43030 + Interview Questions in Mathematics Page 114 InterviewSolution

5651.	Show that the general solution of the differential equation is given by ( x + y + 1) = A (1 – x – y – 2 xy ), where A is parameter
Answer» Show that the general solution of the differential equation is given by ( x + y + 1) = A (1 – x – y – 2 xy ), where A is parameter

Discussion

5652.	For the differential equation dydt+5y=0 with y(0) = 1, the general solution is
Answer» For the differential equation $\frac{d y}{d t} + 5 y = 0$ with y(0) = 1, the general solution is

Discussion

5653.	ntx= cos inverse (8t-8t+1), y=sin inverse (3t-4t) [0
Answer» ntx= cos inverse (8t-8t+1), y=sin inverse (3t-4t) [0

Discussion

5654.	A helicopter is flying along the curve given by y−x32=7, (x≥0). A soldier positioned at the point (12,7) wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is :
Answer» A helicopter is flying along the curve given by $y - x^{\frac{3}{2}} = 7, (x \geq 0)$ . A soldier positioned at the point $(\frac{1}{2}, 7)$ wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is :

Discussion

5655.	If →a,→b,→c are three vectors such that [→a→b→c]=5, then the value of [→a×→b,→b×→c,→c×→a] is :
Answer» If $\to a, \to b, \to c$ are three vectors such that $[\to a \to b \to c] = 5$ , then the value of $[\to a \times \to b, \to b \times \to c, \to c \times \to a]$ is :

Discussion

5656.	if tan(A-B)=1/root3 and tan(A+B)=1 then find A
Answer» if tan(A-B)=1/root3 and tan(A+B)=1 then find A

Discussion

5657.	If (1 + k)x2 – 4x – 1 + k = 0 has real roots tanα and tanβ, then
Answer» If (1 + k)x2 – 4x – 1 + k = 0 has real roots tanα and tanβ, then

Discussion

5658.	Let z=x+iy be a complex number. If 1+¯¯¯zz is a real number, then
Answer» Let $z = x + i y$ be a complex number. If $\frac{1 + ¯ ¯ ¯ z}{z}$ is a real number, then

Discussion

5659.	If Z=5x+3y, subject to 3x+5y≤15,5x+2y≤10,x≥0,y≥0, then Zmax is equal to
Answer» If $Z = 5 x + 3 y,$ subject to $3 x + 5 y \leq 15, 5 x + 2 y \leq 10, x \geq 0, y \geq 0,$ then $Z_{max}$ is equal to

Discussion

5660.	If∣∣a sin2 θ+b sin θ cos θ+c cos2 θ−12(a+c)∣∣≤12k, then k2 is equal to
Answer» $I f ∣ ∣ a s i n^{2} θ + b s i n θ c o s θ + c c o s^{2} θ - \frac{1}{2} (a + c) ∣ ∣ \leq \frac{1}{2} k, t h e n k^{2} i s e q u a l t o$

Discussion

5661.	Show that the differential equations (x−y)dydx=x+2y is homogeneous and solve it also. OR Find the differential equations of the family of curves (x−h)2+(y−k)2=r2, where h and k are arbitrary constants.
Answer» Show that the differential equations $(x - y) \frac{d y}{d x} = x + 2 y$ is homogeneous and solve it also. OR Find the differential equations of the family of curves $(x - h)^{2} + (y - k)^{2} = r^{2},$ where h and k are arbitrary constants.

5661.

Show that the differential equations (x−y)dydx=x+2y is homogeneous and solve it also. OR Find the differential equations of the family of curves (x−h)2+(y−k)2=r2, where h and k are arbitrary constants.

Answer»

Show that the differential equations $(x - y) \frac{d y}{d x} = x + 2 y$ is homogeneous and solve it also.

OR

Find the differential equations of the family of curves $(x - h)^{2} + (y - k)^{2} = r^{2},$ where h and k are arbitrary constants.

