43030 + Interview Questions in Mathematics

1.	Let →a and →b be two unit vectors such that →a⋅→b=0. For some x,y∈R, let →c=x→a+y→b+(→a×→b). If \|→c\|=2 and the vector →c is inclined at the same angle α to both →a and →b, then the value of 8cos2α is .
Answer» Let $\to a$ and $\to b$ be two unit vectors such that $\to a \cdot \to b = 0$ . For some $x, y \in R$ , let $\to c = x \to a + y \to b + (\to a \times \to b)$ . If $\| \to c \| = 2$ and the vector $\to c$ is inclined at the same angle $α$ to both $\to a$ and $\to b$ , then the value of $8 {cos}^{2} α$ is .

Discussion

2.	If the equation 4sin(x+π3)cos(x−π6)=a2+√3sin2x−cos2x has a solution, then the value of a can be
Answer» If the equation $4 sin (x + \frac{π}{3}) cos (x - \frac{π}{6}) = a^{2} + \sqrt{3} sin 2 x - cos 2 x$ has a solution, then the value of $a$ can be

Discussion

3.	All the letters of the word EAMCOT are arranged in a different possible ways. Find the number of arrangments in which no two vowels are adjacent to each other.
Answer» All the letters of the word EAMCOT are arranged in a different possible ways. Find the number of arrangments in which no two vowels are adjacent to each other.

Discussion

4.	The number of different ways in which 8 persons can stand in a row so that between two particular persons A and B there are always two persons, is
Answer» The number of different ways in which 8 persons can stand in a row so that between two particular persons A and B there are always two persons, is

Discussion

5.	Which of the following functions are homogeneous?
Answer» Which of the following functions are homogeneous?

Discussion

6.	From origin, chords are drawn to the circle x2+y2−2y=0.The locus of the middle points of these chords is
Answer» From origin, chords are drawn to the circle $x^{2} + y^{2} - 2 y = 0$ .The locus of the middle points of these chords is

Discussion

7.	In △ABC, the incircle touches the sides BC, CA and AB, respectively, D, E, and F. If the radius of the incircle is 4 units and BD, CE and AF are consecutive integers, then the value of s3, where s is a semi-perimeter of triangle, is
Answer» In $△ A B C$ , the incircle touches the sides $B C, C A$ and $A B$ , respectively, $D, E,$ and $F$ . If the radius of the incircle is $4$ units and $B D, C E$ and $A F$ are consecutive integers, then the value of $\frac{s}{3}$ , where $s$ is a semi-perimeter of triangle, is

Discussion

8.	If asked to examine the consistencyof the equations should we find the solution or verification alone is enough?
Answer» If asked to examine the consistencyof the equations should we find the solution or verification alone is enough?

Discussion

9.	Four unit circles pass through the origin and have their centres on the coordinate axes. The area of the quadrilateral whose vertices are the points of intersection (in pairs) of the circle, is
Answer» Four unit circles pass through the origin and have their centres on the coordinate axes. The area of the quadrilateral whose vertices are the points of intersection (in pairs) of the circle, is

Discussion

10.	Let * be any binary operation on a set A. An element a' ϵ A is called an inverse of a ϵ A for the binary operation *, if
Answer» Let * be any binary operation on a set A. An element a' ϵ A is called an inverse of a ϵ A for the binary operation *, if

Discussion

11.	Total number of ordered pairs (x,y) satisfying \|x\|+\|y\|=2, sin(πx23)=1 is
Answer» Total number of ordered pairs (x,y) satisfying $\| x \| + \| y \| = 2, s i n (\frac{π x^{2}}{3}) = 1$ is

Discussion

12.	For three sets A, B and C, show that (i) A∩B=A∩C need not imply B = C. (ii) A⊂B⇒C−B⊂C−A
Answer» For three sets A, B and C, show that (i) $A \cap B = A \cap C$ need not imply B = C. (ii) $A \subset B \Rightarrow C - B \subset C - A$

12.

For three sets A, B and C, show that (i) A∩B=A∩C need not imply B = C. (ii) A⊂B⇒C−B⊂C−A

Answer»

For three sets A, B and C, show that

(i) $A \cap B = A \cap C$ need not imply B = C.

