28278 + Interview Questions in Maths in Class 12 Page 251 InterviewSolution

12501.	Let p, q and r be the statements: p: Mathura is in U.P. q: Mathura is in lndia. r : p rarrq Contra-positive of r is
Answer» If MATHURA is not in INDIA then Mathura is not in U.P. Mathura is NEITHER in U.P. nor in India Mathura is in India but not in U.P. NONE of these ANSWER :A

Discussion

12502.	int xsqrt(x + 2) dx
Answer» ANSWER :`2/5(x+2)^(5/2)-4/3(x+2)^(3/2)+C`

Discussion

12503.	For theta in (0, (pi)/(2)). Sec h^(-1)(cos theta) =
Answer» `log\|tan((pi)/(6)+(THETA)/(2))\|` `log\|tan((pi)/(3)+(theta)/(2))\|` `log\|tan((pi)/(4)+(theta)/(2))\|` `log\|tan((pi)/(4)-(theta)/(2))\|` ANSWER :C

Discussion

12504.	Ifq_(1), _(2), q_(3)are roots of the equation x^(3)+64=0,then the value of \|(q_(1),q_(2),q_(3)),(q_(2),q_(3), q_(1)),(q_(3),q_(1),q_(2))\| is :-
Answer» 1 4 16 0 Solution :`\|(q_(1),q_(2),q_(3)),(q_(2),q_(3),q_(1)),(q_(3),q_(1),q_(2))\|rarrC_(1)+C_(2)+C_(3)` `=(q_(1)+q_(2)+q_(3))\|(1, q_(2),q_(3)),(1,q_(3),q_(1)),(1,q_(1),q_(2))\|` `=0(because"SUM of ROOTS is zero")`

Discussion

12505.	One of the most important techniques of counting is the principle of exlcusion and inclusion. Let A_(1),A_(2),……………,A_(m) be m sets and n(A_(1)) represents the cardinality of the set A_(1) (the number of elements in the set A_(1)) then according to the principle of exlusion and inclusion. n(A_(1)uuA_(2)uu.........uuA_(m)) =sum_(i=1)^(m)n(A_(1))-sum_(i=j)n(A_(i)nnA_(j))+sum_(iltjltk)n(A_(i)nnA_(j)nnA_(k))-...........+(-1)^(m+1)n(A_(1)nnA_(2)nn........nnA_(m)) In particular if A,B,C are three sets, then n(AuuBuuC)=n(A)+n(B)+n(C)-n(AnnB)-n(BnnC)-n(CnnA)+n(AnnBnnC). Principle of exclusion and inclusion must be applied whenever there is a chance of repeated counting of some of the samples. The number of numbers from 1 to 100, whcih are neither divisible by 3 nor by 5 nor by 7 is67
Answer» 55 45 33 Answer :C

Discussion

12506.	Integrate the following rational functions : int(tantheta+tan^(3)theta)/(1+tan^(3)theta)
Answer» Answer :`-(1)/(3)log\|1+tantheta\|+(1)/(6)log` `\|TAN^(2)theta-tantheta+1\|+(1)/(SQRT3)tan^(-1)((2tantheta-1)/(sqrt3))+c`

Discussion

12507.	One of the most important techniques of counting is the principle of exlcusion and inclusion. Let A_(1),A_(2),……………,A_(m) be m sets and n(A_(1)) represents the cardinality of the set A_(1) (the number of elements in the set A_(1))) then according to the principle of exlusion and inclusion. n(A_(1)uuA_(2)uu.........uuA_(m)) =sum_(i=1)^(m)n(A_(1))-sum_(i=j)n(A_(i)nnA_(j))+sum_(iltjltk)n(A_(i)nnA_(j)nnA_(k))-...........+(-1)^(m+1)n(A_(1)nnA_(2)nn........nnA_(m)) In particular if A,B,C are three sets, then n(AuuBuuC)=n(A)+n(B)+n(C)-n(AnnB)-n(BnnC)-n(CnnA)+n(AnnBnnC). Principle of exclusion and inclusion must be applied whenever there is a chance of repeated counting of some of the samples. A six letters word is formed using the letters of the wrod ALMIGHTY with or without repetition. The number of words that contain exactly three different letters is
Answer» 15600 30240 `.^(8)P_(6)-.^(8)P_(3)` NONE of these Answer :b

