43030 + Interview Questions in Mathematics

1.	Find the equation of the straight line which passes through the point P(2, 6) and cuts the coordinate axes at the point A and B respectively so that APBP=23.
Answer» Find the equation of the straight line which passes through the point P(2, 6) and cuts the coordinate axes at the point A and B respectively so that $\frac{A P}{B P} = \frac{2}{3} .$

Discussion

2.	If a < b < c < d , then the roots of the equation (x – a) (x – c ) + 2 (x – b) (x – d) = 0 are
Answer» If a < b < c < d , then the roots of the equation (x – a) (x – c ) + 2 (x – b) (x – d) = 0 are

Discussion

3.	For any integer n, the argument of z=(√3+i)4n+1(1−i√3)4n is
Answer» For any integer $n,$ the argument of $z = \frac{(\sqrt{3} + i)^{4 n + 1}}{(1 - i \sqrt{3})^{4 n}}$ is

Discussion

4.	If the arcs of the same length in two circles substend angles 65∘ and 110∘ at the centre, find the ratio of their radii.
Answer» If the arcs of the same length in two circles substend angles $65^{\circ} a n d 110^{\circ}$ at the centre, find the ratio of their radii.

Discussion

5.	Prove that: 1−cos 2θ+sin 2θ1+cos 2θ+sin 2θ=tan θ
Answer» Prove that: $\frac{1 - c o s 2 θ + s i n 2 θ}{1 + c o s 2 θ + s i n 2 θ} = t a n θ$

5.

Prove that: 1−cos 2θ+sin 2θ1+cos 2θ+sin 2θ=tan θ

Answer»

Prove that:

$\frac{1 - c o s 2 θ + s i n 2 θ}{1 + c o s 2 θ + s i n 2 θ} = t a n θ$

Discussion

6.	∫10e2 In xdx= [MP PET 1990]
Answer» $\int_{0}^{1} e^{2 I n x} d x =$ [MP PET 1990]

Discussion

7.	A letter is known to have come either from LONDON or CLIFTON; on the postmark only the two consecutive letters ON are visible. The probability that it came from LONDON is
Answer» A letter is known to have come either from LONDON or CLIFTON; on the postmark only the two consecutive letters ON are visible. The probability that it came from LONDON is

Discussion

8.	If 2x+y=5 is tangent at (2,1) on hyperbola intersects asymptotes at A and B such that AB=4√5. If the center of the hyperbola is (−1,−3), then the least possible value of sum of the semi-transverse axis and the semi-conjugate axis is p√q, where HCF(p,q)=1 and q is a prime number, then the value of p+q is
Answer» If $2 x + y = 5$ is tangent at $(2, 1)$ on hyperbola intersects asymptotes at $A$ and $B$ such that $A B = 4 \sqrt{5}$ . If the center of the hyperbola is $(- 1, - 3),$ then the least possible value of sum of the semi-transverse axis and the semi-conjugate axis is $p \sqrt{q}$ , where $H C F (p, q) = 1$ and $q$ is a prime number, then the value of $p + q$ is

Discussion

9.	The general solution of tan2x=−cot(x+π3)is(nϵZ)
Answer» The general solution of $t a n 2 x = - c o t (x + \frac{π}{3}) i s (n ϵ Z)$

Discussion

10.	The tangent at α on the ellipse x2a2+y2b2=1 meets the auxiliary circle at two points which subtend a right angle at the centre. Then e =
Answer» The tangent at $α$ on the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ meets the auxiliary circle at two points which subtend a right angle at the centre. Then e =

Discussion

11.	The equation of the line parallel to Y−axis and 3 units to the right of it, is
Answer» The equation of the line parallel to $Y -$ axis and $3$ units to the right of it, is

Discussion

12.	The equation of the tangent to the curve y=2x2+3 sin x at x=0 is
Answer» The equation of the tangent to the curve $y = 2 x^{2} + 3 s i n x a t x = 0$ is

Discussion

13.	The values of parameter a such that the line (log2(1+5a−a2))x−5y−(a2−5)=0 is a normal to the curve xy=1, may be in the interval
Answer» The values of parameter a such that the line $(l o g_{2} (1 + 5 a - a^{2})) x - 5 y - (a^{2} - 5) = 0$ is a normal to the curve $x y = 1$ , may be in the interval

Discussion

14.	If the standard deviation of number 3,4,8,14,16 is 5.21, then the standard deviation of number 6,7,11,17,19 is
Answer» If the standard deviation of number $3, 4, 8, 14, 16$ is $5.21,$ then the standard deviation of number $6, 7, 11, 17, 19$ is

Discussion

15.	If sinθ1sinθ2−cosθ1cosθ2−1=0, where θ1+θ2∈(0,2π), then the value of (1+tanθ14)(1+tanθ24) is
Answer» If $sin θ_{1} sin θ_{2} - cos θ_{1} cos θ_{2} - 1 = 0,$ where $θ_{1} + θ_{2} \in (0, 2 π),$ then the value of $(1 + tan \frac{θ_{1}}{4}) (1 + tan \frac{θ_{2}}{4})$ is

Discussion

16.	The sum of three numbers which are consecutive terms an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.
Answer» The sum of three numbers which are consecutive terms an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.

