28278 + Interview Questions in Maths in Class 12 Page 102 InterviewSolution

5051.	Equation of the line through the point (2,-1,1) and the intersection of the lines 2x-y-4=0=y+2z,x+3z-4=0=2x+5z-8 is
Answer» `x+y+z=2,x+2y=0` `x+y+z=2,x+2z=4` `x+2y+z=1,x+2z=4` `x+2y+z=1,x+2y=0` ANSWER :B

Discussion

5052.	The set of allvalues of x satisfying {x}=x[xx] " where " [xx] represents greatest integer function {xx} represents fractional part of x
Answer» 0 `-(1)/(2)` `-1 LT x lt 1` Both A and B Solution :`{x}=[x]x` `{x}=x(x-{x})=x^(2)-x{x}` `implies {x}=(x^(2))/(1+x)` Now `0 le {x} lt 1` `implies (x^(2))/(1+x) ge 0` `implies 1+x gt 0` `implies x gt -1` `(x^(2))/(1+x) lt 1` `implies x^(2) lt 1 +x` `implies x^(2) -x-1 lt 0` `implies (1-sqrt(5))/(2) lt x lt (1+sqrt(5))/(2)` Given equationagain can be written as `x-[x]=x[x]` `(x)/(1+x)=[x]=LAMBDA` `x=(lambda)/(1-lambda)` `(1-sqrt(5))/(2) lt (lambda)/(1-lambda) lt (1+sqrt(5))/(2)` ` implies (-1-sqrt(5))/(2) lt lambda lt (sqrt(5)-1)/(2)` `lambda=0 implies x=0` `lambda= -1 implies x=(-1)/(2)`

Discussion

5053.	Every person in the room has shake hands with a certain number of other persons. The probability that count of the number of people who have shaken hands an odd number of times is even is .
Answer» 0 `1/2` `2/3` 1 Answer :D

Discussion

5054.	Howmanywaysarethereto arrangethelettersin thewordGARDEN withthevowelsinalphabeticalorder?
Answer» 360 240 120 480 Answer :A

Discussion

5055.	Matchthe following
Answer» `{:(I,II,III,IV),(d,a,E,C):}` `{:(I,II,III,IV),(d,a,c,b):}` `{:(I,II,III,IV),(d,c,a,e):}` `{:(I,II,III,IV),(a,d,b,c):}` Answer :A

Discussion

5056.	If a,b,c are in G.P and a^(1/x) = b^(1/y) = c^(1/z), prove that x,y,z are in A.P.
Answer» ANSWER :HENCE, `X, y` and `Z` are in A.P.

Discussion

5057.	Let A = {x : x is a multiple of 3} and B = {x/x is a multiplc of 5}. Then A nn B is given by
Answer» `{3, 6, 9, ….}` `{5, 10, 15, 20, ….}` `{15, 30, 45, ….}` `{30, 60, 90, ….}` Answer :C

Discussion

5058.	If the position vector of three points are a - 2b + 3c, 2a + 3b - 4c, - 7b + 10 c, then the three points are
Answer» collinear non-coplanar non-collinear None of these Solution : Let P, Q, R be the GIVEN points. Then, we have QR =- 2PQ HENCE, P Q and R are collinear.

Discussion

5059.	A,B,C are aiming to shoot a balloon. A will succeed 4 times out of 5 attempts. The change of B to shoot the balloon is 3 out of 4 and that of C is 2 out of 3. If three aim the balloon simultaneously, then find the probability that atleast two of them hit the balloon.
Answer» ANSWER :`(5)/(6)`

Discussion

5060.	Evaluate the integerals.int sqrt(e ^(x) -4) dx on [log _(e) 4, oo].
Answer» `TAN^(-1)((sqrt(E^(x)-4))/(2))+sqrt(e^(x)-4)+c` `2sqrt(e^(x)-4)-4 Tan^(-1) ((sqrt(e^(x)-4))/(2))+c` `2sqrt(e^(x)-4)-4 Cos^(-1) ((sqrt(e^(x)-4))/(2))+c` `sqrt(e^(x)-4)-4 Tan^(-1) (sqrt(e^(x)-4))+c` ANSWER :B

