Explore topic-wise InterviewSolutions in Current Affairs.

This section includes 7 InterviewSolutions, each offering curated multiple-choice questions to sharpen your Current Affairs knowledge and support exam preparation. Choose a topic below to get started.

1.

A bag consists of 10 balls each marked with one e 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
2.

Equation of one of the two lines x^(2)+2xy+cos theta-y^(2)=0 is

Answer»

`x-y.cot THETA=0`
`x+y.tan theta=0`
`x.sin theta+y(1+cos theta)=0`
`x.cos theta+y(1+sin theta)=0`

ANSWER :C
Discussion
3.

If 2alpha= -1-isqrt(3)" and"2beta= -1+isqrt(3), then 5alpha^(4)+5beta^(4)+7alpha^(-1)beta^(-1) is equal to

Answer»

`-1`
`-2`
`0`
`2`

ANSWER :D
Discussion
4.

Find points at which the tangent to the curve y = x^(3) – 3x^(2) – 9x + 7 is parallel to the x-axis.

Answer»


ANSWER :(3,-20) and (-1,12)
Discussion
5.

The vector perpendicular to the vectors 4hat(i)-hat(j)+3hat(k) and -2hat(i)+hat(j)-2k whose magnitude is 9, is

Answer»

`3HAT(i)+6hat(j)-6hat(k)`
`3hat(i)-6hat(j)+6hat(k)`
`-3hat(i)+6hat(j)+6hat(k)`
None of these

Solution :Let `a =4hati - hatj+3 HATK, b=-2 hati+ hatj-2 hatk`
and `c = x hati + y hatj +z hatk`
Given, `a*c=0`
i.e., ` 4x-y+3z=-0""…(i)`
and `b*c=0`
i.e, `x^(2)+y^(2)+z^(2)=0""…(ii)`
Also, `|c|=9`
i.e.,` x^(2)+y^(2)+z^(2)=81""...(iii)`
Now, from Eqs. (i) and (ii), we get
`2x+z=0impliesz=-2x`
On putting this value in Eq. (ii) by 3 and then adding, we get
`({:(8x-2y+6z=0),(-6x+3y-6z=0):})/(2x+y=0impliesy=-2x)`
On putting this value in Eq. (iv) we get
`5x^(2)+4x^(2)=81`
`=?9x^(2)=81impliesx^(2)=9`
`impliesx = pm 3`
`therefore y= pm 6 and z = pm 6`
`therefore` REQUIRED vector, `c=x hati+y hatj +z hatk = pm 3 hatipm 6 hatj pm 6 hatk=3 hati-6 hatj - 6 hatkor -3 hati+6 hatj+6 hatk`
Discussion
6.

Choose the correct answer The mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face is

Answer»

1
2
5
`8/3`

ANSWER :B
Discussion
7.

int(1+x^(2))/(sqrt(1-x^(2)))dx=...

Answer»

`(3)/(2)SIN^(-1)x-(1)/(2)x SQRT(1-x^(2))+C`
`(3)/(2)sin^(-1)x+(1)/(2)x sqrt(1-x^(2))+c`
`(3)/(2)cos^(-1)e-(1)/(2)x sqrt(1-x^(2))+c`
`(3)/(2)cos^(-1)x+(1)/(2)x sqrt(1-x^(2))+c`

Answer :A
Discussion
8.

If A=[{:(4,x+2),(2x-3,x+1):}]is symmetric matrix then find x.

Answer»


ANSWER :`x=5`
Discussion
9.

The product of lenghts of perpendicular from any point on the hyperbola x^(2) - y^(2) = 16 to its asymptotes, is

Answer»

2
4
8
16

Answer :C
Discussion
10.

Find the rate of change of the area of circle w.r.t.r when r=8 cm.

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SOLUTION :Let radius =R cm. `therefore` AREA of the CIRCLE `A=pir^2rArr(dA)/dr=2pir` The RATE of change of area for r=8 is `(dA)/(dr)]_(r=8)=[2pir]_(r=8)=16picm^2//cm`
Discussion
11.