Discussion

5662.	Integrate: 2xx−1
Answer» Integrate: $\frac{2 x}{x - 1}$

Discussion

5663.	Each of the circles x2+y2−2x−2y+1=0 and x2+y2+2x−2y+1=0 touches internally a circle of radius 2. The equation of circles touching all the three circles, is
Answer» Each of the circles $x^{2} + y^{2} - 2 x - 2 y + 1 = 0$ and $x^{2} + y^{2} + 2 x - 2 y + 1 = 0$ touches internally a circle of radius $2$ . The equation of circles touching all the three circles, is

Discussion

5664.	Let [x] denote the greatest integer function of x. If the domain of the function 1[x]2−7[x]+12 is R−[a,b), then the value of a+b is
Answer» Let $[x]$ denote the greatest integer function of $x$ . If the domain of the function $\frac{1}{[x]^{2} - 7 [x] + 12}$ is $R - [a, b),$ then the value of $a + b$ is

Discussion

5665.	How many integers satisfy the condition 0≤23[x]≤1 __
Answer» How many integers satisfy the condition $0 \leq \frac{2}{3} [x] \leq 1$ __

Discussion

5666.	If point P(0,1) on the curve y=x+esinx is at the shortest distance from y=mx+c. Then the value of m2 is
Answer» If point $P (0, 1)$ on the curve $y = x + e^{sin x}$ is at the shortest distance from $y = m x + c .$ Then the value of $m^{2}$ is

Discussion

5667.	Two tangents are drawn to the parabola y2=8x which meets the tangent at vertex at P and Q respectively. If PQ=4 units, then the locus of the point of intersection of the two tangents is
Answer» Two tangents are drawn to the parabola $y^{2} = 8 x$ which meets the tangent at vertex at $P$ and $Q$ respectively. If $P Q = 4 units$ , then the locus of the point of intersection of the two tangents is

Discussion

5668.	If y = ln (xa+bx)x, then x3d2ydx2 is equal to
Answer» If y = ln $(\frac{x}{a + b x})^{x}$ , then $x^{3} \frac{d^{2} y}{d x^{2}}$ is equal to

Discussion

5669.	1:- 9-sec2A-9tan2A=? 2:-(1+tan theeta +sec theeta)(1+cot theeta- cosec theeta)=? 3:- (Sec A+ tanA)(1-sin A)=?
Answer» 1:- 9-sec2A-9tan2A=? 2:-(1+tan theeta +sec theeta)(1+cot theeta- cosec theeta)=? 3:- (Sec A+ tanA)(1-sin A)=?

Discussion

5670.	Find the sum to n terms of a G.P. √7,√21,3√7...
Answer» Find the sum to $n$ terms of a G.P. $\sqrt{7}, \sqrt{21}, 3 \sqrt{7} . . .$

Discussion

5671.	The coordinates of the vertex of a parabola represented by y=ax2+bx+c is, . Take D as discriminant;
Answer» The coordinates of the vertex of a parabola represented by $y = a x^{2} + b x + c$ is, . Take D as discriminant;

Discussion

5672.	If the middle term in the expansion of (p2+2)8 is 1120, then the value of p is
Answer» If the middle term in the expansion of ${(\frac{p}{2} + 2)}^{8}$ is $1120$ , then the value of $p$ is

Discussion

5673.	limx→π2acotx−acosxcot x−cos x
Answer» ${lim}_{x \to \frac{π}{2}} \frac{a^{c o t x} - a^{c o s x}}{c o t x - c o s x}$

Discussion

5674.	15.sin (tam1x), lxl< is equal to2 (B)2 (C)+x2v1+ x2
Answer» 15.sin (tam1x), lxl< is equal to2 (B)2 (C)+x2v1+ x2

Discussion

5675.	38.4sinxsin2xsin4x=sin3x Find the value if x?
Answer» 38.4sinxsin2xsin4x=sin3x Find the value if x?

