43030 + Interview Questions in Mathematics

1.	Q9/15 The value ofd/d(sin x cosx)
Answer» Q9/15 The value ofd/d(sin x cosx)

Discussion

2.	sin 4x sin 8x
Answer» sin 4x sin 8x

Discussion

3.	If * and O are two binary operations defined by a * b = a + b and a O b = ab, then
Answer» If * and ^Oare two binary operations defined by a * b = a + b and a ^O b = ab, then

Discussion

4.	tan–1(tan 19) is equal to
Answer» tan–1(tan 19) is equal to

Discussion

5.	If U = {x : x ϵ Z, -3 < x ≤ 9}, A = {x : x = 2P + 1, P ϵ Z , -2 ≤ P ≤ 3}, B = {x : x = q + l, q ϵ Z, 0 ≤ q ≤ 3}, verify De Morgan’s laws for complementation.
Answer» If U = {x : x $ϵ$ Z, -3 < x $\leq$ 9}, A = {x : x = 2P + 1, P $ϵ$ Z , -2 $\leq$ P $\leq$ 3}, B = {x : x = q + l, q $ϵ$ Z, 0 $\leq$ q $\leq$ 3}, verify De Morgan’s laws for complementation.

Discussion

6.	Who do you think is the naughtiest child in your class? Describe her/him in five lines._____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Answer» Who do you think is the naughtiest child in your class? Describe her/him in five lines. _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________

Discussion

7.	Which of the following set of points are non- collinear [MP PET 1990]
Answer» Which of the following set of points are non- collinear [MP PET 1990]

Discussion

8.	Findthe values of aand b suchthat the function defined byis acontinuous function.
Answer» Find the values of a and b such that the function defined by is a continuous function.

8.

Findthe values of aand b suchthat the function defined byis acontinuous function.

Answer»

Find
the values of a
and b such
that the function defined by

is a
continuous function.

Discussion

9.	∫-π2π2-π2cosxsin2xdx
Answer» $\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{- \frac{π}{2}}{\sqrt{\cos x \sin^{2} x}} d x$

Discussion

10.	If the sum of the first 3 terms of an ap is 15 and that of next 3 terms is 33 then find the sum of first 9 terms of the AP
Answer» If the sum of the first 3 terms of an ap is 15 and that of next 3 terms is 33 then find the sum of first 9 terms of the AP

Discussion

11.	If cos 2θ = 0, then 0cos θsin θcos θsin θ0sin θ0cos θ2=_________________.
Answer» If cos 2θ = 0, then ${\|\begin{matrix} 0 & \cos θ & \sin θ \\ \cos θ & \sin θ & 0 \\ \sin θ & 0 & \cos θ \end{matrix}\|}^{2} =$ _________________.

Discussion

12.	76. If equation of a projectile projected at angle θ with horizontal are following x = 15t (m); y = 15t – 5t2 (m), where x and y co-ordinate then angle of projection θ will be?
Answer» 76. If equation of a projectile projected at angle θ with horizontal are following x = 15t (m); y = 15t – 5t2 (m), where x and y co-ordinate then angle of projection θ will be?

Discussion

13.	Find theposition vector of a point R which divides the line joining twopoints P and Q whose position vectors areexternallyin the ratio 1: 2. Also, show that P is the mid point of the linesegment RQ.
Answer» Find the position vector of a point R which divides the line joining two points P and Q whose position vectors areexternally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.

Discussion

14.	I=∫(xcosx−sinxxsinx)dx=ln(\|f(x)\|)+c.The value of limx→0f(x) is
Answer» $I = \int (\frac{x cos x - sin x}{x sin x}) d x = ln (\| f (x) \|) + c$ . The value of $lim x \to 0 f (x)$ is

Discussion

15.	Find thedegree measure of the angle subtended at the centre of a circle ofradius 100 cm by an arc of length 22 cm.
Answer» Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 cm.