16.

The sum of three numbers which are consecutive terms an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.

Answer»

The sum of three numbers which are consecutive terms an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.

Discussion

17.	If distance between the directrices be thrice the distance between the foci, then the eccentricity of ellipse is
Answer» If distance between the directrices be thrice the distance between the foci, then the eccentricity of ellipse is

Discussion

18.	ABCD is a parallelogram and A1 and B1 are the midpoints of sides BC and CD, respectively. If −−→AA1+−−→AB1=λ−−→AC, then λ is equal to
Answer» $A B C D$ is a parallelogram and $A_{1}$ and $B_{1}$ are the midpoints of sides $B C$ and $C D$ , respectively. If $- - \to A A_{1} + - - \to A B_{1} = λ - - \to A C$ , then $λ$ is equal to

Discussion

19.	Let 0<θ<π2. If the eccentricity of the hyperbola x2cos2θ−y2sin2θ=1 is greater than 2, then the length of its latus rectum lies in the interval:
Answer» Let $0 < θ < \frac{π}{2}$ . If the eccentricity of the hyperbola $\frac{x^{2}}{{cos}^{2} θ} - \frac{y^{2}}{{sin}^{2} θ} = 1$ is greater than $2$ , then the length of its latus rectum lies in the interval:

Discussion

20.	Let f(x)=x+lnx−xlnx, x∈(0,∞) Column 1Column 2Column 3(I)f(x)=0 for some x∈(1,e2)(i)limx→∞f(x)=0(P)f is increasing in (0,1)(II)f′(x)=0 for some x∈(1,e) (ii)limx→∞f(x)=−∞ (Q)f is decreasing in (e,e2)(III)f′(x)=0 for some x∈(0,1) (iii)limx→∞f′(x)=−∞ (R)f′ is increasing in (0,1)(IV)f′′(x)=0 for some x∈(1,e) (iv)limx→∞f′′(x)=0 (S)f′ is decreasing in (e,e2) Which of the following options is the only CORRECT combination?
Answer» Let $f (x) = x + ln x - x ln x, x \in (0, \infty)$ $\begin{matrix} C o l u m n 1 & C o l u m n 2 & C o l u m n 3 (I) f (x) = 0 for some x \in (1, e^{2}) & (i) lim x \to \infty f (x) = 0 & (P) f is increasing in (0, 1) (I I) f^{'} (x) = 0 for some x \in (1, e) & (i i) lim x \to \infty f (x) = - \infty & (Q) f is decreasing in (e, e^{2}) (I I I) f^{'} (x) = 0 for some x \in (0, 1) & (i i i) lim x \to \infty f^{'} (x) = - \infty & (R) f^{'} is increasing in (0, 1) (I V) f^{''} (x) = 0 for some x \in (1, e) & (i v) lim x \to \infty f^{''} (x) = 0 & (S) f^{'} is decreasing in (e, e^{2}) \end{matrix}$ Which of the following options is the only CORRECT combination?

Discussion

21.	How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit s repeated?
Answer» How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit s repeated?

Discussion

22.	limx→0x(1+acosx)−bsinxx3=1, then
Answer» $l i m x \to 0 \frac{x (1 + a c o s x) - b s i n x}{x^{3}} = 1$ , then

Discussion

23.	If Dk=∣∣∣∣∣1nn2kn2+n+1n2+n2k−1n2n2+n+1∣∣∣∣∣ and n∑k=1Dk=56, where n∈N, then n is equal to
Answer» If $D_{k} = ∣ ∣∣∣ ∣ \begin{matrix} 1 & n & n 2 k & n^{2} + n + 1 & n^{2} + n 2 k - 1 & n^{2} & n^{2} + n + 1 \end{matrix} ∣ ∣∣∣ ∣$ and $n \sum k = 1 D_{k} = 56,$ where $n \in N,$ then $n$ is equal to

Discussion

24.	If a, b, c are in G.P. write the area of the triangle formed by the line ax+by+c=0 with the coordinates axes.
Answer» If a, b, c are in G.P. write the area of the triangle formed by the line $a x + b y + c = 0$ with the coordinates axes.