Discussion

5676.	What is e vs x graph for positive hargec at origin.?
Answer» What is e vs x graph for positive hargec at origin.?

Discussion

5677.	If x (\|x\|<π) and y are the solutions of the equation 12sinx+5cosx=2y2−8y+21, then the value of 12cot(xy2) is
Answer» If $x$ $(\| x \| < π)$ and $y$ are the solutions of the equation $12 sin x + 5 cos x = 2 y^{2} - 8 y + 21,$ then the value of $12 cot (\frac{x y}{2})$ is

Discussion

5678.	∫sin4x dx is equal to
Answer» $\int {sin}^{4} x d x$ is equal to

Discussion

5679.	2a∫0x3√2ax−x2 dx is equal to
Answer» $2 a \int 0 x^{3} \sqrt{2 a x - x^{2}} d x$ is equal to

Discussion

5680.	Show that the lines x−12=y−23=z−34 and x−45=y−12=z intersect. Also, find their point of intersection.
Answer» Show that the lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} a n d \frac{x - 4}{5} = \frac{y - 1}{2} = z$ intersect. Also, find their point of intersection.

Discussion

5681.	If f: R → Ris defined by f(x) = x2 − 3x+ 2, find f(f(x)).
Answer» If f: R → R is defined by f(x) = x² − 3x + 2, find f(f(x)).

Discussion

5682.	ntFind the Bi-quadratic equation with rational coefficients whose one of the root is 2+-3n
Answer» ntFind the Bi-quadratic equation with rational coefficients whose one of the root is 2+-3n

Discussion

5683.	Find the value of k, which equation real and equal root : kx2 + kx +1 = 4x2 - x
Answer» Find the value of k, which equation real and equal root : kx2 + kx +1 = 4x2 - x

Discussion

5684.	A die is thrown once.Determine the vertex which contains a right angle in ΔABC, where A(4,-2), B(7,9) and C(7,-2).
Answer» A die is thrown once.Determine the vertex which contains a right angle in $Δ A B C,$ where A(4,-2), B(7,9) and C(7,-2).

Discussion

5685.	3x+8y=-1, 1x-2y=2 x≠0, y≠0
Answer» $\frac{3}{x} + \frac{8}{y} = - 1, \frac{1}{x} - \frac{2}{y} = 2 (x \neq 0, y \neq 0)$

Discussion

5686.	Write the first fiveterms of the sequences whose nth term is
Answer» Write the first five terms of the sequences whose n^th term is

Discussion

5687.	If(A) 0 (B) (C) notdefined (D) 1
Answer» If (A) 0 (B) (C) not defined (D) 1

5687.

If(A) 0 (B) (C) notdefined (D) 1

Answer»

If

(A) 0 (B)

(C) not
defined (D) 1

Discussion

5688.	∫-π4π4 11+cos 2xdx is equal to(a) 1(b) 2(c) 3(d) 4
Answer» $\int_{\frac{- π}{4}}^{\frac{π}{4}} \frac{1}{1 + \cos 2 x} d x$ is equal to (a) 1 (b) 2 (c) 3 (d) 4

Discussion

5689.	Let P be a variable point on the ellipse x2100+y264=1 with foci F1 and F2. If A is the area of triangle PF1F2, then the maximum possible value of A is
Answer» Let $P$ be a variable point on the ellipse $\frac{x^{2}}{100} + \frac{y^{2}}{64} = 1$ with foci $F_{1}$ and $F_{2}$ . If $A$ is the area of triangle $P F_{1} F_{2}$ , then the maximum possible value of $A$ is

