28278 + Interview Questions in Maths in Class 12

1.	Find the correct statement :
Answer» If `rho[A]=rho[A\|B]=` no. of UNKNOWNS then the system has unique solution. The homogeneous system of linear equations AX = 0 has a trivial solution of \|A\| = 0. `ADJ(lambda A)=lambda(adj A)` `\|adj (adj A)\|=\|A\|^(n^(2))` Answer :A

Discussion

2.	IfS and T are foci of x^(2)/(16)-y^(2)/(9)=1. If P is a point on the hyperbola then \|SP-PT\|=
Answer» 8 3 6 12 Answer :a

Discussion

3.	((1+sin theta+icos theta)/(1+sin theta - icos theta))^(n)=
Answer» `cos ((N pi)/(2)-n theta)+ISIN((npi)/(2)-n theta)` `cos ((n pi)/(2)+n theta)+isin((npi)/(2)+n theta)` `sin ((n pi)/(2)-n theta)+icos((npi)/(2)-n theta)` `cosn (( pi)/(2)+ 2theta)+isin n((pi)/(2)+ 2theta)` Answer :A

Discussion

4.	The value of f(e^(x)(1+x)dx)/(cos^(2)(e^(x)x)) is equal to
Answer» `-COT(ex^(x))+C` `TAN(xe^(x))+C` `tan(E^(x))+C` `cot(e^(x))+C` Answer :B

Discussion

5.	If f(x)=[x](sinx+cosx-1) (where [.] denotes the greatest integer function). then f'(x)=[x](cosx-sinx) for any x in integer. Statement II f'(x) does not exist for any x in integer.
Answer» STATEMENT I is TRUE, Statement II is also true, Statement II is the CORRECT EXPLANATION of statement I. Statement I is true, Statement II is also true, Statement II is not the correct explanation of Statement I. Statement I is true, Statement II is false Statement I is false, Statement II is true Answer :B

Discussion

6.	Evalute the following integrals int (e^(x))/(2e^(2x) + 3e^(x)-2)dx
Answer» ANSWER :`(1)/(5)log\|(e^(X)-(1)/(2))/(e^(x)+(1)/(2))\|+C`

Discussion

7.	If PN is the ordinate of a point P on the ellipse x^2/a^2+y^2/b^2=1and the tangent at P meets the X-axis at T then show (CN) (CT) =a^2 where C is the centre of the ellipse.
Answer» Answer :`therefore (CN)(CT)=(a COS THETA) (a/costheta)=a^2`

Discussion

8.	Prove that product of parameters of four concyclic points on the hyperbola xy=c^(2) is 1. Also, prove that the mean of these four concyclic points bisects the distance between the centres of the hyperbola and the circle.
Answer» Solution :Given equation of hyperbola is `xy=C^(2)` Let four conyclic point on the hyperbola be `(x_(i),y_(i)),i=1,2,3,4`. Let the equation of the circle through point A, B, C and D be `x^(2)+y^(2)+2gx+2fy+d=0"(1)"` Solving circle and hyperbola, we get `x^(2)+(c^(4))/(x^(2))+2gx+2f*(c^(2))/(x)+d=0` `rArr""x^(4)+2gx^(3)+dx^(2)+2fc^(2)x+c^(4)=0"(2)"` `therefore"Product of roots,"x_(1)x_(2)x_(3)x_(4)=c^(4)` Now, `(x_(i),y_(i))-=(ct_(i),(c)/(t_(i)))` `therefore""(ct_(1))(ct_(2))(ct_(3))(ct_(4))=c^(4)` `rArr""t_(1)t_(2)t_(3)t_(4)=1` Now, MEAN of points `(x_(i),y_(i)),i=1,2,3,4," is "((sum_(i=1)^(4)x_(i))/(4),(sum_(i=1)^(4)y_(i))/(4))`. From Eq. (2), `x_(1)+x_(2)+x_(3)+x_(4)=-2G` `rArr""(sum_(i=1)^(4)x_(i))/(4)=-(g)/(2)` Also, `sum_(i=1)^(4)y_(i)=sum_(i=1)^(4)(c^(2))/(x_(i))=(c^(2)sumx_(1)x_(2)x_(3))/(x_(1)x_(2)x_(3)x_(4))=(c^(2))/(c^(4))(-2fc^(2))=-2f` `therefore""(sumy_(i))/(4)=-(f)/(2)` Thus, `((sum_(i=1)^(4)x_(i))/(4),(sum_(i=1)^(4)y_(i))/(4))-=(-(g)/(2),-(f)/(2))`.