Discussion

12508.	One of the most important techniques of counting is the principle of exlcusion and inclusion. Let A_(1),A_(2),……………,A_(m) be m sets and n(A_(1)) represents the cardinality of the set A_(1) (the number of elements in the set A_(1))) then according to the principle of exlusion and inclusion. n(A_(1)uuA_(2)uu.........uuA_(m)) =sum_(i=1)^(m)n(A_(1))-sum_(i=j)n(A_(i)nnA_(j))+sum_(iltjltk)n(A_(i)nnA_(j)nnA_(k))-...........+(-1)^(m+1)n(A_(1)nnA_(2)nn........nnA_(m)) In particular if A,B,C are three sets, then n(AuuBuuC)=n(A)+n(B)+n(C)-n(AnnB)-n(BnnC)-n(CnnA)+n(AnnBnnC). Principle of exclusion and inclusion must be applied whenever there is a chance of repeated counting of some of the samples. The number of natural numbers less than or equal to 2985984, which are neither perfect squares nor perfect cubes is
Answer» 2984124 2984244 2959595 None of these Answer :a

Discussion

12509.	If vec(a),vec(b),vec(c ) be three vector of magnitude sqrt(3), 1, 2, such thatvec(a)xx(vec(a)xx vec(c ))+3vec(b)=vec(0), if theta is theangle between , then cos theta is equal to :
Answer» `3//4` `1//2` `1//4` None of these SOLUTION :We have `VEC(a)xx(vec(a)xx vec(c ))+3vec(B)=vec(0)` `rArr (vec(a).vec(c ))vec(a)-(vec(a)vec(a))vec(c )+3vec(b)=vec(0)` `rArr (2 sqrt(3)cos THETA)vec(a)-3vec(c )+ 3vec(b)=vec(0)` `rArr (2 cos theta)vec(a)-sqrt(3)vec(c )+ sqrt(3)vec(b)=vec(0)` `rArr \|(2 cos theta)vec(a)-sqrt(3)vec(c )\|=\|-sqrt(3)vec(b)\|` `rArr \|2cos theta vec(a)-sqrt(3)vec(c )\|^(2) = \|-sqrt(3)vec(b)\|^(2)` `rArr 4cos^(2)theta\|vec(a)\|^(2)+3\|vec(c )\|^(2)-4sqrt(3)cos theta (vec(a).vec(c ))=3\|vec(b)\|^(2)` `rArr 12cos^(2)theta + 12-4sqrt(3)cos theta xx sqrt(3)xx 2 cos theta = 3` `rArr 12 cos^(2)theta + 9-24 cos^(2)theta = 0` `rArr 12 cos^(2)theta = 9 rArr cos^(2)theta = (9)/(12)` `rArr cos theta = (3)/(4)`

Discussion

12510.	Organic materials present in food helps in providing :-
Answer» Energy Growth of body Repair of tissues All of the above Answer :A

Discussion

12511.	Let A and B be two events such that P(barAuuB)=(1)/(6) , P(AnnB)=(1)/(4) and P(barA)=(1)/(4) where barA stands for complement of event A. Then events A and B are
Answer» EQUALLY likely and mutually EXCLUSIVE equally likely but not INDEPENDENT independent but not equally likely mutually exclusive and independent ANSWER :C

Discussion

12512.	If n is an even integer, then: C_(0)^(2)-C_(1)^(2)+C_(2)^(2)-C_(3)^(2)+ . . .+(-1)^(n)C_(n)^(2) is:
Answer» `""^(2N)C_(n)` `(-1)^(n)""^(2n)C_(n)` `(-1)^(n)""^(2n)C_(n-1)` NONE of these Answer :D

Discussion

12513.	Let the sequence x_n converge and the sequence y_n diverge. What can be said about convergence of the sequences (a) x_n+y_n , (b) x_ny_n?
Answer» Answer :(a) It DIVERGES (B) it may be either CONVERAGE or DI VERGE.