Discussion

5690.	(i) If A=1-202 130-21, find A−1. Using A−1, solve the system of linear equationsx − 2y = 10, 2x + y + 3z = 8, −2y + z = 7(ii) A=3-422 351 01, find A−1 and hence solve the following system of equations:3x − 4y + 2z = −1, 2x + 3y + 5z = 7, x + z = 2(iii) A=1-202130-21 and B=72-6-21-3-42 5, find AB. Hence, solve the system of equations:x − 2y = 10, 2x + y + 3z = 8 and −2y + z = 7(iv) If A=120-2 -1-20-11, find A−1. Using A−1, solve the system of linear equationsx − 2y = 10, 2x − y − z = 8, −2y + z = 7(v) Given A=22-4-42-42-1 5, B=1-10234012, find BA and use this to solve the system of equationsy + 2z = 7, x − y = 3, 2x + 3y + 4z = 17(vi) If A=2311 22–3 1-1, find A–1 and hence solve the system of equations 2x + y – 3z = 13, 3x + 2y + z = 4, x + 2y – z = 8.(vii) Use product 1-1202-33-24-20192-361-2 to solve the system of equations x + 3z = 9, −x + 2y − 2z = 4, 2x − 3y + 4z = −3.
Answer» (i) If $A = [\begin{matrix} 1 & - 2 & 0 \\ 2 & 1 & 3 \\ 0 & - 2 & 1 \end{matrix}]$ , find A⁻¹. Using A⁻¹, solve the system of linear equations x − 2y = 10, 2x + y + 3z = 8, −2y + z = 7 (ii) $A = [\begin{matrix} 3 & - 4 & 2 \\ 2 & 3 & 5 \\ 1 & 0 & 1 \end{matrix}]$ , find A⁻¹ and hence solve the following system of equations: 3x − 4y + 2z = −1, 2x + 3y + 5z = 7, x + z = 2 (iii) $A = [\begin{matrix} 1 & - 2 & 0 \\ 2 & 1 & 3 \\ 0 & - 2 & 1 \end{matrix}] and B = [\begin{matrix} 7 & 2 & - 6 \\ - 2 & 1 & - 3 \\ - 4 & 2 & 5 \end{matrix}]$ , find AB. Hence, solve the system of equations: x − 2y = 10, 2x + y + 3z = 8 and −2y + z = 7 (iv) If $A = [\begin{matrix} 1 & 2 & 0 \\ - 2 & - 1 & - 2 \\ 0 & - 1 & 1 \end{matrix}]$ , find A⁻¹. Using A⁻¹, solve the system of linear equations x − 2y = 10, 2x − y − z = 8, −2y + z = 7 (v) Given $A = [\begin{matrix} 2 & 2 & - 4 \\ - 4 & 2 & - 4 \\ 2 & - 1 & 5 \end{matrix}], B = [\begin{matrix} 1 & - 1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{matrix}]$ , find BA and use this to solve the system of equations y + 2z = 7, x − y = 3, 2x + 3y + 4z = 17 (vi) If $A = (\begin{array}{rrr} 2 & 3 & 1 \\ 1 & 2 & 2 \\ - 3 & 1 & - 1 \end{array})$ , find A^–1 and hence solve the system of equations 2x + y – 3z = 13, 3x + 2y + z = 4, x + 2y – z = 8. (vii) Use product $[\begin{matrix} 1 & - 1 & 2 \\ 0 & 2 & - 3 \\ 3 & - 2 & 4 \end{matrix}] [\begin{matrix} - 2 & 0 & 1 \\ 9 & 2 & - 3 \\ 6 & 1 & - 2 \end{matrix}]$ to solve the system of equations x + 3z = 9, −x + 2y − 2z = 4, 2x − 3y + 4z = −3.

Discussion

5691.	The larger of 9950+10050 and 10150 is . . .
Answer» The larger of $99^{50} + 100^{50} a n d 101^{50}$ is . . .

Discussion

5692.	Given M=[1123+1223+⋯+1100023] (where [.] denotes greatest integer function), (M - 20) is equal to ___
Answer» Given $M = [\frac{1}{1^{\frac{2}{3}}} + \frac{1}{2^{\frac{2}{3}}} + \dots + \frac{1}{1000^{\frac{2}{3}}}]$ (where [.] denotes greatest integer function), (M - 20) is equal to ___

