43030 + Interview Questions in Mathematics

1.	find the value of:( 1+ cos(pi/8))(1+ cos(3pi/8))(1+ cos(5pi/8))(1+ cos(7pi/8)) = ?
Answer» find the value of: ( 1+ cos(pi/8))(1+ cos(3pi/8))(1+ cos(5pi/8))(1+ cos(7pi/8)) = ?

Discussion

2.	The value of the integral is I=∫π/40cos2xdx
Answer» The value of the integral is $I = \int_{0}^{π / 4} c o s^{2} x d x$

Discussion

3.	The value of the integral ∫∞0 x dx(1+x)(1+x2) is equal to
Answer» The value of the integral $\int_{0}^{\infty} \frac{x dx}{(1 + x) (1 + x^{2})}$ is equal to

Discussion

4.	The population P=P(t) at time ‘t′ of a certain species follows the differential equation dPdt=0.5P−450. If P(0)=850, then the time at which population becomes zero is :
Answer» The population $P = P (t)$ at time $‘ t^{'}$ of a certain species follows the differential equation $\frac{d P}{d t} = 0.5 P - 450.$ If $P (0) = 850,$ then the time at which population becomes zero is :

Discussion

5.	nC0 + 2.nC1 + 3.nC2 +..............(n+1)nCn =
Answer» $^{n} C_{0}$ + 2. $^{n} C_{1}$ + 3. $^{n} C_{2}$ +..............(n+1) $^{n} C_{n}$ =

Discussion

6.	tan -1
Answer» tan -1

Discussion

7.	Prove the following by using the principle of mathematical induction for all n∈N1⋅3+3⋅5+5⋅7+⋯+(2n−1)(2n+1)=n(4n2+6n−1)3
Answer» Prove the following by using the principle of mathematical induction for all $n \in N$ $1 \cdot 3 + 3 \cdot 5 + 5 \cdot 7 + \dots + (2 n - 1) (2 n + 1)$ $= \frac{n (4 n^{2} + 6 n - 1)}{3}$

Discussion

8.	The vectors →a+→b,→a−k→b, where k is scalar, are collinear for
Answer» The vectors $\to a + \to b, \to a - k \to b,$ where $k$ is scalar, are collinear for

Discussion

9.	The equation of the plane which makes with the coordinate axes, a triangle with centroid (α,β,γ) is given by
Answer» The equation of the plane which makes with the coordinate axes, a triangle with centroid $(α, β, γ)$ is given by

Discussion

10.	Let f(x)={x+a,if x≥1ax2+1,if x<1 , then f(x) is differentiable at x = 1 if
Answer» Let $f (x) = {\begin{matrix} x + a, & i f x \geq 1 a x^{2} + 1, & i f x < 1 \end{matrix}$ , then f(x) is differentiable at x = 1 if

Discussion

11.	The value of ∫c[(3x−8y2)dx+(4y−6xy)dy], where C is the boundary of the region bounded by x = 0, y = 0 and x + y = 1 is_____1.67
Answer» The value of $\int_{c} [(3 x - 8 y^{2}) d x + (4 y - 6 x y) d y]$ , where C is the boundary of the region bounded by x = 0, y = 0 and x + y = 1 is_____ 1.67

Discussion

12.	If tan-ı _ + tan-1x+2-1 1+1 Tx2 415.then find the value of r
Answer» If tan-ı _ + tan-1x+2-1 1+1 Tx2 415.then find the value of r

Discussion

13.	From an aeroplane vertically above a horizontal road, the angles of depression of two friends standing on either side of the aeroplane are observed to be α & β. The distance between the two friends is 1 m. The height (in metres) of the aeroplane from the ground is
Answer» From an aeroplane vertically above a horizontal road, the angles of depression of two friends standing on either side of the aeroplane are observed to be $α & β$ . The distance between the two friends is 1 m. The height (in metres) of the aeroplane from the ground is

