43030 + Interview Questions in Mathematics Page 53 InterviewSolution

2601.	Q) 7x-6-2x^2
Answer» Q) 7x-6-2x^2

Discussion

2602.	The equation of tangent to the curve y(1 + x2) = 2 - x, where it crosses x-axis is (a) x + 5y = 2 (b) x - 5y = 2 (c) 5x - y = 2 (d) 5x + y = 2
Answer» The equation of tangent to the curve y(1 + x²) = 2 - x, where it crosses x-axis is (a) x + 5y = 2 (b) x - 5y = 2 (c) 5x - y = 2 (d) 5x + y = 2

Discussion

2603.	The number of distinct real roots of the equation ∣∣∣∣sinxcosxcosxcosxsinxcosxcosxcosxsinx∣∣∣∣=0, in the interval −π4≤x≤π4 is
Answer» The number of distinct real roots of the equation $∣ ∣∣ ∣ \begin{matrix} sin x & cos x & cos x cos x & sin x & cos x cos x & cos x & sin x \end{matrix} ∣ ∣∣ ∣ = 0,$ in the interval $- \frac{π}{4} \leq x \leq \frac{π}{4}$ is

Discussion

2604.	3. A real valued function f(x) satisfies the equation f(x+y) = f(x) f(a-y) + f(y) f(a-x), where x,y belongs to R , where a is a constant and f(0) = 0 , then f(2a-x) is (1) -f(x) (2) f(x) (3) f(a-x) (4) f(-x)
Answer» 3. A real valued function f(x) satisfies the equation f(x+y) = f(x) f(a-y) + f(y) f(a-x), where x,y belongs to R , where a is a constant and f(0) = 0 , then f(2a-x) is (1) -f(x) (2) f(x) (3) f(a-x) (4) f(-x)

Discussion

2605.	Number of solutions of the equation \|sinx\|=sinx−2cosx in [0,4π] is
Answer» Number of solutions of the equation $\| sin x \| = sin x - 2 cos x$ in $[0, 4 π]$ is

Discussion

2606.	101-226.x (1-2x)
Answer» 101-226.x (1-2x)

Discussion

2607.	The number of divisors of 2100 is ___
Answer» The number of divisors of 2100 is ___

Discussion

2608.	The equation of the circle circumscribing the triangle whose sides are the lines y = x + 2, 3y = 4x and 2y = 3x, is __________.
Answer» The equation of the circle circumscribing the triangle whose sides are the lines y = x + 2, 3y = 4x and 2y = 3x, is __________.

Discussion

2609.	Find integration of:X√(X+X^2) dx
Answer» Find integration of: X√(X+X^2) dx

Discussion

2610.	If 2n+1Pn−1: 2n−1Pn=3:5, then the value of n is :
Answer» If $^{2 n + 1} P_{n - 1} :^{2 n - 1} P_{n} = 3 : 5$ , then the value of $n$ is :

Discussion

2611.	The figure shows a relation between the sets P and Q. Write this relation(i) In set builder form(ii) In roster formWhat is its domain and range?
Answer» The figure shows a relation between the sets $P$ and $Q$ . Write this relation $(i)$ In set builder form $(i i)$ In roster form What is its domain and range?

Discussion

2612.	Which among the following won't represent a function ?
Answer» Which among the following won't represent a function ?

Discussion

2613.	Prove that the curves x = y 2 and xy = k cut at right angles if 8 k 2 = 1. [ Hint : Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other. ]
Answer» Prove that the curves x = y 2 and xy = k cut at right angles if 8 k 2 = 1. [ Hint : Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other. ]

Discussion

2614.	The number of ways in which 20 letters a1,a2,a3,…,a10,b1,b2,b3,…,b10 can be arranged in a line so that suffixes of the letters a and also those of b are respectively in ascending order of magnitude is
Answer» The number of ways in which $20$ letters $a_{1}, a_{2}, a_{3}, \dots, a_{10}, b_{1}, b_{2}, b_{3}, \dots, b_{10}$ can be arranged in a line so that suffixes of the letters $a$ and also those of $b$ are respectively in ascending order of magnitude is

