43030 + Interview Questions in Mathematics Page 104 InterviewSolution

5151.	What is the rate of change of volume of a cube with respect to an edge when the diagonal of the cube is 6√3 ?___
Answer» What is the rate of change of volume of a cube with respect to an edge when the diagonal of the cube is $6 \sqrt{3}$ ? ___

Discussion

5152.	Prove that: (i) tan8θ−tan6θ−tan2θ=tan8θtan6θtan2θ (ii) tan15∘+tan30∘+tan15∘tan30∘=1 (iii) tan36∘+tan9∘+tan36∘tan9∘=1 (iv) tan13θ−tan9θ−tan4θ=tan13θtan9θtan4θ
Answer» Prove that: (i) $t a n 8 θ - t a n 6 θ - t a n 2 θ = t a n 8 θ t a n 6 θ t a n 2 θ$ (ii) $t a n 15^{\circ} + t a n 30^{\circ} + t a n 15^{\circ} t a n 30^{\circ} = 1$ (iii) $t a n 36^{\circ} + t a n 9^{\circ} + t a n 36^{\circ} t a n 9^{\circ} = 1$ (iv) $t a n 13 θ - t a n 9 θ - t a n 4 θ = t a n 13 θ t a n 9 θ t a n 4 θ$

Discussion

5153.	If one of the 0 of polynomial 3 x square minus 8 x + 2 k plus one ,7 times the other. find the zeros and the value of k
Answer» If one of the 0 of polynomial 3 x square minus 8 x + 2 k plus one ,7 times the other. find the zeros and the value of k

Discussion

5154.	Mean and standard deviation of 100 observations were found to be 40 and 10 respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27 respectively, Find the correct standard deviation.
Answer» Mean and standard deviation of 100 observations were found to be 40 and 10 respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27 respectively, Find the correct standard deviation.

Discussion

5155.	A slip of paper is given to A, who marks it with either a plus or a minus sign; the probability of his writing a plus is 13. He then passes the slip to B, who may either leave it or change the sign before passing it on to C. Next C passes the slip to D after perhaps changing the sign; finally D passes it to an honest judge after perhaps changing the sign. The judge sees a plus sign on the slip. It is known that B,C and D each change the sign with probability 23. Then probability that A originally wrote a plus is ab (where a,b are coprime numbers), then the value of a+b is
Answer» A slip of paper is given to $A$ , who marks it with either a plus or a minus sign; the probability of his writing a plus is $\frac{1}{3}$ . He then passes the slip to $B$ , who may either leave it or change the sign before passing it on to $C$ . Next $C$ passes the slip to $D$ after perhaps changing the sign; finally $D$ passes it to an honest judge after perhaps changing the sign. The judge sees a plus sign on the slip. It is known that $B, C$ and $D$ each change the sign with probability $\frac{2}{3}$ . Then probability that $A$ originally wrote a plus is $\frac{a}{b}$ (where $a, b$ are coprime numbers), then the value of $a + b$ is

Discussion

5156.	The equation of the line through the point (0,1,2) and perpendicular to the linex−12=y+13=z−1−2 is :
Answer» The equation of the line through the point $(0, 1, 2)$ and perpendicular to the line $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{- 2}$ is :

Discussion

5157.	Integrate: Sin2x/Sin5xSin3x dx
Answer» Integrate: Sin2x/Sin5xSin3x dx

5157.

Integrate: Sin2x/Sin5xSin3x dx

Answer»

Integrate:

Sin2x/Sin5xSin3x dx

Discussion

5158.	Two dices are rolled. If both dices have six faces numbered 1,2,3,5,7 and 11, then the probability that the sum of the numbers on the top faces is less than or equal to 8 is:
Answer» Two dices are rolled. If both dices have six faces numbered $1, 2, 3, 5, 7$ and $11,$ then the probability that the sum of the numbers on the top faces is less than or equal to $8$ is:

Discussion

5159.	x1,x2,…,x34 are numbers such that xi=xi+1=150 for all i∈{1,2,3,…,9} and xi+1−xi+2=0 for all i∈{10,11,12,…,33}. Then median of x1,x2,…,x34 is
Answer» $x_{1}, x_{2}, \dots, x_{34}$ are numbers such that $x_{i} = x_{i + 1} = 150$ for all $i \in {1, 2, 3, \dots, 9}$ and $x_{i + 1} - x_{i} + 2 = 0$ for all $i \in {10, 11, 12, \dots, 33} .$ Then median of $x_{1}, x_{2}, \dots, x_{34}$ is

