28278 + Interview Questions in Maths in Class 12 Page 234 InterviewSolution

11651.	inte^(ax)sinbxdx
Answer» Answer :`(e^(ax))/(SQRT(a^(2)+b^(2)))sin(bx-tan^(-1)""(b)/(a))+C`

Discussion

11652.	Find (d^(2)y)/(dx^(2)), if y= x^(3) + tan x
Answer» ANSWER :`6X + 2 SEC^(2)x.tanx`

Discussion

11653.	Let f(n) =[1/3 + (3n)/100]n,whre [x] denotes the greatest integer less than or equal to x. Then sum_(n=1)^(56) f(n) is equal to
Answer» 689 1399 1287 56 Answer :B

Discussion

11654.	If two vectors are parallel vectors then :
Answer» They MUST be in same direction. They must have EQUAL magnitude They must be in EXACTLY OPPOSITE direction. none of above

Discussion

11655.	int (e^(3x) + e^(x))/(e^(4x) -e^(2x)+1)dx =
Answer» `(1)/(4) LOG (e^(4X)- e^(2x) + 1) + C ` ` tan^(-1) (e^(x) - e^(-x)) + C` `tan^(-1) (e^(x) - e^(-x))+ C` `tan^(-1) (e^(-x) - e^(x))+ C` Answer :B

Discussion

11656.	If alpha,beta,gamma and delta are the equation x^(4)-1 = 0, then the value of(aalpha+b beta+cgamma+ddelta)/(agamma+bdelta+calpha+dbeta)+(agamma+bdelta+calpha+dbeta)/(aalpha+b beta+cgamma+ddelta), is
Answer» `3beta` 0 `2gamma` `3alpha` ANSWER :d

Discussion

11657.	Match th statements given in Column-I with theintervals//union of intervalsgivenin column -II
Answer» ANSWER :A::B::C::D

Discussion

11658.	Let ABC is be a fixed triangle and P be veriable point in the plane of triangle ABC. Suppose a,b,c are lengths of sides BC,CA,AB opposite to angles A,B,C, respectively. If a(PA)^(2) +b(PB)^(2)+c(PC)^(2) is minimum, then point P with respect to DeltaABC is
Answer» centroid circumcentre orthocenter INCENTRE Solution :Let `A(x_(1),y_(1)), B(x_(2),y_(2)),C(x_(3),y_(3))` and `P(h,k)` be the POINTA. Now, `aAP^(2)+b BP^(2)+c CP^(2)` `=a[(h-x_(1))^(2)+(k-y_(1))^(2)] +b[(h-x_(2))^(2)+(k-y_(2))^(2)] +c [(h-x_(2))^(2)+(k-y_(3))^(2)]` `= [h^(2)(a+b+c) -2H(ax_(1)+bx_(2)+cx_(3))+(ax_(1)^(2)+bx_(2)^(2)+cx_(3)^(2))]` `+[k^(2)(a+b+c)-2k(ay_(1)+by_(2)+cy_(3))+(ay_(1)^(2)+by_(2)^(2)+cy_(3)^(2))]` which is minimum when ` = (ax_(1)+bx_(2)+cx_(3))/(a+b+c), k =(ay_(1)+by_(2)+cy_(3))/(a+b+c)` So, P is incentre of `DeltaABC`.

Discussion

11659.	If alpah,beta are the roots of x^2-x+1=0 then the quadratic equation whose roots are alpha^(2015), beta^(2015) is
Answer» `x^2-x+1=0` `x^2+x+1=0` `x^2+x-1=0` `x^2-x-1=0` ANSWER :A

Discussion

11660.	If int(cos^(4)x)/(sin^(4)x)dx=Kcotx+Msin2x+L(x)/(2) + C then
Answer» L = 1 K = - 2 M = -14 none of these Answer :A

Discussion

11661.	Using matrices, solve the following system of linear equations: x-y+z=4,2x+y-3z=0,x+y+z=2
Answer» ANSWER :x=2,y=-1,z=1

Discussion

11662.	If a_(1), a_(2) …… a_(n) = n a_(n - 1), for all positive integer n gt= 2, then a_(5) is equal to
Answer» a. 125 B. 120 c. 100 d. 24 Answer :B

