposite angles of a quadrilateral are supplementary, then the quadrilateral is called …….

(xv) There non-collinear points describe a ………. circle.

16. In Fig. 1.11, O is the centre of the circle and BC = AO. Which of the following relationship between

x and y is correct?

Ax = 3y always

Bx = 2y always

Cx = 4y always

Dx = 2y or x = 3y.


A

B D

C

2 3 1 4

A CB

HG

E B A

D C

66°

80°Fig. 1.8 Fig. 1.9 Fig. 1.10

17. In fig 1.12, is an isosceles triangle with AB = AC and Find

18. With reference to Fig. 1.14 determine angles a, b and c.19. ABCD is a cyclic quadrilateral. AE is drawn parallel to CD and BA is produced to F. If

(See fig. 1.15)

20. ABCD is a cyclic quadrilateral. AE is drawn parallel to CD and BA is produced to F. If

21. If two sides of a cyclic quadrilateral are parallel, prove that (i) the remaining two sides are equal and (ii) both the diagonals are equal.

22. In a cyclic quadrilateral ABCD, if 23. If the two sides of a pair of opposite sides of a cyclic quadrilateral are equal, prove that its diagonals

are equal.24. ABCD is a cyclic quadrilateral, in which AB and CD when produced meet in E and EA = ED. Prove

that(i) AD || BC (ii) EB = EC.

25. Circles are described on the sides of a triangle as diameters. Prove that the circles on any two sides intersect each other on the third side (or third side produced).

26. Prove that the circles described on the our sides of a rhombus as diameter, pass through the point of intersection of its diagonals.

27. Prove that the perpendicular bisectors of the sides of a cyclic quadrilateral are concurrent.28. Prove that the centre of the circle circumscribing the cyclic rectangle ABCD is the point of

intersection of its diagonals.29. The diagonals of a cyclic quadrilateral are at right angles. Prove that the perpendicular from the

point of their intersection on any side, when produced backward bisects the opposite side.30. ABCD is a cyclic quadrilateral AB and DC are produced to meet in E. Prove that 31. In fig.1.16, AB is a diameter of the circle. Also, AD Prove that CE = AD.32. In Fig. 1.17, AB = AD = PB and Determine (i)

Hence, or otherwise, prove that AP is a parallel to DB.


O X

D

CY

BA

B C

E

D A

50° 62°

a

c

b

43°

Fig. 1.12 Fig. 1.13 Fig. 1.14

CD

E

F25°

AB

85°

Fig. 1.15.

D C P D

33. AB and CD are two parallel chords of a circle, which are on opposite sides of the centre, such that AB = 10 cm, CD = 24 cm and the distance between AB and CD is 17 cm. Find the radius of the circle

34. In in fig. 1.18, is a right angle. A semicircle is drawn on AB as diameter. P is any point of AC produced. When joined, BP meets the semicircle in point D. Prove that

AB² = AC.AP + BD.BP.35. X and Y are centres of circles of radius 9 cm and 2 cm

and XY = 17 cm. Z is the centre of a circle of radius r cm. which touches the above circles externally. Given that write an equation in r and solve it for r.

36. is right angled at B. On the side AC, a point D is taken such that AD = DC and AB = BD. Find the measure of

37. In fig. 1.20. O is the centre of the circle. Determine (i) 38. AB is a chord of a circle, whose centre is O. If p is any point on the mirror arc AB, prove that

(See Fig. 1.20)

39. ABCD is a rectangle, Prove that the centre of the circle through A, B, C, D is the point of intersection of its diagonals.

40. In Fig. 1.22, AB = CD. Prove that BE = DE and AE = CE. Where E is the point of intersection of AD and BC.

41. In fig. 1.23, a diameter AB of a circle bisects a chord PQ. If AQ || PB, prove that chord PQ is also a diameter of the circle.


A E

D

D

A B

D C

PC

A B

Fig. 1.16 Fig. 1.17 Fig. 1.18

M O

B Q

P

A

75°

Fig. 1.19

Y

Z

X Q P

A Q

B D

E

A Q

P B

A Q

Fig. 1.20 Fig. 1.21 Fig. 1.22

A Q

P B

42. Two circles are drawn with sides AB and AC of a triangle ABC as diameters. The circles intersect at a point D. Prove that D lies on BC.

43. Two diameters of a circles intersect each other at right angles. Prove that the quadrilateral formed by joining their end points is a square.

44. ABCD is a cyclic quadrilateral with AD || BC. Prove that AB = DC.

45. Prove that bisectors of the sides of a cyclic quadrilateral are concurrent.46. Prove that the circle drawn with any side of a rhombus as a diameter, passes through the point of

intersection of its diagonals. 47. D is the mid point of side BC of an isosceles triangle ABC with AB = AC. Prove that the circle

drawn with either of the equal sides as a diameter passes through the point D.48. The circle passing through the vertices A, B and C of a parallelogram ABCD intersects side CD (or

CD produced) at the point P. Prove that AP = AD.49. Prove that the quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic.50. D and E are respectively, the points on equal sides AB and AC of an isosceles triangle ABC such

that AD = AE. Prove that the points B, C, E and D are concylic.51. The diagonals AC and BD of a cyclic quadrilateral ABCD

intersect at right angles at E. A line l through E and perpendicular to AB meets CD at F. Prove that F is the mid-point of CD. (see Fig. 1.24)(i) AD.BC = AE.DB (ii) AB.DE = CE.DB.

MATHEMATICSCLASS X

TANGENTS TO A CIRCLE


Fig. 1.23


1. AB is a chord of a circle with center O. The tangent at B meets Ao produced P. If

2. In given Fig. 2.1, PT is a tangent to a circle. If and m

3. Two Chords AB and CD of a circle intersect each other at O internally. If AO = 3.5 cm, CO = 5 cm and DO = 7 cm, find OB.

4. Two chords AB and CD of a circle intersect each other at P outside the circle. If AB = 5 cm, BP = 3 cm and PD = 7 cm, find CD.

5. Find the length of the tangent drawn from a point whose distance from the centre of the circle of the circle is 25 cm, Given that the radius of the circle is 7 cm.

6. In given fig. 2.2, to the circle intersect each other at A and B. The common tangne meet the two circles at C and D. Prove that

.7. If PA and PB are tangents from an outside point P such that PA = 10 cm and Find the

length of chord AB.[ Hint : PA = PB therefore

8. In given Fig. 2.3, circles C(O, r) and C (O’, r/2) touch internally at a point A and AB is a chord of the circle C (O, r) intersecting C(O’, r/2) at C. Prove that AC = CB.

9. In a right the perpendicular BD on the hypotenuse AC is drawn. Prove that(i) AC × AD = AB²(ii) AC × CD = BC²

10. Two circles touch internally at a point P and a chord AB to the circle of larger radius intersects the other circles at C and D. Prove that

11. Two circles C(O, r) and C(O’, s) touch each other at P, externally or internally. A line is drawn to pass through P intersecting the two circles at Q and R respectively. Prove that OQ || O’R.

12. In the given fig. 2.4, the diameters of two wheels have measures 2 cm and 4 cm. Determine the lengths of the belts AD and BC that pass around the wheels if it is given that belts cross each other at right angles. [Hint : Joint


T

P

A B

45°

C D

B

A

O O’

Fig. 2.2 Fig. 2.3

O B

C

A

O’

OO

OB

C D

Fig. 2.4

OO’. If P is the point of intersection of belt calculate PA, similarly

calculate PD. Also AD = CB]13. In the given fig. 2.5. Find x if 14. If PAB is a secant to a circle intersecting the circle at A and B and PT is a tangent. Prove that PA ×

PB = PT². On the above theorem prove the following. Two circles intersect each other at P and Q. From a point R on PQ produced, two tangents RB and RC are drawn to the two circles touchng then at B and C. Prove that RB = RC.