Discussion

5061.	The partial fractions of (x^(2))/((x^(2)+a^(2))(x^(2)+b^(2))) are
Answer» `(1)/(a^(2)+B^(2))[(a^(2))/(x^(2)+a^(2))-(b^(2))/(x^(2)+b^(2))]` `(1)/(b^(2)-a^(2))[(a^(2))/(x^(2)+a^(2))-(b^(2))/(x^(2)+b^(2))]` `(1)/(a^(2)-b^(2))[(a^(2))/(x^(2)+a^(2))-(b^(2))/(x^(2)+b^(2))]` `(1)/(a^(2)-b^(2))[(1)/(x^(2)+a^(2))-(1)/(x^(2)+b^(2))]` ANSWER :C

Discussion

5062.	Find (dy)/(dx) of the functions given in Exercises 12 to 15. x^(y)=1.
Answer»

Discussion

5063.	If the areabounded by y=x, y=sinx and x=(pi)/(2) is ((pi^(2))/(k)-1) sq. units then the value of k is equal to
Answer» 2 3 6 8 Answer :D

Discussion

5064.	Integrate the following functions : int(sinx)/(sqrt((3+4cosx)^(2)))dx
Answer» ANSWER :`(1)/(4(x+4cosx))+C`

Discussion

5065.	ABC is a triangular park with all sides equal. If a pillar at A subtends an angle of 45^(@) at C, the angle of elevation of the pillar at D, the middle point of BC is
Answer» `tan^(-1) (sqrt(3)//2)` `tan^(-1) (2//sqrt(3))` `COT^(-1) sqrt(3)` `tan^(-1) sqrt(3)` ANSWER :B

Discussion

5066.	If f(x)={([x]+[-x],x!=2),(lamda, x=2):}is continuous at x=2 then lamda= (where [.] denotes greatest integer)
Answer» `-1` 0 1 2 Solution :`lamda=lim_(xto2)[X]+[-x]` `[x]-[x]-1` `=-1`

Discussion

5067.	If alpha, beta , gamma are the roots of x^(3) + 2x^(2) - 5x + 3 = 0 then the equation whose roots alpha - (1)/(beta gamma), beta - (1)/(gamma alpha), gamma - (1)/(alpha beta ) is
Answer» `9X^(3) + 8X^(2) + 80x + 64 = 0` `9x^(3) + 8x^(2) + 80x - 64 = 0 ` `9x^(3) + 8x^(2) - 80x + 64 = 0` `9x^(2) - 8x^(2) + 80x - 64 = 0` ANSWER :3

Discussion

5068.	An electron orbiting around the nucleus of an atom
Answer» has a magnetic dipole moment exerts an electric force on the nucleusequal to that on it by the nucleus does produce a magnetic INDUCTION at the nucleus has a net energy inverselyproportional to its distance from the nucleus. Solution :`(mv^(2))/(R)=(kZe^(2))/(r^(2))rArr(mv^(2))/(2)=(kZe^(2))/(2R),M.E.=(mv^(2))/(2)-(Ze^(2))/(r)=(-Ze^(2))/(2r)prop(1)/(r)`

Discussion

5069.	Evalute the following integrals int (2x + 3)/(x^(2) + x + 1) dx
Answer» Answer :LOG `\|X^(2) + x + 1\| + (4)/(SQRT(3)) TAN^(-1) ((2x + 1)/(sqrt(3)) ) +c `

Discussion

5070.	Ifzsatisfiesthe conditionarg(z + i)=(pi)/(4) . Then theminimumvalue of\|z +1 - i\| +\|z - 2 +3 i\| is _______.
Answer» Solution :Locus given by equation `arg(z+i) =(pi)/(3)` is a ray EF as SHOW in the FOLLOWINGFIGURE. COMPLEX number z moves on ray Ef. Now,for any position of complex number P(z) on the ray EF, `\|z-(-1+i)\|+\|z-(2-3i)\|=PA+ PB` Clearly, PA+PB is minimum when POINTP isa point C, where point A,C and B are COLLINER. So, `(PA +PB)_("min") = AB = 5`