Discussion

11663.	""^(2n)C_(2)-2""^(n)C_(2)=
Answer» `N^(2)` `(n-1)^(2)` `(n+1)^(2)` `2N^(2)` ANSWER :A

Discussion

11664.	If the lines ax + ky + 10 = 0, bx + (k + 1) y + 10 = 0 and cx + (k+2)y + 10 = 0 are concurrent, then
Answer» a, B, C are in G.P. a, b, c are in H.P a, b, c are in A.P `(a + b)^(2) = c` ANSWER :C

Discussion

11665.	x(dy)/(dx) - y + x sin ((y)/(x)) = 0
Answer» ANSWER :`X [ 1 - cos ((y)/(x))] = C sin ((y)/(x))`

Discussion

11666.	Identify the type of conic and find centre, foci, vertices, and directices of each of the following: 9x^(2)-y^(2)-36x-6y+18=0
Answer» ANSWER :`SQRT(10)`

Discussion

11667.	If f(x)= (x^(2))/(x+a), then f''(a) is equal to
Answer» 4A `(1)/(8A)` `(1)/(4a)` 8a ANSWER :C

Discussion

11668.	A=[{:(1,2,2),(2,1,2),(2,2,1):}], then A^(3) - 4A^(2) -6A is equal to
Answer» 0 A `-A` I Answer :C

Discussion

11669.	The value of f(0), so that f(x)= (sqrt(a^(2)-ax + x^(2))-sqrt(a^(2) + ax + x^(2)))/(sqrt(a +x)- sqrt(a-x)) becomes continuous for all x, is given by ………..
Answer» `a sqrta` `sqrta` `-sqrta` `-a sqrta` ANSWER :C

Discussion

11670.	If A =[{:(1,2,3),(3,-2,1),(4,2,1):}] then show that A^(3)-23A - 40 I =0
Answer» ANSWER :`A^(3)-23 A - 40 I=0`

Discussion

11671.	If x-15=\|-5\| , then x = ?
Answer» `-20` `-10` `2/3` 20 Answer :D

Discussion

11672.	Letvec(a),vec(b),vec(c )be unit such thathat(a) + hat(b ) + hat(c ) = vec(alpha)andhat(a).hat(b)=hat(b).hat(c ) . hat(a) = 1/2\| (hat(a) xx hat(b)) xx hat(c)\|is equal to -
Answer» 0 `1/2` 1 2 Answer :B

Discussion

11673.	If [sinx]+[x/(2pi)]+[(2x)/(5pi)]=(9x)/(10pi), where [*] denotes the greatest integer function, the number of solutions in the interval (30,40) is ………… .
Answer» ANSWER :1

Discussion

11674.	Number of straight lines represented by x^5+y^5 = 0is ……
Answer» ANSWER :1

Discussion

11675.	The value (cos (A+B+C)+cos (A-B-C))/(2 cos (B+C))
Answer» `COS A` `SIN A` `2 COSA` `2 sin A` ANSWER :A

Discussion

11676.	There are 10 intermediate stations on a railway line between two particular stations. The number of ways that a train can be made to stop at 3 of these intermediate stations so that no two of these halting stations are consecutive, is
Answer» 56 126 20 120 Answer :A

Discussion

11677.	If P (A) = 0.4,P (B \| A)= 0.3 and P (B^c \| A^c)=0.2. find P(A^c)
Answer» <P> SOLUTION :`P(A^c)=1-P(A)=1-0.4=0.6=6/10=3/5`

Discussion

11678.	Ifint(sintheta-costheta)/((sintheta+costheta)sqrt(sinthetacostheta+sin^(2)thetacos^(2)theta))d theta = "cosec"^(_1)(f(theta))+C , then
Answer» `F(THETA)=sin2theta+1` `f(theta)=1-sin 2 theta` `f(theta)=sin2theta-1` NONE of these Answer :a

Discussion

11679.	Le A = {1,2, 3,4, 5}. Let {1,2,3} and {4, 5} be two equivalence classes of a relation R on A. The number of elements in R is …………….
Answer» ANSWER :13

Discussion

11680.	Write the negation of the following statement: i. p: for every positive real number x, the number x-1 is also positive. ii. q: All cats scratch. r: For every real number x, either x>1 or x
Answer» ANSWER :that 0`LT` X`lt` L.