15. AB and CD are two chords which when produced meet at P and if AP = CP, show that AB = CD.(see fig 2.6)

MATHEMATICSCLASS X

LINEAR EQUATIONS IN TWO VARIABLES



AD

E B C

xA

B

P

CD

Solve each of the following (from 1 to 5)

1. 3x – 5y + 1 = 0 x – y + 1 = 0

2. 3.

4.

5. Solve for u and v:3(2u + v) = 7 uv 3(u + 3v) = 11 uv.

6. Solve :

and hence find a for which y = ax – 4.7. Solve graphically the following system of equation :

2x + y – 3 = 0, 2x – 3y – 7 = 08. Determine graphically the coordinates of the vertices of a triangle the equations of whose sides are y

= x, y = 2x, x + y = 6.9. Show graphically that the following system of equations has infinitely many solutions :

2y = 4x – 6, 2x = y + 3 10. Show graphically that the following system of equations in inconsistent :

3x – 5y = 206x – 10y = – 40

11. Solve graphically the following system of linear equations :2x – y = 24x – y = 8

Also, find the coordinates of the points where the lines meet the axis of x.12. Show that the following system of equations has no solution :

x – 2y = 6 3x – 6y = 0

13. For what value of K, the following equations are in consistent :x – 4y = 6

3x + ky = 514. Determine the value of for which the following system of equations has infinitely many solutions:

x + 3y = – 312 x + y =

15. Find the value of K for which the system of equations :8 x + 5y = 9Kx + 10 y = 15 has no solution.

16. There are two examination rooms A and B. If 10 candidates are sent from A to B, the number of students in each room is the same. If 20 candidates are sent from B to A the number of students A is double the number of students in B. Find the number of students in each rooms.

17. Ten years ago, father was twelve times as old as his son and ten years hence he will be twice as old as his on will be. Find their present ages.

18. A and B are friends and A is elder to B by 2 years. A’s father D is twice as old as A nad B is twice as old as his sister. If the ages of D and C differ by 40 years, find the age of A.


19. Two numbers are in the ratio 2 : 3. If 5 is added to each number, the ratio becomes 5 : 7, find the number.

20. Five years ago, A was twice as old as B and 10 years later A shall be twice as old as B. What are the present ages of A and B ?

21. Ram can row a boat 8 km downstream and return in 1 hour 40 minutes. If the speed of the stream is 2 km/hr, find the speed of the boat in still water.

22. A man has only 20 paisa coins and 25 paisa coins in his purse. If he has 50 coins in all totaling Rs. 11.25, how many coins of each does he have ?

23. 4 chairs and 3 tables cost Rs. 2100 and 5 chairs and 2 tables cost Rs. 1750, Find the cost of 1 chair and 1 table separately.

24. A man rowing at the rate of 5 km an hour in still water takes thrice as much time in going 40 km upstream and 55 km downstream. Find the speed of the stream and that of the boat in still water.

25. A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours it can go 40 km upstream and 55 km, downstream. Find the speed of the stream and that of the boat in still water.

26. In a two digit number, unit’s digit is twice the ten’s digit. If the digits are reversed, ew number is 27 more than the original number. Find the number .

27. A fraction becomes 4/5 if 1 is added to both numerator and denominator. If however 5 is subtracted from both numerator and denominator, the fraction becomes ½. What is the friction?

28. The ratio of incomes of two persons is 9 : 7 and the ration oftheir expenditures is 4 : 3. If each of them saves Rs. 200 per month, Find their monthly incomes.

29. A man sold a chair and a table together for Rs. 1520 thereby making a profit of 25% on chair and 10% on table. By selling them together for Rs. 1535 he would have made a profit of 10% on the chair and 25% on the table. Find the cost price of each.

30. Ved travels 600 km to his home partly by train and partly by car. He takes 8 hours if he travels 120 km by train and the ret by car. He takes 20 minutes longer if he travels 200 km by train and the rest by car. Find the speed of the train and the car.

31. A part of monthly exprenses of a family is constant and the remaining varies with the price of wheat. When the rate of wheat is Rs. 250 a quintal, the total monthly expenses of the family are Rs. 1000 and when it is Rs. 240 a quintal, the total monthly expenses are Rs. 980. Find the total monthly expenses of the family when the cost of wheat is Rs. 350 a quintal.

32. A person invested some amount at the rate of 12% simple interest and some other amount at the rate of 10% simple interest. He received yearly interest of Rs. 130. But if he has interchanged the amounts invested, he would have received Rs. 4 more as interest. How much amount did he invest at different rates ?

33. Students of a class are made to stand in rows. If 4 students are extra in arrow ,there would be 2 rows less. If 4 students are less in row, there would be 4 more rows. Find the number of students in the class.

34. The area of a rectangle gets reduced by 80 sq.units if its length is reduced by 5 units and the breadth is increased by 2 units. If we increase the length and the breadth by 5 units, the area is increased by 50 sq. units. Find the length and the breadth of the rectangle.

35. The ages of two girls are in the ratio of 5 : 7. Eight years ago their ages were in the ratio of 7 : 13. Find their present ages.

36. A part of monthly hostel charges are fixed and the remaining depend in the number of days one has taken food in the mess. When a student A takes food for 20 days, he has pay 1000 as hostel charges whereas a student B whose takes food for 26 days, pays Rs. 1180 on hostel charge. Find the fixed charge and the cost of food per day.

37. The taxi charges in a city comprise of a fixed charge together with the charge for the distance covered. For a journey of 10 kms, the charges paid are Rs. 75 and for a journey of 15 km, the charges paid are Rs. 110. What will a person have to pay for travelling a distance of 25 km?


38. An aeroplane takes 1 hour less for a journey of 1200 km if its speed is increased by 100 km/hr from its usual speed. Find its usual speed.

39. Ten years hence, a man’s age will be twice the age of his son. Ten years ago, the man was four times as old as his son. Find their presents ages.

40. Solve the following system of equation graphically. Also find the area of the triangle enclosed between lines and x – axis.

X + y = 5 , 3x – y = 3.

MATHEMATICSCLASS X

G.C.D AND L.C.M. OF POLYNOMIALS



Find the G.C.D. of each of the following (1-6) :

1. 2.

3. 4.

5. 6.

Find the L.C.M. of each of the following (7-9)7. 8.9.10. The G.C.D. of polynomials P(x) and Q(x) is x – 3.

If P(x) = (x – 3) (x² + x – 2) and Q(x) = x² – 5x , find the L.C.M. of P(x) and Q(x).11. The G.C.D. and L.C.M. of polynomials P(x) and Q(x) are

12. Find the L.C.M. pf 13. Find the G.C.D. and L.C.M. of polynomials

verify that p(x) × q(x) = G.C.D. × L.C.M.14. Find the G.C.D. of

15. Find the G.C.D. of 16. Find the LCM of the polynomials : 17. Find the values of a and b so that the polynomials P(x) and q(x) have (x + 1) (x – 4) as their G.C.D.

18 Find the values of a and b so that the polynomials P(x) and q(x) have (x – 1) (x + 4) as their G.C.D.

19. Find the G.C.D. of 20. Find the value of k for which the G.C.D. of

MATHEMATICSCLASS X

RATIONAL EXPRESSIONS



Reduce the following to the lowest term: (Q. 1 to 14)

23. Find the multiplicative inverse of the sum of


24. Which rational expression should be added to the multiplicative inverse of to get the

additive inverse of

25. Which rational expression should be added to the multiplicative inverse of additive

inverse of

26. Simplify :

27. Simplify :

28. Simplify :

29. Simplify :

30. Simplify : .

MATHEMATICSCLASS X

QUADRATIC EQUATIONS



1. Find the value of C for which the quadratic equation has real roots.2. Find the value of k, so that the quadratic equation has real roots.3. Find the value of k, if the equation has equal roots.4. For what value of k, does the following equation have equal roots :

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.


20.

21.

22.

23.

24.

25.26.27.28.29.30.

31.

32.

33.

34.

35. 36.37.

38.

39.

40.

41.

42.

43.

44. The time taken by a person to cover 150 km was hours more than the time taken in return

journey. If he returned at a speed 10 km/hr more than the speed of going, what wathe speed per hours in each direction ?

