28278 + Interview Questions in Maths in Class 12

1.	The position vectors of the vertices A,B,C of DeltaABC are hati-hatj-3hatk,2hati+hatj-2hatk and -5hati+2hatj-6hatk respectively. The length of the bisector AD of the angle /_BAC where D is on the line segment BC is
Answer» `15/2` `3/4sqrt(10)` `4/3sqrt(10)` None of these Solution :We have `VEC(AB)=hati+2hatj+hatk, vec(AC)-6hati+3hatj-3hatk` `implies\|vec(AB)\|=sqrt(6)` and `\|vec(AC)\|=3sqrt(6)` Clearly, point D divides BC in the ratio `AB:AC` i.e. `1:3`. `:.` Position vector ofDis `((-5hati+2hatj-6hatk)+3(2hati+hatj-2hatk))/(1+3)` `implies` Position vector of D is `=1/4(hati+5hatj-12hatk)` `:.vec(AD)=1/4(hati+5hatj-12hatk)-(hati-hatj-3hatk)` `vec(AD)=3/4(-hati+3hatj)` `implies\|vec(AD)\|=3/4sqrt(10)`

Discussion

2.	A square matrix where every element is unity is called an identity matrix.
Answer» ANSWER :FALSE

Discussion

3.	A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F_(1) and F_(2) are available. Food F_(1) costs Rs. 4 per unit food and F_(2) costs Rs. 6 per unit. One unit of food F_(1) contains 3units of vitamin A and 4 units of minerals. One unit of food F_(2) contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.
Answer» Answer :The MINIMUM VALUE of Z = 4x + 6Y is Rs. 104. A diet CONTAINS 24 units of food `F_(1)"a nd "4/3` unit of food `F_(2)`.

Discussion

4.	Let [x] denotes the greatest integer function. Draw a rough sketch of the portions of the curves x^(2)=4[sqrt(x)]y and y^(2)=4[sqrty]x that lie within the square {(x,y)\|1lexle4, 1 leyle4}. Find the area of the part of the square that is enclosed by the two curves and the line x+y=3.
Answer» ANSWER :`19/6` SQ UNITS

Discussion

5.	For each of the functions find the f _(x), f _(y), and show that f _(xy) = f _(yx). f (x,y) = (3x)/( y + sin x)
Answer» ANSWER :`THEREFORE F _(XY)= f _(YX)`

Discussion

6.	Has the function f(x)=x^(n) minimum value at x=(1)/(e)?
Answer» ANSWER :YES.

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7.	Transpose of a column matrix is a column matrix .
Answer» ANSWER :FALSE

Discussion

8.	If the bisector of the angles between the lines in the two pairs 3x^(2)-4xy+5y^(2)=0 and 5x^(2)+4xy+3y^(2)-0 are same then the angle made by the first pair with the second is
Answer» `30^(@)` `45^(@)` `60^(@)` `90^(@)` ANSWER :D

Discussion

9.	Thereare two values of 'a' which makes the value of determinant \|(1,-2,5),(2,a,-1),(0,4,2a)\| = 86, find the sum of these values of 'a'.
Answer» 4 `-1` 1 None of these Answer :B

Discussion

10.	Evaluate int sqrt(1 - x - x^(2))dx
Answer»

Discussion

11.	If x^(2) + 9y^(2)= 1, then minimum and maximum value of 3x^(2) - 27y^(2) + 24xy respectively
Answer» `0, 5` `-5, 5` `-5, 10` `0, 10` Solution :Let `X = cos theta, y = (1)/(3) sin theta` `Z = 3cos^(2)theta - 27(1)/(9) sin^(2)theta + 8 sinthetacostheta` `= 3 COS2THETA+ 4sin2theta rArr -5 le Z le 5`

Discussion

12.	A general linearprogramming problemis to maximize or minimizea function f= px +qy, p^(2)+q^(2) ne 0 subject ot (i) x ge 0, y ge 0,(ii) a_(1)x+b_(1)ygec_(1),(iii)a_(2)x+b_(2)ylec_(2) etcthen f and (i) (ii) , (iii) etc are defined as
Answer» objectivefunction non negativityconstraints negativity constraits production FUNCTION ANSWER :A::B