The region represented by the inequation x-y le -1, x-y le 0, x le 0, y le 0 is …………….

Answer»

BOUNDED
unbounded
do not EXIST
TRIANGULAR region

Answer :C
Discussion
12.

f(x)=-1+kx+k neither touches nor intecepts the curve f(x)= In x, then minimum value of k in

Answer»

`((1)/(e),(1)/SQRT(e))`
`(e,e^(2))`
`((1)/sqrt(e),e)`
NONE of these

Answer :A
Discussion
13.

Evaluate : int (sin x +cos x ) sqrt(9 +16 sin 2x) dx

Answer»


SOLUTION :N/A
Discussion
14.

Consider the function f(x)=x^(2)+bx+c, where D=b^(2)-4cgt0, then match the follwoing columns.

Answer»

SOLUTION :
Discussion
15.

Solution of the differential equation x-tan^(-1)y-1c e^(-tan^(-1)y) is

Answer»

`y=(c-X)E^(x)`
`y=(c-x)e^(-x)`
`y=(c+x)e^(x)`
`y=(c+x)e^(-x)`

ANSWER :D
Discussion
16.

Letvec v_(0) be a fixedvector and vecv_(0)=[(1)/(0)]. Then for n ge 0 a sequence is definedvec v_(n+1)=vec v_(n)+((1)/(2))^(n+1)[(0,-1),(1,0)]^(n+1) vec v_(0) then lim_(n to oo) vecv_(n)=[(alpha), (beta)]. Find (alpha)/(beta) .

Answer»


ANSWER :2
Discussion
17.

Let theta=(a_(1),a_(2),a_(3),...,a_(n)) be a given arrangement of n distinct objects a_(1),a_(2),a_(3),…,a_(n). A derangement of theta is an arrangment of these n objects in which none of the objects occupies its original position. Let D_(n) be the number of derangements of the permutations theta. There are 5 different colour balls and 5 boxes of colours same as those of the balls. The number of ways in which one can place the balls into the boxes, one each in a box, so that no ball goes to a box of its own colour is

Answer»

`40`
`44`
`45`
`60`

SOLUTION :`(b)` `(D_(N))/(n!)-(D_(n-1))/((n-1)!)=((-1)^(n))/(n!)` gives
`(D_(n))/(n!)=sum_(r=2)^(n)((D_(r ))/(r!)-(D_(r-1))/((r-1)!))=sum_(r=2)^(n)((-1)^(r ))/(r!)`
`impliesD_(n)=n!sum_(r=2)^(n)((-1)^(r ))/(r!)`
`:. D_(5)=5!((1)/(2!)-(1)/(3!)+(1)/(4!)-(1)/(5!))=44`
Discussion
18.

Let theta=(a_(1),a_(2),a_(3),...,a_(n)) be a given arrangement of n distinct objects a_(1),a_(2),a_(3),…,a_(n). A derangement of theta is an arrangment of these n objects in which none of the objects occupies its original position. Let D_(n) be the number of derangements of the permutations theta. The relation between D_(n) and D_(n-1) is given by

Answer»

`D_(n)-nD_(n-1)=(-1)^(n)`
`D_(n)-(n-1)D_(n-1)=(-1)^(n-1)`
`D_(n)-nD_(n-1)=(-1)^(n-1)`
`D_(n)-D_(n-1)=(-1)^(n-1)`

Solution :`(a)` `D_(n)-nD_(n-1)=(-1)(D_(n-1)-(n-1)D_(n-2))`
By implied induction on `n`, we OBTAIN
`D_(n)-nD_(n-1)=(-1)^(n-2)(D_(2)-2D_(1))`, Where `D_(1)=0` and `D_(2)=1`
`=(-1)^(n)`
Discussion
19.

