28278 + Interview Questions in Maths in Class 12

1.	(a) Find the intervals in whichthe function f(x)=log(1+x)-x/(1+x) is (i) increasing, (ii) decreasing function. (b) Find the intervals in which the function f(x)= x/(log_(e) x),x gt 0 is increasing or decreasing.
Answer» ANSWER :`(a)(i)[0,infty[""(ii)]-infty,0]-{-1}` (b) Decreases in `]0,E]-{1}" and inceases in "[e, infty[`

Discussion

2.	Let f(x)=1(1-x)^(2) sin^(2)x + x^(2) for all x in IR, and let g(x)={:(x),(int),(1):} ((2(t-1))/(t+1)- ln t) f(t)dt for all x in (1,oo)Which of the folloiwng is true ?
Answer» G is INCREASING on `(1, OO)` g is DECREASING on`(1, oo)` g is increasing on (1, 2) and decreasing on`(2, oo)` g is decreasing on (1, 2) and increasingon`(2, oo)` ANSWER :B

Discussion

3.	If the pairs of straight lines represented by 3x^(2) + 2hxy - 3y^(2) = 0 and 3x^(2) + 2hxy - 3y^(2) + 2x - 4y + c = 0 form a square, then (h, c) =
Answer» (4,-1) (-1, 4) (-4 ,1) (1, -4) ANSWER :A

Discussion

4.	Let p,q, r denote respectively the statements :" you are honest ", "you are laborious ",and " you will receive a promotion " Translate (p vv q) rarr rstatements into English language .
Answer» SOLUTION :`(p VV q) rarrr` . Its ENGLISH LANGUAGE is "If you are honest of laborious then you will receive a PROMOTION."

Discussion

5.	Evaluation of definite integrals by subsitiution and properties of its : int_(-1)^(1)(x-[x])dx=........ where [.] denotes maximum integer function.
Answer» 0 1 2 3 Answer :B

Discussion

6.	How many of the functions Suppose A is a set of n elements and B is a set with m elementsare one - one with (i)m=n ,(ii) m < n, (iii) m>n
Answer» Solution :"We have" `\|A\|=n,\|B\|=m` `:.` "The number of one-one FUNCTIONS from" `"A to B is" ""^mP_n=(m!)/((m-n)!) "When "m gt n.` `"if "m=n,"the number of one-one functions is "(m!)/((m-m)!)=(m!)/(0!)=m! =n!` `"If" m LT n," then there is no POSSIBILITY of one-one functions".

Discussion

7.	Integrate the rational functions (2x-3)/((x^(2)-1)(2x+3))
Answer» ANSWER :`(5)/(2)log\|x+1-(1)/(10)log\|x-\|-(12)/(5)log\|2x+3\|+c`

Discussion

8.	I: The solution of x^(2) + y^(2) (dy)/(dx) = 4 " is " x^(3) + y^(3) = 12x + c II: The solution of2xy (dy)/(dx) = 1 + y^(2) " is " x^(2) + y^(2) = cx
Answer» only I is TRUE only II is true both I and II are true NEITHER I nor II true Answer :A

Discussion

9.	Choose the correct answer int(dx)/(x^(2)+2x+2) equals
Answer» ANSWER :`TAN^(-1)(x+1)+C`

Discussion

10.	Find values of k if the points (3,5), (k,3) and (1,1) are collinear
Answer» ANSWER :2

Discussion

11.	IF l and m are variable real numbers such that 5l^2-4lm+6m^2+3l=0, then the variable line lx+my=1 always touches a fixed parabola,whose axis is parallel to the X-axis. If (c,d) is the focus of the parabola , then the value of 2^\|d-c\| is
Answer» 1 2 4 8 Answer :B

Discussion

12.	IF l and m are variable real numbers such that 5l^2-4lm+6m^2+3l=0, then the variable line lx+my=1 always touches a fixed parabola,whose axis is parallel to the X-axis. If ex+f=0 is directrix of the parabola and e,f are prime numbers , then the value of \|e-f\| is
Answer» 2 4 6 8 Answer :D

Discussion

13.	A straight line through P(1, 2) is such that its intercept between the axes is bisected at P. Its equation is
Answer» X + y = -1 x + y = 3 x + 2Y = 5 2x + y = 4 Answer :D

