43030 + Interview Questions in Mathematics

1.	If the system of equations x−ky−z=0,kx−y−z=0 and x+y−z=0has a non-trivial solution, then k can be
Answer» If the system of equations $x - k y - z = 0, k x - y - z = 0$ and $x + y - z = 0$ has a non-trivial solution, then $k$ can be

Discussion

2.	Find the sum of first 10 terms of following seriesS=3(1)1+5(13+23)12+22+7(13+23+33)12+22+32+....
Answer» Find the sum of first $10$ terms of following series $S = \frac{3 (1)}{1} + \frac{5 (1^{3} + 2^{3})}{1^{2} + 2^{2}} + \frac{7 (1^{3} + 2^{3} + 3^{3})}{1^{2} + 2^{2} + 3^{2}} + . . . .$

Discussion

3.	Question 10If sin θ+cos θ=p and sec θ+cosec θ=q, then prove that q(p2−1)=2p.
Answer» Question 10 If $s i n θ + c o s θ = p$ and $s e c θ + c o s e c θ = q$ , then prove that $q (p^{2} - 1) = 2 p$ .

Discussion

4.	Derive v=u+at S=ut+1/2at square 2as=v square -u square
Answer» Derive v=u+at S=ut+1/2at square 2as=v square -u square

Discussion

5.	Suppose the Earth is following an elliptical path x29+y24=1 and the Sun is located at point (1,1). If P(θ) is the nearest point to the Sun, then which of the following is correct
Answer» Suppose the Earth is following an elliptical path $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$ and the Sun is located at point $(1, 1) .$ If $P (θ)$ is the nearest point to the Sun, then which of the following is correct

Discussion

6.	If z=(4sin2θ−1)+i(cos2θ+1) is purely imaginary number, then the number of value(s) of θ∈[0,2nπ] where n∈I is
Answer» If $z = (4 {sin}^{2} θ - 1) + i ({cos}^{2} θ + 1)$ is purely imaginary number, then the number of value(s) of $θ \in [0, 2 n π]$ where $n \in I$ is

Discussion

7.	Prove that: cot4x(sin5x+sin3x)=cotx(sin5x–sin3x)
Answer» Prove that: $cot 4 x (sin 5 x + sin 3 x) = cot x (sin 5 x - sin 3 x)$

Discussion

8.	Let PQ be a chord of the ellipse x2a2+y2b2=1 which subtends right angle at the centre (0,0). Then its distance from the centre is equal to
Answer» Let $P Q$ be a chord of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ which subtends right angle at the centre $(0, 0) .$ Then its distance from the centre is equal to

Discussion

9.	If three positive real numbers a, b, c are in A.P. such that abc =4, then the minimum value of b is
Answer» If three positive real numbers a, b, c are in A.P. such that abc $= 4$ , then the minimum value of b is

Discussion

10.	If I1=∫102x2 dx, I2=∫102x3dx, I3=∫212x2dx and I4=∫212x3 dx then
Answer» $I f I_{1} = \int_{0}^{1} 2^{x^{2}} d x, I_{2} = \int_{0}^{1} 2^{x^{3}} d x, I_{3} = \int_{1}^{2} 2^{x^{2}} d x a n d I_{4} = \int_{1}^{2} 2 x^{3} d x t h e n$

Discussion

11.	Which of the following is the graph of cosec x, 0≤x≤2π?
Answer» Which of the following is the graph of cosec $x, 0 \leq x \leq 2 π$ ?

Discussion

12.	two vectors,each of magnitude A have a resul†an t of same magnitude A. the angle between the two vectors is
Answer» two vectors,each of magnitude A have a resul†an t of same magnitude A. the angle between the two vectors is

Discussion

13.	Minimum number of unequal vectors which can give zero resul†an t are
Answer» Minimum number of unequal vectors which can give zero resul†an t are

Discussion

14.	If 5−2x3≤x6−5, then x lies in
Answer» If $\frac{5 - 2 x}{3} \leq \frac{x}{6} - 5$ , then $x$ lies in

Discussion

15.	The value of P(A∪B)if 2P(A)=P(B)=513 and P(A/B)=25 is:
Answer» The value of $P (A \cup B)$ if $2 P (A) = P (B) = \frac{5}{13}$ and $P (A / B) = \frac{2}{5}$ is:

Discussion

16.	Evaluate each of the following integrals:∫-π2π2cos2x1+exdx
Answer» Evaluate each of the following integrals: $\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{\cos^{2} x}{1 + e^{x}} d x$

