Question bank xi

ASSIGNMENT FOR --XI

TRIGONOMETRIC FUNCTIONS

Some valuable points to remember:

Law of sine :

Law of cosine :

Q .1 Find in degrees and radians the angle subtended b/w the hour hand and the minute hand Of a clock at half past three. [answer is 750 , 5π/12]

Q.2 Prove that: tan300+ tan150+ tan300. tan150 =1.

[hint: take tan450 = tan(300+150)

http://upload.wikimedia.org/wikipedia/commons/4/4c/Unit_circle_angles_color.svg

http://en.wikipedia.org/wiki/File:Triangle_with_notations_2.svg

Q.3 Prove that (1+cos π8 ) (1+ cos 3π8 ) (1+ cos 5π

8 ) (1+ cos 7π8 ) = 1

8

.

[Hint : cos 7π8 = cos( π - π8 ) = - cos π8 , cos 5π

8 = cos ( π - 3π8 ) = -

cos 3π8 ]

Q.4 (i) Prove that sin200 sin400 sin600 sin800 = 316

[Hint : L.H.S. sin200 sin400 sin600 sin800

(√3/2 ¿

⇨ √32×2 (2sin200 sin400 sin800) ⇨ √3

4 [(cos200 – cos600) sin800

(ii) Prove that: cosπ7 cos2π7 cos4 π

7 = - 18 .

[Hint: let x = π7 , then 12 sinx (2sinx cosx cos2x cos4x)]

Sin2x

(iii) Prove that: tan200 tan400 tan800 = tan600.

[hint: L.H.S. sin 200 sin 400sin 800

cos200 cos 400cos 800 solve as above method.]

Q.5 If cos(A+B) sin(C-D) = cos(A-B) sin(C+D) , then show that tanA tanB tanC + tanD = 0

[Hint: We can write above given result as cos (A+B)cos(A−B) = sin(C+D)

sin(C−D)

By C & D cos ( A+B )+cos (A−B)cos ( A+B )−cos(A−B) = sin (C+D )+sin(C−D)

sin (C+D )−sin(C−D) ]

Q.6 If 𝛂, are the acute angles and cos2𝛂 = 3 cos2β−13−cos2 β , show that

tan 𝛂 = √ 2 tan𝛃.

[Hint: According to required result , we have to convert given part into tangent function By using cos2𝛂 = 1−tan ²α

1+tan ²α ∴ we will

get 1−tan ²α1+tan ²α =

3( 1− tan² β1+tan ² β

)−1

3−1−tan ² β1+ tan ² β

= 3−3 tan ²β−1−tan ²β3+3 tan ²β−1+ tan ² β [

Q.7 Solve the equation: sin3θ + cos2θ = 0

{ Answer: -θ = nπ+ π2 and 5θ = nπ- π2 , ∀ nєz. }

Q.8 Solve: 3cos2x - 2 √3 sinx .cosx – 3sin2x = 0. [ans. X= nπ+π /6 or X= nπ-π /3 ∀ nєz. ]

Q.9 In △ ABC, if acosA = bcosB , show that the triangle is either isosceles or right angled.

[hint: ksinAcosA = k sinBcosB ⇨ sin2A = sin2B ]

Q.10 Prove that c−bcosAb−ccosA = cosBcosC [use cos A =b2+c2−a2

2bc on L.H.S]

SETS

Some valuable points to remember: 1. If A and B are finite sets, and A B = then number of elementsin the union of two setsn(AUB) = n(A) + n(B)2. If A and B are finite sets, A B ≠ thenn(AU B ) = n(A) + n(B) - n(A ∩B)3. n(A B) = n(A – B) + n(B – A) + n(A B)4. n(A B C) = n(A) + n(B) + n(C) – n(B∩C) – n(A∩B) – n(A∩C) +n(A∩B∩C)(A) Number of elements in the power set of a set with n elements =2n.(B)Number of Proper subsets in the power set = 2n-2

Venn diagram of only A or A – B= A∩B’

A ⊆B (subset), Set A is a subset of set B because all members of set A are in set B. If they are not, we have a proper subset.A ⊂B (proper subset)Question bank for recapitulation:

Q. 1 If U = {1,2,3......,10} , A = {1,2,3,5}, B = {2,4,6,7}, then find (A-B)’.

Q.2 In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only, Product A and C but not product B , atleast one of three products. [Answer n(A’∩B’∩C’)=11, n(A∩B’∩C)=4 , n(AUBUC)=44.]