Discussion

11681.	Consider f(x)=tan^(-1)(2/x),g(x)=sin^(-1)(2/(sqrt(4+x^(2)))) and h(x)=tan(cos^(-1)(sin)), then show that (h(f(x))+h(g(x))={(0,xlt0),(x,xgt0):}
Answer» SOLUTION :NA

Discussion

11682.	The minimum distance of origin from the plane passing through the point with position vector P and perpendicular to the lineL_(2) is:
Answer» `SQRT14` `(7)/(sqrt14)` `(11)/(sqrt14)` NONE of these Answer :C

Discussion

11683.	Let A=[{:(7,5), (4, 8):}], B=[{:(4, 3), (7, 5):}] " and "C=[{:(-5, 3), (7, -4):}] IF Tr(S) denotes the trace of a square matrix S then sum_(k=0)^(infty)1/(3^(k))Tr{A(BC)^(k)}=
Answer» `45/2` 36 `81/2` 9 Answer :A

Discussion

11684.	Evaluate{:\|( cos alpha cos beta , cos alpha sin beta , -sin alpha ),( -sin beta , cos beta, 0),( sin alpha cos beta, sin alpha sin beta, cos alpha ) \|:} =0
Answer» ANSWER :`1`

Discussion

11685.	If x+y+z=3,x+2y+3z,x+4y+9z=6," then: "(y,z)equiv
Answer» `(-1,0)` `(1,0)` `(1,-1)` `(-1,1)` ANSWER :B

Discussion

11686.	If tan^(2)theta, sin^(2)theta are the roots of ax^(2)+bx+c=0 then b^(2)- c^(2) =
Answer» 4AC `a^(2)` `4BC` 4ab Answer :A

Discussion

11687.	If f(x) = (x+2)/(2x+3). Then int(f(x)/(x^(2)))^(1//2)dxis equal to (1)/(sqrt(2))g((1+sqrt2f(x))/(1-sqrt(2f(x))))-sqrt((2)/(3))h((sqrt(3f(x))+sqrt(2))/(sqrt(3f(x)-sqrt(2)))) +C where
Answer» `g(x) = tan^(-1)x, h(x) = LOG \|X\|` `g(x) - log\|x\|, h(x) = tan ^(-1)x` `g(x) = h(x) = tan^(-1)x` `g(x) = log\|x\|,h(x) = log \|x\|` ANSWER :D

Discussion

11688.	IF3x^2 +8xy+5y^2 +14 x+22 y+8is resolvableintotwolinearfactorsthen thefactorsare
Answer» `(2X + 3y+4 )(3X +5Y +2)` `(3y+x+9)(y-3x -2)` `(2x+3y-4)(2x -y-2)` `(x-y+4 ), (x-2y +5)` Answer :A

Discussion

11689.	The orthocenter of triangle whose vertices are A(a,0,0),B(0,b,0) and C(0,0,c) is (k/a,k/b,k/c) then k is equal to
Answer» `(1/a^(2)+1/b^(2)+1/c^(2))^(-1)` `(1/a+1/b+1/c)^(-1)` `(1/a^(2)+1/b^(2)+1/c^(2))` `(1/a+1/b+1/c)` Solution :Let `O(alpha,BETA,GAMMA)` be orthocenter. Now, `AO BOT BC` Similarly, `a(alpha)=b(beta)=cgamma=k` `THEREFORE alpha=k/a, beta=k/b,gamma=k/c` The plane `x/a+y/b+z/c=1` contains `(alpha,beta,gamma)` `therefore k(1/a^(2)+1/b^(2)+1/c^(2))=1`

Discussion

11690.	If the line y = x touches the curve y=x^(2)+bx+c at (1, 1), then …………
Answer» B = 1, C = 2 `b=-1, c=1` b = 1, c = 1 b = 0, c = 1 Answer :B