Discussion

5071.	The value of (.^(n)C_(0))/(n)+(.^(n)C_(1))/(n+1)+(.^(n)C_(2))/(n+2)+"..."+(.^(n)C_(2))/(2n)
Answer» `UNDERSET(0)OVERSET(1)intx^(n-1)(1-x)^(n)dx` `underset(1)overset(2)intx^(n)(x-1)^(n-1)dx` `underset(1)overset(2)int(1+x)^(n)dx` `underset(0)overset(1)int(1-x)^(n)x^(n-1)dx` SOLUTION :Let, `S = (.^(n)C_(0))/(n)+(.^(n)C_(1))/(n+1)+(.^(n)C_(2))/(n+2)+"...."+(.^(n)C_(n))/(2n)` `= .^(n)C_(0)underset(0)overset(1)intx^(n-1)+.^(n)C_(1)underset(0)overset(1)intx^(n)dx+"....."+.^(n)C_(n)underset(0)overset(1)intx^(2n-1)dx` `=underset(0)overset(1)int[.^(n)C_(0)x^(n-1)+.^(n)C_(1)x^(n)+"..."+.^(n)C_(n)x^(2n-1)]dx` `= underset(0)overset(1)intx^(n-1)(1+x)^(n)dx` `= underset(1)overset(2)intx^(n)(x-1)^(n-1)dx`

Discussion

5072.	Find lambda and mu if (2hati+6hatj+27hatk)xx(hati+lambda hatj+mu hatk)=vec(0).
Answer» ANSWER :`MU=(27)/(2),lambda=3`

Discussion

5073.	Which point on the coordinate plane shown here is indicated by the following coordinates?
Answer» `(2, -1)` `(-1.5, -3)` `(-1, 2)` `(3, 2)` ANSWER :B::C

Discussion

5074.	There are 10 stations on a railway. A train has to stop at three of these stations. The probability that no two of them are consecutive is
Answer» `(2)/(5)` `(7)/(15)` `(1)/(15)` `(1)/(3)` ANSWER :B

Discussion

5075.	If f is integrable on [a, b] then int_(a)^(b) [f(x)-f_("ave") ] dx is equal to
Answer» `F(b) -f(a)` `(1)/(2) (f(b)-(a) )` `f(a) - f(b)` `0` Answer :D

Discussion

5076.	int (1)/(" tan x tan" (60^(@) - x) tan (60^(@) + x) )dx =
Answer» `(1)/(3) `log \| sin 3X \| + C 3 log \| sin 3x\| + C `(1)/(2) ` log\| sin 3x \| + C `(1)/(3)` log \|COS 3x \| +C Answer :A

Discussion

5077.	Let a, b c be given positive numbers, then values of x,y and z in R^(+) which satisfies equations x+y+z=a+b+x and 4xyz=-(a^(2)x+b^(2) +c^(2)z)= abc are respectively.
Answer» `(b+c)/(2), (a+c)/(2),(a+b)/(2)` `a/2, b/2, c/2` `(a+b)/(2), (a+c)/(2), (b+c)/(2)` NONE of these ANSWER :A

Discussion

5078.	Let f (x) = {{:( x "," , x lt 1), ( 2-x \|"," , 1 lt x le 2"then" f (x) is), ( -2 + 3x -x ^(2)",", x gt 2 ):}
Answer» differentiable at `x =1` differentiable at `x =2` differentiable at `x=1 and x =2` NONE of these Answer :B

Discussion

5079.	The sum of infinite terms of the geometric progression (sqrt(2) + 1)/(sqrt(2) - 1),1/(2 - sqrt(2)),1/2....... Is
Answer» `SQRT(2)(sqrt(2) + 1)^(2)` `(sqrt(2) + 1)^(2)` `5sqrt(2)` `3sqrt(2) + sqrt(5)` ANSWER :A