Let theta=(a_(1),a_(2),a_(3),...,a_(n)) be a given arrangement of n distinct objects a_(1),a_(2),a_(3),…,a_(n). A derangement of theta is an arrangment of these n objects in which none of the objects occupies its original position. Let D_(n) be the number of derangements of the permutations theta. D_(n) is equal to

Answer»

`(N-1)D_(n-1)+D_(n-2)`
`D_(n-1)+(n-1)D_(n-2)`
`n(D_(n-1)+D_(n-2))`
`(n-1)(D_(n-1)+D_(n-2))`

Solution :`(d)` For every choice of `r=1,2,3,….(n-1)` when the `n^(TH)` object `a_(n)` goes to the `RTH` place, there are `D_(n-1)+D_(n-2)` WAYS of the other `(n-1)` OBJECTS `a_(1)`, `a_(2)`, ….,`a_(n-1)` to be deranged.
Hence `=D_(n)=(n-1)(D_(n-1)+D_(n-2))`
Discussion
20.

Solve 24x lt 100, when (i) x is a natural number, (ii) x is an integer.

Answer»


ANSWER :The solution SET of the INEQUALITY is {1,2,3,4}
ii. The solution set of the inequality is {………..-3, -2, -1,0,1,2,3,4}
Discussion
21.

If p has truth value T, what can be said about the truth values of ~~p ^^qrarr p vv q.

Answer»

SOLUTION :
Discussion
22.

If a line lies in the octant OXYZ and it makes equal angles with the axes, then

Answer»

`l=m=n=(1)/(sqrt(3))`
`l=m=n=pm(1)/(sqrt(3))`
`l=m=n=-(1)/(sqrt(3))`
`l=m=n=pm(1)/(sqrt(2))`

ANSWER :B
Discussion
23.

If angle bisector of overline(a) = 2hat(i) + 3 hat(j) + 4 hat(k) and overline(b) = 4hat(i) - 2 hat(j) + 3 hat(k) is overline(c ) = alpha hat(i) + 2 hat(j) + beta hat(k) then

Answer»

`vec(c ).hat(k) + 7 = 0`
`vec(c ) .hat(k) - 15 = 0`
`vec(c ) .hat(k) + 15 = 0`
`vec(c ) .hat(k) - 7 = 0`

Solution :The ANGLE bisector of `OVERLINE(a)` and `overline(B)` is `overline(p)`
`overline(p) = lambda (hat(a) + hat(b))`
`= lambda (((2 hat(i) + 3 hat(j) + 4 hat(k)) + (4 hat(i) - 2 hat(j) + 3 hat(k)))/(sqrt(4 + 9 + 16))) = (lambda)/(sqrt(29)) [6 hat(i) + hat(j) + 7 hat(k)]`
`= (lambda)/(2 sqrt(29)) [12 hat(i) + 2 hat(j) + 14 hat(k)]`
then `overline(p) = overline(c ) IMPLIES alpha = 12 beta = 14`
Now `overline(c ). hat(k) = 14 = 0`
Discussion
24.

Find the numerically greatest term (s) in the expansion of (4a-6b)^(13)" when "a=3,b=5

Answer»


ANSWER :`T_11, T_10`
Discussion
25.

lim_(xto0^(-))(1)/(3-2^((1)/(x))) is equal to

Answer»

0
1
`1/2`
`1/3`

ANSWER :D
Discussion
26.

Consider an infinite geometric series with first term a and common ratio r. If its sum is 4 and the second term is ( 3)/( 4) , then :

Answer»

`a = 2, R = (3)/(8)`
`a = (4)/(7), r = (3)/(7)`
`a = (3)/(2), r = (1)/(2)`
`a = 3, r = (1)/(4)`

Answer :D
Discussion
27.

Solve system of linear equations, using matrix method in examples 7 to 14 x-y+z=4 2x+y-3z=0 x+y+z=2

Answer»


ANSWER :x=2 ,y=-1 and z=1
Discussion
28.

