28278 + Interview Questions in Maths in Class 12

1.	If bara,barb,barc are non-coplaner, then show that the vectors bara -bar b , barb + barc ,bar c + baraare coplanar
Answer» ANSWER :HENCE, the GIVEN VECTORS are COPLANAR.

Discussion

2.	A sociologist chose 300 students at random from each of two schools and asked each student how many siblings he or she has. The results are shown in the table below. There are a total of 2,400 students at Lincoln School and 3,300 students at Washington School. Based on the survey data, which of the following most accurately compares the expected total number of students with 4 siblings at the two schools?
Answer» The total number of STUDENTS with 4 siblings is expected to be equal at the two schools. The total number of students with 4 siblings at Lincoln School is expected to be 30 more than at Washington School. The total number of students with 4 siblings at Washington School is expected to be 30 more than at Lincoln School. The total number of students with 4 siblings at Washington School is expected to be 900 more than at Lincoln School. Solution :Choice C is correct. When survey participants are selected at RANDOM from a larger POPULATION, the sample statistics calculated from the survey can be generalized to the larger population. Since 10 of 300 students surveyed at Lincoln School have 4 siblings, one can estimate that this same ratio holds for all 2,400 students at Lincoln School. Also, since 10 of 300 students surveyed at Washington School have 4 siblings, one can estimate that this same ratio holds for all 3,300 students at Washington School. Therefore, APPROXIMATELY`(10)/(30)xx 2,400 = 80` students at Lincoln School and `(10)/(30)xx 3,300 = 110` students at Washington School are expected to have 4 siblings. Thus, the total number of students with 4 siblings at Washington School is expected to be 110 − 80 = 30 more than the total number of students with 4 siblings at Lincoln School. Choices A, B, and D are incorrect and may result from either conceptual or calculation errors. For example, choice A is incorrect, EVEN though there is the same ratio of survey participants from Lincoln School and Washington School with 4 siblings, the two schools have a different total number of students, and thus, a different expected total number of students with 4 siblings.

Discussion

3.	A sociologist chose 300 students at random from each of two schools and asked each student how many siblings he or she has. The results are shown in the table below. There are a total of 2,400 students at Lincoln School and 3,300 students at Washington School. What is the median number of siblings for all the students surveyed?
Answer» 0 1 2 3 Solution :Choice B is correct. The MEDIAN of a data set is the middle value when the data points are sorted in either ascending or descending ORDER. There are a total of 600 data points PROVIDED, so the median will be the average of the 300th and 301st data points. When the data points are sorted in order: Values 1 through 260 will be 0. Values 261 through 450 will be 1. Values 451 through 540 will be 2. Values 541 through 580 will be 3. Values 581 through 600 will be 4. Therefore, both the 300th and 301st values are 1, and hence the median is 1. CHOICES A, C, and D are incorrect and may result from either a calculation error or a conceptual error.

Discussion

4.	Find the values of a, for which the quadratic expression ax^2 + (a - 2) x - 2 is negative for exactly two integral values of x.
Answer» ANSWER :`[1,2)`

Discussion

5.	Findint(dx )/(x^2 -a^2) . Henceevaluateint(dx)/(x^2 -16)
Answer» ANSWER :`=>int (dx)/(x^2 - 16 ) = int (dx)/(x^2-4^2) =1/8 log""\|(x-4)/(x+4)\|+C`

Discussion

6.	Let a,b and c be three distinct real roots of the cubic x ^(3) +2x ^(2)-4x-4=0. If the equation x ^(3) +qx ^(2)+rx+le =0 has roots 1/a, 1/b and 1/c, then the vlaue of (q+r+s) is equal to :
Answer» `3/4` `1/2` `1/4` `1/6` ANSWER :C

Discussion

7.	int_(2)^(3) (x dx)/(x^(2)+1)
Answer» SOLUTION :`"Let I "=int_(2)^(3)(x)/(x^(2)+1)dx` `underset(rArr""2x dx=DT rArr x dx=(dt)/(2))(" Let " x^(2)+1=t)` x=2 then t =5 and x=3 then t=0 `I=(1)/(2)int_(0)^(10) (1)/(t) dt=(1)/(2)[log \|t\|]_(5)^(10)` `=(1)/(2)[log \|10log \|5\|]` `=(1)/(2)log\|(10)/(5)\|=(1)/(2)log 2`