Discussion

13.	The position vectors of the points (1, -1) and (-2, m) are vec(a) and vec(b) respectively. If vec(a) and vec(b) are collinear then find the value of m.
Answer» ANSWER :m = 2

Discussion

14.	Examine the continuity of the following function at given point : f(x)=(log x-log 8)/(x-8)", for "x ne 8 "8,for "x=8 "at,"x=8
Answer» SOLUTION :Given `f(8)=8"…(i)"` `UNDERSET(xrarr8)(lim)f(x)=underset(xrarr8)(lim)(logx-LOG8)/(x-8)` Putting `x=8+h,` then `x-8=h` and as `xrarr8, hrarr0.` `therefore""underset(xrarr8)(lim)f(x)=underset(hrarr0)(lim)(log(8+h)-log8)/(h)` `=underset(hrarr0)(lim)(log((8+h)/(8)))/(h)` `=underset(hrarr0)(lim)(log(1+(h)/(8)))/((h)/(8))xx(1)/(8)` `=(1)/(8)XX1(because underset(xrarr0)(lim)(log(1+x))/(x)=1)"...(ii)"` From equation (i) and (ii), `underset(xrarr8)(lim)f(x) ne f(8)` `therefore"f is discontinuous at x = 8."`

Discussion

15.	If z=x+ iy is a complex number such that z^(1//3)=a +ib, thent he value of (1)/(a^2 +b^2)((x)/(a)+(y)/(b))=
Answer» -1 -2 0 2 Answer :B

Discussion

16.	I: If z_1and z_2 are two nonzero complex numbers such that \|z_1+z_2\|=\|z_1\|+\|z_2\| " then " agz_(1)-argz_(2)" is " pi//2 II : If z_1 and z_2 are two complex numbers such that \|z_1z_2\|=1 and arg z_1 - arg z_2=pi//2" then " bar(z)_1 barz_2=-i
Answer» only I is TRUE only II is true both I and II are true NEITHER I nor II are true ANSWER :B

Discussion

17.	Find the number of palindromes with 6 digits that can be formed using the digits (i) 0,2,4,6,8 (ii) 1,3,5,7,9
Answer» ANSWER :`5^3`

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18.	cos ((dy)/(dx)) = a (a ne R), y = 1 when x = 0
Answer» ANSWER :`COS((y - 2)/(X)) = a`

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19.	A manufacturer has three machine operators A, B and C. The first operator A produces. 1% defective items, whereas the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced. What is the probability that it was produced by A?
Answer» ANSWER :`(5)/(34)`

Discussion

20.	The sum of 10 item is 12 and sum of their squares is 18, then find the standard deviation ?
Answer» ANSWER :`3//5`

Discussion

21.	Let A is a matrix of order 100xx 50 andB is matrix of order50xx 75 and AB =C if matrix D is obtained by eliminating n columns and n+25rows of C and \|D\| ne 0,then n can be
Answer» 80 50 20 10 Answer :B

Discussion

22.	If a, b and n are positive find the value of 1+(na)/(a+b)+(n(n+1))/(2!) ((a)/(a+b))^(2)+…
Answer» SOLUTION :N/A

Discussion

23.	A random variable X has the following probability distribution: Determine (i) K (ii) P(X lt 3) (iii) P(X gt 6) (iv) P(0 lt X lt 3)
Answer» Answer :`(i)(1)/(10), (ii) (3)/(10), (III) (17)/(100), (iv) (3)/(10)`