Discussion

16.	Let an denote the number of all n−digit positive integers formed by the digits 0,1 or both such that noconsecutive digits in them are 0. Let bn= the number of such n−digit integers ending with digit 1 andcn= the number of such n-digit integers ending with digit 0.Which of the following is correct ?
Answer» Let $a_{n}$ denote the number of all $n -$ digit positive integers formed by the digits $0, 1$ or both such that no consecutive digits in them are $0$ . Let $b_{n} =$ the number of such $n -$ digit integers ending with digit $1$ and $c_{n} =$ the number of such n-digit integers ending with digit $0$ . Which of the following is correct $?$

Discussion

17.	A tangent is drawn at the point P(α,β) on the parabola y2=4ax. This tangent meets a second parabola y2=4a(x+k) at A and B, what is the midpoint of AB?
Answer» A tangent is drawn at the point $P (α, β)$ on the parabola $y^{2} = 4 a x$ . This tangent meets a second parabola $y^{2} = 4 a (x + k)$ at A and B, what is the midpoint of AB?

Discussion

18.	Given a non-empty set X , let : P( X ) × P( X ) → P( X ) be defined as A B = ( A − B ) ∪ ( B − A ), &mnForE; A , B ∈ P( X ). Show that the empty set Φ is the identity for the operation * and all the elements A of P( X ) are invertible with A −1 = A . (Hint: ( A − Φ ) ∪ ( Φ − A ) = A and ( A − A ) ∪ ( A − A ) = A * A = Φ ).
Answer» Given a non-empty set X , let : P( X ) × P( X ) → P( X ) be defined as A B = ( A − B ) ∪ ( B − A ), &mnForE; A , B ∈ P( X ). Show that the empty set Φ is the identity for the operation * and all the elements A of P( X ) are invertible with A −1 = A . (Hint: ( A − Φ ) ∪ ( Φ − A ) = A and ( A − A ) ∪ ( A − A ) = A * A = Φ ).

Discussion

19.	if z is a complex number such that \|z-1\|=1 z is not equal to 2 then the real part of z-2/z is
Answer» if z is a complex number such that \|z-1\|=1 z is not equal to 2 then the real part of z-2/z is

Discussion

20.	If x is real and satisfies x + 2 > √x+4, then
Answer» If x is real and satisfies x + 2 > $\sqrt{x + 4}$ , then

Discussion

21.	The range of the function f x is equal to X square + 1 upon x if x is greater than zero is
Answer» The range of the function f x is equal to X square + 1 upon x if x is greater than zero is

Discussion

22.	A scalar value function is defined as f(x)=xTAx+bTx+c, where A is a symmetric positive definite matrix with dimension n×n;b and x are vectors of dimension n×1. The minimum value of f(x) will occur when x equals.
Answer» A scalar value function is defined as $f (x) = x^{T} A x + b^{T} x + c$ , where $A$ is a symmetric positive definite matrix with dimension $n \times n; b$ and $x$ are vectors of dimension $n \times 1$ . The minimum value of $f (x)$ will occur when $x$ equals.

Discussion

23.	The value of 5cot(5∑k=1cot−1(k2+k+1)) is equal to
Answer» The value of $5 cot (5 \sum k = 1 {cot}^{- 1} (k^{2} + k + 1))$ is equal to

Discussion

24.	The range of θ for which the point (√3sinθ,√4cosθ) lies outside x24−y25=1 is
Answer» The range of $θ$ for which the point $(\sqrt{3} sin θ, \sqrt{4} cos θ)$ lies outside $\frac{x^{2}}{4} - \frac{y^{2}}{5} = 1$ is

Discussion

25.	z1 (3, 2), z2 (1, 5) and origin O form a triangle. Find the angle between the sides Oz1 and Oz2
Answer» $z_{1}$ (3, 2), $z_{2}$ (1, 5) and origin O form a triangle. Find the angle between the sides O $z_{1}$ and O $z_{2}$

Discussion

26.	Let α, β be the roots of the quadratic equation x² + px + p³ = 0. If (α, β) is a pt. on the parabola y² = x, then what are the roots of the qudratic equation?
Answer» Let α, β be the roots of the quadratic equation x² + px + p³ = 0. If (α, β) is a pt. on the parabola y² = x, then what are the roots of the qudratic equation?

Discussion

27.	The value of I=∫x3+xx4−9dx is(where C is constant of integration)
Answer» The value of $I = \int \frac{x^{3} + x}{x^{4} - 9} d x$ is (where $C$ is constant of integration)

Discussion

28.	34 Factorize aa- 7ab-18 bb+a+2b
Answer» 34 Factorize aa- 7ab-18 bb+a+2b

Discussion

29.	The perpendicular from the origin to the line y = mx + c meets it at the point ( – 1, 2). Find the values of m and c .
Answer» The perpendicular from the origin to the line y = mx + c meets it at the point ( – 1, 2). Find the values of m and c .