Discussion

11691.	Given that lim_(n to oo)sum_(r=1)^(n)(log(n^(2)+r^(2))-2logn)/(n)=log2+(pi)/(2)-2, then lim_(n to oo) (1)/(n^(2m))[(n^(2)+1^(2))^(m)(n^(2)+r^(2))^(m)......(n^(2))^(m)]^(1//n) is equal to
Answer» `2^(m)E^(m((PI)/(2)-2))` `2^(m)e^(m(2-(pi)/(2)))` `e^m((pi)/(2)-2)` `e^(2M((pi)/(2)-2))` ANSWER :A

Discussion

11692.	Let P(x) =x^(2)+ 1/2x + b and Q(x) = x^(2) + cx + dbe two polynomials with real coefficients such that P(x) Q(x) = Q(P(x)) for all real x. Find all the real roots of P(Q(x)) =0.
Answer» ANSWER :`1/2,-1`

Discussion

11693.	(i) If a, b, c, are in A.P. then show that ax….+….by +c=0 passes through a fixed point. Find then fixed point. (ii)If 9a^(2)+16b^(2)-24ab-25c^(2)=0, then the family of straight lines ax+by+0 is concurrent at the point whose co-ordinates are given by "________" (iii) If 3a+4b-5c=0, then the family of straight lines ax+by+c=0 passes through a fixed point. Find the coordinates of the point.
Answer» Solution :(i) a,b,c are in A.P. `impliesa-2b+c=0` Comparing the given equation `ax+by+c=0` with `a-2b+c=0` we find that the straight line `ax+by6+c=0` passes through the fixed point `(1,-2).` (ii) We have, `9a^(2)+16b^(2)-24ab-25c^(2)=0` `implies(3a-4b)^(2)-(5c)^(2)=0` `implies(3a-4b+5c)(3a-4b-5c)=0` `implies((3)/(5)a-(4)/(5)b+c)((-3)/(5)a+(4)/(5)b+c)=0` `implies` The FAMILY of lines `ax+by+c=0` is either concurrent at `((3)/(5),(-4)/(5))or (-(3)/(5),(4)/(5))` (III) We have, `3a+4b-5c=0` `implies-(3)/(5)a-(4)/(5)b+c=0` `implies` The given family of lines `ax+by+x=0` passes through the fixed point `((-3)/(5),(-4)/(5)).`

Discussion

11694.	A box contains tickets numbered from 1 to 20. If 3 tickets are drawn one by one with replacement then the probability of getting prime number exactly 2 times is
Answer» `(36)/(125)` `(12)/(125)` `(1)/(125)` `(4)/(125)` ANSWER :A

Discussion

11695.	Show that the relation R is R defined as R = {(a,b) : a le b} is reflexive and transitive but not symmetric.
Answer» SOLUTION :N/A

Discussion

11696.	Find the order and degree (if defined) of the following differential equations. ((d^2y)/(dx^2))^2 + cos ((dy)/(dx)) = 0
Answer» Solution :The HIGHEST ORDER derivative in the DIFFERENTIAL EQUATION is `(d^2y)/(dx^2)` . `therefore` The order of the differential equation = 2. The DEGREE of the differential equation is not defined.

Discussion

11697.	For positive l, m and n, if the planes x = my + mz, y =1z+ nx, z=mx+ 1y intersect in a straight, line, then: 1, m and n satisfy the equation:
Answer» `1 ^(2) + m^(2) + N^(2) =2` ` I ^(2) + m^(2) + n^(2) + 2ln =1` `I ^(2) + m^(2) +n^(2) =1` NONE of these ANSWER :B

Discussion

11698.	Find the number of even proper divisors of 2^3""3^2""5^3
Answer» ANSWER :35

Discussion

11699.	Sum of the series sum_(k=0)^(n)""^(n)C_(k)(-1)^(k)1/(a_(k)) where a_(k)=sum_(i=0)^(k)""^(k)C_(i)b_(i) where b_(i)=sum_(j=0)^(i)""^(i)C_(j)((-2)/3)^(j) is
Answer» `1/(2^(N))` `1/(3^(n))` `1/(4^(n))` `(3/4)^(n)` ANSWER :C