Discussion

5080.	If P(AnnB)=7//10andP(B)=17//20, where P stands for probability then P(A\|B) is equal to
Answer» 1)`7//8` 2)`17//20` 3)`14//17` 4)`1//8` Answer :C

Discussion

5081.	If thereis amultiplerootoforder3 fortheequationx^4- 2x ^3 + 2x -a=0then theotherrootis
Answer» `-1` `0` 1 2 Answer :A

Discussion

5082.	If a_(n) = ( log_(e ) 3)^(n) sum_(k!(n-k)!)^(k^(2))then a_(1) + a_(2) +a_(3) +"...."oo is equal to
Answer» `3 log_(E ) 9` `9 log_(e ) 3` `9 log_(e ) 3 ( log_(e )3+1)` `( log_(e ) 9)^(2)` Answer :C

Discussion

5083.	If the equation of the circle passing throught the points (3,4), (3,2), (1,4) is x^(2)+y^(2)+2ax+2by+c=0 then the ascending order of a,b,c is
Answer» a,B,C b,c,a a,c,b b,a,c Answer :A

Discussion

5084.	If y=sin[sin(log3x)],"find "dy/dx.
Answer» ANSWER :`1/xcos[SIN(LOG3X)]COS(log3x),XGT0`

Discussion

5085.	Let f:[-1,1] to R_(f) be a function defined by f(x)=(x)/(x+2). Show that f is invertible.
Answer» ANSWER :`

Discussion

5086.	For any non-zero real value of m, the equation of the parabola to which the line mx - y + 10 + m^(2) = 0 is a tangent, is
Answer» `X^(2) = y - 10` `y^(2)= 4(x-2)` `x^(2) = -4(y-10)` `x^(2) = -4Y` Answer :C

Discussion

5087.	Find the shortest distance and the equation of the shortest distance between the following two lines : vec(r) = (-hati + hatj + 9 hatk) + lambda (2 hati + hatj - 3 hatk) and vec(r) = (3 hati - 15 hatj + 9 hatk) + mu (2 hati - 7 hatj + 5 hatk).
Answer» Answer :`4 SQRT(3) ; VEC(r) = ( 3 HATI + 3 hatj - 3 hatk) - mu (4 hati + 4 hatj + 4 hatk)`

Discussion

5088.	Find the second order derivatives of the following functions: e^(ax)
Answer» ANSWER :`a^(2) E^(AX)`

Discussion

5089.	The points (5,-4,2),(4,-3,1),(7,-6,4),(8,-7,5) are vertices of a:
Answer» Rhombus Square Parallelogram Rectangle Answer :C

Discussion

5090.	Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E cup F) = 0.7, find P(E\|F) and P(F\|E).
Answer» <P> ANSWER :`P(E\|F)=(2)/(3), P(F\|E)=(1)/(3)`

Discussion

5091.	IFalpha, betaare therootsofx^2-p(x+1)-c=0 then(alpha ^2 + 2 alpha+1)/(alpha^2+2alpha +c)+(beta ^2+2beta +1)/(beta ^2 + 2 beta +c)=
Answer» 3 2 1 0 Answer :C

Discussion

5092.	int e^(x). " sin x( sin x+ 2 cos x) dx "
Answer» ANSWER :`E^(X) . SIN^(2) x+c`

Discussion

5093.	The value of (4cos^2 75^@-3) sin15^@ is :-
Answer» `2-sqrt3` `2+sqrt3` `1/SQRT2` `(-1)/sqrt2`

Discussion

5094.	Oneof theprincipalsolutionsofsqrt(3) secx=-2isequalto
Answer» `(2 pi )/(3)` `pi/6` `(5PI)/(6)` `pi/3` ANSWER :C

Discussion

5095.	The position vectors of vertices of a Delta ABC are 4hat(i)-2hat(j), hat(i)-3hat(k) and -hat(i)+5hat(j)+hat(k) respectively, then angle ABC is equal to
Answer» `pi/6` `pi/4` `pi/3` `pi/2` SOLUTION :Here, `AB=-3 hati+6hatj-3hatk` `BC = -2 hati+hatj+4hatk` and `AB*BC=6+6-12=0` `RARR""/_ABC=pi/2`