Discussion

5160.	45. The largest Set of real values of x for which f(x)=(x+2)(5-x) - 1/x-4 is a real function is 1)(2,5] 2)[3,4]
Answer» 45. The largest Set of real values of x for which f(x)=(x+2)(5-x) - 1/x-4 is a real function is 1)(2,5] 2)[3,4]

Discussion

5161.	20. Find the value of y for which the distance between the points P(2,3)and O(10,y)is 10 units
Answer» 20. Find the value of y for which the distance between the points P(2,3)and O(10,y)is 10 units

Discussion

5162.	The value of f(0) such that the function f(x)=2x−sin−1x2x+tan−1x is continuous at every point in its domain, is equal to
Answer» The value of $f (0)$ such that the function $f (x) = \frac{2 x - {sin}^{- 1} x}{2 x + {tan}^{- 1} x}$ is continuous at every point in its domain, is equal to

Discussion

5163.	If the equation of the plane through the points (2,2,1) and (9,3,6) and perpendicular to the plane 2x+6y+6z=9 is px+qy+rz=9, then p+q+r=
Answer» If the equation of the plane through the points $(2, 2, 1)$ and $(9, 3, 6)$ and perpendicular to the plane $2 x + 6 y + 6 z = 9$ is $p x + q y + r z = 9,$ then $p + q + r =$

Discussion

5164.	If the magnitude of cross product of vector A and B =root 3 * dot product of A and B , then what is the value of \|A+B\| ?
Answer» If the magnitude of cross product of vector A and B =root 3 * dot product of A and B , then what is the value of \|A+B\| ?

Discussion

5165.	The area enclosed between the ellipse 9x2+4y2−36x+8y+4=0 and the line 3x+2y–10=0 in first Quadrant is
Answer» The area enclosed between the ellipse $9 x^{2} + 4 y^{2} - 36 x + 8 y + 4 = 0$ and the line $3 x + 2 y - 10 = 0$ in first Quadrant is

Discussion

5166.	If esin x-e-sin x-4=0, then x =(a) 0(b) sin-1 loge 2-5(c) 1(d) none of these
Answer» If $e^{\sin x} - e^{- \sin x} - 4 = 0$ , then x = (a) 0 (b) $\sin^{- 1} \{\log_{e} (2 - \sqrt{5})\}$ (c) 1 (d) none of these

Discussion

5167.	Convertthe following in the polar form:(i) , (ii)
Answer» Convert the following in the polar form: (i) , (ii)

5167.

Convertthe following in the polar form:(i) , (ii)

Answer»

Convert
the following in the polar form:

(i) , (ii)

Discussion

5168.	If the value of integral ∫sin4xcos2xdx=xp+sin2xq+sin4xr+sin6xs+C, for fixed constants p,q,r and s. Then the value of p+q+r+s4=(where C is integration constant)
Answer» If the value of integral $\int {sin}^{4} x {cos}^{2} x d x = \frac{x}{p} + \frac{sin 2 x}{q} + \frac{sin 4 x}{r} + \frac{sin 6 x}{s} + C,$ for fixed constants $p, q, r$ and $s .$ Then the value of $\frac{p + q + r + s}{4} =$ (where $C$ is integration constant)

Discussion

5169.	If x = a (θ − sin θ), y = a (1 + cos θ) prove that, find d2ydx2.
Answer» If x = a (θ − sin θ), y = a (1 + cos θ) prove that, find $\frac{d^{2} y}{d x^{2}}$ .

Discussion

5170.	Find the value of a for which the equation has coincident roots a^2x^2+2(a+1)x+4=0
Answer» Find the value of a for which the equation has coincident roots a^2x^2+2(a+1)x+4=0

5170.

Find the value of a for which the equation has coincident roots a^2x^2+2(a+1)x+4=0

Answer»

Find the value of a for which the equation has coincident roots

a^2x^2+2(a+1)x+4=0

Discussion

5171.	the number of real solutions of equation is: sqrt(1 + cos2x) = sqr(2).sin^-1(sinx) x belongs to [-pi, pi] options: 0, 1, 2, infinite
Answer» the number of real solutions of equation is: sqrt(1 + cos2x) = sqr(2).sin^-1(sinx) x belongs to [-pi, pi] options: 0, 1, 2, infinite

Discussion

5172.	If m and M respectively denote the minimum and maximum values of f(x) = (x + 1)2 + 3 in the interval [−3, 1], then the ordered pair (m, M) = _________.
Answer» If m and M respectively denote the minimum and maximum values of f(x) = (x + 1)² + 3 in the interval [−3, 1], then the ordered pair (m, M) = _________.