45. Some student planned a picnic. The budget for food was Rs. 24. Because 4 of the group failed to go, the cost of food to each member got increased by Re. 1. How many students attended the picnic.


46. In a flight of 600 km an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/hr and the time of flight increased by 30 minutes. Find the duration of the flight.

47. A two digit number is such that the product of the digits is 14. When 45 is added to the number, then the digits are reversed. Find the number.

48. The sum of the squares of two consecutive natural numbers is 313. Find the numbers.49. The angry Arjun carried some arrows for fighting with Bheeshm. With half the arrows he cut down

the arrows thrown by Bheeshm on him and with six other arrows he killed the rath driver of Bheeshm. With one arrow each he knocked down respectively the rath, flag and bow of Bheeshm. Finally with one more than four times the square root of arrows he laid Bheeshm unconscious on an arrow bed. Find the total number of arrows Arjun had.

50. Rs. 6,500 were divided equally among a certain number of persons. Had there been 15 more persons, each would have got Rs. 30 less. Find the original number of persons.

51. Rs. 6,500 were divided equally among a certain number of persons. Had there been 15 more persons, each would have got Rs. 30 less. Find the original number of persons.

52. The side of square exceeds the side of another square by 4 cm and the sum of the areas of two squares is 400 sq. cm. Find the dimensions of the squares.

53. In a flight of 2800 km, an aircraft was slowed down due to bad weather. Its average speed was reduced by 100 km/hr and time increased by 30 minutes. Find the original duration of the flight.

54. The area of a rectangle gets reduced by 80 sq units if its length is reduced by 5 units and the breadth is increase by 2 units. If we increase the length by 10 units and decrease the breadth by 5 units the area is increased by 50 sq units. Find the length and breadth of the rectangle.

55. A plane left 30 minutes later than the scheduled time and in order to reach the destination 1500 km away in time, it has to increase the speed by 250 km/hr from the usual speed.

56. Find two consecutive numbers whose squares have the sum 85.57. The hypotenuse of a right triangle is 20 m. If the difference between the lengths of the other sides is

4 m, find the other sides.58. The hypotenuse of a right triangle is 6 m more than twice the shortest side. If the third side is 2 m

less than the hypotenuse, find the sides of the triangle.59. Sum of ages (in years) of a son and his father is 35 and the product of their ages is 150. Find their

ages.

MATHEMATICSCLASS X


ARITHMETIC PROGRESSION


1. Find out which of the following sequence are in arithmetic progression. For those which are Arithmetic progression, find out the common difference.

(a) 3, 6, 12, 24 (b) 0, – 5, – 10, – 15 (c)

(d) 6, 6, 6, 6, ……. (e) 1, 0, 1, 7, 2, 4, 3, 12. Find the common difference of the AP and write the next two terms.

(a) 51, 59, 67, 75….. (b) 75, 67, 59, 51, ……

(c) 0, (d) 119, 136, 153, 170.

3. Find the 10th term of – 40, – 15, 10, 35 …….

4. Find the 9th term of

5. Find the 12th term of the A.P. with first term 9 and common difference 10.6. Find the 12th term of the A.P. with first term – 20 and common difference 4.7. Show that 8. Find k, if the given value of x is the kth term of the given A.P. 25, 50, 75, 100, ……….. : x = 1,000.9. Two AP’s have the same common difference. The first term of one of there is 13 and that of the

other is 18. What is the difference between their(a) 12th terms (b) 100th terms (c)

10. The AP’s have the same common difference. The difference between their 50 th term is 223355. What is the difference between their 200th terms.

11. Find the 15th, 22nd , and nth term of an AP given by 3, 9, 15, 21, ……12. Find the general term of the following A.P.

13. Find the sum of :

(a) 1 + 7 + 13 + 19 + …….. 40 terms (b) 2 +

14. Evaluate : 4 + 3 + 8 + 5 + 12 + 7 + …… 32 terms.15. Find the sum of all integers between 92 and 186 which an multiplies of 9.

16. Find k, if the given value of x is the kth term of the given A.P.

17. Which term of the A.P. ; 3, 15, 27, 39, ………… will be 132 more than its 54th terms?18. Find the sum : 3 + 11 + 19 …….. +80319. Find the sum : – 5 + (–8) + (– 11) + …….. + (– 230).20. Prove that no matter what the real number a and b are, the sequence with the nth term a + bn is

always an A.P. What is the common difference? What is the sum of first 20 terms?

MATHEMATICSCLASS X

INSTALMENTSGraphics By:- Pradeep Tokas 17 Written By:- Raj Kumar Badhan


1. An electric iron is sold for Rs. 110 cash or Rs. 50 cash down payment followed by Rs. 62 after a month. Find the rate of interest charged under instalment plan.

2. A pressure cooker is available on Rs.180 cash or Rs. 70 cash down payment followed by Rs. 60 a month for 2 months. Find the rate of interest charged under instalment plan.

3. A house is sold for Rs. 30,000 cash or Rs. 17500 as cash down payment and instalment of Rs. 1600 per month for 8 month. Determine the rate of interest correct to one decimal place, under instalment plan.

4. A radio is available for Rs. 950 cash or Rs. 2000 cash down payment and 10 monthly instalments of Rs. 80 each. Find the rate of interest charged.

5. I got a loan of Rs. 3000 for the repair of my house. The loan is to paid back in 3 annual instalments. How much is each instalment if the interest is compounded annually on the balance at 5% and is included in each instalment.

6. A sum of Rs. 1500 is to be paid back in three annual instalments. How much is each instalment if the interest compounded annually on the balance of 4% and is to be included in each instalment.

7. A loan of Rs. 2500 is to be paid back two equal half-yearly instalments. How much is each instament if the interest is compounded half-yearly at 8% p.a.

8. A man borrows some money on compound interest and returns it in two years in equal annual instalments. If the rate of interest is 5% and yearly instalment is Rs. 441, find the principal.

9. Kusum borrowed money and returned in 3 equal instalments of Rs. 4630.50 each. What sum did she borrow if the rate of interest was 20% a compounded quarterly? Find also the total interest charged.

10. A shopkeeper advertises a table fan for Rs. 150 cash or Rs. 35 as cash down payment and Rs. 20 a month for 6 months. What is the rate of interest?

11. A man borrows money on compound interest and returns two equal annual instalments. If the rate of interest is 15% p.a. and yearly interest isn Rs. 10.58, find the principal interest charged with each instalment.

12. A house building society offeres a flat for Rs. 55000 cahs or on the terms that the buyer should pay Rs. 4275 as cash down payment and the rest in three equal half yearly instalment. The interest charged by the society is at the rate of 16% per year compounded half-yearly. Find the value of each instalment of the flat is purchased under instalment plan.

13. A shopkeeper sells a sofa set for Rs. 500 cash or Rs. 200 cahs down and rs. 55 per month for 6 months. Find the rate of interest.

14. A sum of Rs. 50,440 is borrowed to be paid is 3 years in 3 equal instalment. What is the annual instalment if the rate of interest.

15. A sum of Rs. 6000 is to be paid back in 3 annual instalments. How much is each instalment if the interest is compounded annually on the balance at 5% and is to be included in each instalment?

16. A steel container is sold at Rs. 120 cash or Rs. 25 as each down payment and Rs. 25 a month for 4 months. Find the rate of interest charged under the instalment plan.

17. A house is sold for Rs. 12000 cash or Rs. 7000 as cash down payment and Rs. 630 a month for 4 months. What is the rate of interest ?

18. A sum of Rs. 7500 is to be paid back in three equal instalments. How much is each instalment if the interest is compounded annually at 4% ?

19. An electric heater is sold at Rs. 150 cahs or Rs. 36 as cash down payment and Rs. 25 a month for 5 months. Find the rate of interest for this instalment plan.

20. A mixi is marked at Rs. 1000 cash or Rs. 250 cash down payment followed by Rs. 200 a month for 4 months. Find the rate of interest for this instalment plan.