Integrate the following functions (xcos^-1x)/sqrt(1-x^2)

Answer»

Solution :`int (x cos^-1 x)/SQRT(1-x^2) dx`
=`1/2 int (-2X)/sqrt(1-x^2) cos^-1 x dx`
=`-1/2 [cos^-1 x XX 2 sqrt(1-x^2) - int -1/sqrt(1-x^2) 2 sqrt(1-x^2) dx]`
=`-1/2 [2sqrt(1-x^2) cos^-1 x+2 int dx]`
=`-1/2[2sqrt(1-x^2) cos^-1x +2x]+c`
=`-[sqrt(1-x^2) cos^-1 x+x]+c`
Discussion
29.

Form the differential equation of the family of hyperbolas havig foci on x-axis and centre at origin.

Answer»


ANSWER :`xyy" + X(y')^(2) - YY' = 0`
Discussion
30.

(sqrt(3)-1)+(1)/(2)(sqrt(3)-1)^(2)+(1)/(3)(sqrt(3)-1)^(3)+….oo

Answer»

`log(3+sqrt(2))`
`log2`
0
`log(2+sqrt(3))`

ANSWER :D
Discussion
31.

Given four points A(2, 1, 0), B(1, 0, 1), C(3, 0, 1) and D(0, 0, 2). Point D lies on aline L orthogonal to the plane determined by the points A, B and C.

Answer»

`sqrt(2)`
`1//2`
2
`1//sqrt2`

SOLUTION :
`""|{:(x-2,,y-1,,z),(1-2,,0-1,,1-0),(3-2,,0-1,,1-0):}|=0`
`""(x-2)[(-1)-(-1)]-(y-1)[(-1)-1]+z[1+1]=0`
or `""2(y-1)+2z=0`
or `""y+z-1=0`
The vector NORMAL to the PLANE is `vecr= 0hati+hatj+ hatk`
The equation of the line through `(0, 0, 2)` and PARALLEL to `vecn` is `vecr = 2hatk+lamda(hatj+hatk)`
The perpendicular distance of `D(0, 0, 2)` from plane
`ABC` is `|(2-1)/(sqrt(1^(2)+1^(2))|=(1)/(sqrt2)|`
Discussion
32.

Given four points A(2, 1, 0), B(1, 0, 1), C(3, 0, 1) and D(0, 0, 2). Point D lies on aline L orthogonal to the plane determined by the points A, B and C. The equation of the line L is

Answer»

`vecr=2hatk+lamda(hati+HATK)`
`vecr= 2hatk+lamda(2hatj+hatk)`
`vecr=2hatk+lamda(hatj+hatk)`
none

SOLUTION :
`""|{:(x-2,,y-1,,z),(1-2,,0-1,,1-0),(3-2,,0-1,,1-0):}|=0`
`""(x-2)[(-1)-(-1)]-(y-1)[(-1)-1]+z[1+1]=0`
or `""2(y-1)+2z=0`
or `""y+z-1=0`
The vector normal to the plane is `vecr= 0hati+hatj+ hatk`
The equation of the line through `(0, 0, 2)` and parallel to `vecn` is `vecr = 2hatk+lamda(hatj+hatk)`
The perpendicular DISTANCE of `D(0, 0, 2)` from plane
`ABC` is `|(2-1)/(sqrt(1^(2)+1^(2))|=(1)/(sqrt2)`
Discussion
33.

If A and B are two events such that P(A)= (1)/(2), P(B)= (1)/(3) and P(A//B)= (1)/(4) then P(A' cap B') equals ………..

Answer»

`(1)/(12)`
`(3)/(4)`
`(1)/(4)`
`(3)/(16)`

ANSWER :C
Discussion
34.

Equation of the hyperbola with one focus at the origin and directrix x+3=0 and eccentricity sqrt3 is

Answer»

`X^(2)-2y^(2)+18x+27=0`
`2x^(2)-y^(2)+18x+27=0`
`x^(2)-2y^(2)-18x-27=0`
`2x^(2)-y^(2)-18x+27=0`

ANSWER :B
Discussion
35.

The variance of 6,5,8,10,3,4,9,11 is

Answer»

8
9
7.5
10

Answer :C
Discussion
36.