Discussion

25.	limx→0{sin(α+β)x+sin(α−β)x+sin 2αx}cos2βx−cos2αx
Answer» ${lim}_{x \to 0} \frac{{s i n (α + β) x + s i n (α - β) x + s i n 2 α x}}{c o s^{2} β x - c o s^{2} α x}$

Discussion

26.	If the roots of equation x2−bxax−c=m−1m+1 are equal but opposite in sign, the value of m will be
Answer» If the roots of equation $\frac{x^{2} - b x}{a x - c} = \frac{m - 1}{m + 1}$ are equal but opposite in sign, the value of m will be

Discussion

27.	limh→0√x+h−√xh,x≠0
Answer» ${lim}_{h \to 0} \frac{\sqrt{x + h} - \sqrt{x}}{h}, x \neq 0$

Discussion

28.	∫e√xcos e√x√xdx=
Answer» $\int \frac{e^{\sqrt{x}} c o s e^{\sqrt{x}}}{\sqrt{x}} d x =$

Discussion

29.	Differentiate the following functions with respect to x : ex−tan xcot x−xn
Answer» Differentiate the following functions with respect to x : $\frac{e^{x} - t a n x}{c o t x - x^{n}}$

29.

Differentiate the following functions with respect to x : ex−tan xcot x−xn

Answer»

Differentiate the following functions with respect to x :

$\frac{e^{x} - t a n x}{c o t x - x^{n}}$

Discussion

30.	If S=1+22x+32x2+42x3+......∞ where x=15. Find the value of 32 S.___
Answer» If $S = 1 + 2^{2} x + 3^{2} x^{2} + 4^{2} x^{3} + . . . . . . \infty$ where $x = \frac{1}{5}$ . Find the value of 32 S.___

Discussion

31.	If q is false and p∧q↔r is true, then which one of the following statements is a tautology?
Answer» If $q$ is false and $p \land q \leftrightarrow r$ is true, then which one of the following statements is a tautology?

Discussion

32.	The probability that a couple will have first 3 daughters then 1 son is
Answer» The probability that a couple will have first 3 daughters then 1 son is

Discussion

33.	Find the derivative of f(x) from the first principle of where i(x) = tan x + sec x. (ii) Evaluate limx→a (x+2)32−(a+2)32x−a
Answer» Find the derivative of f(x) from the first principle of where i(x) = tan x + sec x. (ii) Evaluate $l i m_{x \to a} \frac{(x + 2)^{\frac{3}{2}} - (a + 2)^{\frac{3}{2}}}{x - a}$

33.

Find the derivative of f(x) from the first principle of where i(x) = tan x + sec x. (ii) Evaluate limx→a (x+2)32−(a+2)32x−a

Answer»

Find the derivative of f(x) from the first principle of where i(x) = tan x + sec x.

(ii) Evaluate $l i m_{x \to a} \frac{(x + 2)^{\frac{3}{2}} - (a + 2)^{\frac{3}{2}}}{x - a}$

Discussion

34.	limx->(\|x\|)sinx ?
Answer» lim_x->(\|x\|)^sinx?

Discussion

35.	If the product of two twin prime numbers and the absolute difference of smallest composite number and the smallest prime number is 646, then larger prime number is
Answer» If the product of two twin prime numbers and the absolute difference of smallest composite number and the smallest prime number is $646$ , then larger prime number is

Discussion

36.	Using the method of slope, show that the following points are collinear : (i) A(4, 8), B(5, 12), C(9, 28)(ii) A(16, −18), B(3, −6), C(−10, 6)
Answer» Using the method of slope, show that the following points are collinear : $(i) A (4, 8), B (5, 12), C (9, 28) (i i) A (16, - 18), B (3, - 6), C (- 10, 6)$

36.

Using the method of slope, show that the following points are collinear : (i) A(4, 8), B(5, 12), C(9, 28)(ii) A(16, −18), B(3, −6), C(−10, 6)

Answer»

Using the method of slope, show that the following points are collinear :

$(i) A (4, 8), B (5, 12), C (9, 28) (i i) A (16, - 18), B (3, - 6), C (- 10, 6)$

Discussion

37.	Find the sum of the following series to n terms : (1-7) 13+33+53+73+....
Answer» Find the sum of the following series to n terms : (1-7) $1^{3} + 3^{3} + 5^{3} + 7^{3} + . . . .$

Discussion

38.	If the integral of the function sin(lnx)x=f(x), then find the value of f(1) (take constant of integration equal to zero) ___
Answer» If the integral of the function $\frac{s i n (l n x)}{x} = f (x)$ , then find the value of f(1) (take constant of integration equal to zero) ___