Discussion

14.	The values ofsum_(m=1)^(10)[sin(2pim//11)-icos(2pim//11)] is
Answer» i `-i` 1 -1 Answer :A

Discussion

15.	Form the differential equation by eliminating the arbitrary constant from the equation (y-b)^(2) = 4 (x-a)
Answer» `2 (d^(2)y)/(dx^(2)) + ((dx)/(DY))^(3) = 0` `(d^(2)y)/(dx^(2)) + ((dy)/(dx))^(3) = 0` `2(d^(2)y)/(dx^(2)) + ((dy)/(dx))^(3)= 0` `2 (d^(2)y)/(dx^(2)) - ((dy)/(dx))^(3) = 0` ANSWER :C

Discussion

16.	If two consecutive terms in the expansion of (x+a)^n are equal to where n is a positive integer then ((n+1)a)/(x+a) is
Answer» Negative integer RATIONAL NUMBER a real number a POSITIVE integer Answer :D

Discussion

17.	If the vectors a hati + 2 hatj + 3 hatk and - hati + 5 hatj + a hatk are perpendicular to each other, then a equals :
Answer» 6 `-6` 5 `-5.` ANSWER :D

Discussion

18.	One kind of cake requires 200 g of flour and 25 g of fat and another kind of cake requires 100g of flour and 50 g of fat. Find the maximum number of cakes which can be in make from 5 kg of flour and 1 kg of fat, assuming that there is no shortage of other in gradients used in making the cakes. Formulate the above as a linear programming problem and solve it graphically.
Answer» ANSWER :MAXIMUM number of CAKES = 30, 20 cakes of `1^(st)` KIND and 10 cakes of `2^(nd)` kind be prepared.

Discussion

19.	If P (A) = P (B) = x and P(AnnB)=P(A'nnB')=1//3, then x = ?
Answer» `1//2` `1//3` `1//4` `1//6` ANSWER :A

Discussion

20.	Find the integral of (1)/(sqrt(a^(2)-x^(2)) with respect to x and hence find int (1)/(sqrt(7-6x-x^(2))dx
Answer» ANSWER :`SIN^(-1)((x+3)/4)+C`

Discussion

21.	int_(- pi//2)^(pi//2) ( x^(14) sin^(11) x+ sin^(2) x) dx is equal to
Answer» `0` `PI` `pi/2` `-pi` ANSWER :C

Discussion

22.	The straight lines x+ 3y -4=0,x+y-4=0and 3x+y - 4 = 0
Answer» from an ISOSCELES TRIANGLE are concurrent from an equilateral triangle from a right angled isosceles triangle Answer :A

Discussion

23.	If e^(ax) +e^(-bx) = p_0 +p_1x +p_2x^2 + ....oo then (p_0 , p_1,p_2) =
Answer» `(2,a -B , (a^2+b^2)/2) ` `(2,a + b , (a^2+b^2)/2) ` `(2, a-b,a^2+b^2)` `(2,a - b , (a^2- b^2)/2) ` ANSWER :A

Discussion

24.	A unit vector vec(a) is perpendicualr to the vectors hati+2hatj-hatk and 3hati-hatj+hatk then find the components of vec(a).
Answer» Answer :`(3)/(SQRT(83))I,(-5)/(sqrt(83))HATJ,(-7)/(sqrt(83))hatk` OR `(-3)/(sqrt(83))HATI,(5)/(sqrt(83))hatj,(7)/(sqrt(83))hatk`

Discussion

25.	Which of the following limits does not exist ?
Answer» `lim _(x to OO) cosec ^(-1)((x)/(x+7))` `lim _(x to 1) SEC ^(-1) (SIN ^(-1) x)` `lim _(x to 0^(+)) x ^(1/x)` `lim _(x to 0) (TAN ((pi)/(8) +x))^(cot x)` Answer :A::D

Discussion

26.	If (sqrt(5)+sqrt(3)i)^(33)= 2^(49)z, then modulus of the complex number z is equal to
Answer» `1` `SQRT(2)` `2sqrt(2)` `4` ANSWER :B

Discussion

27.	Let z and omega be two complex numberssuch that \|z\|le 1, \|omega\| le 1 and \|z+ iomega\| = \|z_(1)-z_(2)\|is equal to
Answer» `(2)/(3)` `(sqrt(5))/(3)` `(3)/(2)` `(2sqrt(5))/(3)` Solution :We have `2=\|z+iomega\|le\|z\|+\|omega\|""(THEREFORE \|z_(1)+z_(2)\|le\|z_(1)\|+\|z_(2)\|)` `therefore \|z\|+\|omega\|le2""...(1)` But it is given that `\|z\|le1 and \|omega\|le1`. `rArr \|z\|+\|omega\|le2""...(2)` From (1) and (2), `\|z\|=\|omega\|=1` Also, `\|z+iomega\|=\|z-(ibaromega)\|` `rArr \|z-(-iomega)\|=\|z-(ibaromega)\|` This means that z lies on perpendicularbisector of theline segment JOINING `(-iomega) and (ibaromega)`, which is real axis, as `(-iomega) and (ibaromega)` are CONJUGATE to each other. For`z, Im(z) = 0` If `z =x, " then " \|z\|le1` `rArr x^(2)le1` `rArr -1le x le 1` Therefore, (3) is the correct option.