21. A room cooler is marked at Rs. 2000 cash or Rs. 400 cash down payment followed by Rs. 300 per month for 6 months. Determine the rate of interest charged under this instalment plan.

22. A sum of Rs. 3,310 is to be paid in two equal annual instalments. How much is each instalment if the interest is compounded annually at 10%.

23. A sum of money borrowed at 5% per annum compounded interestis paid back in three equal annual instalment. If each instalment is of Rs. 18,522. Determine the sum borrowed.

24. A sum of money is borrowed and paid back in two equal annual instalments of Rs. 882 each allowing 5% compounded interest. What was the sum borrowed ?

25. A shirt is sold for Rs. 60 cash or Rs. 20 as cash down payment and Rs. 8 per months. What is the rate of interest ?

26. A loan of Rs. 8400 is to be paid in two equal annual instalments, the interest being charged at 10% per annum compounded annually. Find each instalment.

MATHEMATICSCLASS X


TAXATION


1. The annual income of Yogesh is Rs. 1,80500 exclusive of HRA. He contributes Rs. 4,000 per month in his provident fund and pays an annual premium of Rs. 12,000 towards life insurance policy. He also invested Rs. 6,000 in NSC. Calculate the income tax paid by Yogesh in the last month of the year, if his earlier deduction for first 11 months for income tax was at the rate of Rs. 650 per month.

Assume the following for calculation income tax:

(a) Standard deduction 1/3 of the annual subject to a maximum of Rs. 20,000(b) Rate of Income Tax :

Slab Income Tax(i) Up to Rs. 50,000 No tax(ii) From Rs. 50,0001 to Rs. 60,000 10 % of the amount exceeding Rs. 50,000(iii) From Rs. 60,0001 to Rs. 1,50,000 Rs. 1,000 + 20% of the amount exceeding Rs. 60,000(iv) From Rs. 1,50,001 onwards Rs. 19000 + 20% of the amount exceeding Rs. 1,50,000

(c) Rebate in tacx 20% of the total savings subject to a maximum of Rs. 12,000.

(d) Surcharge 2% of the net tax payable.

6. The annual income of Poonam (excluding HRA) is Rs. 97,000. Show contributes Rs. 1000 per month in per provident fund and pays an annual premium of Rs. 3000 towards her L.I.C. policy. Calculate the income tax Poonam has to pay on her salary during the financial year.Assume the following for calculating income tax :

(a) Standard deduction 1/3 of the total annual income subject to a maximum of Rs. 20,000 if the income is more than Rs. 1 lakh

(Rs. 25000 in case annual income is less than Rs. one lakh)(b) Rate of Income Tax :

Slab Income Tax(i) Up to Rs. 50,000 No tax(ii) From Rs. 50,001 to Rs. 60,000 10% of the amount exceeding Rs. 50,000(iii) From Rs. 60,001 to Rs. 1,50,000 Rs. 1,000 + 20% of the amount exceeding Rs. 60,000(c) Rebate in tax 20% of the total savings subject to a maximum of Rs.

12,000.(d) Surcharge 2% of the net tax payable.

7. The annual income of Pushpa (excluding HRA) is Rs. 99,000. She contributes rs. 1,000 per month in her provident fund and pays an annual premium of Rs. 2,000 towards her LIC policy. Calculate the income tax Pushpa has to pay on her salary during the financial year.Assume the following for calculating income :(a) Standard deduction 1/3 rd of the total annual income subject to a maximum

of Rs. 20,000 if income is more than Rs. one lakh (Rs. 25,000 in case the annual income tax is less than Rs. one lakh)(b) Rate of Income Tax

Slab Income Tax


(i) Up to Rs. 50,000 No tax(ii) From Rs. 50,001 to Rs. 60,000 10% of the amount exceeding Rs. 50,000(iii) From Rs. 60,0001 tp Rs. 1,50,000 Rs. 1000 + 20% of the amount exceeding Rs. 60,000(c) Rebate in tax 20% of the total savings subject to a maximum of Rs.

12,000(d) Surcharge 2% of the tax payable.

8. The annual income of Anamika (excluding HRA) is Rs. 96,000. She contributes Rs. 1,200 per month in her provident fund and pays an annual premium of Rs. 1600 towards her L.I.C. policy. Calculate the Income tax Anamika has to pay on her salary during the financial year.

9. The annual income of Mandira (excluding HRA) is Rs. 1,60,000. She contributes Rs. 4,000 per month in her provident fund and pays an annual premium of Rs. 16,000 towards her L.I.C. policy Calculate the income tax Mandira has to pay on her salary during the financial year.

10. The total salary of Manjeet singh is Rs. 1,25,000 excluding HRA during the year. He pays a premium of Rs. 10,800 annually towards LIC and contributes Rs. 2000 per month towards GPF Rs. 250 are deducted each month from his salary as income tax. Calculate the income tax Manjeet Singh be to pay in the last month of the financial year.Assume the condition of Q 7 for calculating the income tax.

11. Radhika has total income of Rs. 98000 during a year (HRA not included). She contributes Rs. 800 per month towards provident fund account and pays an annual premium of Rs. 1200 towards LIC. Calculate the income tax payable in the last month. If she has been paying Rs. 120 per month towards income tax for he first 11 months.Assume the conditions for calculating income tax for Q. 7.

12. The annual income of Mr. Shiv kumar (excluding HRA) is Rs. 1,65,000. He contributes Rs. 4,000 per month in his provident fund and pays an annual premium of Rs. 16,000 towards his life Insurance Policy. Calculate the income tax paid by Shiv Kumar in the last month of the year, if his earlier deductions for first 11 months for income tax were at the rate of Rs. 400 per month. Assume the condition for calculating the income tax as that of Q . 7.

13. The total salary o Yogendra Arora is Rs. 1,46,200 excluding HRA during the year. He pays a premium of Rs. 14,400 annually towards LIC and contributes Rs. 2,00 per month towards GPF. Rs.500 is deducted each month from his salary as income tax. Calculate the income tax Ygender. Assume the conditions for calculating income tax as of Q. 7.

14. The total salary of Mr. M.P. Singh is Rs. 1,15,700 excluding HRA during the year. He pays a premium of Rs. 9600 annually towards LIC and cotributes Rs/. 1500 per month towards GPF. M.P. Singh has to pay in the 1st month of the financial year.

15. Kusum has an annual income of Rs. 1,51,500 (excluding HRA). During the year, she contributed Rs. 2,500 per month towards provident fund and paid an annual premium of Rs. 8,400 to LIC. Calculate the income tax paid by Kusum in the last month of the year, if her earlier deduction for 11 months of income tax were at the rate of Rs. 500 per month.Assume the following for calculating income tax :(a) Standard deduction 1/3 of the total annual income subject to a maximum of

Rs. 20,000 (25,000 if income is less than Rs. 1 lac)(b) Rate of Income Tax :

Slab Income Tax(i) Up to Rs. 50,000 No tax(ii) From Rs. 50,001 to Rs. 60,000 10% of the amount exceeding Rs. 50,000(iii) From Rs. 60,001 to Rs. 1,50,000 Rs. 1,000 + 20% of the amount exceeding Rs. 60,000(iv) From Rs. 1,50,001 onwards Rs. 19,000 + 30% of the exceeding Rs. 1,50,000


(c) Rebate in Income tax 20% of the total savings subject to a maximum of Rs. 12,000.

(d) Surcharge 2% of the tax payable.

16. The total salary of Surinder Pal is Rs. 1,53,100 excluding HRA during the year. He denoted Rs. 8000 to National Defence Fund. He pays a premium of Rs. 12,000 annually towards LIC and contributes Rs. 1,200 per month towards GPF. Rs. 500 is deducted each month from his salary as income tax. Calculate the income tax Surinder Pal has to pay in the last month of the financial year.Assume the following for calculating income :(a) Standard deduction 1/3 of the total income subject to a maximum of Rs.