Discussion

2615.	If p=∣∣∣x11x∣∣∣ and Q=∣∣∣∣x111x111x∣∣∣∣ then dQdx=.......
Answer» If $p = ∣ ∣ ∣ \begin{matrix} x & 1 1 & x \end{matrix} ∣ ∣ ∣$ and $Q = ∣ ∣∣ ∣ \begin{matrix} x & 1 & 1 1 & x & 1 1 & 1 & x \end{matrix} ∣ ∣∣ ∣$ then $\frac{d Q}{d x}$ =.......

Discussion

2616.	limx→0(1+x−1−xx)
Answer» ${lim}_{x \to 0} (1 + x - 1 - \frac{x}{x})$

Discussion

2617.	If all the letters of the word COMMONT be arranged as in a dictionary, then the 1000th word is
Answer» If all the letters of the word COMMONT be arranged as in a dictionary, then the 1000th word is

Discussion

2618.	The solution of the differential equation (2x−y+2)dx+(4x−2y−1)dy=0 is(where k is integration constant)
Answer» The solution of the differential equation $(2 x - y + 2) d x + (4 x - 2 y - 1) d y = 0$ is (where $k$ is integration constant)

Discussion

2619.	A function f(x) satisfies the relation ∫x2f(t)=x22+∫2xt2f(t)dt Then answer the following The range of f(x) is
Answer» A function f(x) satisfies the relation $\int_{2}^{x} f (t) = \frac{x^{2}}{2} + \int_{x}^{2} t^{2} f (t) d t$ Then answer the following The range of f(x) is

Discussion

2620.	The sum of integral values of a, where aϵ[0,10] for which f(x)=loga+1(2ax−x2) is strictly increasing in [12,1] is 'b'. The value of 'b/11' is
Answer» The sum of integral values of a, where $a ϵ [0, 10]$ for which $f (x) = {log}_{a + 1} (2 a x - x^{2})$ is strictly increasing in $[\frac{1}{2}, 1]$ is 'b'. The value of 'b/11' is

Discussion

2621.	Let f(x) be a polynomial of degree three, if the curve y=f(x) has relative extrema at x=±2√3=4x2+y2=4 in two parts. Then find the integral part of areas of these two parts.
Answer» Let $f (x)$ be a polynomial of degree three, if the curve $y = f (x)$ has relative extrema at $x = \pm \frac{2}{\sqrt{3}} = 4 x^{2} + y^{2} = 4$ in two parts. Then find the integral part of areas of these two parts.

Discussion

2622.	The number of circles which passes through the origin and makes intercept of length 8 units and 6 units on the coordinate axes respectively, is
Answer» The number of circles which passes through the origin and makes intercept of length $8$ units and $6$ units on the coordinate axes respectively, is

Discussion

2623.	A vector p is added with vector Q= 4i+3j yields a resul†an t vector that is in positive y direction and has a magnitude equal to that of magnitude of Q then magnitude of p vecto
Answer» A vector p is added with vector Q= 4i+3j yields a resul†an t vector that is in positive y direction and has a magnitude equal to that of magnitude of Q then magnitude of p vecto

Discussion

2624.	Let E1 and E2 be the two ellipses centred at origin. The major axis of E1 and E2 lie along the x− axis and y− axis respectively. Let S be the circle x2+(y−1)2=2, the straight line x+y=3 touches the curve S,E1 and E2 at P,Q and R respectively such that PQ=PR=2√23. If e1 and e2 are the eccentricities of E1 and E2, then which of the following is/are correct
Answer» Let $E_{1}$ and $E_{2}$ be the two ellipses centred at origin. The major axis of $E_{1}$ and $E_{2}$ lie along the $x -$ axis and $y -$ axis respectively. Let $S$ be the circle $x^{2} + (y - 1)^{2} = 2,$ the straight line $x + y = 3$ touches the curve $S, E_{1}$ and $E_{2}$ at $P, Q$ and $R$ respectively such that $P Q = P R = \frac{2 \sqrt{2}}{3} .$ If $e_{1}$ and $e_{2}$ are the eccentricities of $E_{1}$ and $E_{2},$ then which of the following is/are correct