Discussion

24.	The minimum value of \|alpha bomega+comega^2\| , where a, b and c are all not equal integer and omega( ne 1) is a cube root of unity , is
Answer» `sqrt(3)` `1/2` 1 0 Solution :Let`Z\|alpha+bomega +comega^2\|` `rArrz^2 = \|alpha + b omega+ comega^2\|=(a^2 + b^2 +c^2-ab-bc -ca)or z^2=1/2{(a-b)^2+(b-c)^2+(c-b)^2}` since a,b,care all intergers but not all simulataneously EQUAL `RARR ` If a=bthen `a ne c and be ne c ` Because DIFFERENCE of intergerse = interger ` rArr(b-c)^2 le 1 ` and we have taken `a= b rArr (a -b)^2 =0` From EQ.(i)`z^2ge 1/2 (0+1+1)` `rArr z^2ge 1` Hence minimum value of \|Z\| is 1

Discussion

25.	If (veca + vecb).(veca-vecb) = 0 show that \|veca\| = \|vecb\|.
Answer» SOLUTION :`(veca+vecb).(veca-vecb) = 0` `implies veca.(veca-vecb)+vecb(veca-vecb) = 0` [because DOT product is distributed over VECTOR addition.] `implies veca.veca-veca.vecb+vecb.veca-vecb.vecb = 0` `implies veca.veca-vecb.vecb = 0 [because vecavecb = vecb.veca = 0]` `implies \|veca\|^2 = \|vecb\|^2 implies \|veca\| = \|vecb\|` (Proved)

Discussion

26.	If OA is equally inclined to OX,OY and OZ and if A is sqrt(3) unit from the origin , then A is
Answer» `(3,3,3)` `(-1,1,-1)` `(-1,1,1)` `(1,1,1)` ANSWER :D

Discussion

27.	The minimum value of \|z_(1)-z_(2)\| as z_(1) and z_(2) vary over the curves \|sqrt(3)(1-2z)+2i\|=2sqrt(7) and \|sqrt(3)(-1-z)-2i\|=\|sqrt(3)(9-z)+18i\| respectively is
Answer» `(7sqrt(7))/(2sqrt(3))` `(5sqrt(7))/(2sqrt(3))` `(14sqrt(7))/(sqrt(3))` `(7sqrt(7))/(5sqrt(3))` Solution :We have, `\|sqrt(3)(1-2z)+2i\|=2sqrt(7)` `\|sqrt(3)(1-2z)+2i\|=2sqrt(7)` `rArr \|-2sqrt(3z)+(sqrt(3)+2i)\|=2sqrt(7)` `rArr \|Z-(1/2+1/sqrt(3)i)\|=sqrt(7/3)` Clearly, it represents a CIRCLE haivng center at `(1//2,1//sqrt(3))` and radius `r_(1)=sqrt(7/3)`. it is given that `z_(1)` LIES on (i) The equation of another curve is `\|sqrt(3)(-1-z)-2i\|=\|sqrt(3)(9-z)+18i\|` or `\|-1-z-2/sqrt(3)i\|=\|9-z+6sqrt(3)i\|` or `,\|z+1+2/sqrt(3)i\|=\|z-9-6sqrt(3)i\|` or `,\|z-(-1-2/sqrt(3)i)\|=\|z-(9+6sqrt(3)i)\|` This, represents perpendicular bisector of the line segement joining point `A(-1,-2/sqrt(3))` and `B(9,6sqrt(3))`. The coordinates of the mid-point C of AB are `(4,8//sqrt(3))`. Clearly, A,B and the center of the circle are collinear.

Discussion

28.	If pth, qth, rth terms of an A.P are a,b,c then a(q - r ) + b(r - p ) + c(p - q) =
Answer» 0 1 pqr abc ANSWER :A

Discussion

29.	Heinrich must buy at least 100 shares of stock for his portfolio. The shares he buys will be from stock X. Which costs $22 pershare and Stock Y. Which costs $35 per share. His budget for buying stock is no more than $4,500. He must buy at least 20 shares of Stock X and 15 shares of Y. WHich of the following represents the situation described if a is the number of shares of Stock X purchased and b is the number of shares of stock Y purchased?
Answer» `22a+35b le4,500` `a+B ge100` `a LE 20` `b le15` `22a+35b le4,500` `a+b le100` `a le 20` `b le15` `22a+35b le4,500` `a+b le100` `a GE 20` `b ge15` `22a+35b le4,500` `a+b ge100` `a ge 20` `b ge15` ANSWER :D