Discussion

17.	The value of ∫2xx2+3x+2dx is(where C is integration constant)
Answer» The value of $\int \frac{2 x}{x^{2} + 3 x + 2} d x$ is (where $C$ is integration constant)

Discussion

18.	Deduce Boyle's law
Answer» Deduce Boyle's law

Discussion

19.	Let , show that , where I is the identity matrix of order 2 and n ∈ N
Answer» Let , show that , where I is the identity matrix of order 2 and n ∈ N

Discussion

20.	Let fx=kcosxπ-2x,where x≠π23,where x=π2 and if limx→π2 fx=fπ2, find the value of k.
Answer» Let $f (x) = \{\begin{cases} \frac{k \cos x}{π - 2 x}, & where x \neq \frac{π}{2} \\ 3, & where x = \frac{π}{2} \end{cases}$ and if $\lim_{x \to \frac{π}{2}} f (x) = f (\frac{π}{2})$ , find the value of k.

Discussion

21.	If the function f(x) = x4 - 62x2 + ax + 9 attains a local maximum at x = 1, then a = _________________.
Answer» If the function f(x) = x⁴ - 62x² + ax + 9 attains a local maximum at x = 1, then a = _________________.

Discussion

22.	Let f:R→R be a continuous function such that f(x)+f(x+1)=2, for all x∈R. If I1=∫80f(x)dx and I2=∫3−1f(x)dx then the value of I1+2I2 is equal to
Answer» Let $f : R \to R$ be a continuous function such that $f (x) + f (x + 1) = 2,$ for all $x \in R .$ If $I_{1} = \int_{0}^{8} f (x) d x$ and $I_{2} = \int_{- 1}^{3} f (x) d x$ then the value of $I_{1} + 2 I_{2}$ is equal to

Discussion

23.	The equations of bisectors of angles between YZ plane and XZ plane is:
Answer» The equations of bisectors of angles between $Y Z$ plane and $X Z$ plane is:

Discussion

24.	If a circle passes through the point (a, b) and cuts the circle x2+y2=4 orthogonally, then the locus of its centre is
Answer» If a circle passes through the point (a, b) and cuts the circle $x^{2} + y^{2} = 4$ orthogonally, then the locus of its centre is

Discussion

25.	Question 2 (i)Find the sums given below(i) 7+1012+14+……+84
Answer» Question 2 (i) Find the sums given below (i) $7 + 10 \frac{1}{2} + 14 + \dots \dots + 84$

Discussion

26.	The value of determinant Δ=∣∣∣∣abca+2xb+2yc+2zxyz∣∣∣∣ is
Answer» The value of determinant $Δ = ∣ ∣∣ ∣ \begin{matrix} a & b & c a + 2 x & b + 2 y & c + 2 z x & y & z \end{matrix} ∣ ∣∣ ∣$ is

Discussion

27.	limx→0log(1+x+x2)+log(1−x+x2)secx−cosx=
Answer» $lim x \to 0 \frac{log (1 + x + x^{2}) + log (1 - x + x^{2})}{sec x - cos x} =$

Discussion

28.	A vertical line passing through the point (h,0) intersects the ellipse x24+y23=1 at the points P and Q. Let the tangents to the ellipse at P and Q meet at the point R. If Δ(h)=area of the triangle PQR, Δ1=max1/2 ≤ h ≤ 1Δ(h) and Δ2=min1/2 ≤ h ≤ 1Δ(h), then 8√5Δ1−8Δ2=
Answer» A vertical line passing through the point $(h, 0)$ intersects the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1$ at the points $P$ and $Q$ . Let the tangents to the ellipse at $P$ and $Q$ meet at the point $R$ . If $Δ (h) = area of the triangle P Q R$ , $Δ_{1} = max 1 / 2 \leq h \leq 1 Δ (h)$ and $Δ_{2} = min 1 / 2 \leq h \leq 1 Δ (h)$ , then $\frac{8}{\sqrt{5}} Δ_{1} - 8 Δ_{2} =$

Discussion

29.	Difference between Hunds rule and Pauli exclusion principle
Answer» Difference between Hunds rule and Pauli exclusion principle

Discussion

30.	82.A dictionary is printed consisting of 7-lettered words that can be made with the letters of the word CRICKET. If tge words are printed in the alphabetic order, as in an ordinary dictionary, Find the position of the word CRICKET in the dictionary.
Answer» 82.A dictionary is printed consisting of 7-lettered words that can be made with the letters of the word CRICKET. If tge words are printed in the alphabetic order, as in an ordinary dictionary, Find the position of the word CRICKET in the dictionary.