Discussion

11700.	Let f (x) =2-\|x-3\| , 1 le x le 5 andfor rest of the values f (x) can be obtained by unsing the relation f (5x)=alpha f (x) AA x in R.The vlaue of f (2007), taking alpha =5, is :
Answer» 1118 2007 1050 132 Answer :A

Discussion

Explore topic-wise InterviewSolutions in .

inte^(ax)sinbxdx

Find (d^(2)y)/(dx^(2)), if y= x^(3) + tan x

Let f(n) =[1/3 + (3n)/100]n,whre [x] denotes the greatest integer less than or equal to x. Then sum_(n=1)^(56) f(n) is equal to

If two vectors are parallel vectors then :

int (e^(3x) + e^(x))/(e^(4x) -e^(2x)+1)dx =

If alpha,beta,gamma and delta are the equation x^(4)-1 = 0, then the value of(aalpha+b beta+cgamma+ddelta)/(agamma+bdelta+calpha+dbeta)+(agamma+bdelta+calpha+dbeta)/(aalpha+b beta+cgamma+ddelta), is

Match th statements given in Column-I with theintervals//union of intervalsgivenin column -II

Let ABC is be a fixed triangle and P be veriable point in the plane of triangle ABC. Suppose a,b,c are lengths of sides BC,CA,AB opposite to angles A,B,C, respectively. If a(PA)^(2) +b(PB)^(2)+c(PC)^(2) is minimum, then point P with respect to DeltaABC is

If alpah,beta are the roots of x^2-x+1=0 then the quadratic equation whose roots are alpha^(2015), beta^(2015) is

If int(cos^(4)x)/(sin^(4)x)dx=Kcotx+Msin2x+L(x)/(2) + C then

Using matrices, solve the following system of linear equations: x-y+z=4,2x+y-3z=0,x+y+z=2

If a_(1), a_(2) …… a_(n) = n a_(n - 1), for all positive integer n gt= 2, then a_(5) is equal to

""^(2n)C_(2)-2""^(n)C_(2)=

If the lines ax + ky + 10 = 0, bx + (k + 1)﻿ y + 10 = 0 and cx + (k+2)y + 10 = 0 are﻿ concurrent, then﻿

x(dy)/(dx) - y + x sin ((y)/(x)) = 0

Identify the type of conic and find centre, foci, vertices, and directices of each of the following: 9x^(2)-y^(2)-36x-6y+18=0

If f(x)= (x^(2))/(x+a), then f''(a) is equal to

A=[{:(1,2,2),(2,1,2),(2,2,1):}], then A^(3) - 4A^(2) -6A is equal to

The value of f(0), so that f(x)= (sqrt(a^(2)-ax + x^(2))-sqrt(a^(2) + ax + x^(2)))/(sqrt(a +x)- sqrt(a-x)) becomes continuous for all x, is given by ………..

If A =[{:(1,2,3),(3,-2,1),(4,2,1):}] then show that A^(3)-23A - 40 I =0

If x-15=|-5| , then x = ?

Letvec(a),vec(b),vec(c )be unit such thathat(a) + hat(b ) + hat(c ) = vec(alpha)andhat(a).hat(b)=hat(b).hat(c ) . hat(a) = 1/2| (hat(a) xx hat(b)) xx hat(c)|is equal to -

If [sinx]+[x/(2pi)]+[(2x)/(5pi)]=(9x)/(10pi), where [*] denotes the greatest integer function, the number of solutions in the interval (30,40) is ………… .

Number of straight lines represented by x^5+y^5 = 0is ……

The value (cos (A+B+C)+cos (A-B-C))/(2 cos (B+C))

There are 10 intermediate stations on a railway line between two particular stations. The number of ways that a train can be made to stop at 3 of these intermediate stations so that no two of these halting stations are consecutive, is

If P (A) = 0.4,P (B | A)= 0.3 and P (B^c | A^c)=0.2. find P(A^c)

Ifint(sintheta-costheta)/((sintheta+costheta)sqrt(sinthetacostheta+sin^(2)thetacos^(2)theta))d theta = "cosec"^(_1)(f(theta))+C , then

Le A = {1,2, 3,4, 5}. Let {1,2,3} and {4, 5} be two equivalence classes of a relation R on A. The number of elements in R is …………….