Discussion

12514.	In a basket, there are 4 apples, 2 mangoes and 5 bananas. Fruits of same kind are identical. Find the total number of selections. atleast one banana
Answer» ANSWER :75

Discussion

12515.	In a basket, there are 4 apples, 2 mangoes and 5 bananas. Fruits of same kind are identical. Find the total number of selections without any restrictio
Answer» ANSWER :90

Discussion

12516.	Evaluate the definite integrals int_(0)^((pi)/(2))sin2xtan^(-1)(sinx)dx
Answer» ANSWER :`pi/2-1`

Discussion

12517.	If Ais square matrix such that A^(2)=A, show that (I+A)^(3)=7A+I.
Answer» ANSWER :`=I+7A`

Discussion

12518.	The nearest point on the circle x^(2)+y^(2)-6x+4y-12=0" from "(-5,4)" is "
Answer» only I is true only II is true both I and II are true neither I nor II true Answer :B

Discussion

12519.	The solution of the differential equaiton 3xy'-3y+(x^2-y^2)^(1//2)=0, satisfying the condition y(1)=1 is
Answer» `3COS^(-1)((y)/(x))=LN\|x\|` `3cos((y)/(x))=ln \|x\|` `3 cos^(-1)((y)/(x))=2ln \|x\|` `3 SIN^(-1)((y)/(x))=ln \|x\|` ANSWER :A

Discussion

12520.	Find the sum of the series to n terms whose n ^(th) term is 3n +2.
Answer» ANSWER :`(3N + 7)`

Discussion

12521.	The unit vector perpendicualr to both the vectors hati+2hatj-2hatk and -hati+2hatj+2hatk is ………………
Answer» `(1)/(SQRT(5))(2hati-hatk)` `(1)/(sqrt(5))(-2hati+hatk)` `(1)/(sqrt(5))(2hati+hatj+hatk)` `(1)/(sqrt(5))(2hati+hatk)` ANSWER :D

Discussion

12522.	Selectthe INCORRECT order of atomic size ?
Answer» LI LT K Ti `~~` Zr Cu lt Zn C lt AL

Discussion

12523.	Check the injectivity and surjectivity of the following functions . f: R rarr R , f(x) = x^(2) -2
Answer» SOLUTION :N/A

Discussion

12524.	(i) {:\|( cos theta , -sin theta ),( sin theta , cos theta ) \|:}"" (ii) {:\|( x^(2) -x+1,x-1),( x+1,x+1) \|:}
Answer» ANSWER :` 1` ` X^(3) , -x^(2) +2`

Discussion

12525.	Two persons A and B are rolling die on the condition that the person who gets 3 will win the game. If A starts the game, then find the probabilities of A and B respectively to win the game.
Answer» `(6)/(11),(5)/(11)` `(5)/(11),(6)/(11)` `(8)/(11),(3)/(11)` `(3)/(11),(8)/(11)` ANSWER :A

Discussion

12526.	Examine the consistency of the system of equations x+ 3y = 52x+6y =8
Answer» ANSWER :INCONSISTENT

Discussion

12527.	If \|{:(x+1,x-1),(x-3,x+2):}\|=\|{:(4,-1),(1,3):}\| then find x
Answer» ANSWER :2

Discussion

12528.	int e^(log x)/x dx=
Answer» `LOG x+C` `e^logx+c` `xlog x +c` `xe^logx+c` ANSWER :B

Discussion

12529.	Let A and B be two sets such that A xx B has 6 elements. If three elements of A xx B are {(1,4),(2,6),(3,6)}, then
Answer» A = {1, 2} and B = {3, 4, 6} A = {4, 6} and B = {1, 2, 3} A = {1, 2, 3} and B = {4, 6} A = {1, 2, 4} and B = {3, 6} Answer :C

Discussion

12530.	Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also find the maximum volume.
Answer» ANSWER :`= 864 PI (CM)^(3)`

Discussion

12531.	Find the derivative of the following functions with respect to x sin 3x. sin3x
Answer» Answer :`3 SIN^(2)x sin 4x`

Discussion

12532.	Let Papoint denoting a comples number z on the complex plane. i.e.""z=Re(z)+i Im(z)," where "i=sqrt(-1) if""Re(z)=xand Im (z)=y,then z=x+iy If Pmovew such that \|Re(z)\|+\|Im(z)=a(ainR^(+)) The locus of P is
Answer» a PARALLELOGRAM which is not arhombus a RHOMBUS which is not a SQUARE a RECTANGLE which is not a square a square Answer :d