Discussion

39.	If a plane cuts off intercepts OA = a, OB = b, OC = c from the co-ordinate axes, then the area of the triangle ABC=
Answer» If a plane cuts off intercepts OA = a, OB = b, OC = c from the co-ordinate axes, then the area of the triangle ABC=

Discussion

40.	If z3+(3+2i)z+(−1+ia)=0 has one real root, then the value of the a lies in the interval a ϵ R
Answer» If $z^{3} + (3 + 2 i) z + (- 1 + i a) = 0$ has one real root, then the value of the $a$ lies in the interval $a ϵ R$

Discussion

41.	The number of integral values of k for which the equation (k+1)x2+2(k−1)xy+y2−x+2y+3=0 represents an ellipse, is
Answer» The number of integral values of $k$ for which the equation $(k + 1) x^{2} + 2 (k - 1) x y + y^{2} - x + 2 y + 3 = 0$ represents an ellipse, is

Discussion

42.	Solve for x:\|x−\|3−x\|\|−3x=8
Answer» Solve for $x : \| x - \| 3 - x \| \| - 3 x = 8$

Discussion

43.	Least value of n, (n∈N) for which 2n+1 is not a prime is
Answer» Least value of $n, (n \in N)$ for which $2 n + 1$ is not a prime is

Discussion

44.	Rishabh is picking out balls randomly from a box containing 60 colored balls: 15 green,12 red, 11 blue, 10 yellow, 8 black and 4 white."What is the minimum number of balls Rishabh needs to pick to ensure that he has at least 9 balls of the same color?
Answer» Rishabh is picking out balls randomly from a box containing 60 colored balls: 15 green,12 red, 11 blue, 10 yellow, 8 black and 4 white."What is the minimum number of balls Rishabh needs to pick to ensure that he has at least 9 balls of the same color?

Discussion

45.	The set of values of x for which tan3x−tan2x1+tan3x⋅tan2x=1 is (where n∈Z)
Answer» The set of values of $x$ for which $\frac{tan 3 x - tan 2 x}{1 + tan 3 x \cdot tan 2 x} = 1$ is (where $n \in Z$ )

Discussion

46.	If (h, k) is a point from which the tangents to the three circles x2 + y2 − 4x + 7 = 0, 2x2 + 2y2 − 3x + 5y + 9 = 0 and x2 + y2 + y = 0 are equal in length. Find the value of h + k ___ .
Answer» If (h, k) is a point from which the tangents to the three circles $x^{2} + y^{2} - 4 x + 7 = 0$ , $2 x^{2} + 2 y^{2} - 3 x + 5 y + 9 = 0$ and $x^{2} + y^{2} + y = 0$ are equal in length. Find the value of h + k ___ .

Discussion

47.	The range of 3x2+9x+173x2+9x+7 is
Answer» The range of $\frac{3 x^{2} + 9 x + 17}{3 x^{2} + 9 x + 7}$ is

Discussion

48.	Given that 4x+1 + 4x = 3y+4 - 3y, where x and y are integers, the value of x - y is
Answer» Given that 4^x+1 + 4^x = 3^y+4 - 3^y, where x and y are integers, the value of x - y is

Discussion

49.	How many triangles could be formed by taking 3 vertices of a hexagon?
Answer» How many triangles could be formed by taking 3 vertices of a hexagon?

Discussion

50.	If all the letters of the word "SECRET" are arranged in all possible ways and written out in alphabetical(dictionary) order, then the rank of the given word "SECRET" will be
Answer» If all the letters of the word $" S E C R E T "$ are arranged in all possible ways and written out in alphabetical(dictionary) order, then the rank of the given word $" S E C R E T "$ will be

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

Find the equation of the straight line which passes through the point P(2, 6) and cuts the coordinate axes at the point A and B respectively so that APBP=23.

If a &lt; b &lt; c &lt; d , then the roots of the equation (x – a) (x – c ) + 2 (x – b) (x – d) = 0 are

For any integer n, the argument of z=(√3+i)4n+1(1−i√3)4n is

If the arcs of the same length in two circles substend angles 65∘ and 110∘ at the centre, find the ratio of their radii.