Discussion

9.	The vectors 2 hat(i) - 2 hat(j) + hat(k), hat(i) - 2 hat(j) + 3 hat(k) and 3 hat(i) + hat(j) - 2hat(k)
Answer» are LINEARLY dependent are linearly independent form SIDES of a triangle are coplanar Answer :B

Discussion

10.	Three numbers are chosen at random from the numbers 1, 2. ... 20. The probability that the arithmetic mean of these numbers is 4 is
Answer» `7//2280` `3//4` `2//7281` `0` ANSWER :A

Discussion

11.	Two persons A and B are playing a game. A is tossing two coins simultaneously and B is rollinga die. A will win if he gets tail on both the coins, B will win if he gets a prime number on the die. If they take their turns alternately and A starts the game find their respective probabilities of wining.
Answer» <P> Answer :P(A win ) `=(2)/(5); P(B)` (win)= `(3)/(5)`

Discussion

12.	Evalute the following integrals int (e^(x)(x - 1))/((x + 1)^(3)) dx
Answer» ANSWER :`(e^(x))/((x + 1)^(2)) + C `

Discussion

13.	From a point (C,0) three normals are drawn to the parabola y^2=x. Then
Answer» `C LT (1)/(2)` `C=(1)/(2)` `C GT (1)/(2)` `(1)/(2) gt C gt (1)/(4)` ANSWER :C

Discussion

14.	For positive l, m and n, if the points x=ny+mz, y=lz+nx, z=mx+ly intersect in a straight line, when Q. l, m and n satisgythe equation
Answer» `l^2+m^2+n^2=2` `l^2+m^2+n^2+2mln=1` `l^2+m^2+n^2=1` NONE of these Answer :(B)

Discussion

15.	A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.
Answer» ANSWER :`(11)/(50)`

Discussion

16.	If A=[{:(cosalpha,-sinalpha),(sinalpha,cosalpha):}]andA+A=I,then the value of alpha is ........ (A) (pi)/(6) (B) (pi)/(3) (C ) pi (D) (3pi)/(2)
Answer» ANSWER :(B) `(PI)/(3)`

Discussion

17.	vec(r ) =(hat(i)+2hat(j) -4hat(k)) + lambda(2hat(i) +3hat(j) +6hat(k)) vec(r )=(3hat(i) +3hat(j) -5hat(k)) + mu (-2hat(i) +3hat(j) +8hat(k))
Answer» ANSWER :`(14sqrt(241))/(241)` UNITS

Discussion

18.	int_(1)^(2)logxdx=..........
Answer» `LOG((E)/(2))` `log((2)/(e))` `log((e^(2))/(4))` `log_(e)((4)/(e))` ANSWER :D

Discussion

19.	Find the maximum and minimum values, if any of the following functions given by :f(x)=\|sin(4x)+3\|
Answer» ANSWER :4, 2

Discussion

20.	Person living in mountain have rosy cheeks because :-
Answer» They are ADAPTED to that environment. They TAKE apple daily Their RBC concentration becomes high due to LOW oxygen PRESSURE All the above Answer :A

Discussion

21.	Find the rate of change of the area of a circle with respect to its radius r when r=3cm.
Answer»

Discussion

22.	Find the points of extremum of the following functions : (a)F(x) = int_(1)^(x) e^(-(t^(2))/(2))(1 - t^(2)) dt(b)F(x) = int_(0)^(x^(2)) (t^(2) - 5t + 4)/( 2 + e^(t)) dt
Answer» Answer :(a)The maximum is at X = 1 , the minimum at x = - 1 (b)Theminimaare at x = - 2; 0; 2, the maxima at `x = PM 1`