20,000(b) Rate of Income Tax

Slab Income Tax(i) Up to Rs. 50,000 No tax(ii) From Rs. 50,001 to Rs. 60,000 10% of the amount exceeding Rs. 50,000(iii) From Rs. 60,001 to Rs. 1,50,000 Rs. 1,000 + 20% of the amount exceeding Rs. 60,000(iv) From Rs. 1,50,001 onwards Rs. 19,000 + 30% of the exceeding Rs. 1,50,000 (c) Rebate in Income tax 20% of the total savings subject to a maximum of Rs.

60,000.(d) Surcharge 2% of the tax payable.

17. The annual income of Mr. Arun (excluding HRA is Rs. 1,65,000. He contributes Rs. 4000 her month in his provident fund and pays an annual premium of Rs. 10,000 towards his life insurance policy. He also purchase NSC for Rs. 7000.Calculate the income tax paid by Arun in the last month of the year, if his earlier deduction for the first 11 months for income tax were at the rate of Rs. 500 per month. Calculate the income tax considering conditions of Q. 7.


MATHEMATICSCLASS X

SIMILAR TRIANGLES


1. In fig. 1, DE is parallel to BC. If

2. In are points on the sides AB and AC respectively, such that DE||BC. If AD = 5 cm, DB = 8 cm and AC = 6.5 cm, find AE.

3. In fig. 2. DE is parallel to BC. If = 4.8 cm, find EC.

4. In fig 3. If DE||BC and AD = 4x – 3, AE = 8x – 7, BD = 3x – 1 and CE = 5x – 3, find x.5. In the given fig. 4 AB || MN. If PA = x – 2, PM = x, PB = x – 1 and PN = x + 2, find the value of x.

6. In given fig. 5, find x if DE || BC.7. In DE || BC. If AD = x, DE = x – 2, AE = x + 2 and EC = x – 1, find the value of x.8. In fig. 6, LM || AB. If AL = x – 3, AC = 2x,

BM = x – 2, BC = 2x + 3, find the value of x. 9. If D and E are respectively, the points on the sides

AB and AC of a triangle ABC, such that AD = 6 cm BD = 9 cm, AE = 1.8 cm, show that DE || BC.

10. D and E are respectively, the points on the sidesAB and AC of a triangle ABC. If AB = 5.6 cm, AD = 1.4 cm, AC = 7.2 cm and AE = 1.8 cm, show that DE || BC.


A

D E

B C

A

D E

B C

A

D E

B C

4x – 3 8x – 7

3x – 1 5x – 3

Fig. 1 Fig. 2 Fig. 3

P

A B

M N

x –

2

x – 1x + 2

4 8D E

B C

x – 3 3x – 19

C

Fig. 4 Fig. 5

A B

C

L M2x2x + 3

x – 2x – 3

11. In AD is the bisector of find BC, if(i) AB = 5.6 cm, AC = 4 cm, DC = 3 cm.(ii) AB = 5 cm, AC = 4 cm, BD = 2.5 cm.(iii) AB = 5.6 cm, AC = 6 cm, DC = 3 cm.

12. In AD is the bisector of find AC, if(i) AB = 8 cm, BD = 5 cm, DC = 4 cm.(ii) AB = 5 cm, BD = 2 cm, DC = 3 cm.(iii) AB = 5 cm, BD = 2 cm, BC = 5.4 cm.

13. In AD is the bisector of find AB, if (i) BC = 10 cm, BD = 6 cm, AC = 6 cm.(ii) AC = 4.2 cm, DC = 6 cm, BD = 10 cm.

14. If AE is the bisector of the exterior BC produced in E.If AB = 5 cm, BC = 5 cm and AC = 3 cm. Find CE.

15. In

16. the bisector of meets AC at D. A line PQ || AC meets AB, BC and BD at P, Q, and R respectively. Show that PR. BQ = QR.BP.

17. The bisector of the angles B and C of triangle ABC, meet the opposite sides in D and E respectively. If DE || BC, prove that the triangles is isosceles.

18. In are points on the sides AB and AC respectively, such that BD = EC. Prove that DE || BC.

19. In D and E are points on sides AB and AC respectively, such that AD × EC = AE × DE. Prove that DE || BC.

20. ABCD is a parallelogram, P is a point on side BC. DP when produces meets AB produced at L, prove that

(i) (ii)

21. Draw a line segment of length 12 cm and divide it internally in the ratio 3 : 4.

22. A vertical stick 12 m long casts a shadow 8 m long on the ground. At the same time, a tower casts the shadow 40 m long on the ground. Determine, the height of the tower.

23. is right angled at B. BD is perpendicular to AC. Prove that (see fig. 8.)


D C

P

A B L

A D C

B

B C

E D

A

P

Fig. 8. Fig. 9.

24. D is a point on the side of BC of a such that

25. In the given fig. 9, considering triangled BEP and CPD, prove that BP × PD = EP × PC.26. In a BD and CE are the altitudes. Prove that are similar. Is

27. In the given fig. 10. QA and PB are perpendiculars to AB. If AO = 10 cm, BO = 6 cm and PB = 9 cm, find AQ.

28. In Fig. 11, AB || QR. Find the length of PB.29. In Fig. 8.91, XY || BC. Find the length of XY.30. The perimeters of two similar triangles are 30 cm and 20 cm respectively. If one side of the first

triangle is 12 cm, determine the corresponding side of the second triangle.31. In the given fig. 13. ABC is an right angled triangle at C. Prove that ABC ~ ADE and find the

lengths of AE and DE.

32. In the given Fig. 14A, ABC = 90° and BD AC. If BD = 8 cm and AD = 4 cm, find CD.33. In a triangle ABC, B = 90° and BD AC. If AB = 5.7 cm, BD = 3.8 cm and CD = 5.4 cm, find BC.34. In the given fig. 15, CAB = 90° and AD AC. If AC = 7.5 cm, AB = 1 m and BD = 1.25 m, find

AD.35. In ABC, ray AD bisect A and intersects BC in D. If BC = a, AD = b and AB = c, prove that

36. D is a point on the side BC of a ABC such that ADC = BAC, prove that

37. In a Fig. 16., AP = 3 cm, AR = 4.5 cm, AQ = 6 cm, AB = 5 cm and AC = 10 cm. Find the length of AD.


A

Q

O B

P

10 cm 9 cm A B

Q R

3 cm

9 cm

P

xY

A

B C 6 cm

3 cm

1 cm

Fig. 10 Fig. 11 Fig. 12

B C

E

A

D

3 cm 2 cm

A

B C

D4 cm

8 cm

C

A B

75

cm D

1.25 m

Fig. 13 Fig. 14 Fig. 15

P Q B D C

A

R A BD E

C

9 cm7 cm

8 cm 10 cm

Fig. 16 Fig. 17

38. In the fig. 17, if A = CED, AB = 9 cm, AD = 7 cm, CD = 8 cm and CE = 10 cm, find DE.39. ABC is a triangle in which AB = AC and D is any point in BC.

Prove that (AB)² – (AD)² = BD.CD.

40. A point D is on the side BC of an equilateral triangle ABC, such that DC = Prove that (AD)²

= 13(CD)².41. BO and CO are respectively the bisector of B and C of ABC. AO produced meets BC at P.

Show that.

42. Express x in terms of a,b and c. (Fig. 18)

43. In fig. 19, If AB = x unit CD = y units and PQ = z unit, prove that

44. In Fig. 20, 1 = 2 and 3 = 4, show that PT.QR = PR.ST.45. In the given Fig, 21, Prove that

46. In the fig. 22. ABD ~ PQR. Also area ( ABC) = 4. Area ( PQR). If BC = 12 cm, find QR.


A

x

BV M b N c K

46° 46°

x

L

a

QD

zy

C

P

P

T R

Q

3 2

4

Fig. 18 Fig. 19 Fig. 20

D

BE

A

C

A

B C

P

Q R Fig. 21 Fig. 22

47. In the fig 23. triangles ABC and DEF are similar. The area of ABC is 9 sq.cm and that of DEF is 16 sq. cm. If EF = 4.2 cm, find BC.