Question: 3 If U = {x :x ≤ 10, x∈ N}, A = {x :x ∈ N, x is prime}, B = {x : x∈ N, x is even}. Write A ∩B’ in roster form.

Question: 4 In a survey of 5000 people in a town, 2250 were listed as reading English Newspaper, 1750 as reading Hindi Newspaper and 875 were listed as reading both Hindi as well as English. Find how many People do not read Hindi or English Newspaper. Find how many people read only English Newspaper. [Answer: People read only English Newspaper n(E’∩H) = n(E) – n(E∩ H) = 1375. ]

Q.5 A and B are two sets such that n(A-B) = 14 + x, n(B-A) = 3x and n(A ∩B) =x. If n(A) = n(B) then find the value of x.[Answer n(A-B) = 14 + x= n(A ∩B’) = n(A) - n(A ∩B)⇨ x=7]

Q.6 (i) Write roster form of {x: nn ²+1 and 1≤ n ≤3 , n∈ N}

(ii) Write set-builder form of {-4,-3,-2,-1,0,1,2,3,4}

Q.7 If A ={1}, find number of elements in P[P{P(A)}].

RELATIONS & FUNCTIONS

Some valuable points to remember: 1. AXB = { (a,b): a ε A, b ε B} for non-empty sets A,B otherwise

AXB=φ, this is the set of all ordered pairs of elements from A and B. n(AXB)=n(A)Xn(B)

2. AXB,BXA are not equal (not commutative).3. If R is relation on a finite set having n elements, then the number

of relations on A is 2nxn .4. Is the relation a function?{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}

Since x = 2 gives me two possible destinations, then this relation is not a function.

5. Inverse Function is a relation from B to A defined by { (b,a) :(a,b)εR}

for modulas function(|x|).

6.Graph of f : R → R such that f(x) = |x+1|

Question bank for recapitulation:

Q.1 Find the domain and range of f(x) = ¿ x−4∨ ¿x−4

¿ . [ans. Domain of f = R – {4}, Range of f = {-1, 1}]

Q.2 Let f(x) = , find f(-1) , f(3).

Q.3 If Y= f(x)= , then find x=f(y).

Q.4 Let R be the relation on the set N of natural numbers defined by R = {(a, b): a+3b = 12, a, bЄ N }.Find R, domain of R and range of R. {ans. (9,1),(6,2),3,3)}

Q.5 Write the domain & range of f(x) = 1/ (5x – 7).

Q.6 If f : R → R Be defined by f(x) = x2 +2x+1 then find

(i) f(-1) x f(1) ,is f(-1)+f(1)=f(0)

(ii) f(2) x f(3) , is f(2) x f(3)=f(6)

Q.7 If f(x)= x2 – 1X2, then find the value of : f(x) +f(1/x) . [ans.

0]

Q.8 Find the domain and Range of the function

f(x) = 12−sin 3 x . {Domain = R , Range = [1/ 3 , 1] }

Q.9 Draw a graph of f : R → R defined by f(x) = |x-2|

Q.10 If R is a relation from set A={11,12,13} to set B={8,10,12} defined by y=x-3,then write R-1. {Ans.(8,11),(10.13)}

COMPLEX NUMBERS & QUADRATIC EQUATIONS


A complex number z is a number of the form z = x + yi. Its conjugate is a number of the form = x - yi. z = (x + yi)(x - yi) = x2+ y2 = |z|2

Triangle Inequality:

1. |z1 + z2| |z1| + |z2|

2. |z1 + z2| |z1| - |z2|

3. |z1 - z2| |z1| - |z2|

4. |z1 + z2 + z3| |z1| + |z2| + |z3|

5. |z1z2| = |z1||z2|

Evaluate i203. We have to replace 203 with its remainder on division

by 4. 203 = 4X50 + 3; i203 = i3 = – i

ϴ, x>0, y>0

π - ϴ, x<0, y>0

arg (z) = ϴ - π , x<0, y<0

- ϴ, x>0, y<0

Can be used in finding principal argument of complex numbers. Question bank for recapitulation:

Q.1 Solve X2 – (3√2 – 2i) x - 6√2i = 0 (3√2, -2i)

Q.2 If z = 1(1−i )(2+3 i) , then |z| is [ans. 1/√(26) ]

Q.3 If z = ( 1+i1−i ), then z4 is [ans. 1]

Q.4 If z = 11−cosφ−isinφ , then Re(z) is [ans. ½]

Q.5 If (1 - i) (1 - 2i)(1 - 3i)...........(1 - ni) = (x - yi) , show that 2.5.10...........(1+n2) = x2+y2

Q.6Express in polar form: −21+ i√ 3 . [ ans. cos2π/3 +isin2 π/3]

Q.7 If x = - 5 +2√(-4) , find the value of x4+9x3+35x2 – x+4.