Construct a 2 xx 2 matrix A=[a_(ij)], whose elements are given by a_(ij)= (i)/(j)

Answer»


ANSWER :`A= [[1,(1)/(2)],[2,1]]`
Discussion
37.

Draw the graph of y=(3x-x^(3))/(1-3x^(2)) and hence the graph of y=tan^(-1).(3x-x^(3))/(1-3x^(2)).

Answer»

Solution :We have `y=F(x)=(3x-x^(3))/(1-3x^(2))`, which is an odd function.
Domain of the function is `R-{+-(1)/(sqrt(3))}`
1. y-intercept
f(0)=0
So the graph cuts the y-axis at (0,0).
2. x-intercept (zeros)
Put y=0 or `3x-x^(3)=0 :. x=0, +-sqrt(3)`
So the graph meets the x-axis at (0,0) and `(+-sqrt(3),0)`.
3. Asymptotes
Vertical asymptotes
Clearly , the graph has vertical asymptote `x=+-(1)/(sqrt(3))`, where the denominator becomes zero.
Horizontal asymptotes
Clearly, the graph has no horizontal asymptote as the degree of the NUMERATOR is higher than the degree of the numerator is higher than the degree of the denominator.
Oblique asymptotes ltbr. `y=(x^(3)-3x)/(3x^(2)-1)=(1)/(3)x-(-(8)/(9)x)/(3x^(2)-1)`
Hence the oblique asymptote is `y=(1)/(3)x`.
THUS, importantpoints ans lines are as shown in the following figure.

4. Monotinicity/Extremum
`f'(x)=((3-3x^(2))(1-3x^(2))+6X(3x-x^(3)))/((1-3x^(2))^(2))`
`=3((x^(2)+1)^(2))/((1-3x^(2))^(2)) gt0`
Hence the function is increasing throughout.
`underset(xrarr-(1^(-))/(sqrt(3)))lim(3x-x^(3))/(1-3x^(2))=oo` and `underset (xrarr-(1^(+))/(sqrt(3)))lim(3x-x^(3))/(1-3x^(2))=-oo`
f(x) approaches asymptote `y=x//3` as `xrarr-oo`.
Thus, in `(-oo,-(1)/(sqrt(3)))`, f(x) increases from `'-oo'` to `'oo'` intersecting the x-axis at `(-sqrt(3),0)`
`underset(xrarr-(1^(+))/(sqrt(3)))lim(3x-x^(3))/(1-3x^(2))=-oo` and `underset(xrarrsqrt(3)^(-))lim(3x-x^(3))/(1-3x^(2))=oo`
Thus, in `(-(1)/(sqrt(3)),(1)/(sqrt(3)))`, f(x) increases from `'-oo'` to `'oo'` intersecting the x-axis at (0,0).
`underset(xrarrsqrt(3)^(+))lim(3x-x^(3))/(1-3x^(2))=-oo` and `underset(xrarroo)lim (3x-x^(3))/(1-3x^(2))=oo`
Thus, in `((1)/(sqrt(3)),oo),f(x)` increases from `'-oo'` to `'oo'` intersecting the x-axis at `(sqrt(3),0)`.
f(x) approaches asymptote `y=x//3` as `xrarroo`.
From the above discussion, the graph of `y=f(x)` can be drawn as follows.

Now in each of the intervals `(-oo,-(1)/(sqrt(3))),(-(1)/(sqrt(3)),(1)/(sqrt(3))), ((1)/(sqrt(3)),oo), f(x)` takes values `(-oo,oo)`.
So in each of the intervals, `tan^(-1).(3x-x^(3))/(1-3x^(2))` takes values `(-(pi)/(2),(pi)/(2))`.
So the graph of `g(x)=tan^(-1).(3x-x^(3))/(1-3x^(2))` can be drawn as follows.

Here `y=+-(pi)/(2)` are asymptotes.
Discussion
38.

A batsman can score 0, 2, 3, or 4 runs for each ball he receives. If N is the number of ways of scoring a total of 20 runs in one over of six balls, then N is divisible by:

Answer»

5
7
14
16

Answer :D
Discussion
39.