Discussion

30.	Find the probability of getting Two dice are rolled. What is the probability that none of the dice shows the number 2 ?
Answer» ANSWER :`((5^(2))/(6^(2)))`

Discussion

31.	Find the volume of the solid generated by revolving, about the x-axis, the infinite branch of the curve y=2((1)/(x)-(1)/(x^(2)))"for"xge1.
Answer» ANSWER :`(4PI)/(3)`

Discussion

32.	Let n and k be positive integers such that n gt (k(k+1))/2. The number of solutions (x_(1),x_(2), . ..x_(k)),x_(1) ge 1, x_(2) ge 2,, . . .x_(k) ge k for all integers satisfying x_(1)+x_(2)+ . . .+x_(k)=n is:
Answer» ANSWER :`(1)/(2)(2n-k^(2)+k-2)C_(k-1)`

Discussion

33.	The range of x for which the expansionof (1-3/x)^(-3//4) is valid is
Answer» `\|x\| LT 1` `\|x\| lt 3//4` `\|x \| lt -3//4` `\|x\| gt 3` ANSWER :D

Discussion

34.	9^7 + 7^9is divisible by
Answer» 6 24 64 72 Answer :C

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35.	int_(0)^(pi//2)(tan^(7)x)/(cot^(7)x + tan^(7)x)dx is equal to
Answer» 1.`(PI)/(2)` 2.`(pi)/(4)` 3.`(pi)/(6)` 4.`(pi)/(3)` ANSWER :B

Discussion

36.	y=x(x-3)^(2) decreases for the values of x given by …………..
Answer» `1LT x LT 3` `x lt 0` `x gt 0` `0 lt x lt (3)/(2)` Answer :A

Discussion

37.	If the plane 2x+4y+z+2=0 and x-2y+kz+5=0 are perpendicular to each other what is the value of k ?
Answer» SOLUTION :EQUATION of x-axis is `x/1=y/0=z/0`

Discussion

38.	If the function f(x) = x^(3)+2px^(2)+27x+16is stricly increasing for all x in R then the range of p is
Answer» `(-OO,(-9)/(2))cup((9)/(2),oo)` `(-oo,-9)cup(9,oo)` `((-9)/(2),(9)/(2))` `(-9,9)` Answer :C

Discussion

39.	If the roots of the equation (a)/(x-a)+(b)/(x-b)=1 are equal in magnitude and opposite in sign, then
Answer» `a=b` `a+b=1` `a-b=1` `a+b=0` ANSWER :D

Discussion

40.	Find a particular solution of the differential equation (dy)/(dx)+ycotx=4xcosecx(xne0). Given that y=0 when x=(pi)/(2).
Answer» ANSWER :`y SIN x = 2x^(2) - (pi^(2))/(2)(sin x NE 0)`

Discussion

41.	Two dice are thrown together and the total score is noted. The events E, F and G are 'a total of 4', 'a total of 9 or more', and 'a total is divisible by 5', respectively. Calculate P(E), P(F) and P(G) and decide which pairs of events, if any, are independent.
Answer» ANSWER :E, F and G are not INDEPENDENT.

Discussion

42.	An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red?
Answer» ANSWER :`(1)/(2)`

Discussion

43.	Assume that the chances of a patient having a heart attack is 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?
Answer» ANSWER :`(14)/(29)`

Discussion

44.	If the coefficients ""^nC_4, ""^nC_5, ""^nC_6 of (1 +x)^n are in A.P. then n is equal to
Answer» 12 11 7 8 Answer :C

Discussion

45.	Evaluate: int(xdx)/((x^2+2x+1))
Answer» ANSWER :`1/4((x^2-2)/(x^2+2x+2))-1/2tan^(-1)(x+1)+C`

Discussion

46.	underset(x to pi//2)lim (sqrt(1-sin x))/((pi//2-x)sqrt(1+sin x))=
Answer» `1//2` `pi//2` 0 `-1//2` ANSWER :A