Discussion

5096.	If Delta ABC is such that angleA = 90^(@), angleB != angleC, then (b^(2) + c^(2))/(b^(2) - c^(2))sin(B-C) =
Answer» `1//3` `1//2` 1 `3//2` ANSWER :C

Discussion

5097.	The unit vector in the direction 6hati-2hatj+3hatk is ………….
Answer» `(6)/(7)HATI+(2)/(7)HATJ+(3)/(7)hatk` `(6)/(7)hati-(2)/(7)hatj+(3)/(7)hatk` `(-6)/(7)hati+(2)/(7)hatj+(3)/(7)hatk` `(6)/(7)hati+(2)/(7)hatj-(3)/(7)hatk` Answer :B

Discussion

5098.	Let each side of smallest square of chess board in one unit in length. Find the sum of area of all possible squares whose side parallel to side of chess board.
Answer» ANSWER :1968

Discussion

5099.	If the circle x^(2)+y^(2)+2ax+4ay-3a^(2)=0 and x^(2)+y^(2)-8ax-6ay+7a^(2)=0 touch each other externally, the point of contact is
Answer» (a,a) (0,a) (a,0) (-a,0) ANSWER :C

Discussion

5100.	Write the vale of intsqrt(1-cos2x)dx
Answer» SOLUTION :`intsqrt(1-cos2x)DX=intsqrt(2sin^2x)dx`=sqrt2intsinx.dx=-sqrt2cosx+c`

Discussion

Explore topic-wise InterviewSolutions in .

Equation of the line through the point (2,-1,1) and the intersection of the lines 2x-y-4=0=y+2z,x+3z-4=0=2x+5z-8 is

The set of allvalues of x satisfying {x}=x[xx] " where " [xx] represents greatest integer function {xx} represents fractional part of x

Every person in the room has shake hands with a certain number of other persons. The probability that count of the number of people who have shaken hands an odd number of times is even is .

Howmanywaysarethereto arrangethelettersin thewordGARDEN withthevowelsinalphabeticalorder?

Matchthe following

If a,b,c are in G.P and a^(1/x) = b^(1/y) = c^(1/z), prove that x,y,z are in A.P.

Let A = {x : x is a multiple of 3} and B = {x/x is a multiplc of 5}. Then A nn B is given by

If the position vector of three points are a - 2b + 3c, 2a + 3b - 4c, - 7b + 10 c, then the three points are

A,B,C are aiming to shoot a balloon. A will succeed 4 times out of 5 attempts. The change of B to shoot the balloon is 3 out of 4 and that of C is 2 out of 3. If three aim the balloon simultaneously, then find the probability that atleast two of them hit the balloon.

Evaluate the integerals.int sqrt(e ^(x) -4) dx on [log _(e) 4, oo].

The partial fractions of (x^(2))/((x^(2)+a^(2))(x^(2)+b^(2))) are

Find (dy)/(dx) of the functions given in Exercises 12 to 15. x^(y)=1.

If the areabounded by y=x, y=sinx and x=(pi)/(2) is ((pi^(2))/(k)-1) sq. units then the value of k is equal to

Integrate the following functions : int(sinx)/(sqrt((3+4cosx)^(2)))dx

ABC is a triangular park with all sides equal. If a pillar at A subtends an angle of 45^(@) at C, the angle of elevation of the pillar at D, the middle point of BC is

If f(x)={([x]+[-x],x!=2),(lamda, x=2):}is continuous at x=2 then lamda= (where [.] denotes greatest integer)

If alpha, beta , gamma are the roots of x^(3) + 2x^(2) - 5x + 3 = 0 then the equation whose roots alpha - (1)/(beta gamma), beta - (1)/(gamma alpha), gamma - (1)/(alpha beta ) is

An electron orbiting around the nucleus of an atom

Evalute the following integrals int (2x + 3)/(x^(2) + x + 1) dx

Ifzsatisfiesthe conditionarg(z + i)=(pi)/(4) . Then theminimumvalue of|z +1 - i| +|z - 2 +3 i| is _______.