Discussion

30.	Let A be a finite set of size n. The number of elements in the power set of A×A is
Answer» Let $A$ be a finite set of size $n$ . The number of elements in the power set of $A \times A$ is

Discussion

31.	Let S(3,4) and S′(9,12) be two foci of an ellipse. If the coordinates of the foot of the perpendicular from focus S to a tangent of the ellipse is (1,−4), then the eccentricity of the ellipse is
Answer» Let $S (3, 4)$ and $S^{'} (9, 12)$ be two foci of an ellipse. If the coordinates of the foot of the perpendicular from focus $S$ to a tangent of the ellipse is $(1, - 4),$ then the eccentricity of the ellipse is

Discussion

32.	Find the equation of the ellipse whose foci are (0,±5) and the length of whose major axis is 20.
Answer» Find the equation of the ellipse whose foci are $(0, \pm 5)$ and the length of whose major axis is 20.

Discussion

33.	If a and b are distinct integers, prove that a – b is a factor of a n – b n , whenever n is a positive integer. [ Hint: write a n = ( a – b + b ) n and expand]
Answer» If a and b are distinct integers, prove that a – b is a factor of a n – b n , whenever n is a positive integer. [ Hint: write a n = ( a – b + b ) n and expand]

Discussion

34.	For each binary operation * defined below, determine whether * is commutative or associative. (i) On Z , define a * b = a − b (ii) On Q , define a * b = ab + 1 (iii) On Q , define a * b (iv) On Z + , define a * b = 2 ab (v) On Z + , define a * b = a b (vi) On R − {−1}, define
Answer» For each binary operation * defined below, determine whether * is commutative or associative. (i) On Z , define a * b = a − b (ii) On Q , define a * b = ab + 1 (iii) On Q , define a * b (iv) On Z + , define a * b = 2 ab (v) On Z + , define a * b = a b (vi) On R − {−1}, define

Discussion

35.	If 3 tan θ=3 sin θ, find the value of sin θ.
Answer» If $\sqrt{3} \tan θ = 3 \sin θ$ , find the value of sin θ.

Discussion

36.	48. (SecA-CosA)square+(cosecA-SinA)Square-(CotA-TanA)Square is
Answer» 48. (SecA-CosA)square+(cosecA-SinA)Square-(CotA-TanA)Square is

Discussion

37.	If limx→3(3k+2x2)=0, then k is equal to
Answer» If $lim x \to 3 (3 k + 2 x^{2}) = 0,$ then $k$ is equal to

Discussion

38.	Let s be the sum of the digits of the number 152×518 in base 10. Then
Answer» Let $s$ be the sum of the digits of the number $15^{2} \times 5^{18}$ in base $10$ . Then

Discussion

39.	27. [2 cOs- x dr0 cosx 4 sin2x
Answer» 27. [2 cOs- x dr0 cosx 4 sin2x

Discussion

40.	Write the relation R = {(x, x3): x is a prime number less than 10} in roster form.
Answer» Write the relation R = {(x, x³): x is a prime number less than 10} in roster form.

Discussion

41.	If ω is a complex cube root of unity and (1+ω)7 = A + Bω , then (A, B) is
Answer» If ω is a complex cube root of unity and ${(1 + ω)}^{7}$ = A + B $ω$ , then (A, B) is

Discussion

42.	Mark the correct alternative in the following question:In a college 30% students fail in Physics, 25% fail in Mathematics and 10% fail in both. One student is chosen at random. The probability that she fails in Physics if she failed in Mathematics isa 110 b 13 c 25 d 920
Answer» Mark the correct alternative in the following question: In a college 30% students fail in Physics, 25% fail in Mathematics and 10% fail in both. One student is chosen at random. The probability that she fails in Physics if she failed in Mathematics is $(a) \frac{1}{10} (b) \frac{1}{3} (c) \frac{2}{5} (d) \frac{9}{20}$

Discussion

43.	Pair the statements that have the same answer.
Answer» Pair the statements that have the same answer.

Discussion

44.	Question 14(ii)50 students enter for a school javelin throw competition. The distance (in metres) thrown are recorded below :Distance (inm)0−2020−4040−6060−8080−100Number of students 6 11 17 12 4(ii) Draw a cumulative frequency curve (less than type) and calculate the median distance thrown by using this curve.
Answer» Question 14(ii) 50 students enter for a school javelin throw competition. The distance (in metres) thrown are recorded below : $\begin{matrix} D i s t a n c e (i n m) & 0 - 20 & 20 - 40 & 40 - 60 & 60 - 80 & 80 - 100 N u m b e r o f s t u d e n t s & 6 & 11 & 17 & 12 & 4 \end{matrix}$ (ii) Draw a cumulative frequency curve (less than type) and calculate the median distance thrown by using this curve.