Discussion

31.	The set of real values of x, for which h(x)=1+2x2+4x4+6x6+⋯+100x100 is concave downward is
Answer» The set of real values of $x$ , for which $h (x) = 1 + 2 x^{2} + 4 x^{4} + 6 x^{6} + \dots + 100 x^{100}$ is concave downward is

Discussion

32.	If f:R→R be given by f(x)=4x4x+2 for all xϵR. Then,
Answer» If $f : R \to R$ be given by $f (x) = \frac{4^{x}}{4^{x} + 2}$ for all $x ϵ R$ . Then,

Discussion

33.	The set of values of x for which sin−1(sin5)>x2−4x is
Answer» The set of values of $x$ for which ${sin}^{- 1} (sin 5) > x^{2} - 4 x$ is

Discussion

34.	If 12, 13, n are the direction cosines of a line, then the values of n are ________________.
Answer» If $\frac{1}{2}, \frac{1}{3}, n$ are the direction cosines of a line, then the values of n are ________________.

Discussion

35.	If tangent at point (1,2) on the curve y=ax2+bx+72 is parallel to normal at (−2,2) on the curve y=x2+6x+10, then
Answer» If tangent at point $(1, 2)$ on the curve $y = a x^{2} + b x + \frac{7}{2}$ is parallel to normal at $(- 2, 2)$ on the curve $y = x^{2} + 6 x + 10,$ then

Discussion

36.	sec θ-tan θsec θ+tan θ=1+2tan2θ-2sec θ tan θ
Answer» $\frac{(\sec θ - \tan θ)}{(\sec θ + \tan θ)} = (1 + 2 \tan^{2} θ - 2 \sec θ \tan θ)$

Discussion

37.	Let f(x)=⎧⎨⎩x2−[x]2,x<2a,x=2x2−[x]2+3,x>2. If f(x) is continuous at x=2, then the value of a is(where [.] represents the greatest integer function and a∈R)
Answer» Let $f (x) = ⎧ ⎨ ⎩ \begin{matrix} x^{2} - [x]^{2}, & x < 2 a, & x = 2 x^{2} - [x]^{2} + 3, & x > 2 \end{matrix}$ . If $f (x)$ is continuous at $x = 2$ , then the value of $a$ is (where $[.]$ represents the greatest integer function and $a \in R$ )

Discussion

38.	Evaluate the definite integrals. ∫π20cos2xdx.
Answer» Evaluate the definite integrals. $\int_{0}^{\frac{π}{2}} c o s 2 x d x .$

Discussion

39.	4. (ax +b)(cr+d)i
Answer» 4. (ax +b)(cr+d)i

Discussion

40.	(sin230° – sec260° + 4cot245°) = ?(a) 4(b) 2(c) 1(d) 14
Answer» (sin²30° – sec²60° + 4cot²45°) = ? (a) 4 (b) 2 (c) 1 (d) $\frac{1}{4}$

Discussion

41.	The number of vectors of unit length perpendicular to vectors a→=i^+j^ and b→=j^+k^ is _________________.
Answer» The number of vectors of unit length perpendicular to vectors $\vec{a} = \hat{i} + \hat{j} and \vec{b} = \hat{j} + \hat{k} is_________________.$

Discussion

42.	Let S1 and S2 are the unit circles with centres at C1(0,0) and C2(1,0) respectively. Let S3 is another circle of unit radius, passes through C1 and C2 and its centre is above the x -axis. If equation of common tangent to S1 and S3, which does not cut S2, is ax+by+2=0 then find the value of (a2−b).
Answer» Let $S_{1}$ and $S_{2}$ are the unit circles with centres at $C_{1} (0, 0)$ and $C_{2} (1, 0)$ respectively. Let $S_{3}$ is another circle of unit radius, passes through $C_{1}$ and $C_{2}$ and its centre is above the $x$ -axis. If equation of common tangent to $S_{1}$ and $S_{3},$ which does not cut $S_{2},$ is $a x + b y + 2 = 0$ then find the value of $(a^{2} - b)$ .