Discussion

14.	Evaluate [i18+(1i)25]3
Answer» Evaluate ${[i^{18} + {(\frac{1}{i})}^{25}]}^{3}$

Discussion

15.	If the point (x1+t(x2−x1),y1+t(y2−y1),z1+t(z2−z1)) divides the line segment joining (x1,y1,z1) and (x2,y2,z2) internally in the ratio t1−t, then the possible values of t lies in the interval:
Answer» If the point $(x_{1} + t (x_{2} - x_{1}), y_{1} + t (y_{2} - y_{1}), z_{1} + t (z_{2} - z_{1}))$ divides the line segment joining $(x_{1}, y_{1}, z_{1})$ and $(x_{2}, y_{2}, z_{2})$ internally in the ratio $\frac{t}{1 - t}$ , then the possible values of $t$ lies in the interval:

Discussion

16.	Two independent random varibles X and Yare identical and uniformly distributed in the range [0,2].The probability that min[X,Y] is less than 1 is equal to___________ 0.75
Answer» $T w o i n d e p e n d e n t r a n d o m v a r i b l e s X a n d Y a r e i d e n t i c a l a n d u n i f o r m l y d i s t r i b u t e d i n t h e r a n g e [0, 2] . T h e p r o b a b i l i t y t h a t m i n [X, Y] i s l e s s t h a n 1 i s e q u a l t o$ ___________ 0.75

Discussion

17.	If nC4,nC5 and nC6 are in A.P., then find n.
Answer» If $^{n} C_{4},^{n} C_{5}$ and $^{n} C_{6}$ are in A.P., then find n.

Discussion

18.	If A and B are two independent events such that P(A′∩B)=215 and P(A∩B′)=16, then P(A) is
Answer» If $A$ and $B$ are two independent events such that $P (A^{'} \cap B) = \frac{2}{15}$ and $P (A \cap B^{'}) = \frac{1}{6},$ then $P (A)$ is

Discussion

19.	If √a+ib = x+iy, then possible value of √a+ib is
Answer» If $\sqrt{a + i b}$ = x+iy, then possible value of $\sqrt{a + i b}$ is

Discussion

20.	if p+iq be one of the roots of the equation x^3+ax+b=0, then 2p is one of the roots of the equation (1) x^3+ax+b=0 (2) x^3-ax-b=0 (3) x^3+ax-b=0 (4) x^3+bx+a=0
Answer» if p+iq be one of the roots of the equation x^3+ax+b=0, then 2p is one of the roots of the equation (1) x^3+ax+b=0 (2) x^3-ax-b=0 (3) x^3+ax-b=0 (4) x^3+bx+a=0

Discussion

21.	Prove:(†anθ+2)(2†anθ+1)=5†anθ+2sec^2 θ
Answer» Prove:(†anθ+2)(2†anθ+1)=5†anθ+2sec^2 θ

Discussion

22.	Differentiate the function sin−1(x√x),0≤x≤1
Answer» Differentiate the function ${sin}^{- 1} (x \sqrt{x}), 0 \leq x \leq 1$

Discussion

23.	The value of the expression 47C4+∑5j=1 52−jC3 is
Answer» The value of the expression $^{47} C_{4} + \sum_{j = 1}^{5}^{52 - j} C_{3}$ is

Discussion

24.	Determinant a. 1. 1 a. b. c. =(a-b)(b-c)(c-a). (a+b+c) a^3 b^3 c^3Prove using properties LHS=RHS
Answer» Determinant a. 1. 1 a. b. c. =(a-b)(b-c)(c-a). (a+b+c) a^3 b^3 c^3 Prove using properties LHS=RHS

Discussion

25.	If (p2−1)x2+(p−1)x=0 is an identity in x, then the value of p is
Answer» If $(p^{2} - 1) x^{2} + (p - 1) x = 0$ is an identity in $x$ , then the value of $p$ is

Discussion

26.	If 3cos θ = 2 then (2sec2 θ + 2tan2 θ – 7) = ?(a) 0(b) 1(c) 3(4) 4
Answer» If 3cos θ = 2 then (2sec² θ + 2tan²θ – 7) = ? (a) 0 (b) 1 (c) 3 (4) 4

Discussion

27.	Let S={a∈N,a≤100}. If the equation [Tan2x]−Tanx−a=0 has real roots then number of elements in S is (where [] is step functin).
Answer» Let $S = {a \in N, a \leq 100}$ . If the equation $[T a n^{2} x] - T a n x - a = 0$ has real roots then number of elements in S is (where [] is step functin).