Discussion

47.	Let f : (-1,1) toIRbeadifferentiablefunction withf(0)=- 1and f'(0)=1IFg(x)={f(2f(x)+2)}^2 ,theng'(0) =
Answer» 0 `-2` `4` `-4` ANSWER :D

Discussion

48.	Prove that the function f given byf(x) = log sin x" is increasing on ")0,pi/2) and decreasing on (pi/2,pi).
Answer» Solution :f(X) = log (sin x) ` rArrf'(x) = (COS x)/(sin x) = cos x` (a)f(x) is increasing. ` rArrf,(x) GT 0` ` rArrcot x gt 0` `RARRX in (0,pi/2)` `:.F(x)" is increasing in "(0,pi/2)`. (b)f(x) is decreasing. `rArrf'(x) lt 0` ` rArrcotx lt 0` ` rArrx in (pi/2, pi)` ` :. f(x)" is decreasing in "(pi/2, pi)`.

Discussion

49.	Evalute the following integrals int (x + 4)/(6x - 7 - x^(2)) dx
Answer» Answer :`- (1)/(2) log \| 6X - 7 - x^(2) \| - (7)/(2 sqrt(2)) log \| (x - 3- sqrt(2))/(x - 3+ sqrt(2))\| + c `

Discussion

50.	If vecaandvecb are two collinear vectors , then which of the following are incorrect :
Answer» `vecb=lambdaveca`,for some SCALAR `lambda` `veca=+-vecb` the respective COMPONENTS of `vecaandvecb` are not proportional both the vectors `vecaandvecb` have same DIRECTION , but different magnitudes. Answer :B::C::D

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

The position vectors of the vertices A,B,C of DeltaABC are hati-hatj-3hatk,2hati+hatj-2hatk and -5hati+2hatj-6hatk respectively. The length of the bisector AD of the angle /_BAC where D is on the line segment BC is

A square matrix where every element is unity is called an identity matrix.

Let [x] denotes the greatest integer function. Draw a rough sketch of the portions of the curves x^(2)=4[sqrt(x)]y and y^(2)=4[sqrty]x that lie within the square {(x,y)|1lexle4, 1 leyle4}. Find the area of the part of the square that is enclosed by the two curves and the line x+y=3.

For each of the functions find the f _(x), f _(y), and show that f _(xy) = f _(yx). f (x,y) = (3x)/( y + sin x)

Has the function f(x)=x^(n) minimum value at x=(1)/(e)?

Transpose of a column matrix is a column matrix .

If the bisector of the angles between the lines in the two pairs 3x^(2)-4xy+5y^(2)=0 and 5x^(2)+4xy+3y^(2)-0 are same then the angle made by the first pair with the second is

Thereare two values of 'a' which makes the value of determinant |(1,-2,5),(2,a,-1),(0,4,2a)| = 86, find the sum of these values of 'a'.

Evaluate int sqrt(1 - x - x^(2))dx

If x^(2) + 9y^(2)= 1, then minimum and maximum value of 3x^(2) - 27y^(2) + 24xy respectively

A general linearprogramming problemis to maximize or minimizea function f= px +qy, p^(2)+q^(2) ne 0 subject ot (i) x ge 0, y ge 0,(ii) a_(1)x+b_(1)ygec_(1),(iii)a_(2)x+b_(2)ylec_(2) etcthen f and (i) (ii) , (iii) etc are defined as

The position vectors of the points (1, -1) and (-2, m) are vec(a) and vec(b) respectively. If vec(a) and vec(b) are collinear then find the value of m.

Examine the continuity of the following function at given point : f(x)=(log x-log 8)/(x-8)", for "x ne 8 "8,for "x=8 "at,"x=8

If z=x+ iy is a complex number such that z^(1//3)=a +ib, thent he value of (1)/(a^2 +b^2)((x)/(a)+(y)/(b))=

I: If z_1and z_2 are two nonzero complex numbers such that |z_1+z_2|=|z_1|+|z_2| " then " agz_(1)-argz_(2)" is " pi//2 II : If z_1 and z_2 are two complex numbers such that |z_1z_2|=1 and arg z_1 - arg z_2=pi//2" then " bar(z)_1 barz_2=-i

Find the number of palindromes with 6 digits that can be formed using the digits (i) 0,2,4,6,8 (ii) 1,3,5,7,9

cos ((dy)/(dx)) = a (a ne R), y = 1 when x = 0

The sum of 10 item is 12 and sum of their squares is 18, then find the standard deviation ?