Discussion

43.	Suppose that the foci of the ellipse x29+y25=1 are (f1,0) and (f2,0) where f1>0 and f2<0. Let P1 and P2 be two parabolas with a common vertex at (0,0) and with foci at (f1,0) and (2f2,0) respectively. Let T1 be a tangent to P1 which passes through (2f2,0) and T2 be a tangent to P2 which passes through (f1,0). If m1 is the slope of T1 and m2 is the slope of T2, then the value of (1m21+m22) is
Answer» Suppose that the foci of the ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{5} = 1$ are $(f_{1}, 0)$ and $(f_{2}, 0)$ where $f_{1} > 0$ and $f_{2} < 0$ . Let $P_{1}$ and $P_{2}$ be two parabolas with a common vertex at $(0, 0)$ and with foci at $(f_{1}, 0)$ and $(2 f_{2}, 0)$ respectively. Let $T_{1}$ be a tangent to $P_{1}$ which passes through $(2 f_{2}, 0)$ and $T_{2}$ be a tangent to $P_{2}$ which passes through $(f_{1}, 0)$ . If $m_{1}$ is the slope of $T_{1}$ and $m_{2}$ is the slope of $T_{2}$ , then the value of $(\frac{1}{m_{1}^{2}} + m_{2}^{2})$ is

Discussion

44.	A triangle with vertices (4, 0), (-1, -1), (3, 5) is
Answer» A triangle with vertices (4, 0), (-1, -1), (3, 5) is

Discussion

45.	If A and B are independent events such that P(A)>12, P(A∩¯B)=325and P(¯A∩B)=825,
Answer» If A and B are independent events such that $P (A) > \frac{1}{2}, P (A \cap ¯ B) = \frac{3}{25} and P (¯ A \cap B) = \frac{8}{25},$

Discussion

46.	The domain of the function sin−1[log2(x22)] is
Answer» The domain of the function ${sin}^{- 1} [{log}_{2} (\frac{x^{2}}{2})]$ is

Discussion

47.	Let y=y(x) be the solution of the differential equation dydx=1+xey−x, −√2<x<√2, y(0)=0, then the minimum value of y(x), x∈(−√2,√2) is equal to
Answer» Let $y = y (x)$ be the solution of the differential equation $\frac{d y}{d x} = 1 + x e^{y - x},$ $- \sqrt{2} < x < \sqrt{2},$ $y (0) = 0$ , then the minimum value of $y (x),$ $x \in (- \sqrt{2}, \sqrt{2})$ is equal to

Discussion

48.	For the system of equations 7x-5y=2 and 2x+my=5, find the :(A) Value of k (B) the condition of k for which the given system of equations has a unique solution.
Answer» For the system of equations 7x-5y=2 and 2x+my=5, find the : (A) Value of k (B) the condition of k for which the given system of equations has a unique solution.

Discussion

49.	5. an =(-l)"‘15"’“1
Answer» 5. an =(-l)"‘15"’“1

Discussion

50.	Rate constant varies with temperature by the equation log10K=5–2000/T. We can conclude that (R=8.314Jmol−1K−1):-
Answer» Rate constant varies with temperature by the equation $l o g_{10} K = 5 - 2000 / T$ . We can conclude that $(R = 8.314 J m o l^{- 1} K^{- 1})$ :-

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

If the system of equations x−ky−z=0,kx−y−z=0 and x+y−z=0has a non-trivial solution, then k can be

Find the sum of first 10 terms of following seriesS=3(1)1+5(13+23)12+22+7(13+23+33)12+22+32+....

Question 10If sin θ+cos θ=p and sec θ+cosec θ=q, then prove that q(p2−1)=2p.

Derive v=u+at S=ut+1/2at square 2as=v square -u square

Suppose the Earth is following an elliptical path x29+y24=1 and the Sun is located at point (1,1). If P(θ) is the nearest point to the Sun, then which of the following is correct

If z=(4sin2θ−1)+i(cos2θ+1) is purely imaginary number, then the number of value(s) of θ∈[0,2nπ] where n∈I is

Prove that: cot4x(sin5x+sin3x)=cotx(sin5x–sin3x)

Let PQ be a chord of the ellipse x2a2+y2b2=1 which subtends right angle at the centre (0,0). Then its distance from the centre is equal to

If three positive real numbers a, b, c are in A.P. such that abc =4, then the minimum value of b is

If I1=∫102x2 dx, I2=∫102x3dx, I3=∫212x2dx and I4=∫212x3 dx then

Which of the following is the graph of cosec x, 0≤x≤2π?

two vectors,each of magnitude A have a resul†an t of same magnitude A. the angle between the two vectors is

Minimum number of unequal vectors which can give zero resul†an t are

If 5−2x3≤x6−5, then x lies in

The value of P(A∪B)if 2P(A)=P(B)=513 and P(A/B)=25 is:

Evaluate each of the following integrals:∫-π2π2cos2x1+exdx

The value of ∫2xx2+3x+2dx is(where C is integration constant)

Deduce Boyle's law

Let , show that , where I is the identity matrix of order 2 and n ∈ N

Let fx=kcosxπ-2x,where x≠π23,where x=π2 and if limx→π2 fx=fπ2, find the value of k.