Discussion

28.	If pnea,qneb,rnec and \|{:(p,b,c),(a,q,c),(a,b,r):}\|=0 then prove that p/(p-a)+q/(q-b)+r/(r-c)=2
Answer» <P> ANSWER :`p/(p-a)+Q/(q-b)+R/(r-c)=2`

Discussion

29.	A statement among the following is
Answer» EVERY rectangle is a parallelogram Who is the FATHER of GEOMETRY? Is 1 a PRIME number? Sit down Answer :A

Discussion

30.	Find (dy)/(dx) in the following : sin^(2)y+cos xy=k.
Answer» ANSWER :`(y SIN XY)/(sin 2y-x sin xy)`

Discussion

31.	If A is a square matrix of order 3xx3, then \|KA\| is equal to
Answer» 1.K\|A\| 2.`K^(2)\|A\|` 3.`K^(3)\|A\|` 4.3 K\|A\| Answer :C

Discussion

32.	If alpha, beta, gamma are the roots of x^(3)+2x^(2)-3x-1=0 then alpha^(-2)+beta^(-2)+gamma^(-2) is equal to
Answer» 12 13 14 15 Answer :B

Discussion

33.	If 0 lt x lt 1, the first negative term in the expansion of (1+x)^(27//5) is
Answer» 5TH TERM 8th term 6th term 7TH term Answer :B

Discussion

34.	int 2^(x)(f'(x)+f(x)log2)dx is equal to
Answer» `2^(X)F'(x)+C` `2^(x)log2+c` `2^(x)f(x)+c` `2^(x)+c` ANSWER :C

Discussion

35.	LetC_1 and C_2 be respectively, the parabolas x^2=y=-1 and y^2=x-1 Let P be any point on C_1 and Q be any point on C_2 . Let P_1 and Q_1 be the refelections of P and Q, respectively with respect to the line y=x. Arithemetic mean of PP_1and Q Q_1 is always less than
Answer» PQ `1/2`PQ 2PQ `3/2`PQ Answer :A

Discussion

36.	Differentiate the functions w.r.t. x. x^(sin x)+(sin x)^(cos x).
Answer» ANSWER :`X^(sin x)[(SINX)/(x)+cosxlogx]+(sinx)^(COS x)[cos xcot x-sin x log sin x]`

Discussion

37.	LetC_1 and C_2 be respectively, the parabolas x^2=y=-1 and y^2=x-1 Let P be any point on C_1 and Q be any point on C_2 . Let P_1 and Q_1 be the refelections of P and Q, respectively with respect to the line y=x. If the point p(pi,pi^2+1)and Q(pi^2+1mu)then P_1 and Q_1 are
Answer» `(PI^2+1,pi)` and `(MU^2+1,mu)` `(mu^2+1,mu)` and `(mu,mu^2+1)` `(mu,mu^2+1)` =and `(mu,mu^2+1)` `(pi,pi^2+1)` =and `(mu^2,1+mu)` ANSWER :B

Discussion

38.	How many numbers between 10 and 10,000 can be formed by using the digits 1, 2, 3, 4, 5 if No digit is repeated in any number
Answer» ANSWER :200

Discussion

39.	The slope of the normal to the curve y = 2x^(2) + 3 sin x at x = 0 is
Answer» 3 `(1)/(3)` `-3` `-(1)/(3)` ANSWER :D

Discussion

40.	A line makes the same angle theta, with each of the x and axis. If the angle beta, which it makes with y-axis, is such that sin^(2)beta=3sin^(2)theta, then cos^(2)theta equals:
Answer» `(3)/(5)` `(1)/(5)` `(2)/(3)` `(2)/(5)` ANSWER :A

Discussion

41.	In a certain college, 25% of the boys and 10% of the girls are studying mathematics. The girls constitute 60% of the student strength. If a student selected at random is found studying mathematics, find the probability that the students is a girl.
Answer» ANSWER :`(3)/(8)`

Discussion

42.	2. C_2 + 6. C_3 + 12. C_4 +….. + n (n-1) . C_n=
Answer» `n (n - 1). 2^(n-1)` `2N (n-1).2^(n-2)` `n(n-1).2^(n-2)` `2n (n+1).2^(n-1)` Answer :C