Discussion

2625.	If sin2θ=x2−4x+5,then x=
Answer» $If {sin}^{2} θ = x^{2} - 4 x + 5, then x =$

Discussion

2626.	The equation of the plane which passes through the z− axis and is perpendicular to the line x−acosθ=y+2sinθ=z−30 is
Answer» The equation of the plane which passes through the $z -$ axis and is perpendicular to the line $\frac{x - a}{cos θ} = \frac{y + 2}{sin θ} = \frac{z - 3}{0}$ is

Discussion

2627.	integration of ^pie/4 sinxdx
Answer» integration of ^pie/4 sinxdx

Discussion

2628.	If P=(3,4,5),Q=(4,6,3),R=(−1,2,4) and S=(1,0,5) are four points then the projection of segment RS on line PQ is
Answer» If $P = (3, 4, 5), Q = (4, 6, 3), R = (- 1, 2, 4)$ and $S = (1, 0, 5)$ are four points then the projection of segment $R S$ on line $P Q$ is

Discussion

2629.	Number of ways in which 200 people can be divided into 100 pairs is
Answer» Number of ways in which $200$ people can be divided into $100$ pairs is

Discussion

2630.	A hyperbola has focus at origin, its eccentricity is √2 and corresponding directrix is x+y+1=0. The equation of its asymptotes is/are:
Answer» A hyperbola has focus at origin, its eccentricity is $\sqrt{2}$ and corresponding directrix is $x + y + 1 = 0$ . The equation of its asymptotes is/are:

Discussion

2631.	A particle moves along the curve . Find the points on the curve at which the y -coordinate is changing 8 times as fast as the x -coordinate.
Answer» A particle moves along the curve . Find the points on the curve at which the y -coordinate is changing 8 times as fast as the x -coordinate.

Discussion

2632.	30. P is a point(a,b,c) . Let A ,B ,C be images of P in y_z ,z_x and x_y planes respectively , then the equation of the plane ABC is
Answer» 30. P is a point(a,b,c) . Let A ,B ,C be images of P in y_z ,z_x and x_y planes respectively , then the equation of the plane ABC is

Discussion

2633.	Let f(x)=x3−3x2+2x. If the equation f(x)=k has exactly one positive and one negative solution, then the value of k is equal to
Answer» Let $f (x) = x^{3} - 3 x^{2} + 2 x .$ If the equation $f (x) = k$ has exactly one positive and one negative solution, then the value of $k$ is equal to

Discussion

2634.	The equation of a line passing through the point (4,−2,5) and parallel to the vector 3^i−^j+2^k in Cartesian form is
Answer» The equation of a line passing through the point $(4, - 2, 5)$ and parallel to the vector $3^i -^j + 2^k$ in Cartesian form is

Discussion

2635.	Evaluate the following integrals:∫xx2+1x-1dx
Answer» Evaluate the following integrals: $\int \frac{x}{(x^{2} + 1) (x - 1)} d x$

Discussion

2636.	Let a function f(x)=x33−2x2+3x on the set A={x\|x2+8≤6x}, then the minimum value of f is
Answer» Let a function $f (x) = \frac{x^{3}}{3} - 2 x^{2} + 3 x$ on the set $A = {x \| x^{2} + 8 \leq 6 x},$ then the minimum value of $f$ is

Discussion

2637.	The function f(x) = cot x is discontinuous on the set(a) {x : x = nπ, n ∈ Z}(b) {x : x = 2nπ, n ∈ Z}(C) x : x=2n+1 π2, n∈Z(d) x : x=nπ2, n∈Z
Answer» The function f(x) = cot x is discontinuous on the set (a) {x : x = nπ, n ∈ Z} (b) {x : x = 2nπ, n ∈ Z} (C) $\{x : x = (2 n + 1) \frac{π}{2}, n \in Z\}$ (d) $\{x : x = \frac{n π}{2}, n \in Z\}$