Discussion

23.	If int_(0)^(b)(dx)/(1+x^(2))=int_(b)^(oo)(dx)/(1+x_(2)), then b=
Answer» `TAN^(-1). 1/3` `(sqrt(3))/(2)` `sqrt(2)` 1 Answer :D

Discussion

24.	If f(x)=(x+2)/(2x+3), then int sqrt((f(x))/(x^(2)))dx equals : (1)/(sqrt(2))g((1+sqrt(2f(x)))/(1-sqrt(2f(x))))-sqrt((2)/(3))h((sqrt(3f(x))+sqrt(2))/(sqrt(3f(x))-sqrt(2)))+c, where :
Answer» `g(X)=log\|x\|,H(x)=tan^(-1)x` `g(x)=h(x)=tan^(-1)x` `g(x)=tan^(-1)x,h(x)=log\|x\|` `g(x)=log\|x\|,h(x)=log\|x\|` ANSWER :D

Discussion

25.	For a certain function u_(x), given that u_(0)=3, u_(1)=12, u_(2)=81, u_(3)=200,u_(4)=100,u_(5)=8, thenDelta^(5)u_(x) is equal to
Answer» 750 `778` `765` `755` SOLUTION :Given , `u_(0)=3, u_(1)=12 , u_(2)=81 , u_(3)=200, u_(4)=100,u_(5)=8` Then , Hence, `Delta""^(5)u_(X)=755`

Discussion

26.	If the system of equations 2x+5y+8z=0 x+4y+7z=0 6x+9y-z=0 has nontrivial solution, then is equal to
Answer» 12 -12 0 none Answer :B

Discussion

27.	Show that f(x)=x^3-6x^2+24x+4 has neither a maximum nor a minimum value.
Answer» Solution :`f(x)=x^3-6x^2+24+4=f(x)=3x^2-12x+24=3(x^2-4x+8)=3((x-2)^2+4)` as `f(x)ne`0 for all `x in R` the function has NEITHER a local MAXIMUM nor local minum. But has a ABSOLUTE minimum 12 at x =2.

Discussion

28.	Given the following frequency distribution with some missing frequencies{:("Wages (RS) ",60-70,50-60,40-50,30-40,20-30),("No. of labourers "," "5," "10, " "f_(1)"" ," "5," "f_(2)""):}If the total frequency is 43 and median is 46.75 then the missing frequenciesare
Answer» 18,5 20,3 17,6 15,8 Answer :B

Discussion

29.	For the matrix A=[{:(2,3),(1,2):}] , show that A^2-4A+I_2=0. Hence, find A^(-1)
Answer» ANSWER :`A^(-1)=[{:(2,-3),(-1,2):}]`

Discussion

30.	Integrate the following functions. int(sinx)/(sin4x)dx
Answer» ANSWER :`(1)/(8)LOG\|(sinx-1)/(sinx+1)\|-(1)/(4sqrt2)log\|(sqrt2sinx-1)/(sqrt2sinx+1)\|+C`

Discussion

31.	Differentiate x^2+2x-sinx+5
Answer» SOLUTION :`y=x^2+2x-sinx+5` `dy/dx=2x+2-cosx`

Discussion

32.	Let P, Q, R and S be the points on the plane with position vectors -2hati-hatj, 4hati, 3hati+3hatj and -3hati+2hatj respectively. The quadrilateral PQRS must be a
Answer» parallelogram, which is NEITHER a rhombus nor a ractangle square rectangle, but not a square rhombus, but not a square Solution :Now , `PQ=6hat(i)+hat(J)` `QR=-hat(i)+3hat(j)` `RS=-6hat(i)-hat(j)` `SP=hat(i)-3hat(j)` `rArr \|PQ\|=sqrt(37)=\|RS\|` `\|QR\|=sqrt(10)=\|SP\|` `therefore PQ.QR=-6+3=-3 NE 0` .