(ii) $A \subset B \Rightarrow C - B \subset C - A$

Discussion

13.	The value of limx→x2(secx−tanx) is
Answer» The value of ${lim}_{x \to \frac{x}{2}} (s e c x - t a n x)$ is

Discussion

14.	The coefficient of xn in expansion of (1+x)(1–x)n is
Answer» The coefficient of $x^{n}$ in expansion of $(1 + x) (1 - x)^{n}$ is

Discussion

15.	Let PQ be a focal chord of the parabola y2=4x. The tangents to the parabola at P and Q meet at a point on y = 2x + 1. If PQ subtends an angle θ at the vertex of y2=4x, then tanθ =
Answer» Let PQ be a focal chord of the parabola $y^{2} = 4 x$ . The tangents to the parabola at P and Q meet at a point on y = 2x + 1. If PQ subtends an angle $θ$ at the vertex of $y^{2} = 4 x$ , then $t a n θ$ =

Discussion

16.	Many people ____ his statement as meaning that she intends to resign
Answer» Many people ____ his statement as meaning that she intends to resign

Discussion

17.	Write the coordinates of the orthocentre of the triangle by points (8,0),(4,6) and (0,0).
Answer» Write the coordinates of the orthocentre of the triangle by points (8,0),(4,6) and (0,0).

Discussion

18.	The equation of line which passes through the point of intersection of 2x + 3y – 5 = 0 and x + y = 2 and also which is farthest from (2, 3) is
Answer» The equation of line which passes through the point of intersection of 2x + 3y – 5 = 0 and x + y = 2 and also which is farthest from (2, 3) is

Discussion

19.	The statement (p∧∼q)∨q∨(∼p∧q) is equivalent to
Answer» The statement $(p \land \sim q) \lor q \lor (\sim p \land q)$ is equivalent to

Discussion

20.	The value of (1+i)4+(1−i)4 is
Answer» The value of $(1 + i)^{4} + (1 - i)^{4}$ is

Discussion

21.	If 18C2r= 18Cr+3, then the value of r is/are
Answer» If $^{18} C_{2 r} =^{18} C_{r + 3}$ , then the value of $r$ is/are

Discussion

22.	If y=tan−15x1−6x2, −1√6<x<1√6, then prove that dydx=21+4x2+31+9x2
Answer» If $y = t a n^{- 1} \frac{5 x}{1 - 6 x^{2}}, - \frac{1}{\sqrt{6}} < x < \frac{1}{\sqrt{6}},$ then prove that $\frac{d y}{d x} = \frac{2}{1 + 4 x^{2}} + \frac{3}{1 + 9 x^{2}}$

Discussion

23.	Two unbiased dice are rolled. The number of elements in the sample space of the above experiment will be ––––––––––___
Answer» Two unbiased dice are rolled. The number of elements in the sample space of the above experiment will be $- -------- -$ ___

Discussion

24.	Find the eccentricity of the hyperbola, the length of whose conjugate axis is34 of the length of transverse asxis.
Answer» Find the eccentricity of the hyperbola, the length of whose conjugate axis is $\frac{3}{4}$ of the length of transverse asxis.

Discussion

25.	Let x be the A.M. and y, z be two G.M.s between two positive numbers. Then, y3+z3xyz is equal to
Answer» Let x be the A.M. and y, z be two G.M.s between two positive numbers. Then, $\frac{y^{3} + z^{3}}{x y z}$ is equal to

Discussion

26.	A line joining two points A(2, 0) and B(3, 1) is rotated about A in anticlockwise direction through an angle 15∘. If B goes to C in the new position, then the coordinates of C are
Answer» A line joining two points A(2, 0) and B(3, 1) is rotated about A in anticlockwise direction through an angle $15^{\circ}$ . If B goes to C in the new position, then the coordinates of C are

Discussion

27.	f(x)=12−tan(πx2) [−1<x<1] and g(x) =√(3+4x−4x2) Find domain of (f + g).
Answer» $f (x) = \frac{1}{2} - t a n (\frac{π x}{2}) [- 1 < x < 1]$ and g(x) = $\sqrt{(3 + 4 x - 4 x^{2})}$ Find domain of (f + g).

27.

f(x)=12−tan(πx2) [−1<x<1] and g(x) =√(3+4x−4x2) Find domain of (f + g).

Answer»

$f (x) = \frac{1}{2} - t a n (\frac{π x}{2}) [- 1 < x < 1]$

and g(x) = $\sqrt{(3 + 4 x - 4 x^{2})}$

Find domain of (f + g).

Discussion

28.	The number of values of θ in [0,2π]that satisfy the equation sin2θ−cosθ14
Answer» $The number of values of θ i n [0, 2 π] t h a t s a t i s f y t h e e q u a t i o n s i n^{2} θ - c o s θ \frac{1}{4}$

Discussion

29.	Let the function f(x)=√x2−\|x+2\|+x be a real valued function, then the domain of f(x) is
Answer» Let the function $f (x) = \sqrt{x^{2} - \| x + 2 \| + x}$ be a real valued function, then the domain of $f (x)$ is

Discussion

30.	Find dydxof the functions given in question. (cos x)y=cos y)x.
Answer» Find $\frac{d y}{d x}$ of the functions given in question. $(c o s x)^{y} = c o s y)^{x}$ .