Discussion

2638.	Find the principal and general solutions of the equation secx=2
Answer» Find the principal and general solutions of the equation $sec x = 2$

Discussion

2639.	Differentiate the following functions with respect to x: x2 sin x log x
Answer» Differentiate the following functions with respect to x: $x^{2} s i n x l o g x$

Discussion

2640.	What is torr
Answer» What is torr

Discussion

2641.	If [sinθ−cosθcosθsinθ][sinθcosθ−cosθsinθ]=[abcd], then the value of a+b+c+d is
Answer» If $[\begin{matrix} sin θ & - cos θ cos θ & sin θ \end{matrix}] [\begin{matrix} sin θ & cos θ - cos θ & sin θ \end{matrix}] = [\begin{matrix} a & b c & d \end{matrix}],$ then the value of $a + b + c + d$ is

Discussion

2642.	Calculate value of root 171 using binomial approximation.
Answer» Calculate value of root 171 using binomial approximation.

Discussion

2643.	The continued product of the four values of [cos(π3)+isin(π3)]3/4 is
Answer» The continued product of the four values of ${[cos (\frac{π}{3}) + i sin (\frac{π}{3})]}^{3 / 4}$ is

Discussion

2644.	lf a sin θ + b cos θ = c, then prove that a cos θ -b sin θ =\sqrt{a^2+b^2-c^{
Answer» lf a sin θ + b cos θ = c, then prove that a cos θ -b sin θ =\sqrt{a^2+b^2-c^{

Discussion

2645.	Prove that (p) raise to power 1/n is irrational when p is prime and n >1.
Answer» Prove that (p) raise to power 1/n is irrational when p is prime and n >1.

Discussion

2646.	Find dy/dx of x^yy^x =1
Answer» Find dy/dx of x^yy^x =1

Discussion

2647.	Integrate 1x(logex)n with respect to x.
Answer» Integrate $\frac{1}{x ({log}_{e} x)^{n}}$ with respect to $x$ .

Discussion

2648.	Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {( a , b ): b = a + 1} is reflexive, symmetric or transitive.
Answer» Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {( a , b ): b = a + 1} is reflexive, symmetric or transitive.

Discussion

2649.	1. Define in details the graph of charles law in and K in detail
Answer» 1. Define in details the graph of charles law in and K in detail

Discussion

2650.	The sum of the rational terms in the expansion of (√2+31/5)10 is
Answer» The sum of the rational terms in the expansion of $(\sqrt{2} + 3^{1 / 5})^{10}$ is

Discussion

Explore topic-wise InterviewSolutions in .

Q) 7x-6-2x^2

The equation of tangent to the curve y(1 + x2) = 2 - x, where it crosses x-axis is (a) x + 5y = 2 (b) x - 5y = 2 (c) 5x - y = 2 (d) 5x + y = 2

The number of distinct real roots of the equation ∣∣∣∣sinxcosxcosxcosxsinxcosxcosxcosxsinx∣∣∣∣=0, in the interval −π4≤x≤π4 is

3. A real valued function f(x) satisfies the equation f(x+y) = f(x) f(a-y) + f(y) f(a-x), where x,y belongs to R , where a is a constant and f(0) = 0 , then f(2a-x) is (1) -f(x) (2) f(x) (3) f(a-x) (4) f(-x)

Number of solutions of the equation |sinx|=sinx−2cosx in [0,4π] is

101-226.x (1-2x)

The number of divisors of 2100 is ___

The equation of the circle circumscribing the triangle whose sides are the lines y = x + 2, 3y = 4x and 2y = 3x, is __________.

Find integration of:X√(X+X^2) dx

If 2n+1Pn−1: 2n−1Pn=3:5, then the value of n is :

The figure shows a relation between the sets P and Q. Write this relation(i) In set builder form(ii) In roster formWhat is its domain and range?