Discussion

8.	Find the area of the region bounded byx-axis, the sine curve y=sinx, the lines x=0 and x=3pi.
Answer» ANSWER :4

Discussion

9.	Examine the continuity of the functionf(x)= {(1+x",","if " x le 2),(5-x",","if" x gt 2):} at x= 2
Answer» ANSWER :x=2

Discussion

10.	Given that the events A and B are such that P(A) = (1)/(2), P(A cup B)=(3)/(5) and P(B) = p. Find p if they are (i) mutually exclusive (ii) independent.
Answer» Answer :(i) `p=(1)/(10)`, (II) `p=(1)/(5)`

Discussion

11.	Let overset (-) xand M.D. be the mean and the mean deviations about overset( -)xof n observations x_i , i= 2……,nIf each of the observation is increased by 5, then the new mean and mean deviations about the new mean, respectively are:
Answer» `overset(-) X ,M.D.` ` overset(-) x +5, M.D.` `overset(-)x ,M.D. +5` ` overset(-) x -15, M.D. +5` ANSWER :B

Discussion

12.	For a positive constant a find (dy)/(dx), where y= a^(t + (1)/(t)), and x= (t + (1)/(t))^(a)
Answer» ANSWER :`(a^(t + (1)/(t) -1)LOG a)/((t + (1)/(t))^(a-1))`

Discussion

13.	The circumcentre of the triangle passing through (1,sqrt(3)), (1, -sqrt(3)), (3, -sqrt(3)) is
Answer» only I is TRUE only II is true both I and II are true NEITHER I nor II are true ANSWER :C

Discussion

14.	If 9 digits (1 to 9) are arranged in the spaces of number 1263__________6, then
Answer» PROBABILITY THA the NUMBER is divisible by 9 is 1 probability that the number of divisible by 9 is 1/2. probability that the number is divisible by 10 is 1/3 probability that the number is divisible by 10 is 0 . Answer :A::D

Discussion

15.	The value of (sum_(i=0)^(100).^(k)C_(1).^(m-k)C_(100-i)((k-i)/(m-100)))/(.^(m)C_(100)), whre m-kgt100,kgt100 is
Answer» `K/m` `m/k` `k/(m^(2))` `m/(k^(2))` ANSWER :a

Discussion

16.	Show that the derivatives of Sin^(-1)sqrt((x-beta)/(alpha-beta))"Tan"^(-1)sqrt((x-beta)/(alpha-x)) are equal .
Answer» ANSWER :`(1)/(2sqrt((alpha-x)(x-beta)))`

Discussion

17.	Givne lines are (x-1)/(l) = (y + 1)/(m) = z/n and (x+1)/(m) = (y-3)/(n) = (z-1)/(l) where l gt m gt n l,m, n are roots of the equation x^3 + x^2 -4 x = 4 then the angle between them is ........
Answer» `pi/2` `COS^(-1)(1/4)` `cos^(-1)(-4/9)` `cos^(-1)(5/9)` ANSWER :C

Discussion

18.	Find the sum of barx and bary for the two regression lines: 40x-18y-214=0 and 8x-10y +66=0.
Answer» SOLUTION :N/A

Discussion

19.	Finda unitvectorperpendicularto eachof thevectors ( veca + vecb)and (veca- vecb)whereveca = hati + hatj + hatk,vecb = hati+2 hatj+ 3hatk
Answer» Answer :`= ( -hat(i) + 2 hat(J) + hat(k) )/( sqrt6)`

Discussion

20.	Find (dy)/(dx) in the following : 2x+3y= sin x
Answer» ANSWER :`(cosx-2)/(3)`

Discussion

21.	The point on the ellipse x^(2)+2y^(2)=6which is nearest to the line x-y=7 is
Answer» `((sqrt6)/(sqrt5),(SQRT3)/(sqrt5))` `(2,-1)` `(-2, 1)` `(-2, -1)` SOLUTION :`P(sqrt6 cos theta, sqrt3 sin theta)` lies on ELLIPSE. Normal at P is perpendicular to `x-y=7` `therefore""1=-(sqrt6 cos theta)/(2sqrt3 sin theta), tan theta=(1)/(SQRT2), theta" line in "4^("th") " QUADRANT, "sin theta=-(1)/(sqrt3), cos theta=(sqrt2)/(sqrt3)` `P(2, -1)`