Discussion

5693.	∫e2x−1e2x+1dx is equal to(where C is constant of integration)
Answer» $\int \frac{e^{2 x} - 1}{e^{2 x} + 1} d x$ is equal to (where $C$ is constant of integration)

Discussion

5694.	Prove that if cos a=cos b then a=2n+-b. By the formula cos c - cos d=2sin (c+d/2).sin (d-c/2)
Answer» Prove that if cos a=cos b then a=2n+-b. By the formula cos c - cos d=2sin (c+d/2).sin (d-c/2)

Discussion

5695.	If the ratio of the roots of lx2+nx+n=0 is p:q, then
Answer» If the ratio of the roots of $l x^{2} + n x + n = 0$ is p:q, then

Discussion

5696.	10. Vxvx
Answer» 10. Vxvx

Discussion

5697.	Find dydxin the following questions: y=cos−1(2x1+x2), -1<x<1.
Answer» Find $\frac{d y}{d x}$ in the following questions: $y = c o s^{- 1} (\frac{2 x}{1 + x^{2}})$ , -1<x<1.

5697.

Find dydxin the following questions: y=cos−1(2x1+x2), -1<x<1.

Answer»

Find $\frac{d y}{d x}$ in the following questions:

$y = c o s^{- 1} (\frac{2 x}{1 + x^{2}})$ , -1<x<1.

Discussion

5698.	The differential equation corresponding to primitive y=edx is or The elimination of the arbitrary constant m from the equation y=emx gives the differential equation [MP PET 1995, 2000; Pb. CET 2000]
Answer» The differential equation corresponding to primitive $y = e^{d x}$ is or The elimination of the arbitrary constant m from the equation $y = e^{m x}$ gives the differential equation [MP PET 1995, 2000; Pb. CET 2000]

5698.

The differential equation corresponding to primitive y=edx is or The elimination of the arbitrary constant m from the equation y=emx gives the differential equation [MP PET 1995, 2000; Pb. CET 2000]

Answer»

The differential equation corresponding to primitive $y = e^{d x}$ is

or

The elimination of the arbitrary constant m from the equation $y = e^{m x}$ gives the differential equation

[MP PET 1995, 2000; Pb. CET 2000]

Discussion

5699.	Find the equation of a straight line passing through the origin and through the point of intersection of the lines 5x + 7y = 3 and 2x - 3y = 7.
Answer» Find the equation of a straight line passing through the origin and through the point of intersection of the lines 5x + 7y = 3 and 2x - 3y = 7.

Discussion

5700.	Two players toss 4 coins each. The probability that they both obtain the same number of heads is
Answer» Two players toss 4 coins each. The probability that they both obtain the same number of heads is

Discussion

Explore topic-wise InterviewSolutions in .

Show that the general solution of the differential equation is given by ( x + y + 1) = A (1 – x – y – 2 xy ), where A is parameter

For the differential equation dydt+5y=0 with y(0) = 1, the general solution is

ntx= cos inverse (8t-8t+1), y=sin inverse (3t-4t) [0

A helicopter is flying along the curve given by y−x32=7, (x≥0). A soldier positioned at the point (12,7) wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is :

If →a,→b,→c are three vectors such that [→a→b→c]=5, then the value of [→a×→b,→b×→c,→c×→a] is :

if tan(A-B)=1/root3 and tan(A+B)=1 then find A

If (1 + k)x2 – 4x – 1 + k = 0 has real roots tanα and tanβ, then

Let z=x+iy be a complex number. If 1+¯¯¯zz is a real number, then

If Z=5x+3y, subject to 3x+5y≤15,5x+2y≤10,x≥0,y≥0, then Zmax is equal to

If∣∣a sin2 θ+b sin θ cos θ+c cos2 θ−12(a+c)∣∣≤12k, then k2 is equal to

Show that the differential equations (x−y)dydx=x+2y is homogeneous and solve it also. OR Find the differential equations of the family of curves (x−h)2+(y−k)2=r2, where h and k are arbitrary constants.