Which among the following won't represent a function ?

Prove that the curves x = y 2 and xy = k cut at right angles if 8 k 2 = 1. [ Hint : Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other. ]

The number of ways in which 20 letters a1,a2,a3,…,a10,b1,b2,b3,…,b10 can be arranged in a line so that suffixes of the letters a and also those of b are respectively in ascending order of magnitude is

If p=∣∣∣x11x∣∣∣ and Q=∣∣∣∣x111x111x∣∣∣∣ then dQdx=.......

limx→0(1+x−1−xx)

If all the letters of the word COMMONT be arranged as in a dictionary, then the 1000th word is

The solution of the differential equation (2x−y+2)dx+(4x−2y−1)dy=0 is(where k is integration constant)

A function f(x) satisfies the relation ∫x2f(t)=x22+∫2xt2f(t)dt Then answer the following The range of f(x) is

The sum of integral values of a, where aϵ[0,10] for which f(x)=loga+1(2ax−x2) is strictly increasing in [12,1] is 'b'. The value of 'b/11' is

Let f(x) be a polynomial of degree three, if the curve y=f(x) has relative extrema at x=±2√3=4x2+y2=4 in two parts. Then find the integral part of areas of these two parts.

The number of circles which passes through the origin and makes intercept of length 8 units and 6 units on the coordinate axes respectively, is

A vector p is added with vector Q= 4i+3j yields a resul†an t vector that is in positive y direction and has a magnitude equal to that of magnitude of Q then magnitude of p vecto

If sin2θ=x2−4x+5,then x=

The equation of the plane which passes through the z− axis and is perpendicular to the line x−acosθ=y+2sinθ=z−30 is

integration of ^pie/4 sinxdx

If P=(3,4,5),Q=(4,6,3),R=(−1,2,4) and S=(1,0,5) are four points then the projection of segment RS on line PQ is

Number of ways in which 200 people can be divided into 100 pairs is

A hyperbola has focus at origin, its eccentricity is √2 and corresponding directrix is x+y+1=0. The equation of its asymptotes is/are:

A particle moves along the curve . Find the points on the curve at which the y -coordinate is changing 8 times as fast as the x -coordinate.

30. P is a point(a,b,c) . Let A ,B ,C be images of P in y_z ,z_x and x_y planes respectively , then the equation of the plane ABC is

Let f(x)=x3−3x2+2x. If the equation f(x)=k has exactly one positive and one negative solution, then the value of k is equal to

The equation of a line passing through the point (4,−2,5) and parallel to the vector 3^i−^j+2^k in Cartesian form is

Evaluate the following integrals:∫xx2+1x-1dx

Let a function f(x)=x33−2x2+3x on the set A={x|x2+8≤6x}, then the minimum value of f is

The function f(x) = cot x is discontinuous on the set(a) {x : x = nπ, n ∈ Z}(b) {x : x = 2nπ, n ∈ Z}(C) x : x=2n+1 π2, n∈Z(d) x : x=nπ2, n∈Z

Find the principal and general solutions of the equation secx=2

Differentiate the following functions with respect to x: x2 sin x log x

What is torr

If [sinθ−cosθcosθsinθ][sinθcosθ−cosθsinθ]=[abcd], then the value of a+b+c+d is

Calculate value of root 171 using binomial approximation.

The continued product of the four values of [cos(π3)+isin(π3)]3/4 is

lf a sin θ + b cos θ = c, then prove that a cos θ -b sin θ =\sqrt{a^2+b^2-c^{

Prove that (p) raise to power 1/n is irrational when p is prime and n >1.

Find dy/dx of x^yy^x =1

Integrate 1x(logex)n with respect to x.

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {( a , b ): b = a + 1} is reflexive, symmetric or transitive.

1. Define in details the graph of charles law in and K in detail

The sum of the rational terms in the expansion of (√2+31/5)10 is