[Hint: Divide given poly. By x2 +10x +41=0 as (x+5)2= (4i)2 Remainder is -160 ]

Q.8 Q.10 Find the values of x and y if x2 – 7x +9yi and y2i+20i – 12 are equal.

[ans. x =4, 3 and y =5, 4.]

SEQUENCE & SERIES


S∞ = a1−r , a is first term , r is the common ratio. As n→∞ rn →

0 for |r|<1. Geometric mean between two numbers a & b is √ab i.e., G2 = ab or G = √ab , Harmonic mean (H.M.)= 2ab

a+b , If a , b, c are in G.P. then

b/a = c/b ⇨ b2 = ac. A – G = a+b2 - √ab ≥ 0 ⇨ A ≥ G≥H, R = ¿ , if a,b,c

are in A.P. then 2b = a+c , d = b−an+1 and nth term = Tn = Sn – Sn-1 and last

term in G.P. is arn-1 , A.M. b/w two numbers a & b is (a+b)/2 .


Q. 1 Find k so that 2/3, k, 5k/8 are in A.P.

Q. 2 There are n A.M.’s between 7 and 85 such that (n-3)th mean : nth mean is 11 : 24.Find n. [Hint: 7, a2, a3, a4,…………..an+1, 85 , n=5]

Q. 3 Find the sum of (1+ 122) + (1

2 + 124 ) + ( 1

22 +126) +…….to ∞

[answer is 7/3 ]

Q.4 If a, b, c, d are in G.P. prove that a2-b2, b2-c2, c2-d2 are also in G.P. [HINT: take a, b= ar, c=ar2, d = ar3 ]

Q.5 If one geometric mean G and two arithmetic mean p and q be inserted between two quantities Show that G2 = (2p-q)(2q-p). [Hint: G2 = ab, a, p, q, b are in A.P. d= (b-a)/3 ,p = a+d= 2a+b

3 , q =

a+2d = a+2b3 ]

Q. 6 In an increasing G.P., the sum of first and last term is 66, and product of the second and last but one term is 128. If the sum of the series is 126, find the number of terms in the series. [Ans. r=2 and n = 6]

Permutation & combination


Permutation: An arrangement is called a Permutation. It is the rearrangement of objects or symbols into distinguishable sequences. When we set things in order, we say we have made an arrangement. When we change the order, we say we have changed the arrangement. So each of the arrangement that can be made by taking some or all of a number of things is known as Permutation.Combination: A Combination is a selection of some or all of a number of different objects. It is an un-ordered collection of unique sizes.In a permutation the order of occurence of the objects or the arrangement is important but in combination the order of occurence of the objects is not important.Formula:

Permutation = nPr = n! / (n-r)! Combination = nCr = nPr / r! where, n, r are non negative integers and 0≤ r≤n. r is the size of each permutation. n is the size of the set from which elements are permuted. ! is the factorial operator.


Q.1 A child has 3 pocket and 4 coins. In how many ways can he put the coins in his pocket. [Ans. Use nr , 81]

Q.2 The principal wants to arrange 5 students on the platform such that the boy SALIM occupies the second position and such

that the girl SITA is always adjacent to the girl RITA . How many

such arrangements are possible? [answer is 8]

Q. 3 When a group- photograph is taken, all the seven teachers should be in the first row and all the twenty students should be in the second row. If the two corners of the second row are reserved for the two tallest students, interchangeable only b/w them, and if the middle seat of the front row is reserved for the principal, how many such arrangements are possible? [total no. Of ways = 6!.(18)!.2! ]

Q.4 A team of 8 students goes on an excursion, in two cars, of which one can seat 5 and the other only 4. In how many ways can they travel? [ Hint: 8C3 + 8C4 = 56 + 70 = 126.]

Q.5 If there are six periods in each working day of a school, in how many ways can one arrange 5 subjects such that each subject is allotted at least one period? [ P(6,5)X5 ans. 3600.]