48. The areas of two similar triangles ABC and DEF are 64 cm² and 121 cm² respectively. If EF = 13.2 cm, find BC. (See fig. 24.)

49. The area of two similar triangles ABC and DEF are 64 cm² and 169 cm² respectively. If the length of BC is 4 cm, find the length of EF.

50. In the given fig. 25, triangles ABC and DEF are similar. If area of ABC = 49 sq. cm and AC = 4.8 cm. Find DF.

51. The areas of two similar triangles are 81 cm² and 49 cm² respectively. If the altitude off the bigger triangle is 4.5 cm. Find the corresponding altitude of the smaller triangle.

52. The areas of two similar triangles are 100 cm² and 49 cm² respectively. If the altitude of the bigger triangle is 5 cm. find the corresponding altitude of the other.

53. The areas of two similar triangles are 81 cm² and 49 cm² respectively. If the altitude of the first triangle is 6.3 cm, find the corresponding altitude of the other.

54. In fig. 26, given that ST || PQ, in PQR(a) If PR = 12. RS = 4. QR = 24. then find RT.(b) If PR = 18. SR = 8. QT = 6. find RT(c) If PS = 3. PR = 15. QR = 25. find QT(d) If RS = 4, QT = 9. PS = RT. find PR

55. In fig. 27, given that in ABC, A = E = x°(a) If CD = 7, BC = 21, BE = 10, then find AE(b) If BD = 2AE, BE = 7, CE = 14, then find BC.(c) If BD = AB, BC = 24, bE = 4, then find AB.

(d) If BD = AE, AB = 6, CD = 8, then find BE.


A

B C

D

E F

A

B C

P

Q R Fig. 23 Fig. 24

A D

B C E F

4.8 cm

R

S T

P Q

C

B E B

D

x° x°

Fig. 25

Fig. 26 Fig. 27

56. Determine, whether the triangle having sides (a – 1) cm. cm and (a + 1) cm is a right-angled triangle. (See fig. 28.)

57. Determine the length of AD in terms of b and c. (see Fig. 29)58. In an isosceles triangle ABC, if AC = BC and AB² = 2AC², prove that C is a right angle.59. PQR is an isosceles right triangle, right angled at R. Prove that PQ² = 2PR².60. In an equilateral triangle ABC, AD is drawn perpendicular to BC, meeting BC in D. Prove that AD²

= 3BD².61. A of an isosceles ABC is acute, in which AB = AC and BD AC. Prove that (BC)² = 2AC.CD.62. In a right angled triangle, with sides a and b and

hypotenuse c, the altitude drawn on the hypotenuse is x. Prove that ab = cx.

63. In a right-angled PQR, P = 90°, If M is the mid-point of PQ, prove that QR² = 4MR² – 3PR².

64. In the given Fig. 30, ABC is a right triangle, at B. AD and CE are the two medians drawn from A and C

respectively. If AC = 5 cm and AD = find the

length of CE.65. In the given Fig. 31, < 90° and segment AD side BC,

show that 66. L and M are the mid-points of AB and BC respectively of

ABC, right-angled at B. Prove that 4LC² = AB² + 4BC².67. In a triangle ABC, AD is drawn perpendicular to BC.

Prove that (AB)² – (BD)² = (AC)² – (CD)².68. In a ABC, AD BC. Prove that AB² + CD² = AC² +

DB².

69. In an isosceles triangle ABC, with AB = AC, BD is perpendicular from B to the side AC. Prove that (BD)² – (CD)² = 2 CD.AD.

MATHEMATICS


C

BA (a – 1) cm

2 cm

C

A B

Db

c

Fig. 28 Fig. 29

A

E

B D C

5 cm3

Fig. 30 A

B x D a – x C

c b h

CLASS XGEOMETRICAL CONSTRUCTIONS


1. Draw circles of radii of measure given below. Let there be point A on the circle. Draw a tangent to the circle without using the centre of the circle.(i) Radius = 3 cm (ii) Radius = 2.8 (iii) Radius = 3.4 cm

2. Draw a circle of a diameter 6 cm. Take a point P outside the circle at a distance of 7 cm from the centre. From point P draw two tangents to the circle and verify that their lengths are equal.

3. Construct a ABC when AB = 6.5 cm, A = 70° and C = 45°. Draw circumcircle of a ABC.4. Construct a ABC in which AB = 6 cm, AC = 4.6 cm and A = 75°. Draw the circumcircle of

ABC.5. Construct a ABC in which BC = 7 cm, B = 60° and C = 45°. Also draw the incircle of ABC.6. Construct a ABC in which BC = 7.5 cm, CA = 5.2 cm and AB = 6 cm. Also draw the incricle of

ABC.7. Construct a PQR in which QR = 6.2 cm, P = 60° and PR = 4.4 cm. Also draw the incircle of

PQR.8. Construct a ABC in which AB = 6 cm, C = 60° and median CD = 5 cm. Construct a AB’C’

similar to ABC with base = AB’ = 8 cm.9. Construct a ABC in which AB = 4 cm, BC = 5 cm, CA = 6 cm, Now Construct a triangle similar

to ABC such that each of its sides is two-third of the corresponding sides of ABC. Also prove your assertion.

10. Construct a quadrilateral ABCD with AB = 3 cm, AD = 2.7 cm, DB = 3.6 cm, B = 110°, BC = 4.2 cm. Construct another quadrilateral A’BC’D’ similar to quadrilateral ABCD so that diagonal AD’ = 4.8 cm.

11. Construct a quadrilateral ABCD in which AB = 4.2 cm, BC = 3.8 cm, CD = 4.5 cm, AD = 6 cm and AC = 7.3 cm. Construct another quadrilateral similar to quadrilateral ABCD with the sides equal to 3/5th of the corresponding sides of quadrilateral ABCD.

12. Construct a ABC in which AB = 5 cm, B = 60° and altitude CD = 3 cm. Construct a AQR similar to ABC such that each side of AQR is 1.5 times that of the corresponding side of a ABC.

13. Construct to ABC with BC = 5 cm, A = 50° and foot of perpendicular D on BC fro A is 4 cm away from B.

14. Construct AQR ~ ABC such that each of its sides is 2/3 of the corresponding sides of ABC. It is given that AB = 5 cm, BC = 6 cm and CA = 7 cm.

15. Construct ABC ~ DEF in which EF = 10 cm, DE = 5 cm and D = 70° using a scale factor ½.16. Construct a ABC in which BC = 5.8 cm, A = 60° and altitude through A is 4.2 cm.17. Construct a ABC with base = BC = 6 cm, vertical A = 60° and median through A is 4 cm.18. Construct a ABC in which BC = 6 cm, A = 60° and median through A = 5.5 cm. Find the length

of altitude drawn on BC from A.19. Construct A ABC in which BC = 6.5 cm, A = 65° and the foot of the perpendicular AD on BC is

4 cm away from B.20. A, B, C are three non-collinear points such that AB = 3.5 cm, BC = 4.8 and CA = 6 cm. Construct a

circle passing through A, B and C. Find the measure of circum radius.

MATHEMATICS


CLASS XTRIGONOMETRY


1. Prove that

2. Prove that

3. Prove that

.4. Show that (1 + tan A tan B)² = (tan A – tan B)² = sec² A. sec² B.5. Show that

6. Prove that

7. Prove that 8. Prove that 9. Prove that

10. Prove that

11. If

12. If tan A = n tan B and sin B = m sin B. Prove that cos²A =

13. In

14. If 2

15. If

16. If

17. Solve the equation

18. Solve the equation 19. Find the value of cot 12° cot 38° cot 52°. cot 60°.cos 78°.20. If sin

21. If A.B.C. are the angles of a triangle, prove that tan

22. Prove that

23. Evaluate


24. Given that

25. If

26. If

27. If

28.


MATHEMATICSCLASS X

HEIGHTS AND DISTANCES


1. If tan A = 1,

2. If cosec

3. String of a kite is 100 m long and it makes an angle of 60° with horizontal. Find the height of the string.

4. A tree is broken by wind. The top struck the ground at an angle of 30° and at a distance of 30 m from the root. Find the whole height of tree.