Discussion

22.	Integration by partial fraction : int (x^(2))/((x sinx+cosx)^(2))dx=....+c
Answer» `(sinx+X COSX)/(x sinx+cosx)` `(six-xcosx)/(x sinx+cosx)` `(sinx-x cosx)/(x sinx-cosx)` NONE of these Answer :D

Discussion

23.	Show that the function f: R_()to R _() defined by f (x) = 1/x is one-one and onto, where R _() is the set of all non-zero numbes. Is the result true, if the domain R _() is replaced by N with co-domain being same as R_(**) ?
Answer» ANSWER :NO

Discussion

24.	Differentiate w.r.t x : sin(x^(3))
Answer» Solution :` y= SIN x^(3)` ` (DY)/(DX)= cos x^(3)d/(dx) (x^(3))` ` = cos x^(3) (3X^(2))` ` 3x^(2) cos x^(3)`

Discussion

25.	Let g(x)=a_(0)+a_(1)x+a_(2)x^(2)+a_(3)x^(3) " and " f(x)=sqrt(g(x)), f(x) has its non-zero local minimum and maximum values at -3 and 3, respectively. If a_(3) in the domain of the function h(x)=sin^(-1)((1+x^(2))/(2x)). f(10) is defined for
Answer» `a_(0) gt 830` `a_(0) lt 830` `a_(0)=830` NONE of these ANSWER :d

Discussion

26.	If y = cos^(-1) ((1)/( sqrt(1+t^(2)))), x = sin^(-1) (sqrt((t^(2))/(1 + t^(2)))), "find " (dy)/(dx)
Answer» ANSWER :1

Discussion

27.	If A is square matrix such that A^(2)=A, then (I+A)^(3)-7A is equal to ……..
Answer» A I-A I 3A Answer :C

Discussion

28.	If 5 coins are tossed find the probability that no two or more consecutive heads occur.
Answer» ANSWER :`(13)/(32)`

Discussion

29.	If alpha, beta, gamma are roots of 2x^(3) + 3x^(2) - 6x + 3 =0, then the value of (1)/(alpha^(4)) + (1)/(beta^(4)) + (1)/(gamma^(4))=
Answer» `11//7` `11//7` `10//3` `-10//3` ANSWER :4

Discussion

30.	If S_(n)={(1)/(1+sqrt(n))+(1)/(2+sqrt(2n))+(1)/(3+sqrt(3n))+....+(1)/(n+sqrt(n^(2)))} then {:(" "Lt),(n rarr oo):} S_(n)=
Answer» 2 LOG 2 log 2 3 log 2 `1/2 log 2` ANSWER :A

Discussion

31.	A coin is loaded such that P(H)=3P(T). It is tossed 3 times. Let X be the random variable which indicates the number of heads which occur. Find the mean of X, variance of X.
Answer» ANSWER :`(9)/(4), (9)/(16)`

Discussion

32.	Let E_(1)equiv a x^(2)+ bx + c,E_(2)equiv bx^(2) + cx + a,E_(3) equiv cx^(2) + bx + a and (a^(2))/(bc)+ (b^(2))/(ca)+(c^(2))/(ab)=3. If thesequadratic expressions have a common zero, then the quadratic expression having zeros that are common to E_(3)and different from the zeros of E_(1) is
Answer» `X^(2)- (a(B+c))/(bc) x+ bc` `AX^(2)+bx +c` `x^(2)- b(c+ a) x + AC` `x^(2)- (a(b+c))/(bc)+(a^(2))/(bc)` Answer :D