Integrate: 2xx−1

Each of the circles x2+y2−2x−2y+1=0 and x2+y2+2x−2y+1=0 touches internally a circle of radius 2. The equation of circles touching all the three circles, is

Let [x] denote the greatest integer function of x. If the domain of the function 1[x]2−7[x]+12 is R−[a,b), then the value of a+b is

How many integers satisfy the condition 0≤23[x]≤1 __

If point P(0,1) on the curve y=x+esinx is at the shortest distance from y=mx+c. Then the value of m2 is

Two tangents are drawn to the parabola y2=8x which meets the tangent at vertex at P and Q respectively. If PQ=4 units, then the locus of the point of intersection of the two tangents is

If y = ln (xa+bx)x, then x3d2ydx2 is equal to

1:- 9-sec2A-9tan2A=? 2:-(1+tan theeta +sec theeta)(1+cot theeta- cosec theeta)=? 3:- (Sec A+ tanA)(1-sin A)=?

Find the sum to n terms of a G.P. √7,√21,3√7...

The coordinates of the vertex of a parabola represented by y=ax2+bx+c is, . Take D as discriminant;

If the middle term in the expansion of (p2+2)8 is 1120, then the value of p is

limx→π2acotx−acosxcot x−cos x

15.sin (tam1x), lxl< is equal to2 (B)2 (C)+x2v1+ x2

38.4sinx*sin2x*sin4x=sin3x Find the value if x?

What is e vs x graph for positive hargec at origin.?

If x (|x|&lt;π) and y are the solutions of the equation 12sinx+5cosx=2y2−8y+21, then the value of 12cot(xy2) is

∫sin4x dx is equal to

2a∫0x3√2ax−x2 dx is equal to

Show that the lines x−12=y−23=z−34 and x−45=y−12=z intersect. Also, find their point of intersection.

If f: R → Ris defined by f(x) = x2 − 3x+ 2, find f(f(x)).

ntFind the Bi-quadratic equation with rational coefficients whose one of the root is 2+-3n

Find the value of k, which equation real and equal root : kx2 + kx +1 = 4x2 - x

A die is thrown once.Determine the vertex which contains a right angle in ΔABC, where A(4,-2), B(7,9) and C(7,-2).

3x+8y=-1, 1x-2y=2 x≠0, y≠0

Write the first fiveterms of the sequences whose nth term is

If(A) 0 (B) (C) notdefined (D) 1

∫-π4π4 11+cos 2xdx is equal to(a) 1(b) 2(c) 3(d) 4

Let P be a variable point on the ellipse x2100+y264=1 with foci F1 and F2. If A is the area of triangle PF1F2, then the maximum possible value of A is

The larger of 9950+10050 and 10150 is . . .

Given M=[1123+1223+⋯+1100023] (where [.] denotes greatest integer function), (M - 20) is equal to ___

∫e2x−1e2x+1dx is equal to(where C is constant of integration)

Prove that if cos a=cos b then a=2n+-b. By the formula cos c - cos d=2sin (c+d/2).sin (d-c/2)

If the ratio of the roots of lx2+nx+n=0 is p:q, then

10. Vxvx

Find dydxin the following questions: y=cos−1(2x1+x2), -1&lt;x&lt;1.

The differential equation corresponding to primitive y=edx is or The elimination of the arbitrary constant m from the equation y=emx gives the differential equation [MP PET 1995, 2000; Pb. CET 2000]

Find the equation of a straight line passing through the origin and through the point of intersection of the lines 5x + 7y = 3 and 2x - 3y = 7.

Two players toss 4 coins each. The probability that they both obtain the same number of heads is

38.4sinxsin2xsin4x=sin3x Find the value if x?

If x (|x|<π) and y are the solutions of the equation 12sinx+5cosx=2y2−8y+21, then the value of 12cot(xy2) is

Find dydxin the following questions: y=cos−1(2x1+x2), -1<x<1.