Discussion

12533.	Find \|veca\|and\|vecb\|, if (veca+vecb)*(veca-vecb)=8and\|veca\|=8\|vecb\|.
Answer» ANSWER :`(16sqrt(2))/(3sqrt(7)),(2sqrt(2))/(3sqrt(7))`

Discussion

12534.	If int e^(x) ((x+2)/(x+4))^(2) dx = f(x)arbitrary constant, then f(x) =
Answer» `(XE^(X))/(x+4)` `E^(x)/(x+4)` `xe^(x)/((x+4)^(2))` `e^(x)/((x+4)^(2))` ANSWER :A

Discussion

12535.	The partial fractions of 1/(x^(3)(x+2)) =
Answer» `(1)/(8x)-(1)/(4X^(2))+(1)/(2x^(3))-(1)/(8(x+2))` `(1)/(8x)+(1)/(4x^(2))+(1)/(2x^(3))-(1)/(8(x+2))` `(1)/(8x)-(1)/(4x^(2))-(1)/(2x^(3))+(1)/(8(x+2))` `(1)/(8x)+(1)/(4x^(2))+(1)/(2x^(3))+(1)/(8(x+2))` Answer :A

Discussion

12536.	Evaluate the following integrals. int(1)/(sinx+sqrt(3)cosx)dx
Answer» Answer :`(1)/(2)log\|(1+sqrt(3)TAN((X)/(2)))/(sqrt(3)-tan((x)/(2)))\|+c`

Discussion

12537.	The number of ways in which 11 identical pencils can be distributed among 6 kids each receiving atleast one is
Answer»

Discussion

12538.	Examine the continuity of the following functions at indicated points.f(x)=(g(x)-g(1))/(x-1)atx=1whereg(x)=\|x-1\|
Answer» Solution :G(x)=\|x-1\| Then g(1)=\|1-1\|=0 Now `F(1)=(g(1)-f(1))/(1-1)=0/0` which we cannot DETERMINE. HENCE f(x) is DISCONTINUOUS at x=1

Discussion

12539.	Show that int_(0)^(pi) (x)/(a^(2)-cos^(2)x) dx = (pi^(2))/(2a sqrt(a^(2)-1))(a gt 1)
Answer» Answer :`(PI^(2))/(2A SQRT(a^(2)-1))`

Discussion

12540.	If f (x) = tan ^(-1) [ (log ((e )/( x ^(2))))/(log (ex ^(2)))] + tan ^(-1) [ (3 + 2 log x )/( 1 - 6 log x )] then the vlaue of f''(x) is
Answer» `X ^(2)` `x` `1` `0` ANSWER :D

Discussion

12541.	The vertical straight line passing through the point of intersection of the straight lines x-3y+1=0, 2x+5y-9=0 and at a distance of 2 units from the origin has the equation
Answer» `x=2` `3x+4y-10=0` `y=1` NONE ANSWER :B

Discussion

12542.	Let Z_(1)=x_(1)+iy_(1), Z_(2)=x_(2)+iy_(2) be complex numbers in fourth quadrant of argand plane and \|Z_(1)\|=\|Z_(2)\|=1, Ref(Z_(1)Z_(2))=0. The complex numbers Z_(3)=x_(1)+ix_(2), Z_(4)=y_(1)+iy_(2), Z_(5)=x_(1)+iy_(2), Z_(6)=x_(6)+iy, will always satisfy
Answer» `\|Z_(4)\|=1` `arg(Z_(1)Z_(4))=-pi//2` `(Z_(5))/(cos(argZ_(1)))+(Z_(6))/(sin(argZ_(1)))` is purely REAL `Z_(5)^(2)+(barZ_(6))^(2)` is purely imaginergy Solution :`(a,b,c,d)` `Z_(1)=e^(itheta_(1))`, `Z_(2)=e^(itheta_(2))`, `Re(Z_(1)Z_(2))=0impliestheta_(1)+theta_(2)=-pi//2`, (as `z_(1)`, `z_(2)` lie in fourth quadrant) `Z_(3)=e^(-itheta_(1))`, `Z_(4)=-e^(itheta_(1))`, `Z_(5)=costheta_(1)(1-i)`, `Z_(6)=sintheta_(1)(-1+i)`