If the function f(x) = x4 - 62x2 + ax + 9 attains a local maximum at x = 1, then a = _________________.

Let f:R→R be a continuous function such that f(x)+f(x+1)=2, for all x∈R. If I1=∫80f(x)dx and I2=∫3−1f(x)dx then the value of I1+2I2 is equal to

The equations of bisectors of angles between YZ plane and XZ plane is:

If a circle passes through the point (a, b) and cuts the circle x2+y2=4 orthogonally, then the locus of its centre is

Question 2 (i)Find the sums given below(i) 7+1012+14+……+84

The value of determinant Δ=∣∣∣∣abca+2xb+2yc+2zxyz∣∣∣∣ is

limx→0log(1+x+x2)+log(1−x+x2)secx−cosx=

A vertical line passing through the point (h,0) intersects the ellipse x24+y23=1 at the points P and Q. Let the tangents to the ellipse at P and Q meet at the point R. If Δ(h)=area of the triangle PQR, Δ1=max1/2 ≤ h ≤ 1Δ(h) and Δ2=min1/2 ≤ h ≤ 1Δ(h), then 8√5Δ1−8Δ2=

Difference between Hunds rule and Pauli exclusion principle

82.A dictionary is printed consisting of 7-lettered words that can be made with the letters of the word CRICKET. If tge words are printed in the alphabetic order, as in an ordinary dictionary, Find the position of the word CRICKET in the dictionary.

The set of real values of x, for which h(x)=1+2x2+4x4+6x6+⋯+100x100 is concave downward is

If f:R→R be given by f(x)=4x4x+2 for all xϵR. Then,

The set of values of x for which sin−1(sin5)&gt;x2−4x is

If 12, 13, n are the direction cosines of a line, then the values of n are ________________.

If tangent at point (1,2) on the curve y=ax2+bx+72 is parallel to normal at (−2,2) on the curve y=x2+6x+10, then

sec θ-tan θsec θ+tan θ=1+2tan2θ-2sec θ tan θ

Let f(x)=⎧⎨⎩x2−[x]2,x&lt;2a,x=2x2−[x]2+3,x&gt;2. If f(x) is continuous at x=2, then the value of a is(where [.] represents the greatest integer function and a∈R)

Evaluate the definite integrals. ∫π20cos2xdx.

4. (ax +b)(cr+d)i

(sin230° – sec260° + 4cot245°) = ?(a) 4(b) 2(c) 1(d) 14

The number of vectors of unit length perpendicular to vectors a→=i^+j^ and b→=j^+k^ is _________________.

A triangle with vertices (4, 0), (-1, -1), (3, 5) is

If A and B are independent events such that P(A)&gt;12, P(A∩¯B)=325and P(¯A∩B)=825,

The domain of the function sin−1[log2(x22)] is

Let y=y(x) be the solution of the differential equation dydx=1+xey−x, −√2&lt;x&lt;√2, y(0)=0, then the minimum value of y(x), x∈(−√2,√2) is equal to

For the system of equations 7x-5y=2 and 2x+my=5, find the :(A) Value of k (B) the condition of k for which the given system of equations has a unique solution.

5. an =(-l)"‘15"’“1

Rate constant varies with temperature by the equation log10K=5–2000/T. We can conclude that (R=8.314Jmol−1K−1):-

The set of values of x for which sin−1(sin5)>x2−4x is

Let f(x)=⎧⎨⎩x2−[x]2,x<2a,x=2x2−[x]2+3,x>2. If f(x) is continuous at x=2, then the value of a is(where [.] represents the greatest integer function and a∈R)

If A and B are independent events such that P(A)>12, P(A∩¯B)=325and P(¯A∩B)=825,

Let y=y(x) be the solution of the differential equation dydx=1+xey−x, −√2<x<√2, y(0)=0, then the minimum value of y(x), x∈(−√2,√2) is equal to