Discussion

33.	P = 2008^(2007) - 2008, Q = 2008^(2) + 2009. The remainder when P is divided by Q is:
Answer» ANSWER :4

Discussion

34.	The statement ~ (p ^^ q) vee q) :
Answer» <P>is a TAUTOLOGY is EQUIVALENT to `(p ^^ Q) VEE ~q` is equivalent to `p vee q ` is a contradiction Answer :A

Discussion

35.	The normal to the circule given by x^(2) + y^(2) - 6x + 8y - 144 = 0 at (8,8) meets the circle again at the point
Answer» (2,-16) (2, 16) (-2, 16) (-2, -16) ANSWER :d

Discussion

36.	Find the number of arrangements that can be formed by using all the letters of the word "CONSIDER' so that relative positions of vowels and consonants remain unaltered.
Answer» ANSWER :720

Discussion

37.	If a, b, c and d are unit vectors, then abs(a-b)^(2)+abs(b-c)^(2)+abs(c-d)^(2)+abs(d-a)^(2)+abs(c-a)^(2)+abs(b-d)^(2) does not exceed
Answer» 4 12 8 16 Answer :B

Discussion

38.	int_(0)^(2pi) x sin^(4) x cos^(6) x dx=
Answer» 0 `(3pi^(2))/(128)` `(5PI^(2))/(128)` `(3pi^(2))/(64)` ANSWER :B

Discussion

39.	Let I_(n)=int_(0)^(1)x^(n)tan^(-1)xdx. If a_(n)I_(n+2)+b_(n)I_(n)=c_(n) for all n ge 1, then
Answer» `a_(1), a_(2), a_(3)` are in G.P `b_(1), b_(2), b_(3)` are in A.P `c_(1), c_(2), c_(3)` are in H.P `a_(1), a_(2), a_(3)` are in A.P Answer :B::D

Discussion

40.	if one of the lines given by the equation ax^2+2hxy+by^2=0 coincides with one of the lines given by a'x^2+2h'xy+b'y^2=0 and the other lines representted by them be perpendicular , then . (ha'b')/(b'-a')=(h'ab)/(b-a)=1/2sqrt((-aa'bb').
Answer» ANSWER :`(HA'B')/(b'-a')=(h'ab)/(b-a)=1/2 SQRT(-ABA'b')`

Discussion

41.	If A=[(cos alpha, sin alpha),(-sin alpha, cos alpha)], then verify that A'A=I
Answer» A Zero matrix A' I Answer :D

Discussion

42.	Choose the correct answer Two events A and B will be independent, if
Answer» <P>A and B are MUTUALLY exclusive P(A'B') = [1 – P(A)] [1 – P(B)] P(A) = P(B) P(A) + P(B) = 1 Answer :B

Discussion

43.	A chord of the parabola y^(2) =4ax subtends a right angle at the vertex. The tangents at the extremeties of the chord intersect on
Answer» x+a=0 x+2a=0 x+4a=0 none Answer :C

Discussion

44.	Bag A contains 6 Green and 8 Red balls and bag B contain 9 Green and 5 Red balls . A card is drawn at random from a well shuffled pack of 52 playing cards. IF is a spade, two balls are drawn at random from bag A, otherwisetwo balls are drawn at random from bag B. IF the two balls are found to be of the same colour, then the probability that they are drawn from bag A is
Answer» `43/181` `1/4` `48/131` `43/138` ANSWER :A

Discussion

45.	Find the value of int_(-(pi)/(4))^(pi/4)(x+(pi)/(4))/(2-cos2x)dx.
Answer» ANSWER :`(PI^(2))/(6sqrt3)`

Discussion

46.	If I_(n) = int_(0)^(1) ( cos^(-1) x)^(n) dx then I_(6)- 360 I_(2) is given by
Answer» `6 (pi//2)^(5)` `6 (pi//2)^(5) - 120(pi//2)^(3)` `6 (pi//2)^(5)` `6(pi//2)^(5) - 4 (pi//2)^(3)` ANSWER :B