Discussion

43.	A tower subtends an angle 75^(@) at a point on the same level as the foot of the tower and at another point, 10 meters above the first, the angle of depression of the foot of the tower is 15^(@). The height of the tower is (in meters)
Answer» `10(sqrt3+1)^(2)` `10(sqrt3-1)^(2)` `10(2+sqrt3)^(2)` NONE of these Answer :C

Discussion

44.	If ab=LCM of (a,b) then 46
Answer» 24 12 8 6 Answer :B

Discussion

45.	If a^2+8b^2+2c^2+2d^2-4ab-4bc-4bd=0 ,then the value of \|{:(a,b),(c,d):}\| is
Answer» `(a^2)/4` `B^2` `C^2` `d^2` ANSWER :A::B::C::D

Discussion

46.	Show that the area of the triangle formed by the two tangents through P(x_(1),y_(1)) to the circle S=x^(2)+y^(2)+2gx+2fy+c=0 and the chord of contact of P w.r.t S=0 is (r(S_(11))^(3//2))/(S_(n)+r^(2)), where r is the radius of the circle.
Answer» ANSWER :` ( R (S_11) ^(3//2))/((S_11+r^(2)))`

Discussion

47.	If area of triangle is 35sq units with vertices (2,-6) ,( 5,4) and( k,4) Then k is
Answer» `12` ` -2` ` -12,-2` `12,-2` ANSWER :D

Discussion

48.	If orthocentre of the triangle formed by ax^2+2hxy+by^2= 0 and px + qy =1is (r,s) then prove that r/p-s/q=(a+b)/(aq^2+bq^2-2hpq)
Answer» ANSWER :`(a+b)/(aq^2-2hpq+bq^2)`

Discussion

49.	If function g is defined by g(x)=x-1 and 2g(c)=10, what is the value of g(3c)?
Answer» `6` `9` `15` `17` ANSWER :D

Discussion

50.	If [[3,-4],[1,2]] [[x],[y]]= [[3],[11]], then write the correct answer from· the following :
Answer» 2 5 3 8 Answer :D

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

(a) Find the intervals in whichthe function f(x)=log(1+x)-x/(1+x) is (i) increasing, (ii) decreasing function. (b) Find the intervals in which the function f(x)= x/(log_(e) x),x gt 0 is increasing or decreasing.

Let f(x)=1(1-x)^(2) sin^(2)x + x^(2) for all x in IR, and let g(x)={:(x),(int),(1):} ((2(t-1))/(t+1)- ln t) f(t)dt for all x in (1,oo)Which of the folloiwng is true ?

If the pairs of straight lines represented by 3x^(2) + 2hxy - 3y^(2) = 0 and 3x^(2) + 2hxy - 3y^(2) + 2x - 4y + c = 0 form a square, then (h, c) =

Let p,q, r denote respectively the statements :" you are honest ", "you are laborious ",and " you will receive a promotion " Translate (p vv q) rarr rstatements into English language .

Evaluation of definite integrals by subsitiution and properties of its : int_(-1)^(1)(x-[x])dx=........ where [.] denotes maximum integer function.

How many of the functions Suppose A is a set of n elements and B is a set with m elementsare one - one with (i)m=n ,(ii) m < n, (iii) m>n

Integrate the rational functions (2x-3)/((x^(2)-1)(2x+3))

I: The solution of x^(2) + y^(2) (dy)/(dx) = 4 " is " x^(3) + y^(3) = 12x + c II: The solution of2xy (dy)/(dx) = 1 + y^(2) " is " x^(2) + y^(2) = cx

Choose the correct answer int(dx)/(x^(2)+2x+2) equals

Find values of k if the points (3,5), (k,3) and (1,1) are collinear

IF l and m are variable real numbers such that 5l^2-4lm+6m^2+3l=0, then the variable line lx+my=1 always touches a fixed parabola,whose axis is parallel to the X-axis. If (c,d) is the focus of the parabola , then the value of 2^|d-c| is

IF l and m are variable real numbers such that 5l^2-4lm+6m^2+3l=0, then the variable line lx+my=1 always touches a fixed parabola,whose axis is parallel to the X-axis. If ex+f=0 is directrix of the parabola and e,f are prime numbers , then the value of |e-f| is

A straight line through P(1, 2) is such that﻿ its intercept between the axes is bisected﻿ at P. Its equation is﻿

The values ofsum_(m=1)^(10)[sin(2pim//11)-icos(2pim//11)] is

Form the differential equation by eliminating the arbitrary constant from the equation (y-b)^(2) = 4 (x-a)

If two consecutive terms in the expansion of (x+a)^n are equal to where n is a positive integer then ((n+1)a)/(x+a) is

If the vectors a hati + 2 hatj + 3 hatk and - hati + 5 hatj + a hatk are perpendicular to each other, then a equals :

If P (A) = P (B) = x and P(AnnB)=P(A'nnB')=1//3, then x = ?