5. A man on the tip of a vertical tower observes a car moving at a uniform speed coming directly towards it. If it takes 12 minutes for the angle of depression to change from 30° to 45°, how soon after this will the car reach the tower.

6. Determine the height of a mountain if the elevate of its top at an unknown distance from the base is 30° and at a distance 10 km further off from the mountain along the same line the angle of elevation is 15°. (Use tan 15° = 0.27)

7. The shadow of a flagstaff is three times as long as the shadow of the flagstaff when sun rays meet the ground at the angle of 60°. Find the angled between the sun’s rays and the ground at the time of longer shadow.

8. The pillar of equal height are either side of a road which 100 m wide. The angle of elevation of the top of the pillar are 60° and 30° at a point on the road between the pillars. Find the position of the point between the pillars and the height of each pillar.

9. The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is 60°. At a point Y, 40 m vertically above X, the angle of elevation is 45°. Find the height of tower PQ and the distance XQ.

10. The angle of elevation of a cliff from a fixed point is . After going up a distance of k metre towards the top of the cliff at an angle of , it is found that the angle of elevation is . Show that the height of the cliff is

11. at the foot of the mountain the elevation of summit is 45 °, after ascending 1000 m towards the mountain up a slope is 30 ° inclination, the elevation is found to be 60°. Find the height of the mountain.

12. The angle of elevation of the top of a tower from a point A due south of the tower is and from B

due east is if AB = d. Show that the height of the tower is .

13. From the top of a building AB, 60 m high. The angle of depression of the top and bottom of a vertical lamp CD are observed to be 30° and 60°, Find (i) Horizontal distance AB and CD.(ii) The difference between the height of the building and the lamp-post.

14. A ladder against a wall at an angle to the horizontal. Its foot is pulled away from the wall through a distance ‘a’ so that it slides a distance ‘b’ down the wall making an angle with the horizontal. Show that


15. An observer 1.5 m tall is 20 m away from the tower 30 m high. Determine the angle of elevation of the top of tower from his eye.

16. Two boat approach a light house in mid sea from opposite direction. The angle of elevation of the top of the light house from tow boats are 30° and 45°. If the distance between two boats is 100 m. Find the height of light house.

17. An aeroplane flying horizontally 1 km above the ground is observed at an angle of 60°. After 10 sec its elevation is observed to be 30°. Find the speed of aeroplane is km/hour.

18. A carpenter makes stools for electrician with square top of side 5 m and a height 1.5 m above the ground. Also each leg is inclined at an angle of 60° to the ground. Find the length of each leg and also the length of two steps to be put at equal distances.

19. If the angle of elevation of a cloud from a point h makes above a lake is ‘ ’ and the angle of depression of its depression of its reflection in the lake be prove that the distance of the cloud

from the point of observation is

20. There are two temples one on each bank of a river just opposite to each other. One temple is 50 m high. From the top of this temple the angle of depression of the top and foot of other temple are 30° and 60°. Find the width of river and the height of the other temple.

21. The horizontal distance between two trees of different height is 60 m. The angle of depression of the top of the 1st tree as seen from the top of the second tree is 45° if the height of the second tree is 80 m, find the height of the first tower.

22. The angles of depression of two ships from the top of a light house are 45° and 30° towards east. If the ships are 200 metres apart, find the height of the light house.

23. The horizontal distance between two towers is 70 m. The angle of depression of the top of the first tower when seen from the top of second tower is 30°. If the height of the second tower is 120m, find he height of the first tower.

24. The horizontal distance between two towers is 70 m. The angle of depression of the top of the first tower when seen from the top of the second tower is 30°. If the height of the second tree is 80m, find the height of the first tree.

25. Two men of either side of a temple 75 m high observe the angle of elevation of the top of the temple to be 30° and 60° respectively. Find the distance between the two men.

26. Two men on either side of a cliff 80 m high observe the angles of elevation of the top of the cliff to be 30° and 60° respectively. Find the distance between the two men.

27. From the top of a tower 50 m high the angles of depression of the top and bottom of a pole are observed to be 45° and 60° respectively. Find the height of the pole.

28. A tower in a city is 150 m high and a multistoried hotel at the city centre at a distance of 1.2 km from the tower. Find the value of h if the top of the hotel, the top of the building and the top of the tower are in the straight line. Also, find the distance of the tower from the city centre.

29. A round balloon of radius ‘a’ subtends an angle at the eye of the observer while the angle of elevation of its centre is . Prove that the height of the centre of the balloon is a sin cosec /2.

30. A man is standing on the deck of a ship, which is 8 m above water level. He observes the angle of elevation of the top of a hill as 60° and the angle of depression of the base of the hill as 30°. Calculate the distance of the hill from the ship and the height of the hill.

31. Determine the height of a mountain ifthe elevation of its top at an unknown distance from the bse is 30° and at a distance 10 km further off from the mountain, along the same line, the angle of elevation is 15°. (Use tan = 15° = 0.27)

32. The angle of elevation of a jet plane from a point A on the ground is 60°. After a flight of 15 seconds,


the angle of elevation changes to 30°. If the jet plane is flying at a constant height of 1500 m, Find the speed of the jet plane.

33. A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60°. When he moves 40 m away from the bank, he finds the angle of elevation to be 30°. Find the height of the tree and the width of the river.

34. A man on the deck of a ship is 16 m above water level. He observes that the angle of elevation of the top of a cliff is 45° and the angle of depression of the base is 30°. Calculate the distance of the cliff from the ship and the height of the cliff.

35. An aeroplane, when 3000 m high, passes vertically above another aeroplane at an instant when the angles of elevation of the two aeroplanes from the same point on the ground are 60° and 45° respectively . Find the vertical distance between the two aeroplanes.

36. From a window (h metres high above the ground) of a house in street, the angles of elevation and depression of the top and the foot of another house on the opposite side of the street are and respectively. Show that the height of the opposite house is h(1 + tan cot )

37. A man on the deck of a ship is 12 m above water level. He observes that the angle of elevation of the top of a cliff is 45°, and the angle of depression of the base is 30°. Calculate the distance of the cliff from the ship and the height of the cliff.

38. A pole 5 m high is fixed on the top of a tower. The angle of elevation of thetop of the pole observed from a point ‘A’ on the ground is 60° and the angle of depression of the pont ‘A’ fro the top of the tower is 45°, Find the height of the tower.

39. The angle of two objects on the same side of the tower are found to be . If the distance between the objects is ‘p’ metres, show that the height ‘h’ of the tower is given by

Also determine the height of the tower, if P = 50 m, = 60°, = 30°.40. From the top of a hill, the angle of depression of two consecutive kilometere stones due east are

found to be 30° and 45°. Find the height of the hill.41. A surveyor noted that the angle of elevation of a marker on the top of a hill was 30º. He walked 40 m

towards the foot of the hill along he level ground and found the angle of elevation of marker as 45°. How far from surveyor’s first position was the marker.

42. From a window (h m high above the ground) of a house in a street. The angle of elevation and depression of the top and foot of the other house on the opposite side of the street are Show that height of the opposite house is h(1 + tan .cot )

43. A tower in a city is 150 m high and a multistoreyed hotel at the city centre is 20 m high. The angle of elevation of the top of the tower at the top of hotel is 5°. A building , h m high is situated on straight road connecting the tower with the city centre at a distance of 1.2 km from the tower. Find the value of ‘h’ if the top of the hotel the top of the building and the top of the tower are in a straight line. Also find the distance of tower from city centre. (Use tan 5° m, distance of tower = 1486 m]


MATHEMATICSCLASS X

SURFACE AREAS AND VOLUMES


1. The total surface of a sphere is 3850 cm². Find the diameter of the sphere.2. The surface Area of sphere is 154 cm². Find its volume.3. The trunk of a tree is cylindrical and its circumference is 176 cm. If the length of the trunk is 3 m.