Discussion

5173.	Calculate the semi-angular width of central maxima, if λ=6000 ˚A, a=18×10−5 cm.
Answer» Calculate the semi-angular width of central maxima, if $λ = 6000 ˚ A, a = 18 \times 10^{- 5} cm$ .

Discussion

5174.	The point of inflection for the function f(x)=lnxx is:
Answer» The point of inflection for the function $f (x) = \frac{ln x}{x}$ is:

Discussion

5175.	∫excos 2xdx is equal to.
Answer» $\int e^{x} cos 2 xdx is equal to$ .

Discussion

5176.	Compute the derivative of tanx.
Answer» Compute the derivative of $tan x .$

Discussion

5177.	∫a8x+9 dx,a>0 is equal to(where C is constant of integration)
Answer» $\int a^{8 x + 9} d x, a > 0$ is equal to (where $C$ is constant of integration)

Discussion

5178.	If tan α=1−cos βsin β, then
Answer» If $t a n α = \frac{1 - c o s β}{s i n β}$ , then

Discussion

5179.	If In=∫sinnxsinxdx where n>1 and n∈N, then In−In−2=(where C is integration constant)
Answer» If $I_{n} = \int \frac{sin n x}{sin x} d x$ where $n > 1$ and $n \in N,$ then $I_{n} - I_{n - 2} =$ (where $C$ is integration constant)

Discussion

5180.	The shortest distance between two parabolas y2=x−2 and x2=y−2 is :
Answer» The shortest distance between two parabolas $y^{2} = x - 2$ and $x^{2} = y - 2$ is :

Discussion

5181.	If Z1=1−2i,Z2=3+i, then the value of 10Re(Z1+Z2Z1⋅Z2)is
Answer» If $Z_{1} = 1 - 2 i, Z_{2} = 3 + i,$ then the value of $10 R e (\frac{Z_{1} + Z_{2}}{Z_{1} \cdot Z_{2}})$ is

Discussion

5182.	If a function A is square of function B, then why frequency of A is twice/double the frequency of B?
Answer» If a function A is square of function B, then why frequency of A is twice/double the frequency of B?

Discussion

5183.	Let x,y be positive real numbers and m,n positive integers. The maximum value of the expression xmyn(1+x2m)(1+y2n) is :
Answer» Let $x, y$ be positive real numbers and $m, n$ positive integers. The maximum value of the expression $\frac{x^{m} y^{n}}{(1 + x^{2 m}) (1 + y^{2 n})}$ is :

Discussion

5184.	Convert the given complex number in polar form:
Answer» Convert the given complex number in polar form:

Discussion

5185.	Suppose f(x) is a polynomial of degree four having critical points at −1,0,1. If T={x∈R\|f(x)=f(0)}, then the sum of squares of all the elements of T is:
Answer» Suppose $f (x)$ is a polynomial of degree four having critical points at $- 1, 0, 1.$ If $T = {x \in R \| f (x) = f (0)}$ , then the sum of squares of all the elements of $T$ is:

Discussion

5186.	The critical point(s) of f(x)=2x+3x23 is/are
Answer» The critical point(s) of $f (x) = 2 x + 3 x^{\frac{2}{3}}$ is/are

Discussion

5187.	19. If tanA=5/6 tan B=1/11 then prove that A+B=/4
Answer» 19. If tanA=5/6 tan B=1/11 then prove that A+B=/4

Discussion

5188.	There are 4 cards numbered 1,3,5 and 7, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean and variance of X.
Answer» There are $4$ cards numbered $1, 3, 5 and 7,$ one number on one card. Two cards are drawn at random without replacement. Let $X$ denote the sum of the numbers on the two drawn cards. Find the mean and variance of $X$ .

Discussion

5189.	If ∣∣∣1−\|x\|1+\|x\|∣∣∣≥12, then x∈
Answer» If $∣ ∣ ∣ 1 - \frac{\| x \|}{1 + \| x \|} ∣ ∣ ∣ \geq \frac{1}{2}$ , then $x \in$

Discussion

5190.	Why a cyclist bend towards the centre while taking a turn?
Answer» Why a cyclist bend towards the centre while taking a turn?