Find the integral of (1)/(sqrt(a^(2)-x^(2)) with respect to x and hence find int (1)/(sqrt(7-6x-x^(2))dx

int_(- pi//2)^(pi//2) ( x^(14) sin^(11) x+ sin^(2) x) dx is equal to

The straight lines x+ 3y -4=0,x+y-4=0and 3x+y - 4 = 0

If e^(ax) +e^(-bx) = p_0 +p_1x +p_2x^2 + ....oo then (p_0 , p_1,p_2) =

A unit vector vec(a) is perpendicualr to the vectors hati+2hatj-hatk and 3hati-hatj+hatk then find the components of vec(a).

Which of the following limits does not exist ?

If (sqrt(5)+sqrt(3)i)^(33)= 2^(49)z, then modulus of the complex number z is equal to

Let z and omega be two complex numberssuch that |z|le 1, |omega| le 1 and |z+ iomega| = |z_(1)-z_(2)|is equal to

If pnea,qneb,rnec and |{:(p,b,c),(a,q,c),(a,b,r):}|=0 then prove that p/(p-a)+q/(q-b)+r/(r-c)=2

A statement among the following is

Find (dy)/(dx) in the following : sin^(2)y+cos xy=k.

If A is a square matrix of order 3xx3, then |KA| is equal to

If alpha, beta, gamma are the roots of x^(3)+2x^(2)-3x-1=0 then alpha^(-2)+beta^(-2)+gamma^(-2) is equal to

If 0 lt x lt 1, the first negative term in the expansion of (1+x)^(27//5) is

int 2^(x)(f'(x)+f(x)log2)dx is equal to

LetC_1 and C_2 be respectively, the parabolas x^2=y=-1 and y^2=x-1 Let P be any point on C_1 and Q be any point on C_2 . Let P_1 and Q_1 be the refelections of P and Q, respectively with respect to the line y=x. Arithemetic mean of PP_1and Q Q_1 is always less than

Differentiate the functions w.r.t. x. x^(sin x)+(sin x)^(cos x).

LetC_1 and C_2 be respectively, the parabolas x^2=y=-1 and y^2=x-1 Let P be any point on C_1 and Q be any point on C_2 . Let P_1 and Q_1 be the refelections of P and Q, respectively with respect to the line y=x. If the point p(pi,pi^2+1)and Q(pi^2+1mu)then P_1 and Q_1 are

How many numbers between 10 and 10,000 can be formed by using the digits 1, 2, 3, 4, 5 if No digit is repeated in any number

The slope of the normal to the curve y = 2x^(2) + 3 sin x at x = 0 is

A line makes the same angle theta, with each of the x and axis. If the angle beta, which it makes with y-axis, is such that sin^(2)beta=3sin^(2)theta, then cos^(2)theta equals:

In a certain college, 25% of the boys and 10% of the girls are studying mathematics. The girls constitute 60% of the student strength. If a student selected at random is found studying mathematics, find the probability that the students is a girl.

2. C_2 + 6. C_3 + 12. C_4 +….. + n (n-1) . C_n=

A tower subtends an angle 75^(@) at a point on the same level as the foot of the tower and at another point, 10 meters above the first, the angle of depression of the foot of the tower is 15^(@). The height of the tower is (in meters)

If a*b=LCM of (a,b) then 4*6

If a^2+8b^2+2c^2+2d^2-4ab-4bc-4bd=0 ,then the value of |{:(a,b),(c,d):}| is

Show that the area of the triangle formed by the two tangents through P(x_(1),y_(1)) to the circle S=x^(2)+y^(2)+2gx+2fy+c=0 and the chord of contact of P w.r.t S=0 is (r(S_(11))^(3//2))/(S_(n)+r^(2)), where r is the radius of the circle.

If area of triangle is 35sq units with vertices (2,-6) ,( 5,4) and( k,4) Then k is

If orthocentre of the triangle formed by ax^2+2hxy+by^2= 0 and px + qy =1is (r,s) then prove that r/p-s/q=(a+b)/(aq^2+bq^2-2hpq)

If function g is defined by g(x)=x-1 and 2g(c)=10, what is the value of g(3c)?

If [[3,-4],[1,2]] [[x],[y]]= [[3],[11]], then write the correct answer from· the following :

A straight line through P(1, 2) is such that its intercept between the axes is bisected at P. Its equation is

If ab=LCM of (a,b) then 46