Given the sequence a, ab, aab, aabb, aaabb,aaabbb,…. Upto 2004 terms, the total number of times a's and b' s are used from 1 to 2004 terms are :

Answer»


Answer :`1002 XX 1003 a's and (1002) ^(2) B's`
Discussion
40.

Integrate the following int(cosec^2(Inx))/(x) dx

Answer»

SOLUTION :`int(cosec^2x(Inx))/x(DX)
PUT In x=z then (dx/x)=DZ]
`intcosec^2zdz`
-cotz+C=-cot(Inx)+C
Discussion
41.

A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is (1)/(100). What is the probability that he will win a prize (a) at least once (b) exactly once (c) at least twice?

Answer»


ANSWER :`(a) 1-((99)/(100))^(50) (B) (1)/(2)((99)/(100))^(49) (c ) 1- (149)/(100)((99)/(100))^(49)`
Discussion
42.

int (1 +x -x^(-1)) e^(x + x^(-1))dx =

Answer»

`(x + 1) E^(x +x^(-1)) + C`
`(x -1) e^(x +x^(-1)) + c`
`(1)/(2) XE^(x + x^(-1)) + c`
`x.e^(x +x^(-1)) + c`

Answer :D
Discussion
43.

Evaluate the following determinants: [[1,0,-5863],[-7361,2,7361],[1,0,4137]]

Answer»

Solution :`[[1,0,-5863],[-7361,2,7361],[1,0,4137]]`
=`2[[1,-5863],[1,4137]]`
(expanding along 2ND COLUMN)
=2(4137+5863)
=`2xx10000=20000`
Discussion
44.

For a series the information available is n = 10 sum x = 60, sum x^(2) = 1000. The standard deviation is

Answer»

8
64
24
128

Answer :A
Discussion
45.

If y ^(y ^(y ^(.^(,^(,^(oo)))))) = log _(e) (x + log _(e) (x + .....)), then (dy)/(dx)at (x = e ^(2) -2, ysqrt2 )is

Answer»

`(log (( E )/( 2)))/( 2 SQRT2 (e ^(2) -1 ))`
`(log 2)/(2 sqrt2(e ^(2) -1 ))`
`(sqrt2 log ""(e)/(2))/( (e ^(2) -1 ))`
NONE of these

Answer :A
Discussion
46.

Which one of the following relations on R is an equivalence relation ?

Answer»

`a R_(1) b iff |a|=|b|`
`a R_(2)b iff a GE b`
`a R_(3) b iff a ` DIVIDES b
` R_(4) b iff a LT b`

Answer :A
Discussion
47.

Form the differential equation of the family of circles in the first quadrant whichtouch the coordinate axes.

Answer»


ANSWER :`(X + YY')^(2) = ( x - y)^(2)(1 + (y')^(2))`
Discussion
48.

If , 15 and 45 are in G.P. then x =

Answer»

A.P.
G.P.
arithmetic-geometric series
none of these

Answer :B
Discussion
49.

Prove that :int_(0)^(1)(sin^(-1)x)/(x) dx = (pi)/(2) log 2

Answer»
Discussion
50.

Forthe followingquestion , choosethe correctanswerfrom thecodes(a).(b).(c) and (d)defindasfollows

Answer»

Statements I istrueStatementII is thealso true,
StatementII is thecorrectexplantion ofStatement I
Statement I is trueStatement II is alsotrue ,
Statement II isnot thecorrectexplantion ofStatement I
Statement I istrue , Statement II is false
StatementI is false , StatementII is true

Solution : Since `vec(PQ)` in notparallelto `vec(TR)`
`:' vec(TR)` is resultant of `vec(RS)" and" vec(ST)`
`"VECTORS" rArr vec(PQ) xx (vec(RS + vec(ST)) ne 0`
But forstatementII we have
`vec(PQ) xx vec(RS) =vec(0)`

Whichis notpossibleas `vec(PQ)" notparallel to" vec(RS)`
HenceStatmentI is trueand Statement II is false.
Discussion