Write the negation of the following statement: i. p: for every positive real number x, the number x-1 is also positive. ii. q: All cats scratch. r: For every real number x, either x>1 or x

Consider f(x)=tan^(-1)(2/x),g(x)=sin^(-1)(2/(sqrt(4+x^(2)))) and h(x)=tan(cos^(-1)(sin)), then show that (h(f(x))+h(g(x))={(0,xlt0),(x,xgt0):}

The minimum distance of origin from the plane passing through the point with position vector P and perpendicular to the lineL_(2) is:

Let A=[{:(7,5), (4, 8):}], B=[{:(4, 3), (7, 5):}] " and "C=[{:(-5, 3), (7, -4):}] IF Tr(S) denotes the trace of a square matrix S then sum_(k=0)^(infty)1/(3^(k))Tr{A(BC)^(k)}=

Evaluate{:|( cos alpha cos beta , cos alpha sin beta , -sin alpha ),( -sin beta , cos beta, 0),( sin alpha cos beta, sin alpha sin beta, cos alpha ) |:} =0

If x+y+z=3,x+2y+3z,x+4y+9z=6," then: "(y,z)equiv

If tan^(2)theta, sin^(2)theta are the roots of ax^(2)+bx+c=0 then b^(2)- c^(2) =

If f(x) = (x+2)/(2x+3). Then int(f(x)/(x^(2)))^(1//2)dxis equal to (1)/(sqrt(2))g((1+sqrt2f(x))/(1-sqrt(2f(x))))-sqrt((2)/(3))h((sqrt(3f(x))+sqrt(2))/(sqrt(3f(x)-sqrt(2)))) +C where

IF3x^2 +8xy+5y^2 +14 x+22 y+8is resolvableintotwolinearfactorsthen thefactorsare

The orthocenter of triangle whose vertices are A(a,0,0),B(0,b,0) and C(0,0,c) is (k/a,k/b,k/c) then k is equal to

If the line y = x touches the curve y=x^(2)+bx+c at (1, 1), then …………

Given that lim_(n to oo)sum_(r=1)^(n)(log(n^(2)+r^(2))-2logn)/(n)=log2+(pi)/(2)-2, then lim_(n to oo) (1)/(n^(2m))[(n^(2)+1^(2))^(m)(n^(2)+r^(2))^(m)......(n^(2))^(m)]^(1//n) is equal to

Let P(x) =x^(2)+ 1/2x + b and Q(x) = x^(2) + cx + dbe two polynomials with real coefficients such that P(x) Q(x) = Q(P(x)) for all real x. Find all the real roots of P(Q(x)) =0.

A box contains tickets numbered from 1 to 20. If 3 tickets are drawn one by one with replacement then the probability of getting prime number exactly 2 times is

Show that the relation R is R defined as R = {(a,b) : a le b} is reflexive and transitive but not symmetric.

Find the order and degree (if defined) of the following differential equations. ((d^2y)/(dx^2))^2 + cos ((dy)/(dx)) = 0

For positive l, m and n, if the planes x = my + mz, y =1z+ nx, z=mx+ 1y intersect in a straight, line, then: 1, m and n satisfy the equation:

Find the number of even proper divisors of 2^3""3^2""5^3

Sum of the series sum_(k=0)^(n)""^(n)C_(k)(-1)^(k)1/(a_(k)) where a_(k)=sum_(i=0)^(k)""^(k)C_(i)b_(i) where b_(i)=sum_(j=0)^(i)""^(i)C_(j)((-2)/3)^(j) is

Let f (x) =2-|x-3| , 1 le x le 5 andfor rest of the values f (x) can be obtained by unsing the relation f (5x)=alpha f (x) AA x in R.The vlaue of f (2007), taking alpha =5, is :

If the lines ax + ky + 10 = 0, bx + (k + 1) y + 10 = 0 and cx + (k+2)y + 10 = 0 are concurrent, then