Discussion

47.	Evalute the following integrals int (1 + cos^(2)x )/(1 - cos 2x ) dx
Answer» ANSWER :`-COTX-(X)/(2)+C`

Discussion

48.	It is known that 10% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles, 9 are defective?
Answer» Answer :`(22 XX 9^(3))/(10^(11))`

Discussion

49.	(i) Find the general term in the expansion of (3x^(2) - 1/(3x))^(9). (ii) Find the term independent of x in the above expension.
Answer» Answer :(i)`(-1)^(R)""^(9)C_(r)3^(9 - 2r)X^(18 - 3r)`, (ii)`:.T_(7) = (-1)^(6)""^(9)C_(6) 3^(9 - 12) = 84/27 = 28/9` is independent of `x`

Discussion

50.	If the primitive of f(x) = (1)/(3 sin x + sin^(3) x) is equal to f(x) = (1)/(3sinx+sinx) is equal to (1)/(6)log\|(t-1)/(t+1)\|+(1)/(12)log\|(2+t)/(2-t)\|+C them
Answer» t = cos x t = tan x/2 t = 2 cos x t = sin x Answer :A

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

Find the correct statement :

IfS and T are foci of x^(2)/(16)-y^(2)/(9)=1. If P is a point on the hyperbola then |SP-PT|=

((1+sin theta+icos theta)/(1+sin theta - icos theta))^(n)=

The value of f(e^(x)(1+x)dx)/(cos^(2)(e^(x)x)) is equal to

If f(x)=[x](sinx+cosx-1) (where [.] denotes the greatest integer function). then f'(x)=[x](cosx-sinx) for any x in integer. Statement II f'(x) does not exist for any x in integer.

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

If PN is the ordinate of a point P on the ellipse x^2/a^2+y^2/b^2=1and the tangent at P meets the X-axis at T then show (CN) (CT) =a^2 where C is the centre of the ellipse.

Prove that product of parameters of four concyclic points on the hyperbola xy=c^(2) is 1. Also, prove that the mean of these four concyclic points bisects the distance between the centres of the hyperbola and the circle.

The vectors 2 hat(i) - 2 hat(j) + hat(k), hat(i) - 2 hat(j) + 3 hat(k) and 3 hat(i) + hat(j) - 2hat(k)

Three numbers are chosen at random from the numbers 1, 2. ... 20. The probability that the arithmetic mean of these numbers is 4 is

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

From a point (C,0) three normals are drawn to the parabola y^2=x. Then

For positive l, m and n, if the points x=ny+mz, y=lz+nx, z=mx+ly intersect in a straight line, when Q. l, m and n satisgythe equation

A card from a pack of 52 cards is lost. From the remaining cards of the pack,﻿ two cards are drawn and are found to be both diamonds. Find the probability of﻿ the lost card being a diamond.﻿

If A=[{:(cosalpha,-sinalpha),(sinalpha,cosalpha):}]andA+A=I,then the value of alpha is ........ (A) (pi)/(6) (B) (pi)/(3) (C ) pi (D) (3pi)/(2)

vec(r ) =(hat(i)+2hat(j) -4hat(k)) + lambda(2hat(i) +3hat(j) +6hat(k)) vec(r )=(3hat(i) +3hat(j) -5hat(k)) + mu (-2hat(i) +3hat(j) +8hat(k))

int_(1)^(2)logxdx=..........

Find the maximum and minimum values, if any of the following functions given by :f(x)=|sin(4x)+3|

Person living in mountain have rosy cheeks because :-

Find the rate of change of the area of a circle with respect to its radius r when r=3cm.