Discussion

45.	The value of the definite integral π2∫0√tanxdx is
Answer» The value of the definite integral $\frac{π}{2} \int 0 \sqrt{tan x} d x$ is

Discussion

46.	If 5 distinct balls are placed at random into 5 cells, then the probability that exactly one cell remains empty is
Answer» If $5$ distinct balls are placed at random into $5$ cells, then the probability that exactly one cell remains empty is

Discussion

47.	let f(x)= sin x + cos x , where x belongs to (0,2pi) . find the interval where function is increasing and decreasing .
Answer» let f(x)= sin x + cos x , where x belongs to (0,2pi) . find the interval where function is increasing and decreasing .

Discussion

48.	In the following example find the distance of each of the given points from the corresponding given plane: pointPlane(−6,0,0)2x−3y+6z−2=0
Answer» In the following example find the distance of each of the given points from the corresponding given plane: $\begin{matrix} p o i n t & P l a n e (- 6, 0, 0) & 2 x - 3 y + 6 z - 2 = 0 \end{matrix}$

48.

In the following example find the distance of each of the given points from the corresponding given plane: pointPlane(−6,0,0)2x−3y+6z−2=0

Answer»

In the following example find the distance of each of the given points from the corresponding given plane:

$\begin{matrix} p o i n t & P l a n e (- 6, 0, 0) & 2 x - 3 y + 6 z - 2 = 0 \end{matrix}$

Discussion

49.	Consider f : {1, 2, 3} → { a , b , c } given by f (1) = a , f (2) = b and f (3) = c . Find f −1 and show that ( f −1 ) −1 = f .
Answer» Consider f : {1, 2, 3} → { a , b , c } given by f (1) = a , f (2) = b and f (3) = c . Find f −1 and show that ( f −1 ) −1 = f .

Discussion

50.	(i) If the roots of the equation a2+b2x2-2ac+bdx+c2+d2=0 are equal, prove that ab=cd. [CBSE 2017](ii) If ad ≠ bc then prove that the equation a2+b2x2+2ac+bdx+c2+d2=0 has no real roots. [CBSE 2017]
Answer» (i) If the roots of the equation $(a^{2} + b^{2}) x^{2} - 2 (a c + b d) x + (c^{2} + d^{2}) = 0$ are equal, prove that $\frac{a}{b} = \frac{c}{d}$ . [CBSE 2017] (ii) If ad ≠ bc then prove that the equation $(a^{2} + b^{2}) x^{2} + 2 (a c + b d) x + (c^{2} + d^{2}) = 0$ has no real roots. [CBSE 2017]

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

Q9/15 The value ofd/d(sin x cosx)

sin 4x sin 8x

If * and O are two binary operations defined by a * b = a + b and a O b = ab, then

tan–1(tan 19) is equal to

If U = {x : x ϵ Z, -3 < x ≤ 9}, A = {x : x = 2P + 1, P ϵ Z , -2 ≤ P ≤ 3}, B = {x : x = q + l, q ϵ Z, 0 ≤ q ≤ 3}, verify De Morgan’s laws for complementation.

Which of the following set of points are non- collinear [MP PET 1990]

Findthe values of aand b suchthat the function defined byis acontinuous function.

∫-π2π2-π2cosxsin2xdx

If the sum of the first 3 terms of an ap is 15 and that of next 3 terms is 33 then find the sum of first 9 terms of the AP

If cos 2θ = 0, then 0cos θsin θcos θsin θ0sin θ0cos θ2=_________________.

76. If equation of a projectile projected at angle θ with horizontal are following x = 15t (m); y = 15t – 5t2 (m), where x and y co-ordinate then angle of projection θ will be?