The value of (.^(n)C_(0))/(n)+(.^(n)C_(1))/(n+1)+(.^(n)C_(2))/(n+2)+"..."+(.^(n)C_(2))/(2n)

Find lambda and mu if (2hati+6hatj+27hatk)xx(hati+lambda hatj+mu hatk)=vec(0).

Which point on the coordinate plane shown here is indicated by the following coordinates?

There are 10 stations on a railway. A train has to stop at three of these stations. The probability that no two of them are consecutive is

If f is integrable on [a, b] then int_(a)^(b) [f(x)-f_("ave") ] dx is equal to

int (1)/(" tan x tan" (60^(@) - x) tan (60^(@) + x) )dx =

Let a, b c be given positive numbers, then values of x,y and z in R^(+) which satisfies equations x+y+z=a+b+x and 4xyz=-(a^(2)x+b^(2) +c^(2)z)= abc are respectively.

Let f (x) = {{:( x "," , x lt 1), ( 2-x |"," , 1 lt x le 2"then" f (x) is), ( -2 + 3x -x ^(2)",", x gt 2 ):}

The sum of infinite terms of the geometric progression (sqrt(2) + 1)/(sqrt(2) - 1),1/(2 - sqrt(2)),1/2....... Is

If P(AnnB)=7//10andP(B)=17//20, where P stands for probability then P(A|B) is equal to

If thereis amultiplerootoforder3 fortheequationx^4- 2x ^3 + 2x -a=0then theotherrootis

If a_(n) = ( log_(e ) 3)^(n) sum_(k!(n-k)!)^(k^(2))then a_(1) + a_(2) +a_(3) +"...."oo is equal to

If the equation of the circle passing throught the points (3,4), (3,2), (1,4) is x^(2)+y^(2)+2ax+2by+c=0 then the ascending order of a,b,c is

If y=sin[sin(log3x)],"find "dy/dx.

Let f:[-1,1] to R_(f) be a function defined by f(x)=(x)/(x+2). Show that f is invertible.

For any non-zero real value of m, the equation of the parabola to which the line mx - y + 10 + m^(2) = 0 is a tangent, is

Find the shortest distance and the equation of the shortest distance between the following two lines : vec(r) = (-hati + hatj + 9 hatk) + lambda (2 hati + hatj - 3 hatk) and vec(r) = (3 hati - 15 hatj + 9 hatk) + mu (2 hati - 7 hatj + 5 hatk).

Find the second order derivatives of the following functions: e^(ax)

The points (5,-4,2),(4,-3,1),(7,-6,4),(8,-7,5) are vertices of a:

Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and﻿ P(E cup F) = 0.7, find P(E|F) and P(F|E)﻿.

IFalpha, betaare therootsofx^2-p(x+1)-c=0 then(alpha ^2 + 2 alpha+1)/(alpha^2+2alpha +c)+(beta ^2+2beta +1)/(beta ^2 + 2 beta +c)=

int e^(x). " sin x( sin x+ 2 cos x) dx "

The value of (4cos^2 75^@-3) sin15^@ is :-

Oneof theprincipalsolutionsofsqrt(3) secx=-2isequalto

The position vectors of vertices of a Delta ABC are 4hat(i)-2hat(j), hat(i)-3hat(k) and -hat(i)+5hat(j)+hat(k) respectively, then angle ABC is equal to

If Delta ABC is such that angleA = 90^(@), angleB != angleC, then (b^(2) + c^(2))/(b^(2) - c^(2))sin(B-C) =

The unit vector in the direction 6hati-2hatj+3hatk is ………….

Let each side of smallest square of chess board in one unit in length. Find the sum of area of all possible squares whose side parallel to side of chess board.

If the circle x^(2)+y^(2)+2ax+4ay-3a^(2)=0 and x^(2)+y^(2)-8ax-6ay+7a^(2)=0 touch each other externally, the point of contact is

Write the vale of intsqrt(1-cos2x)dx

Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E cup F) = 0.7, find P(E|F) and P(F|E).