Discussion

33.	A manufacturer of clectronic circuits has a stock of 200 resistors, 120 transistors and 150 capacitors and is required to produce two types of circuits A and B. Type A requires 20 resistors, 10 transistors and 10 capacitors. Type B requires 10 resistors, 20 transistors and 30 capacitors. If the profit on type A circuit is 50 and that on type B circuit is 60, formulate this problem as a LPP do that the manufacturer can maximise his profit. How many of circuits of Type A and of type B, should be produced by the manufacturer so as to maximise his profit? Determine the maximum profit.
Answer» Answer :Type -A : 6 units Type -B : 3 units MAXIMUM PROFIT = रु 480

Discussion

34.	Prove that int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx and hence evaluate int_((pi)/(6))^((pi)/(3))(1)/(1+sqrt(tanx))dx.
Answer» ANSWER :`1= (PI)/(12)`

Discussion

35.	A bag consists of 10 balls each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?.
Answer» ANSWER :`((9)/(10))^(4)`

Discussion

36.	Sum of last 8 coefficients in (1 + x)^16 is
Answer» `[2^15 - 1/2 . ""^16C_8]` `[2^15 + 1/2.""^6C_2]` `[2^15 - 1/2 . ""^6C_2]` `[2^15- 1/4 . ""^6C_2]` ANSWER :A

Discussion

37.	Evaluate the following integrals. int(dx)/((x+1)(x+2))
Answer» ANSWER :`LOG\|(x+1)/(x+2)\|+C`

Discussion

38.	A car is parked among 'N' cars in a row, not at either end. On his return, the owner finds that exactly 'r' of the 'N' places are still occupied. What is the probability that both neighbouring places are empty ?
Answer» ANSWER :`((N-r)(N-r-1))/((N-1)(N-2))`

Discussion

39.	Evaluate the following integrals. intsinxsin2xsin3xdx
Answer» Answer :`(1)/(4)[(cos6x)/(2)-(COS4X)/(4)-(cos2x)/(2)]+C`

Discussion

40.	If A and B are independentevents such that P(A) = 0.3 andP(B) = 0 . 4 then P(bar(A) cup bar(B)) =
Answer» `0 . 88` `0 . 52` `0 . 2 ` `(1)/(3)` Answer :A

Discussion

41.	The position vectors of the points P and Q are (3,1,2) and (1,-2,-4) repectively. The equation of the plane passing through the point Q and perpendicular to bar PQ is...............
Answer» `bar(R). (2 HAT i + 3 hat j + 6 hat k) = 28` `bar r . (2 hat i + 3 hat j + 6 hat k) = 32` `bar r. (2 hat i + 3 hat j + 6 hat k) + 28 = 0` NONE of these Answer :C

Discussion

42.	Evaluate: int(dx)/(sin^4x+sin^2xcos^2x+cos^4x)
Answer» ANSWER :`1/sqrt3tan^(-1)((tan^2x-1)/(sqrt3tanx))+C`

Discussion

43.	A boat is moving in direction of vector -4hati+3hatj with a speed of 10 m/s. Velocity vector of boat can be expressed as :
Answer» `-8hati+6hatj` `-40hati+30hatj` `-4/5hati+3/5hatj` `-6hati+8hatj` Solution :UNIT vector in the direction of velocity = U sin t cector along the given vector `thereforehatv=(-4hati+3hatj)/(SQRT((4)^(2)+(2)^(2)))` `vecv=\|vecv\|vecv=10((-4)/(5)hati+(3)/(5)hatj)=-8hati+6hatjm//s`

Discussion

44.	The coefficient of x^(n) in (1+x)/(1!)+((1+x)^(2))/(2!)+((1+x)^(3))/(3!)+....
Answer» only I is TRUE only II is true both I and II are true NEITHER I nor II true ANSWER :A

Discussion

45.	If \|bar(a)+bar(b)\|=\|bar(a)-bar(b)\|, then the vectors bar(a) and bar(b) are orthogonal.
Answer» ANSWER :TRUE STATEMENT

Discussion

46.	Two cards are drawn at random from the well shuffled pack of 52 cards. If the first card is not replaced before the second card is drawn, then the probability of getting two aces is ………..
Answer» `(1)/(221)` `(1)/(169)` `(1)/(26)` `(1)/(2)` Answer :A