Discussion

12543.	int_(0)^(pi//4) (sin x + cos x)/(7 + 9 sin 2x) dx is equal to
Answer» `(LOG 3)/(4)` `(log 3)/(36)` `(log 7)/(12)` `(log 7)/(24)` Answer :d

Discussion

12544.	There are eight different coloured balls and 8 bags having the same colours as that of the balls. If one ball is placed at random in each one of the bags, then the probability that 5 of the balls are placed in the respective coloured bags is
Answer» `(1)/(120)` `(1)/(160)` `(1)/(180)` `(1)/(360)` ANSWER :D

Discussion

12545.	Integrating factor of the differential equation y dx - ( x-2y^(2) ) dy=0 is …..
Answer» `y` `-y` `- (1)/(y)` `(1)/(y)` ANSWER :D

Discussion

12546.	Show that 7^(n) +5 is divisible by 6, where n is a positive integer
Answer» ANSWER :6

Discussion

12547.	Find the coefficient of x^10 in the expansion of (1+2x)^21 + (1 + 2x)^22 +……+ (1+ 2x)^30
Answer» ANSWER :`2^10 {""^31C_11 - ""^21C_11}`

Discussion

12548.	int(x+3sqrt(x^(2))+6sqrt(x))/(x(1+3sqrt(x)))dx
Answer» `(3)/(2)x^(2//3)+6tan^(-1)x^(1//6)+C` `(3)/(2)x^(2//3)-6tan^(-1)x^(1//6)+C` `-(3)/(2)x^(2//3)-6tan^(-1)x^(1//6)+C` NONE of these Answer :a

Discussion

12549.	A(1,2),B(2,- 3),C(-2,3) are 3 points. A point P moves such that PA^(2)+PB^(2)=2PC^(2) . Show that the equation to the locus of P is 7 x - 7y + 4 = 0 .
Answer» ANSWER :`7x-7y+4=0`

Discussion

12550.	If y = int_(0)^(x) (t^(2))/(sqrt(t^(2)+1))dt then (dy)/(dx) at x=1 is
Answer» `SQRT(2)` `1/2` `(1)/(sqrt(2))` 2 Answer :C

Discussion

Explore topic-wise InterviewSolutions in .

Let p, q and r be the statements: p: Mathura is in U.P. q: Mathura is in lndia. r : p rarrq Contra-positive of r is

int xsqrt(x + 2) dx

For theta in (0, (pi)/(2)). Sec h^(-1)(cos theta) =

Ifq_(1), _(2), q_(3)are roots of the equation x^(3)+64=0,then the value of |(q_(1),q_(2),q_(3)),(q_(2),q_(3), q_(1)),(q_(3),q_(1),q_(2))| is :-

Integrate the following rational functions : int(tantheta+tan^(3)theta)/(1+tan^(3)theta)

If vec(a),vec(b),vec(c ) be three vector of magnitude sqrt(3), 1, 2, such thatvec(a)xx(vec(a)xx vec(c ))+3vec(b)=vec(0), if theta is theangle between , then cos theta is equal to :

Organic materials present in food helps in providing :-

Let A and B be two events such that P(barAuuB)=(1)/(6) , P(AnnB)=(1)/(4) and P(barA)=(1)/(4) where barA stands for complement of event A. Then events A and B are

If n is an even integer, then: C_(0)^(2)-C_(1)^(2)+C_(2)^(2)-C_(3)^(2)+ . . .+(-1)^(n)C_(n)^(2) is:

Let the sequence x_n converge and the sequence y_n diverge. What can be said about convergence of the sequences (a) x_n+y_n , (b) x_ny_n?

In a basket, there are 4 apples, 2 mangoes and 5 bananas. Fruits of same kind are identical. Find the total number of selections. atleast one banana

In a basket, there are 4 apples, 2 mangoes and 5 bananas. Fruits of same kind are identical. Find the total number of selections without any restrictio

Evaluate the definite integrals int_(0)^((pi)/(2))sin2xtan^(-1)(sinx)dx

If Ais square matrix such that A^(2)=A, show that (I+A)^(3)=7A+I.