Find theposition vector of a point R which divides the line joining twopoints P and Q whose position vectors areexternallyin the ratio 1: 2. Also, show that P is the mid point of the linesegment RQ.

I=∫(xcosx−sinxxsinx)dx=ln(|f(x)|)+c.The value of limx→0f(x) is

Find thedegree measure of the angle subtended at the centre of a circle ofradius 100 cm by an arc of length 22 cm.

A tangent is drawn at the point P(α,β) on the parabola y2=4ax. This tangent meets a second parabola y2=4a(x+k) at A and B, what is the midpoint of AB?

if z is a complex number such that |z-1|=1 z is not equal to 2 then the real part of z-2/z is

If x is real and satisfies x + 2 &gt; √x+4, then

The range of the function f x is equal to X square + 1 upon x if x is greater than zero is

A scalar value function is defined as f(x)=xTAx+bTx+c, where A is a symmetric positive definite matrix with dimension n×n;b and x are vectors of dimension n×1. The minimum value of f(x) will occur when x equals.

The value of 5cot(5∑k=1cot−1(k2+k+1)) is equal to

The range of θ for which the point (√3sinθ,√4cosθ) lies outside x24−y25=1 is

z1 (3, 2), z2 (1, 5) and origin O form a triangle. Find the angle between the sides Oz1 and Oz2

Let α, β be the roots of the quadratic equation x² + px + p³ = 0. If (α, β) is a pt. on the parabola y² = x, then what are the roots of the qudratic equation?

The value of I=∫x3+xx4−9dx is(where C is constant of integration)

34 Factorize a*a- 7*a*b-18 b*b+a+2b

The perpendicular from the origin to the line y = mx + c meets it at the point ( – 1, 2). Find the values of m and c .

Let A be a finite set of size n. The number of elements in the power set of A×A is

Let S(3,4) and S′(9,12) be two foci of an ellipse. If the coordinates of the foot of the perpendicular from focus S to a tangent of the ellipse is (1,−4), then the eccentricity of the ellipse is

Find the equation of the ellipse whose foci are (0,±5) and the length of whose major axis is 20.

If a and b are distinct integers, prove that a – b is a factor of a n – b n , whenever n is a positive integer. [ Hint: write a n = ( a – b + b ) n and expand]

For each binary operation * defined below, determine whether * is commutative or associative. (i) On Z , define a * b = a − b (ii) On Q , define a * b = ab + 1 (iii) On Q , define a * b (iv) On Z + , define a * b = 2 ab (v) On Z + , define a * b = a b (vi) On R − {−1}, define

If 3 tan θ=3 sin θ, find the value of sin θ.

48. (SecA-CosA)square+(cosecA-SinA)Square-(CotA-TanA)Square is

If limx→3(3k+2x2)=0, then k is equal to

Let s be the sum of the digits of the number 152×518 in base 10. Then

27. [2 cOs- x dr0 cosx 4 sin2x

Write the relation R = {(x, x3): x is a prime number less than 10} in roster form.

If ω is a complex cube root of unity and (1+ω)7 = A + Bω , then (A, B) is

Mark the correct alternative in the following question:In a college 30% students fail in Physics, 25% fail in Mathematics and 10% fail in both. One student is chosen at random. The probability that she fails in Physics if she failed in Mathematics isa 110 b 13 c 25 d 920

Pair the statements that have the same answer.

The value of the definite integral π2∫0√tanxdx is

If 5 distinct balls are placed at random into 5 cells, then the probability that exactly one cell remains empty is

let f(x)= sin x + cos x , where x belongs to (0,2pi) . find the interval where function is increasing and decreasing .

In the following example find the distance of each of the given points from the corresponding given plane: pointPlane(−6,0,0)2x−3y+6z−2=0

Consider f : {1, 2, 3} → { a , b , c } given by f (1) = a , f (2) = b and f (3) = c . Find f −1 and show that ( f −1 ) −1 = f .

(i) If the roots of the equation a2+b2x2-2ac+bdx+c2+d2=0 are equal, prove that ab=cd. [CBSE 2017](ii) If ad ≠ bc then prove that the equation a2+b2x2+2ac+bdx+c2+d2=0 has no real roots. [CBSE 2017]

If x is real and satisfies x + 2 > √x+4, then

34 Factorize aa- 7ab-18 bb+a+2b