Discussion

47.	sech^(-1) ((1)/(sqrt(2))) + cosech^(-1) (-1)=
Answer» 0 `SQRT(2)+1` `sqrt(2)` `sqrt(2) -1` ANSWER :A

Discussion

48.	A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.
Answer» ANSWER :LENGTH = `(20)/(pi+4)` m , BREADTH = `(10)/(pi+4)` m

Discussion

49.	If \|z-1\|=\|z-3\| then the locus of z is
Answer» A. STRAIGHT LINE paralled to x-axis B. straight line through origin C. straight line PARALLEL to imaginary axis D. straight line passing through origin and EQUALLY inclined to the axis Answer :C

Discussion

50.	Consider the function f(x) = {{:( x^(2) - 1"," , -1 le x le 1) , ( lnx "," , 1 lt x le e):} Let f_(1) (x) = f (\|x\|) f_(2) (x) = \|f(\|x\|)\| f_(3) (x) = f (-x) Now answer the question If f_(4) (x) = log_(27) (f_(3) (x) + 2), then range of f_(4) (x) is
Answer» 1 2 3 4 Answer :C

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

If bara,barb,barc are non-coplaner, then show that the vectors bara -bar b , barb + barc ,bar c + baraare coplanar

Find the values of a, for which the quadratic expression ax^2 + (a - 2) x - 2 is negative for exactly two integral values of x.

Findint(dx )/(x^2 -a^2) . Henceevaluateint(dx)/(x^2 -16)

Let a,b and c be three distinct real roots of the cubic x ^(3) +2x ^(2)-4x-4=0. If the equation x ^(3) +qx ^(2)+rx+le =0 has roots 1/a, 1/b and 1/c, then the vlaue of (q+r+s) is equal to :

int_(2)^(3) (x dx)/(x^(2)+1)

Find the area of the region bounded byx-axis, the sine curve y=sinx, the lines x=0 and x=3pi.

Examine the continuity of the functionf(x)= {(1+x",","if " x le 2),(5-x",","if" x gt 2):} at x= 2

Given that the events A and B are such that P(A) =﻿ (1)/(2), P(A cup B)=(3)/(5) and﻿ P(B) = p. Find p if they are (i) mutually exclusive (ii) independent.

Let overset (-) xand M.D. be the mean and the mean deviations about overset( -)xof n observations x_i , i= 2……,nIf each of the observation is increased by 5, then the new mean and mean deviations about the new mean, respectively are:

For a positive constant a find (dy)/(dx), where y= a^(t + (1)/(t)), and x= (t + (1)/(t))^(a)

The circumcentre of the triangle passing through (1,sqrt(3)), (1, -sqrt(3)), (3, -sqrt(3)) is

If 9 digits (1 to 9) are arranged in the spaces of number 1263__________6, then

The value of (sum_(i=0)^(100).^(k)C_(1).^(m-k)C_(100-i)((k-i)/(m-100)))/(.^(m)C_(100)), whre m-kgt100,kgt100 is

Show that the derivatives of Sin^(-1)sqrt((x-beta)/(alpha-beta))"Tan"^(-1)sqrt((x-beta)/(alpha-x)) are equal .

Givne lines are (x-1)/(l) = (y + 1)/(m) = z/n and (x+1)/(m) = (y-3)/(n) = (z-1)/(l) where l gt m gt n l,m, n are roots of the equation x^3 + x^2 -4 x = 4 then the angle between them is ........

Find the sum of barx and bary for the two regression lines: 40x-18y-214=0 and 8x-10y +66=0.

Finda unitvectorperpendicularto eachof thevectors ( veca + vecb)and (veca- vecb)whereveca = hati + hatj + hatk,vecb = hati+2 hatj+ 3hatk

Find (dy)/(dx) in the following : 2x+3y= sin x

The point on the ellipse x^(2)+2y^(2)=6which is nearest to the line x-y=7 is

Integration by partial fraction : int (x^(2))/((x sinx+cosx)^(2))dx=....+c

Show that the function f: R_(**)to R _(**) defined by f (x) = 1/x is one-one and onto, where R _(**) is the set of all non-zero numbes. Is the result true, if the domain R _(**) is replaced by N with co-domain being same as R_(**) ?