Find the volume of the timber that can be obtained from the trunk.4. The curved surface area of a cylindrical pillar is 264 m² and its volume is 924 m³. Find the diameter

and the height of the pillar.5. The difference between inside and outside surfaces of a cylindrical tube 14 cm long is 88 cm². If the

volume of the tube is 176 cm³. Find the inner and outer radii of the tube.6. A cylindrical tube open at both ends is made of metal. The internal diameter of the tube is 10.4 cm

and its length is 25 cm. The thickness of the metal is 8 mm calculate the volume of the metal.7. A Rectangular sheet of paper 30 cm × 18 cm can be transformed into curved surface of a right

circular cylinder in two ways i.e., either by rolling the paper along its length or by rolling along its length or by rolling along its breadth. Find the ratio of the volumes of the two cylinder so formed.

8. Find the length of 13.2 kg of copper wire of diameter 4 mm when 1 cm³ of cooper weights 8.4 gm.9. A cylindercial vessel with out lid has to be tin coated on its both sides if the radius of the base is 70

cm and its height is 1.4 m. Calculate the cost of tin coating at the rate of Rs. 3.50 per 1000 cm².10. In the middle of a rectangular field measuring 30 m × 20 m a well of 7 m diameter and 10 m depth is

dug. The earth so removed is evenly spread over the remaining part of the field. Find the height through which the level of field is raised.

11. If 1 cm³ of gold weighs 21 gm. Find the weight of the gold pipe of length 1 m with a bore of 3 cm in which the thickness of the metal pipe is 1 cm.

12. A well is 6 m deep and the cost of cementing its inner surface at 50 paise per dm² is Rs. 1320. Determine the diameter of the well.

13. The radius and height of cone are in the ratio of 3 : 4 and its volume is 301.44 cm³. What is its radius what is the slant height ( ).

14. The interior of a building is in the form of clinder of diameter 4.3 and height 3.8 m surmounted by a cone whose vertical angles is a right angle. Find the area of the surface and the volume of the building. ( )

15. A conical tent is to accommodate 11 persons. Each person must have 4 m² of space on the ground and 20 m³ of air to breadth. Find the height of the cone.

16. Water flows at the rate of 10 meter per minute through a cylindrical pipe 5 m.m. in diameter. How long would it take to fill a conical vessel whose diameter of the base is 40 cm and depth 24 cm.

17. Two cones have their heights in the ratio 1 : 3 and the radii of their bases in the ratio 3 : 1. Find the ratio of their volumes.

18. A conical tent has the area of its base is 154 m² and that of the curved surface is 550 m². Find the volume of tent.

19. The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its

volume is of the volume of the given cone, at what height above the base the section has been

made.20. Rain water which falls on a flat rectangular surface of length 6 m and breadth 4 m is transferred

into a cylindrical vessel of internal radius 20 cm. What will be the height of water in the cylindrical vessel if a rain fall of 1 cm has fallen.


21. If the radii of the circular ends of a conical bucket is 45 cm high are 28 cm and 7 cm. Find the capacity of the bucket.

22. The perimeter of the ends of the frustum of a cone are 96 cm and 68 cm. If the height of the frustum be 20 cm. Find its radii, slant height, volume and total surface area.

23. An oil funnel of tin sheet of a cylindrical portion 8 cm long attached to a frustum of a cone. If the total height be 16 cm the diameter of the cylindrical portion 1 cm and diameter of the top of the funnel 10 cm. Find the area of tin required.

24. A hallow cone is cut by a plane parallel to the base and the upper portion is removed. If the curved surface of the remainder is 8/9 of the curved surface of the whole cone. Find the ratio of the line segment into which the cones altitude is divided by the plane.

25. A cylinder of radius 12 cm contains water to a depth of 20 cm. A spherical iron ball is dropped into the cylinder and thus the level of water is raised by 6.75 cm. Find the radius of the ball..

26. A toy is in the form of a cone mounted on a hemisphere of radius 3.5 cm. The total height of the toy is 15.5 cm. Find the total surface area of the volume of the toy.

27. Two solid sphere made of the same metal have weights 5920 gm and 740 gm. Determine the radius of the heavier sphere if the diameter of the lights one is 5 cm.

28. The section of a right circular cone bya plane through its vertex perpendicular to the base of the equilateral triangle of side 30 cm. Find the volume of the cone.

29. Metal spheres each of radius 2 cm are packed into a rectangular box of internal dimention 16 cm × 8 cm × 8 cm. When 16 spheres are packed the box is filled with preservation liquid. Find the value

of this liquid.

30. Find the length of the longest pole that can be placed in a room 10 m long, 10 m wide end 5 m high.31. Three cubes with edges 15 cm, 12 cm, 9 cm are melted together and formed into a simple cube. Find

the diagonal of this cube.32. A vessel is in the form of a hemispherical bowl is full of water. The contents are emptied into a

cylinder. The internal radii if the owl and cylinder 6 cm and 4 cm. Find the height of the water in the cylinder.

33. The radius of a sphere is two times that of other sphere. Determine the ratios of their surface areas and volumes.

34. An iron spherical ball has been melted and recast into smaller balls of equal size. It the radius of

each smaller ball is of the radius of the original ball. How many such balls are made.

35. Find the weight of a lead pipe 3.5 m long. The external diameter of the pipe is 2.4 cm and thickness of the lead is 2 mm and 1 cm³ of bad weighs 11.0 gm.

36. The volume of right circular cone of height 24 m is 1232 cm³. Find the slant height of the cone.37. The ratio of the total areas of the two cones or revolution is 9 : 16. Find the ratio of their volumes.38. A tent is in the form of right circular cone 28 m high the diameter of the base being 4.8 m and if 10

men sleep in the tent. Find the average number of cubic meter of air space per man.39. A cylindrical tank has a capacity 6160 m³. Find the depth if its radius is 14 m calculate the cost of

painting its curved outer surface at the rate of Rs. 3 per m².40. A boy is cycling such that the wheels of cycles are making 140 revolution per minute. If the diameter

of the wheel is 60 cm. Calculate the speed per hour with which the boy is cycling.41. The minute hand of a clock is 10 cm long. Ifnd the area on the face of the clock described by the

minute hand between 9 A.M. and 9.35 A.M.42. Find the area of the sheet metal requiredto make a closed hallow cone of height 24 cm and base

radius is 7 cm. Also find the capacity of this cone.43 The curved surface of a cylinder is 2540 m² and volume is 26400 cm³ find the curved surface of the

cone which has the same base and height of the cylinder.


44. Cones 3 cm high and 3.5 cm radius are made by melting a sphere of 10.5 cm radius. Find the number of cones.

45. Find what length of convas 2 m in width is required to make a conical tent 12 m in diameters and 6.3 m in slant height. Also find the cost of convas at the rate of Rs. 12.50 per metre..

46. A metallic right circular cone 20 cm high and whose vertical angle is 60° is cut into two parts at the middle points of its height by a plane parallel to the base. If the frustum so obtained be drawn into a wire of diameter of 1/16 cm. Find the length of the wire.

47. A vessel in the form of a inverted cone. Its height is 8 cm and radius of its top which is open is 5 cm. It is filled with water up to the rim. When lead shat each of which is a sphere of radius 0.5 cm are dropped into the vessel one fourth of water flows out. Find the number of lead shots dropped in the vessel.


MATHEMATICSCLASS X

STATISTICS


1. Question on Pictorial Representation of Data :1. The numbers of students studying in the primary schools. Middle schools, secondary schools, senior secondary schools and colleges of a town as shown in Fig. 1 in a pie chart. Read the pie-chart and answer the following questions :(i) What is the percentage of students studying in primary schools?(ii) Which two types of institutions have the same number of students ?(iii) What is the difference between the percentages of students in senior secondary schools and secondary schools ?


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MATHEMATICS - Badhan Educationbadhaneducation.in/assignment/class 10th/math/class... · Web viewFind the value of k for which the G.C.D. of MATHEMATICS CLASS X RATIONAL EXPRESSIONS