Q. 6 Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed? [ans. Required number of ways = (210 x 120) = 25200]

Q. 7 How many numbers greater than 4,00,000 can be formed by using the digits 0, 2, 2, 4, 4, 5? [ans. total no. Of ways 90.]

BINOMIAL EXPANSION


Binomial theorem formula

general term = (r+1)th term= Tr+1 = nCr xn-r. ar

(i) nCr + nCr-1 = n+1Cr (ii) nCx = nCy x = y or x + y = n


Q. 1 If the co-efficient of x in (x2 + kx )5 is 270 , then find k.

Q.2 Show that 2(4n+4) – 15n – 16 is divisible by 225 ∀ n∊N.

Q.3 Find the middle term in the expansion of ( 2x2

3+ 3

2 x2 )10

Q.4 Let ‘n’ be a positive integer. If the coefficients of second, third and fourth terms in (1+x)n are in arithmetic progression, then find the value of n.

Q.5 If the coefficients of (2r + 1)th term and (r + 2)th term in the expansion of (1 + x)43 are equal, find ‘r’.

http://www.mathwarehouse.com/algebra/polynomial/binomial-theorem.php#binomialTheoremFormula

http://www.mathwarehouse.com/algebra/polynomial/binomial-theorem.php#binomialTheoremFormula

Q.6 Find the 4th term from the end in the expansion of [ x3

2 - 2x4 ] 7

[ans. 1. k =3 ,3. 252, 4. n= 7, 5. 2r + r + 1 = 43, 6. 70x]

STRAIGHT LINES


1.Slopes of parallel lines: Slopes are equal or

where m1 and m2 are the slopes of the lines L, and L2, respectively.

2. Slopes of perpendicular lines: Slopes are negative reciprocals or

3.Point-slope form of a straight line:

4.Slope-intercept form of a straight line: y = mx + b

where b is the y- intercept.

5.Normal form of a straight line:

where p is the line's perpendicular distance from the origin and Ѳ is the angle between the perpendicular and the X- axis

6.Distance from a point to a line:


Q. 1 Determine the ratio in which the line 3x+y – 9 = 0 divides the segment joining the points (1,3) and (2,7).

[ Hint: use section formula , k=3/4]

Q.2 (i) Find the co-ordinates of the orthocenter of the ∆ whose angular points are (1,2), (2,3), (4,3) (ii) Find the co-ordinates of the circumcenter of the ∆ whose angular points are (1,2), (3,-4), (5,-6). [ answer (i) (1,6) (ii) (11,2)]

Q. 3 Find the equation of the line through the intersection of the lines x -3y+1=0 and 2x+5y -9=0 and whose distant from the origin is √ 5 . [ Hint: (x -3y+1) +k(2x+5y -9)=0 -------(1) , then find distant from (0,0) on the line (1) ,Answer is 2x+y – 5=0]

Q. 4 The points (1,3) and (5,1) are the opposite vertices of a rectangle.The other two vertices lie on the line y = 2x+c. Find c and the remaining vertices.

[Hint: D(α ,β ¿ C(5,1)

y=2x+c

M (3,2)

A(1,3) B(X,2X-4) c = -4 , use pythagoras B(2,0) then D(4,4)]

Q.5 The extremities of the base of an isosceles ∆ are the points (2a,0) & (0,a). The equation of the one of the sides is x=2a. Find the equation of the other two sides and the area of the ∆. [ ans. Given

points A&B then C (2a,5a/2), x+2y-2a=0 & 3x – 4y+4a=0 & area of ∆ACB is 5a2/2 sq.units.]

CONIC SECTION

Some valuable points to remember: A conic section is the locus of all points in a plane whose distance from a fixed point is a constant ratio to its distance from a fixed line. The fixed point is the focus, and the fixed line is the directrix.The ratio referred to in the

definition is called the eccentricity (e)


Q. 1 A circle has radius 3 & its centre lies on the y = x-1. Find the eqn. of the circle if it passes through (7,3).[hint: h=4,7 k = 3,6]

Q.2 Find the eqn. of circle of radius 5 which lies within the circle x2+y2+14x+10y – 26 = 0 and which touches the given circle at the point (-1,3). [ans. (x+4)2 + (y+1)2 = 52]

Q.3 Find the eqn. of circle circumscribing the ∆ formed by the lines x + y = 6, 2x+y = 4 & x+2y = 5. [ans. (7,-1), (-2,8) & (1,2) is x2+y2 -3x-2y -21=0]

Q.4 Reduce the equation to standard form,find it’s vertex

. [eqn. of the parabola with its vertex at (-1,3).]