Let A is a matrix of order 100xx 50 andB is matrix of order50xx 75 and AB =C if matrix D is obtained by eliminating n columns and n+25rows of C and |D| ne 0,then n can be

If a, b and n are positive find the value of 1+(na)/(a+b)+(n(n+1))/(2!) ((a)/(a+b))^(2)+…

A random variable X has the following probability distribution: Determine (i) K (ii) P(X lt 3) (iii) P(X gt 6) (iv) P(0 lt X lt 3)

The minimum value of |alpha bomega+comega^2| , where a, b and c are all not equal integer and omega( ne 1) is a cube root of unity , is

If (veca + vecb).(veca-vecb) = 0 show that |veca| = |vecb|.

If OA is equally inclined to OX,OY and OZ and if A is sqrt(3) unit from the origin , then A is

The minimum value of |z_(1)-z_(2)| as z_(1) and z_(2) vary over the curves |sqrt(3)(1-2z)+2i|=2sqrt(7) and |sqrt(3)(-1-z)-2i|=|sqrt(3)(9-z)+18i| respectively is

If pth, qth, rth terms of an A.P are a,b,c then a(q - r ) + b(r - p ) + c(p - q) =

Find the probability of getting Two dice are rolled. What is the probability that none of the dice shows the number 2 ?

Find the volume of the solid generated by revolving, about the x-axis, the infinite branch of the curve y=2((1)/(x)-(1)/(x^(2)))"for"xge1.

Let n and k be positive integers such that n gt (k(k+1))/2. The number of solutions (x_(1),x_(2), . ..x_(k)),x_(1) ge 1, x_(2) ge 2,, . . .x_(k) ge k for all integers satisfying x_(1)+x_(2)+ . . .+x_(k)=n is:

The range of x for which the expansionof (1-3/x)^(-3//4) is valid is

9^7 + 7^9is divisible by

int_(0)^(pi//2)(tan^(7)x)/(cot^(7)x + tan^(7)x)dx is equal to

y=x(x-3)^(2) decreases for the values of x given by …………..

If the plane 2x+4y+z+2=0 and x-2y+kz+5=0 are perpendicular to each other what is the value of k ?

If the function f(x) = x^(3)+2px^(2)+27x+16is stricly increasing for all x in R then the range of p is

If the roots of the equation (a)/(x-a)+(b)/(x-b)=1 are equal in magnitude and opposite in sign, then

Find a particular solution of the differential equation (dy)/(dx)+ycotx=4xcosecx(xne0). Given that y=0 when x=(pi)/(2).

Two dice are thrown together and the total score is noted. The events E, F and G are 'a total of 4', 'a total of 9 or more', and 'a total is divisible by 5', respectively. Calculate P(E), P(F) and P(G) and decide which pairs of events, if any, are independent.

An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is﻿ noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn﻿ are put in the urn and then a ball is drawn at random. What is the probability that﻿ the second ball is red?﻿

If the coefficients ""^nC_4, ""^nC_5, ""^nC_6 of (1 +x)^n are in A.P. then n is equal to

Evaluate: int(xdx)/((x^2+2x+1))

underset(x to pi//2)lim (sqrt(1-sin x))/((pi//2-x)sqrt(1+sin x))=

Let f : (-1,1) toIRbeadifferentiablefunction withf(0)=- 1and f'(0)=1IFg(x)={f(2f(x)+2)}^2 ,theng'(0) =

Prove that the function f given byf(x) = log sin x" is increasing on ")0,pi/2) and decreasing on (pi/2,pi).

Evalute the following integrals int (x + 4)/(6x - 7 - x^(2)) dx

If vecaandvecb are two collinear vectors , then which of the following are incorrect :

An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red?