The nearest point on the circle x^(2)+y^(2)-6x+4y-12=0" from "(-5,4)" is "

The solution of the differential equaiton 3xy'-3y+(x^2-y^2)^(1//2)=0, satisfying the condition y(1)=1 is

Find the sum of the series to n terms whose n ^(th) term is 3n +2.

The unit vector perpendicualr to both the vectors hati+2hatj-2hatk and -hati+2hatj+2hatk is ………………

Selectthe INCORRECT order of atomic size ?

Check the injectivity and surjectivity of the following functions . f: R rarr R , f(x) = x^(2) -2

(i) {:|( cos theta , -sin theta ),( sin theta , cos theta ) |:}"" (ii) {:|( x^(2) -x+1,x-1),( x+1,x+1) |:}

Two persons A and B are rolling die on the condition that the person who gets 3 will win the game. If A starts the game, then find the probabilities of A and B respectively to win the game.

Examine the consistency of the system of equations x+ 3y = 52x+6y =8

If |{:(x+1,x-1),(x-3,x+2):}|=|{:(4,-1),(1,3):}| then find x

int e^(log x)/x dx=

Let A and B be two sets such that A xx B has 6 elements. If three elements of A xx B are {(1,4),(2,6),(3,6)}, then

Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also find the maximum volume.

Find the derivative of the following functions with respect to x sin 3x. sin3x

Let Papoint denoting a comples number z on the complex plane. i.e.""z=Re(z)+i Im(z)," where "i=sqrt(-1) if""Re(z)=xand Im (z)=y,then z=x+iy If Pmovew such that |Re(z)|+|Im(z)=a(ainR^(+)) The locus of P is

Find |veca|and|vecb|, if (veca+vecb)*(veca-vecb)=8and|veca|=8|vecb|.

If int e^(x) ((x+2)/(x+4))^(2) dx = f(x)arbitrary constant, then f(x) =

The partial fractions of 1/(x^(3)(x+2)) =

Evaluate the following integrals. int(1)/(sinx+sqrt(3)cosx)dx

The number of ways in which 11 identical pencils can be distributed among 6 kids each receiving atleast one is

Examine the continuity of the following functions at indicated points.f(x)=(g(x)-g(1))/(x-1)atx=1whereg(x)=|x-1|

Show that int_(0)^(pi) (x)/(a^(2)-cos^(2)x) dx = (pi^(2))/(2a sqrt(a^(2)-1))(a gt 1)

If f (x) = tan ^(-1) [ (log ((e )/( x ^(2))))/(log (ex ^(2)))] + tan ^(-1) [ (3 + 2 log x )/( 1 - 6 log x )] then the vlaue of f''(x) is

The vertical straight line passing through the point of intersection of the straight lines x-3y+1=0, 2x+5y-9=0 and at a distance of 2 units from the origin has the equation

Let Z_(1)=x_(1)+iy_(1), Z_(2)=x_(2)+iy_(2) be complex numbers in fourth quadrant of argand plane and |Z_(1)|=|Z_(2)|=1, Ref(Z_(1)Z_(2))=0. The complex numbers Z_(3)=x_(1)+ix_(2), Z_(4)=y_(1)+iy_(2), Z_(5)=x_(1)+iy_(2), Z_(6)=x_(6)+iy, will always satisfy

int_(0)^(pi//4) (sin x + cos x)/(7 + 9 sin 2x) dx is equal to

There are eight different coloured balls and 8 bags having the same colours as that of the balls. If one ball is placed at random in each one of the bags, then the probability that 5 of the balls are placed in the respective coloured bags is

Integrating factor of the differential equation y dx - ( x-2y^(2) ) dy=0 is …..

Show that 7^(n) +5 is divisible by 6, where n is a positive integer

Find the coefficient of x^10 in the expansion of (1+2x)^21 + (1 + 2x)^22 +……+ (1+ 2x)^30

int(x+3sqrt(x^(2))+6sqrt(x))/(x(1+3sqrt(x)))dx

A(1,2),B(2,- 3),C(-2,3) are 3 points. A point P moves such that PA^(2)+PB^(2)=2PC^(2) . Show that the equation to the locus of P is 7 x - 7y + 4 = 0 .

If y = int_(0)^(x) (t^(2))/(sqrt(t^(2)+1))dt then (dy)/(dx) at x=1 is