30.

Find dydxof the functions given in question. (cos x)y=cos y)x.

Answer»

Find $\frac{d y}{d x}$ of the functions given in question.

$(c o s x)^{y} = c o s y)^{x}$ .

Discussion

31.	The frequency distribution of discrete data is given below, the frequency x against value 0 is missing If the mean is 2.5, then the missing frequency x will be Variable012345Frequencyx204040204
Answer» The frequency distribution of discrete data is given below, the frequency $x$ against value $0$ is missing If the mean is $2.5$ , then the missing frequency $x$ will be $\begin{matrix} Variable & 0 & 1 & 2 & 3 & 4 & 5 Frequency & x & 20 & 40 & 40 & 20 & 4 \end{matrix}$

Discussion

32.	Let m1 be the slope of the tangent at a point P on the curve y=x3. This tangent at P intersects the curve again at Q. If the slope of tangent at Q is m2 and m2=km1, then the value of k is
Answer» Let $m_{1}$ be the slope of the tangent at a point $P$ on the curve $y = x^{3} .$ This tangent at $P$ intersects the curve again at $Q .$ If the slope of tangent at $Q$ is $m_{2}$ and $m_{2} = k m_{1},$ then the value of $k$ is

Discussion

33.	Prove that: (1−sinA+cosA)2=2(1+cosA)(1−sinA)
Answer» Prove that: $(1 - sin A + cos A)^{2} = 2 (1 + cos A) (1 - sin A)$

33.

Prove that: (1−sinA+cosA)2=2(1+cosA)(1−sinA)

Answer»

Prove that:

$(1 - sin A + cos A)^{2} = 2 (1 + cos A) (1 - sin A)$

Discussion

34.	The range of parameter a for which the variable line y=2x+a lies between the circles x2+y2−2x−2y+1=0 and x2+y2−16x−2y+61=0 without intersecting or touching either circle is
Answer» The range of parameter $a$ for which the variable line $y = 2 x + a$ lies between the circles $x^{2} + y^{2} - 2 x - 2 y + 1 = 0$ and $x^{2} + y^{2} - 16 x - 2 y + 61 = 0$ without intersecting or touching either circle is

Discussion

35.	Let O=(0,0); let A and B be points respectively on x-axis and y-axis such that ∠OBA=60∘. Let D be a point in the first quadrant such that OAD is an equilateral triangle. Then the slope of DB is
Answer» Let $O = (0, 0)$ ; let $A$ and $B$ be points respectively on $x$ -axis and $y$ -axis such that $∠ O B A = 60^{\circ}$ . Let $D$ be a point in the first quadrant such that $O A D$ is an equilateral triangle. Then the slope of $D B$ is

Discussion

36.	The determinant of the matrix∣∣∣∣123456789∣∣∣∣ is …………. ___
Answer» The determinant of the matrix $∣ ∣∣ ∣ \begin{matrix} 1 & 2 & 3 4 & 5 & 6 7 & 8 & 9 \end{matrix} ∣ ∣∣ ∣$ is …………. ___

Discussion

37.	Suppose f(x)=x3+ax2+bx+c satisfies f(–2)=−10 and takes the extreme value 5027 at x=23. Then the value of a+b+c is
Answer» Suppose $f (x) = x^{3} + a x^{2} + b x + c$ satisfies $f (- 2) = - 10$ and takes the extreme value $\frac{50}{27}$ at $x = \frac{2}{3} .$ Then the value of $a + b + c$ is

Discussion

38.	∫π4π4log(sin x+cos x)dx
Answer» $\int_{\frac{π}{4}}^{\frac{π}{4}} l o g (s i n x + c o s x) d x$

Discussion

39.	Find the sum of odd integers from 1 to 2001.
Answer» Find the sum of odd integers from 1 to 2001.

Discussion

40.	Let a+ar1+ar12+... and a+ar2+ar22+... be two infinite series of positive numbers with the same first term, where r1,r2<1. The sum of the first series is r1 and the sum of the second series is r2 then the value of r1+r2 is
Answer» Let $a + a r_{1} + a {r_{1}}^{2} + . . .$ and $a + a r_{2} + a {r_{2}}^{2} + . . .$ be two infinite series of positive numbers with the same first term, where $r_{1}, r_{2} < 1$ . The sum of the first series is $r_{1}$ and the sum of the second series is $r_{2}$ then the value of $r_{1} + r_{2}$ is

Discussion

41.	Find the equation of the tangent line to the curve y=x2−2x+7 which is (a) parallel to the line 2x-y+9=0. (a) parallel to the line 5y-15x=13.
Answer» Find the equation of the tangent line to the curve $y = x^{2} - 2 x + 7$ which is (a) parallel to the line 2x-y+9=0. (a) parallel to the line 5y-15x=13.