Differentiate w.r.t x : sin(x^(3))

Let g(x)=a_(0)+a_(1)x+a_(2)x^(2)+a_(3)x^(3) " and " f(x)=sqrt(g(x)), f(x) has its non-zero local minimum and maximum values at -3 and 3, respectively. If a_(3) in the domain of the function h(x)=sin^(-1)((1+x^(2))/(2x)). f(10) is defined for

If y = cos^(-1) ((1)/( sqrt(1+t^(2)))), x = sin^(-1) (sqrt((t^(2))/(1 + t^(2)))), "find " (dy)/(dx)

If A is square matrix such that A^(2)=A, then (I+A)^(3)-7A is equal to ……..

If 5 coins are tossed find the probability that no two or more consecutive heads occur.

If alpha, beta, gamma are roots of 2x^(3) + 3x^(2) - 6x + 3 =0, then the value of (1)/(alpha^(4)) + (1)/(beta^(4)) + (1)/(gamma^(4))=

If S_(n)={(1)/(1+sqrt(n))+(1)/(2+sqrt(2n))+(1)/(3+sqrt(3n))+....+(1)/(n+sqrt(n^(2)))} then {:(" "Lt),(n rarr oo):} S_(n)=

A coin is loaded such that P(H)=3P(T). It is tossed 3 times. Let X be the random variable which indicates the number of heads which occur. Find the mean of X, variance of X.

Let E_(1)equiv a x^(2)+ bx + c,E_(2)equiv bx^(2) + cx + a,E_(3) equiv cx^(2) + bx + a and (a^(2))/(bc)+ (b^(2))/(ca)+(c^(2))/(ab)=3. If thesequadratic expressions have a common zero, then the quadratic expression having zeros that are common to E_(3)and different from the zeros of E_(1) is

Prove that int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx and hence evaluate int_((pi)/(6))^((pi)/(3))(1)/(1+sqrt(tanx))dx.

A bag consists of 10 balls each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?.

Sum of last 8 coefficients in (1 + x)^16 is

Evaluate the following integrals. int(dx)/((x+1)(x+2))

A car is parked among 'N' cars in a row, not at either end. On his return, the owner finds that exactly 'r' of the 'N' places are still occupied. What is the probability that both neighbouring places are empty ?

Evaluate the following integrals. intsinxsin2xsin3xdx

If A and B are independentevents such that P(A) = 0.3 andP(B) = 0 . 4 then P(bar(A) cup bar(B)) =

The position vectors of the points P and Q are (3,1,2) and (1,-2,-4) repectively. The equation of the plane passing through the point Q and perpendicular to bar PQ is...............

Evaluate: int(dx)/(sin^4x+sin^2xcos^2x+cos^4x)

A boat is moving in direction of vector -4hati+3hatj with a speed of 10 m/s. Velocity vector of boat can be expressed as :

The coefficient of x^(n) in (1+x)/(1!)+((1+x)^(2))/(2!)+((1+x)^(3))/(3!)+....

If |bar(a)+bar(b)|=|bar(a)-bar(b)|, then the vectors bar(a) and bar(b) are orthogonal.

Two cards are drawn at random from the well shuffled pack of 52 cards. If the first card is not replaced before the second card is drawn, then the probability of getting two aces is ………..

sech^(-1) ((1)/(sqrt(2))) + cosech^(-1) (-1)=

A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

If |z-1|=|z-3| then the locus of z is

Consider the function f(x) = {{:( x^(2) - 1"," , -1 le x le 1) , ( lnx "," , 1 lt x le e):} Let f_(1) (x) = f (|x|) f_(2) (x) = |f(|x|)| f_(3) (x) = f (-x) Now answer the question If f_(4) (x) = log_(27) (f_(3) (x) + 2), then range of f_(4) (x) is

Given that the events A and B are such that P(A) = (1)/(2), P(A cup B)=(3)/(5) and P(B) = p. Find p if they are (i) mutually exclusive (ii) independent.

Show that the function f: R_()to R _() defined by f (x) = 1/x is one-one and onto, where R _() is the set of all non-zero numbes. Is the result true, if the domain R _() is replaced by N with co-domain being same as R_(**) ?