Discussion

5191.	The value of α lying between [0,π] for which the inequality tanα>tan3α is valid, is
Answer» The value of $α$ lying between $[0, π]$ for which the inequality $tan α > {tan}^{3} α$ is valid, is

Discussion

5192.	In a class there are 27 boys and 14 girls. The teacher want to select 1 boy and 1 girl to represent the class in a function. In how many ways teacher can make this selection ?
Answer» In a class there are 27 boys and 14 girls. The teacher want to select 1 boy and 1 girl to represent the class in a function. In how many ways teacher can make this selection ?

Discussion

5193.	The condition for which the equations 2x – 3y = 4, 5x – ky = 5 are consistent with unique solution is
Answer» The condition for which the equations 2x – 3y = 4, 5x – ky = 5 are consistent with unique solution is

Discussion

5194.	If a,b are two single digit prime numbers such that their sum is also prime number, then their product can be
Answer» If $a, b$ are two single digit prime numbers such that their sum is also prime number, then their product can be

Discussion

5195.	what is its roster form of(x,y):x=2y,x,y belongs to natural number and y
Answer» what is its roster form of(x,y):x=2y,x,y belongs to natural number and y<4

Discussion

5196.	Integration (root2 to root5) [x^2] gif
Answer» Integration (root2 to root5) [x^2] gif

Discussion

5197.	Find the mean deviation about the median for the data.13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17
Answer» Find the mean deviation about the median for the data. 13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17

5197.

Find the mean deviation about the median for the data.13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17

Answer»

Find the mean deviation about the median for the data.

13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17

Discussion

5198.	[Hint: multiply numerator and denominator by x-1 and put x-t ]
Answer» [Hint: multiply numerator and denominator by x-1 and put x-t ]

Discussion

5199.	Let f(n)=n∑k=1cosec−1√(k2+1)(k2+2k+2). Then the value of limn→∞12f(n)π is
Answer» Let $f (n) = n \sum k = 1 {cosec}^{- 1} \sqrt{(k^{2} + 1) (k^{2} + 2 k + 2)}$ . Then the value of $lim n \to \infty \frac{12 f (n)}{π}$ is

Discussion

5200.	How many of the following integrals are correct? 1. ∫dx√x2+a2=ln\|x+√x2+a2\|+C 2. ∫dx√x2−a2=ln\|x−√x2−a2\|+C 3. ∫dxx2−a2=12aln∣∣x−ax+a∣∣+C 4. ∫dxa2−x2=12aln∣∣x+ax−a∣∣+C___
Answer» How many of the following integrals are correct? 1. $\int \frac{d x}{\sqrt{x^{2} + a^{2}}} = l n \| x + \sqrt{x^{2} + a^{2}} \| + C$ 2. $\int \frac{d x}{\sqrt{x^{2} - a^{2}}} = l n \| x - \sqrt{x^{2} - a^{2}} \| + C$ 3. $\int \frac{d x}{x^{2} - a^{2}} = \frac{1}{2 a} l n ∣ ∣ \frac{x - a}{x + a} ∣ ∣ + C$ 4. $\int \frac{d x}{a^{2} - x^{2}} = \frac{1}{2 a} l n ∣ ∣ \frac{x + a}{x - a} ∣ ∣ + C$ ___

Discussion

Explore topic-wise InterviewSolutions in .

What is the rate of change of volume of a cube with respect to an edge when the diagonal of the cube is 6√3 ?___

Prove that: (i) tan8θ−tan6θ−tan2θ=tan8θtan6θtan2θ (ii) tan15∘+tan30∘+tan15∘tan30∘=1 (iii) tan36∘+tan9∘+tan36∘tan9∘=1 (iv) tan13θ−tan9θ−tan4θ=tan13θtan9θtan4θ

If one of the 0 of polynomial 3 x square minus 8 x + 2 k plus one ,7 times the other. find the zeros and the value of k

Mean and standard deviation of 100 observations were found to be 40 and 10 respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27 respectively, Find the correct standard deviation.