41.

Find the equation of the tangent line to the curve y=x2−2x+7 which is (a) parallel to the line 2x-y+9=0. (a) parallel to the line 5y-15x=13.

Answer»

Find the equation of the tangent line to the curve $y = x^{2} - 2 x + 7$ which is

(a) parallel to the line 2x-y+9=0.

(a) parallel to the line 5y-15x=13.

Discussion

42.	If the area of a circle increases at a uniform rate, then prove that perimeter varies inversely as the radius.
Answer» If the area of a circle increases at a uniform rate, then prove that perimeter varies inversely as the radius.

Discussion

43.	If tan(πcosx)=cot(πsinx), then the absolute value of 4√2cos(x−π4) is
Answer» If $tan (π cos x) = cot (π sin x)$ , then the absolute value of $4 \sqrt{2} cos (x - \frac{π}{4})$ is

Discussion

44.	Integration of f(x) with respect to x where f(x)=1(x−1/2)2+3/4 will be-
Answer» Integration of $f (x)$ with respect to $x$ where $f (x) = \frac{1}{(x - 1 / 2)^{2} + 3 / 4}$ will be-

Discussion

45.	One of the two events must occur. If the chance of one is 23 of the other. then odds in favour of the other are
Answer» One of the two events must occur. If the chance of one is $\frac{2}{3}$ of the other. then odds in favour of the other are

Discussion

46.	Simplify 1−log5=13(log12+logx+13log5)
Answer» Simplify $1 - l o g 5 = \frac{1}{3} (l o g \frac{1}{2} + l o g x + \frac{1}{3} l o g 5)$

Discussion

47.	The area of the loop of the curve y2=x4(x+2) is [in square units]
Answer» The area of the loop of the curve $y^{2} = x^{4} (x + 2)$ is [in square units]

Discussion

48.	Find the distance between the planes 2x−y+2z=5 and 5x−2.5y+5z=20.
Answer» Find the distance between the planes $2 x - y + 2 z = 5$ and $5 x - 2.5 y + 5 z = 20$ .

Discussion

49.	Three circles having radii 2,3 and 10 touch each other at points A,B and C. Then the circumradius of the △ABC is
Answer» Three circles having radii $2, 3$ and $10$ touch each other at points $A, B$ and $C .$ Then the circumradius of the $△ A B C$ is

Discussion

50.	∠APB=x,∠AQB=y and ∠AOB=z Then prove that: ∠x+∠y=∠z Also, find the value of angleCAB
Answer» $∠ A P B = x, ∠ A Q B = y$ and $∠ A O B = z$ Then prove that: $∠ x + ∠ y = ∠ z$ Also, find the value of angleCAB

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

Let →a and →b be two unit vectors such that →a⋅→b=0. For some x,y∈R, let →c=x→a+y→b+(→a×→b). If |→c|=2 and the vector →c is inclined at the same angle α to both →a and →b, then the value of 8cos2α is .

If the equation 4sin(x+π3)cos(x−π6)=a2+√3sin2x−cos2x has a solution, then the value of a can be

All the letters of the word EAMCOT are arranged in a different possible ways. Find the number of arrangments in which no two vowels are adjacent to each other.

The number of different ways in which 8 persons can stand in a row so that between two particular persons A and B there are always two persons, is

Which of the following functions are homogeneous?

From origin, chords are drawn to the circle x2+y2−2y=0.The locus of the middle points of these chords is

In △ABC, the incircle touches the sides BC, CA and AB, respectively, D, E, and F. If the radius of the incircle is 4 units and BD, CE and AF are consecutive integers, then the value of s3, where s is a semi-perimeter of triangle, is

If asked to examine the consistencyof the equations should we find the solution or verification alone is enough?

Four unit circles pass through the origin and have their centres on the coordinate axes. The area of the quadrilateral whose vertices are the points of intersection (in pairs) of the circle, is

Let * be any binary operation on a set A. An element a' ϵ A is called an inverse of a ϵ A for the binary operation *, if

Total number of ordered pairs (x,y) satisfying |x|+|y|=2, sin(πx23)=1 is

For three sets A, B and C, show that (i) A∩B=A∩C need not imply B = C. (ii) A⊂B⇒C−B⊂C−A

The value of limx→x2(secx−tanx) is

The coefficient of xn in expansion of (1+x)(1–x)n is

Let PQ be a focal chord of the parabola y2=4x. The tangents to the parabola at P and Q meet at a point on y = 2x + 1. If PQ subtends an angle θ at the vertex of y2=4x, then tanθ =

Many people ____ his statement as meaning that she intends to resign

Write the coordinates of the orthocentre of the triangle by points (8,0),(4,6) and (0,0).