Discussion

28.	Find the equations of the sides of the triangle, the coordinates of whose vertices are (1, 4), (2, -3) and (-1, -2).
Answer» Find the equations of the sides of the triangle, the coordinates of whose vertices are (1, 4), (2, -3) and (-1, -2).

Discussion

29.	How many 5-letter words can be formed with the letters of the word ‘QUEUE’?___
Answer» How many 5-letter words can be formed with the letters of the word ‘QUEUE’? ___

Discussion

30.	Let n be a positive integer. Let A=n∑k=0(−1)k nCk[(12)k+(34)k+(78)k+(1516)k+(3132)k]. If 63A=1−1230, then n is equal to
Answer» Let $n$ be a positive integer. Let $A = n \sum k = 0 (- 1)^{k}^{n} C_{k} [{(\frac{1}{2})}^{k} + {(\frac{3}{4})}^{k} + {(\frac{7}{8})}^{k} + {(\frac{15}{16})}^{k} + {(\frac{31}{32})}^{k}] .$ If $63 A = 1 - \frac{1}{2^{30}},$ then $n$ is equal to

Discussion

31.	Let a,s,t be nonzero real numbers. Let P(at2,2at), and S(as2,2as) be distinct points on the parabola y2=4ax. If st=1, then the tangent at P and the normal at S to the parabola meet at a point whose ordinate is
Answer» Let $a, s, t$ be nonzero real numbers. Let $P (a t^{2}, 2 a t), and S (a s^{2}, 2 a s)$ be distinct points on the parabola $y^{2} = 4 a x$ . If $s t = 1$ , then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is

Discussion

32.	If A is an orthogonal matrix and B=AB, then BTA (BT is transpose of B) is
Answer» If $A$ is an orthogonal matrix and $B = A B,$ then $B^{T} A (B^{T} is transpose of B)$ is

Discussion

33.	limx→π(1−4tanx)cotx=
Answer» $lim x \to π (1 - 4 tan x)^{cot x} =$

Discussion

34.	f:R→R, f(x)=x\|x\| is
Answer» $f : R \to R, f (x) = x \| x \| i s$

Discussion

35.	The minimum value of ∣∣[μ]∣∣ for which the circle x2+y2=9 and x2+y2−μx−6=0 have two common tangents is (where [.] is greatest integer function)
Answer» The minimum value of $∣ ∣ [μ] ∣ ∣$ for which the circle $x^{2} + y^{2} = 9$ and $x^{2} + y^{2} - μ x - 6 = 0$ have two common tangents is (where [.] is greatest integer function)

Discussion

36.	43. What is the proof that (a+b)2=a2+2ab+b2
Answer» 43. What is the proof that (a+b)2=a2+2ab+b2

Discussion

37.	The equation of common tangent to the parabolas y2=4x and x2=4y is
Answer» The equation of common tangent to the parabolas $y^{2} = 4 x$ and $x^{2} = 4 y$ is

Discussion

38.	√2+√3+√4+√6 is equal to
Answer» $\sqrt{2} + \sqrt{3} + \sqrt{4} + \sqrt{6}$ is equal to

Discussion

39.	∫π013+2 sin x+cos xdx=
Answer» $\int_{0}^{π} \frac{1}{3 + 2 s i n x + c o s x} d x =$

Discussion

40.	For the function f(x)=x100100+x9999+....+x22+1. Prove that f′(1)=100 f′(0)
Answer» For the function $f (x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . . + \frac{x^{2}}{2} + 1$ . Prove that $f^{'} (1) = 100 f^{'} (0)$

Discussion

41.	134. Prove that (1+w)-(1-w) =0 , where w is cube root of unity
Answer» 134. Prove that (1+w)-(1-w) =0 , where w is cube root of unity

Discussion

42.	If →a,→b,→c,→d are non-zero vectors, then [→a×→b →a×→c →d] can be simplified as:
Answer» If $\to a, \to b, \to c, \to d$ are non-zero vectors, then $[\to a \times \to b \to a \times \to c \to d]$ can be simplified as:

Discussion

43.	The value of limx→0(sinxm+cos3xm)2mx is
Answer» The value of $lim x \to 0 {(sin \frac{x}{m} + cos \frac{3 x}{m})}^{\frac{2 m}{x}}$ is

Discussion

44.	If S is the sample space and P(A) = 13 P(B) and S = A ∪ B, where A and B are two mutually exclusive events, then P (A) =(a) 1/4(b) 1/2(c) 3/4(d) 3/8
Answer» If S is the sample space and P(A) = $\frac{1}{3}$ P(B) and S = A ∪ B, where A and B are two mutually exclusive events, then P (A) = (a) 1/4 (b) 1/2 (c) 3/4 (d) 3/8

Discussion

45.	The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1,2,3,4,5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is
Answer» The total number of numbers, lying between $100$ and $1000$ that can be formed with the digits $1, 2, 3, 4, 5,$ if the repetition of digits is not allowed and numbers are divisible by either $3$ or $5$ , is

Discussion

46.	A rectangular steel bar of length 500 mm, width 100 mm, and thickness 15 mm is cantilevered to a 200 mm steel channel using 4 bolts as shown.For an external load of 10 kN applied at the tip of the steel bar, the resultant shear load on the bolt at B, is (round off to one decimal place).16.005
Answer» A rectangular steel bar of length 500 mm, width 100 mm, and thickness 15 mm is cantilevered to a 200 mm steel channel using 4 bolts as shown. For an external load of 10 kN applied at the tip of the steel bar, the resultant shear load on the bolt at B, is (round off to one decimal place). 16.005

Discussion

47.	Q1. The value of sin55^° - sin19^°+ sin53^° - sin17^° is always equal to Q2. Tan9^° - tan27^° - tan63^° +tan81^°=
Answer» Q1. The value of sin55^° - sin19^°+ sin53^° - sin17^° is always equal to Q2. Tan9^° - tan27^° - tan63^° +tan81^°=

Discussion

48.	If sinA+B=cosA-B=32, then cot 2A = _________.
Answer» If $\sin (A + B) = \cos (A - B) = \frac{\sqrt{3}}{2}$ , then cot 2A = _________.

Discussion

49.	Prove that 111111..…..1 (91 digits) is not a prime number.
Answer» Prove that 111111..…..1 (91 digits) is not a prime number.

Discussion

50.	Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is ?
Answer» Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is ?

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

find the value of:( 1+ cos(pi/8))(1+ cos(3pi/8))(1+ cos(5pi/8))(1+ cos(7pi/8)) = ?

The value of the integral is I=∫π/40cos2xdx

The value of the integral ∫∞0 x dx(1+x)(1+x2) is equal to

The population P=P(t) at time ‘t′ of a certain species follows the differential equation dPdt=0.5P−450. If P(0)=850, then the time at which population becomes zero is :

nC0 + 2.nC1 + 3.nC2 +..............(n+1)nCn =

tan -1

Prove the following by using the principle of mathematical induction for all n∈N1⋅3+3⋅5+5⋅7+⋯+(2n−1)(2n+1)=n(4n2+6n−1)3

The vectors →a+→b,→a−k→b, where k is scalar, are collinear for

The equation of the plane which makes with the coordinate axes, a triangle with centroid (α,β,γ) is given by

Let f(x)={x+a,if x≥1ax2+1,if x&lt;1 , then f(x) is differentiable at x = 1 if

The value of ∫c[(3x−8y2)dx+(4y−6xy)dy], where C is the boundary of the region bounded by x = 0, y = 0 and x + y = 1 is_____1.67

If tan-ı _ + tan-1x+2-1 1+1 Tx2 415.then find the value of r

From an aeroplane vertically above a horizontal road, the angles of depression of two friends standing on either side of the aeroplane are observed to be α &amp; β. The distance between the two friends is 1 m. The height (in metres) of the aeroplane from the ground is

Evaluate [i18+(1i)25]3

If the point (x1+t(x2−x1),y1+t(y2−y1),z1+t(z2−z1)) divides the line segment joining (x1,y1,z1) and (x2,y2,z2) internally in the ratio t1−t, then the possible values of t lies in the interval:

Two independent random varibles X and Yare identical and uniformly distributed in the range [0,2].The probability that min[X,Y] is less than 1 is equal to___________ 0.75

If nC4,nC5 and nC6 are in A.P., then find n.