The equation of the line through the point (0,1,2) and perpendicular to the linex−12=y+13=z−1−2 is :

Integrate: Sin2x/Sin5xSin3x dx

Two dices are rolled. If both dices have six faces numbered 1,2,3,5,7 and 11, then the probability that the sum of the numbers on the top faces is less than or equal to 8 is:

x1,x2,…,x34 are numbers such that xi=xi+1=150 for all i∈{1,2,3,…,9} and xi+1−xi+2=0 for all i∈{10,11,12,…,33}. Then median of x1,x2,…,x34 is

45. The largest Set of real values of x for which f(x)=(x+2)(5-x) - 1/x-4 is a real function is 1)(2,5] 2)[3,4]

20. Find the value of y for which the distance between the points P(2,3)and O(10,y)is 10 units

The value of f(0) such that the function f(x)=2x−sin−1x2x+tan−1x is continuous at every point in its domain, is equal to

If the equation of the plane through the points (2,2,1) and (9,3,6) and perpendicular to the plane 2x+6y+6z=9 is px+qy+rz=9, then p+q+r=

If the magnitude of cross product of vector A and B =root 3 * dot product of A and B , then what is the value of |A+B| ?

The area enclosed between the ellipse 9x2+4y2−36x+8y+4=0 and the line 3x+2y–10=0 in first Quadrant is

If esin x-e-sin x-4=0, then x =(a) 0(b) sin-1 loge 2-5(c) 1(d) none of these

Convertthe following in the polar form:(i) , (ii)

If the value of integral ∫sin4xcos2xdx=xp+sin2xq+sin4xr+sin6xs+C, for fixed constants p,q,r and s. Then the value of p+q+r+s4=(where C is integration constant)

If x = a (θ − sin θ), y = a (1 + cos θ) prove that, find d2ydx2.

Find the value of a for which the equation has coincident roots a^2x^2+2(a+1)x+4=0

the number of real solutions of equation is: sqrt(1 + cos2x) = sqr(2).sin^-1(sinx) x belongs to [-pi, pi] options: 0, 1, 2, infinite

If m and M respectively denote the minimum and maximum values of f(x) = (x + 1)2 + 3 in the interval [−3, 1], then the ordered pair (m, M) = _________.

Calculate the semi-angular width of central maxima, if λ=6000 ˚A, a=18×10−5 cm.

The point of inflection for the function f(x)=lnxx is:

∫excos 2xdx is equal to.

Compute the derivative of tanx.

∫a8x+9 dx,a&gt;0 is equal to(where C is constant of integration)

If tan α=1−cos βsin β, then

If In=∫sinnxsinxdx where n&gt;1 and n∈N, then In−In−2=(where C is integration constant)

The shortest distance between two parabolas y2=x−2 and x2=y−2 is :

If Z1=1−2i,Z2=3+i, then the value of 10Re(Z1+Z2Z1⋅Z2)is

If a function A is square of function B, then why frequency of A is twice/double the frequency of B?

Let x,y be positive real numbers and m,n positive integers. The maximum value of the expression xmyn(1+x2m)(1+y2n) is :

Convert the given complex number in polar form:

Suppose f(x) is a polynomial of degree four having critical points at −1,0,1. If T={x∈R|f(x)=f(0)}, then the sum of squares of all the elements of T is:

The critical point(s) of f(x)=2x+3x23 is/are

19. If tanA=5/6 tan B=1/11 then prove that A+B=/4

There are 4 cards numbered 1,3,5 and 7, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean and variance of X.

If ∣∣∣1−|x|1+|x|∣∣∣≥12, then x∈

Why a cyclist bend towards the centre while taking a turn?

The value of α lying between [0,π] for which the inequality tanα&gt;tan3α is valid, is

In a class there are 27 boys and 14 girls. The teacher want to select 1 boy and 1 girl to represent the class in a function. In how many ways teacher can make this selection ?

The condition for which the equations 2x – 3y = 4, 5x – ky = 5 are consistent with unique solution is

If a,b are two single digit prime numbers such that their sum is also prime number, then their product can be

what is its roster form of(x,y):x=2y,x,y belongs to natural number and y

Integration (root2 to root5) [x^2] gif

Find the mean deviation about the median for the data.13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17

[Hint: multiply numerator and denominator by x-1 and put x-t ]

Let f(n)=n∑k=1cosec−1√(k2+1)(k2+2k+2). Then the value of limn→∞12f(n)π is

How many of the following integrals are correct? 1. ∫dx√x2+a2=ln|x+√x2+a2|+C 2. ∫dx√x2−a2=ln|x−√x2−a2|+C 3. ∫dxx2−a2=12aln∣∣x−ax+a∣∣+C 4. ∫dxa2−x2=12aln∣∣x+ax−a∣∣+C___

∫a8x+9 dx,a>0 is equal to(where C is constant of integration)

If In=∫sinnxsinxdx where n>1 and n∈N, then In−In−2=(where C is integration constant)

The value of α lying between [0,π] for which the inequality tanα>tan3α is valid, is