Prove that: 1−cos 2θ+sin 2θ1+cos 2θ+sin 2θ=tan θ

∫10e2 In xdx= [MP PET 1990]

A letter is known to have come either from LONDON or CLIFTON; on the postmark only the two consecutive letters ON are visible. The probability that it came from LONDON is

The general solution of tan2x=−cot(x+π3)is(nϵZ)

The tangent at α on the ellipse x2a2+y2b2=1 meets the auxiliary circle at two points which subtend a right angle at the centre. Then e =

The equation of the line parallel to Y−axis and 3 units to the right of it, is

The equation of the tangent to the curve y=2x2+3 sin x at x=0 is

The values of parameter a such that the line (log2(1+5a−a2))x−5y−(a2−5)=0 is a normal to the curve xy=1, may be in the interval

If the standard deviation of number 3,4,8,14,16 is 5.21, then the standard deviation of number 6,7,11,17,19 is

If sinθ1sinθ2−cosθ1cosθ2−1=0, where θ1+θ2∈(0,2π), then the value of (1+tanθ14)(1+tanθ24) is

The sum of three numbers which are consecutive terms an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.

If distance between the directrices be thrice the distance between the foci, then the eccentricity of ellipse is

ABCD is a parallelogram and A1 and B1 are the midpoints of sides BC and CD, respectively. If −−→AA1+−−→AB1=λ−−→AC, then λ is equal to

Let 0&lt;θ&lt;π2. If the eccentricity of the hyperbola x2cos2θ−y2sin2θ=1 is greater than 2, then the length of its latus rectum lies in the interval:

How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit s repeated?

limx→0x(1+acosx)−bsinxx3=1, then

If Dk=∣∣∣∣∣1nn2kn2+n+1n2+n2k−1n2n2+n+1∣∣∣∣∣ and n∑k=1Dk=56, where n∈N, then n is equal to

If a, b, c are in G.P. write the area of the triangle formed by the line ax+by+c=0 with the coordinates axes.

limx→0{sin(α+β)x+sin(α−β)x+sin 2αx}cos2βx−cos2αx

If the roots of equation x2−bxax−c=m−1m+1 are equal but opposite in sign, the value of m will be

limh→0√x+h−√xh,x≠0

∫e√xcos e√x√xdx=

Differentiate the following functions with respect to x : ex−tan xcot x−xn

If S=1+22x+32x2+42x3+......∞ where x=15. Find the value of 32 S.___

If q is false and p∧q↔r is true, then which one of the following statements is a tautology?

The probability that a couple will have first 3 daughters then 1 son is

Find the derivative of f(x) from the first principle of where i(x) = tan x + sec x. (ii) Evaluate limx→a (x+2)32−(a+2)32x−a

limx-&gt;(|x|)sinx ?

If the product of two twin prime numbers and the absolute difference of smallest composite number and the smallest prime number is 646, then larger prime number is

Using the method of slope, show that the following points are collinear : (i) A(4, 8), B(5, 12), C(9, 28)(ii) A(16, −18), B(3, −6), C(−10, 6)

Find the sum of the following series to n terms : (1-7) 13+33+53+73+....

If the integral of the function sin(lnx)x=f(x), then find the value of f(1) (take constant of integration equal to zero) ___

If a plane cuts off intercepts OA = a, OB = b, OC = c from the co-ordinate axes, then the area of the triangle ABC=

If z3+(3+2i)z+(−1+ia)=0 has one real root, then the value of the a lies in the interval a ϵ R

The number of integral values of k for which the equation (k+1)x2+2(k−1)xy+y2−x+2y+3=0 represents an ellipse, is

Solve for x:|x−|3−x||−3x=8

Least value of n, (n∈N) for which 2n+1 is not a prime is

Rishabh is picking out balls randomly from a box containing 60 colored balls: 15 green,12 red, 11 blue, 10 yellow, 8 black and 4 white."What is the minimum number of balls Rishabh needs to pick to ensure that he has at least 9 balls of the same color?

The set of values of x for which tan3x−tan2x1+tan3x⋅tan2x=1 is (where n∈Z)

If (h, k) is a point from which the tangents to the three circles x2 + y2 − 4x + 7 = 0, 2x2 + 2y2 − 3x + 5y + 9 = 0 and x2 + y2 + y = 0 are equal in length. Find the value of h + k ___ .

The range of 3x2+9x+173x2+9x+7 is

Given that 4x+1 + 4x = 3y+4 - 3y, where x and y are integers, the value of x - y is

How many triangles could be formed by taking 3 vertices of a hexagon?

If all the letters of the word "SECRET" are arranged in all possible ways and written out in alphabetical(dictionary) order, then the rank of the given word "SECRET" will be

If a < b < c < d , then the roots of the equation (x – a) (x – c ) + 2 (x – b) (x – d) = 0 are

Let 0<θ<π2. If the eccentricity of the hyperbola x2cos2θ−y2sin2θ=1 is greater than 2, then the length of its latus rectum lies in the interval:

limx->(|x|)sinx ?