The equation of line which passes through the point of intersection of 2x + 3y – 5 = 0 and x + y = 2 and also which is farthest from (2, 3) is

The statement (p∧∼q)∨q∨(∼p∧q) is equivalent to

The value of (1+i)4+(1−i)4 is

If 18C2r= 18Cr+3, then the value of r is/are

If y=tan−15x1−6x2, −1√6&lt;x&lt;1√6, then prove that dydx=21+4x2+31+9x2

Two unbiased dice are rolled. The number of elements in the sample space of the above experiment will be ––––––––––___

Find the eccentricity of the hyperbola, the length of whose conjugate axis is34 of the length of transverse asxis.

Let x be the A.M. and y, z be two G.M.s between two positive numbers. Then, y3+z3xyz is equal to

A line joining two points A(2, 0) and B(3, 1) is rotated about A in anticlockwise direction through an angle 15∘. If B goes to C in the new position, then the coordinates of C are

f(x)=12−tan(πx2) [−1&lt;x&lt;1] and g(x) =√(3+4x−4x2) Find domain of (f + g).

The number of values of θ in [0,2π]that satisfy the equation sin2θ−cosθ14

Let the function f(x)=√x2−|x+2|+x be a real valued function, then the domain of f(x) is

Find dydxof the functions given in question. (cos x)y=cos y)x.

The frequency distribution of discrete data is given below, the frequency x against value 0 is missing If the mean is 2.5, then the missing frequency x will be Variable012345Frequencyx204040204

Let m1 be the slope of the tangent at a point P on the curve y=x3. This tangent at P intersects the curve again at Q. If the slope of tangent at Q is m2 and m2=km1, then the value of k is

Prove that: (1−sinA+cosA)2=2(1+cosA)(1−sinA)

The range of parameter a for which the variable line y=2x+a lies between the circles x2+y2−2x−2y+1=0 and x2+y2−16x−2y+61=0 without intersecting or touching either circle is

Let O=(0,0); let A and B be points respectively on x-axis and y-axis such that ∠OBA=60∘. Let D be a point in the first quadrant such that OAD is an equilateral triangle. Then the slope of DB is

The determinant of the matrix∣∣∣∣123456789∣∣∣∣ is …………. ___

Suppose f(x)=x3+ax2+bx+c satisfies f(–2)=−10 and takes the extreme value 5027 at x=23. Then the value of a+b+c is

∫π4π4log(sin x+cos x)dx

Find the sum of odd integers from 1 to 2001.

Let a+ar1+ar12+... and a+ar2+ar22+... be two infinite series of positive numbers with the same first term, where r1,r2&lt;1. The sum of the first series is r1 and the sum of the second series is r2 then the value of r1+r2 is

Find the equation of the tangent line to the curve y=x2−2x+7 which is (a) parallel to the line 2x-y+9=0. (a) parallel to the line 5y-15x=13.

If the area of a circle increases at a uniform rate, then prove that perimeter varies inversely as the radius.

If tan(πcosx)=cot(πsinx), then the absolute value of 4√2cos(x−π4) is

Integration of f(x) with respect to x where f(x)=1(x−1/2)2+3/4 will be-

One of the two events must occur. If the chance of one is 23 of the other. then odds in favour of the other are

Simplify 1−log5=13(log12+logx+13log5)

The area of the loop of the curve y2=x4(x+2) is [in square units]

Find the distance between the planes 2x−y+2z=5 and 5x−2.5y+5z=20.

Three circles having radii 2,3 and 10 touch each other at points A,B and C. Then the circumradius of the △ABC is

∠APB=x,∠AQB=y and ∠AOB=z Then prove that: ∠x+∠y=∠z Also, find the value of angleCAB

If y=tan−15x1−6x2, −1√6<x<1√6, then prove that dydx=21+4x2+31+9x2

f(x)=12−tan(πx2) [−1<x<1] and g(x) =√(3+4x−4x2) Find domain of (f + g).

Let a+ar1+ar12+... and a+ar2+ar22+... be two infinite series of positive numbers with the same first term, where r1,r2<1. The sum of the first series is r1 and the sum of the second series is r2 then the value of r1+r2 is