If A and B are two independent events such that P(A′∩B)=215 and P(A∩B′)=16, then P(A) is

If √a+ib = x+iy, then possible value of √a+ib is

if p+iq be one of the roots of the equation x^3+ax+b=0, then 2p is one of the roots of the equation (1) x^3+ax+b=0 (2) x^3-ax-b=0 (3) x^3+ax-b=0 (4) x^3+bx+a=0

Prove:(†anθ+2)(2†anθ+1)=5†anθ+2sec^2 θ

Differentiate the function sin−1(x√x),0≤x≤1

The value of the expression 47C4+∑5j=1 52−jC3 is

Determinant a. 1. 1 a. b. c. =(a-b)(b-c)(c-a). (a+b+c) a^3 b^3 c^3Prove using properties LHS=RHS

If (p2−1)x2+(p−1)x=0 is an identity in x, then the value of p is

If 3cos θ = 2 then (2sec2 θ + 2tan2 θ – 7) = ?(a) 0(b) 1(c) 3(4) 4

Let S={a∈N,a≤100}. If the equation [Tan2x]−Tanx−a=0 has real roots then number of elements in S is (where [] is step functin).

Find the equations of the sides of the triangle, the coordinates of whose vertices are (1, 4), (2, -3) and (-1, -2).

How many 5-letter words can be formed with the letters of the word ‘QUEUE’?___

Let n be a positive integer. Let A=n∑k=0(−1)k nCk[(12)k+(34)k+(78)k+(1516)k+(3132)k]. If 63A=1−1230, then n is equal to

Let a,s,t be nonzero real numbers. Let P(at2,2at), and S(as2,2as) be distinct points on the parabola y2=4ax. If st=1, then the tangent at P and the normal at S to the parabola meet at a point whose ordinate is

If A is an orthogonal matrix and B=AB, then BTA (BT is transpose of B) is

limx→π(1−4tanx)cotx=

f:R→R, f(x)=x|x| is

The minimum value of ∣∣[μ]∣∣ for which the circle x2+y2=9 and x2+y2−μx−6=0 have two common tangents is (where [.] is greatest integer function)

43. What is the proof that (a+b)2=a2+2ab+b2

The equation of common tangent to the parabolas y2=4x and x2=4y is

√2+√3+√4+√6 is equal to

∫π013+2 sin x+cos xdx=

For the function f(x)=x100100+x9999+....+x22+1. Prove that f′(1)=100 f′(0)

134. Prove that (1+w)-(1-w) =0 , where w is cube root of unity

If →a,→b,→c,→d are non-zero vectors, then [→a×→b →a×→c →d] can be simplified as:

The value of limx→0(sinxm+cos3xm)2mx is

If S is the sample space and P(A) = 13 P(B) and S = A ∪ B, where A and B are two mutually exclusive events, then P (A) =(a) 1/4(b) 1/2(c) 3/4(d) 3/8

The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1,2,3,4,5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is

A rectangular steel bar of length 500 mm, width 100 mm, and thickness 15 mm is cantilevered to a 200 mm steel channel using 4 bolts as shown.For an external load of 10 kN applied at the tip of the steel bar, the resultant shear load on the bolt at B, is (round off to one decimal place).16.005

Q1. The value of sin55^° - sin19^°+ sin53^° - sin17^° is always equal to Q2. Tan9^° - tan27^° - tan63^° +tan81^°=

If sinA+B=cosA-B=32, then cot 2A = _________.

Prove that 111111..…..1 (91 digits) is not a prime number.

Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is ?

Let f(x)={x+a,if x≥1ax2+1,if x<1 , then f(x) is differentiable at x = 1 if

From an aeroplane vertically above a horizontal road, the angles of depression of two friends standing on either side of the aeroplane are observed to be α & β. The distance between the two friends is 1 m. The height (in metres) of the aeroplane from the ground is