Q.5 Find the eqn. of ellipse whose centre is at origin , foci are (1,0) & (-1,0) and e=1/2. [hint: PF+PF’ =2a , we get 3x2+4y2-12=0]

Q.6 Find the eqn. of hyp. Whose conjugate axis is 5 and the distance b/w the foci is 13.[ ans. 25x2 – 144y2 = 900]

LIMITS


, limx→0

e x−1x =1 ,lim

x→0

log (1+x)x =1, lim

x→a

xn−an

x−a = nan-1

e-∞ =0, e∞ =∞, limx→0

1x =∞, lim

x→∞

1x =0 , {0/0 form lim

X→ 1

X ²−1X−1 )

c∞ = ∞ if c > 1

= 0 , 0 ≤ c ≤ 1

= 1 , c = 1.


Q.1 lim

x→ 3+¿x

[x ] ¿

¿ and

limx→ 3−¿

x[x ]

¿

¿ where [x] denotes the integral part of x. Are they

equal?

Q.2 Is limx→0

e x−1√1−cosx exist?

Q. 3 Evaluate: (i) limx→π

sin3 x−3 sinx(π−x) ³ [answer is -4]

(ii) limx→π /2

cotx−cosxcos ³ x [ answer is ½]

(iii) limx→0

esin 3x−1log(1+ tan 2 x) [ans. 3/2]

Q.4 Let f(x) = { 3−x ² , x≤−2ax+b ,−2< x<2

x2

2 , x≥2

Find a,b so that limx→2f (x ) and lim

x→−2f (x) exist. [ans. a=3/4,b=1/2]

Q.5 Find k such that following function is continuous at indicated point

f(x) ={1−cos4 x8x ²

, x ≠0

k , x=0 [hint: if L.H.Lt=R.H.Lt= f(a)→f is cts. at x=a,k=1]

DERIVATIVES


First principle(ab-initio)

Leibnitz product rule, Quotient rule

Apply the chain rule in composition of functions


Q.1 Find the derivative of the following functions from first principle:

(i) sin (x + 1)

(ii)

(iii) √sinx Q.2 Find the derivative of

(i)

(ii) sinx+ xcosxxsinx−cosx

(iii) cos3¿)

Q.3 If y = xx+a , prove that x dydx = y(1-y).

Q.4 Write the value of derivative of 1+tanx1−tanx at x = 0 [ans. Is 2]

PROBABILITY


1. P(A∩B’) = P(A-B) = P(A) – P(A∩ B) = P( Only A) 2. P(A’∩ B’) = P(AUB)’ = 1 – P(AUB) and P(A’UB’) = 1 – P(A∩B) 3. P(at least one) = 1 – P(None) = 1 – P(0)4. For any two events A and B, P(A∩B) ≤ P(A)≤P(AUB)≤P(A)+P(B)Question bank for recapitulation:

Q.1 Three squares of chess board are selected at random. Find the porb. Of getting 2 squares of one Colour and other of a different colour. [ans. Is 16/21]Q.2 A box contains 100 bolts and 50 nuts, It is given that 50% bolt and 50% nuts are rusted. Two Objects are selected from the box at random. Find the probability that both are bolts or both are rusted.[ans. Is 260/447.]

Q.3 Fine the probability that in a random arrangement of the letters of the word “UNIVERSITY” the two I’s come together.[ans. Is. 1/5 ] Q.4 A five digit number is formed by the digits 1, 2, 3, 4, 5 without repetition. Find the probability that the number is divisible by 4.[ans. Is. 1/5 ]

Q.5 A pair of dice is rolled. Find the probability of getting a doublet or sum of number to be at least 10.

[ans. is P(AUB) = 5/18.] Q.6 Two unbiased dice are thrown. Find the prob. That neither a doublet nor a total of 10 will appear. [ Answer: is 7/9. ]

Q.7 The prob. Of occurrence of atleast one of the events A & BIs 0.6.If A & B occur simultaneously with a prob. Of 0.2. find P(A’)+P(B’) [ P(A∩B)= 0.2 ans. 1.2]

Education

Question bank xi