Find the points of extremum of the following functions : (a)F(x) = int_(1)^(x) e^(-(t^(2))/(2))(1 - t^(2)) dt(b)F(x) = int_(0)^(x^(2)) (t^(2) - 5t + 4)/( 2 + e^(t)) dt

If int_(0)^(b)(dx)/(1+x^(2))=int_(b)^(oo)(dx)/(1+x_(2)), then b=

If f(x)=(x+2)/(2x+3), then int sqrt((f(x))/(x^(2)))dx equals : (1)/(sqrt(2))g((1+sqrt(2f(x)))/(1-sqrt(2f(x))))-sqrt((2)/(3))h((sqrt(3f(x))+sqrt(2))/(sqrt(3f(x))-sqrt(2)))+c, where :

For a certain function u_(x), given that u_(0)=3, u_(1)=12, u_(2)=81, u_(3)=200,u_(4)=100,u_(5)=8, thenDelta^(5)u_(x) is equal to

If the system of equations 2x+5y+8z=0 x+4y+7z=0 6x+9y-z=0 has nontrivial solution, then is equal to

Show that f(x)=x^3-6x^2+24x+4 has neither a maximum nor a minimum value.

Given the following frequency distribution with some missing frequencies{:("Wages (RS) ",60-70,50-60,40-50,30-40,20-30),("No. of labourers "," "5," "10, " "f_(1)"" ," "5," "f_(2)""):}If the total frequency is 43 and median is 46.75 then the missing frequenciesare

For the matrix A=[{:(2,3),(1,2):}] , show that A^2-4A+I_2=0. Hence, find A^(-1)

Integrate the following functions. int(sinx)/(sin4x)dx

Differentiate x^2+2x-sinx+5

Let P, Q, R and S be the points on the plane with position vectors -2hati-hatj, 4hati, 3hati+3hatj and -3hati+2hatj respectively. The quadrilateral PQRS must be a

P = 2008^(2007) - 2008, Q = 2008^(2) + 2009. The remainder when P is divided by Q is:

The statement ~ (p ^^ q) vee q) :

The normal to the circule given by x^(2) + y^(2) - 6x + 8y - 144 = 0 at (8,8) meets the circle again at the point

Find the number of arrangements that can be formed by using all the letters of the word "CONSIDER' so that relative positions of vowels and consonants remain unaltered.

If a, b, c and d are unit vectors, then abs(a-b)^(2)+abs(b-c)^(2)+abs(c-d)^(2)+abs(d-a)^(2)+abs(c-a)^(2)+abs(b-d)^(2) does not exceed

int_(0)^(2pi) x sin^(4) x cos^(6) x dx=

Let I_(n)=int_(0)^(1)x^(n)tan^(-1)xdx. If a_(n)I_(n+2)+b_(n)I_(n)=c_(n) for all n ge 1, then

if one of the lines given by the equation ax^2+2hxy+by^2=0 coincides with one of the lines given by a'x^2+2h'xy+b'y^2=0 and the other lines representted by them be perpendicular , then . (ha'b')/(b'-a')=(h'ab)/(b-a)=1/2sqrt((-aa'bb').

If A=[(cos alpha, sin alpha),(-sin alpha, cos alpha)], then verify that A'A=I

Choose the correct answer Two events A and B will be independent, if﻿

A chord of the parabola y^(2) =4ax subtends a right angle at the vertex. The tangents at the extremeties of the chord intersect on

Find the value of int_(-(pi)/(4))^(pi/4)(x+(pi)/(4))/(2-cos2x)dx.

If I_(n) = int_(0)^(1) ( cos^(-1) x)^(n) dx then I_(6)- 360 I_(2) is given by

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

It is known that 10% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles, 9 are defective?

(i) Find the general term in the expansion of (3x^(2) - 1/(3x))^(9). (ii) Find the term independent of x in the above expension.

If the primitive of f(x) = (1)/(3 sin x + sin^(3) x) is equal to f(x) = (1)/(3sinx+sinx) is equal to (1)/(6)log|(t-1)/(t+1)|+(1)/(12)log|(2+t)/(2-t)|+C them

A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.

Choose the correct answer Two events A and B will be independent, if