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.

Solve in R and represent the solution on the number line. abs(x-5) lt 1

Answer»

Solution :`abs(x-5) lt 1`
`rArr -1 lt x-5 lt 1`
`rArr 4 lt x lt 6`
If x `in` R then the solution set is S = (4,6)
We can REPRESENTTHE solution on number LINE as
Discussion
2.

At Lincoln High School, approximately 7 percent of enrolled juniors and 5 percent of enrolled seniors were inducted into the National Honor Society last year. If there were 562 juniors and 602 seniors enrolled at Lincoln High School last year, which of the following is closest to the total number of juniors and seniors at Lincoln High School last year who were inducted into the National Honor Society?

Answer»

140
69
39
30

Answer :B
Discussion
3.

A setcontains (12) elements . Thenumberofsubsetsof thesetwhichcontainatmost2 elements is

Answer»

6
15
21
none of these

ANSWER :A
Discussion
4.

No. of distinct terms in (a+b+c+d)^n , n in N is

Answer»

`""^((n+3))C_2`
`""^((n+3))C_3`
`""^((n+2))C_3`
`""^((n+4))C_3`

ANSWER :B
Discussion
5.

int(1)/(e^(x)+1)dx=....

Answer»

`LOG((E^(x))/(e^(x)+1))+c`
`log((e^(x)+1)/(e^(x))+c`
`log(1+e^(x))+c`
`log(1-e^(x))+c`

ANSWER :A
Discussion
6.

Find the slope of the normal to the curve y^(2)=4 axat ((a)/(m^(2)),(2a)/(m))

Answer»


ANSWER :`-(1)/(m)`
Discussion
7.

If I = int(sinx(cosx)^(-5//2)dx)/(sqrt(sinx+3cosx)+sqrt(sinx+4cosx)) =(A)/(4)((tanx+4)^(5//2)-(tanx+3)^(3//2)) -(2)/(4) [4(tanx+4)^(3//2)-3(tanx+3)^(3//2)1+C

Answer»


ANSWER :1.6
Discussion
8.

Find the scalar triple product vecb.(veccxxveca) where veca, vecb and vecc are respectively hati+hatj,hati-hatj, 5hati+2hatj+3hatk.

Answer»

SOLUTION :
Discussion
9.

Given that F(x)=[{:(,cos x,-sin x,0),(,sin x,cos x,0),(,0,0,1):}]."If"in R Then for what values of y, F(x+y)=F(x)F(y)

Answer»


ANSWER :`y in R`
Discussion
10.

The vectors a + 2b + 3c, 2a + b - 2c, 3a - 7c are

Answer»

coplanar
collinear
noncoplanar
none

Answer :A
Discussion
11.

squareABCD is a parallelogram. (A_(1)) and B_(1) are midpoints of the sides bar(BC) and bar(AD) respectively. If vec("AA")_(1)+vec(AB)_(1)=lambda vec(AC) then lambda = …………

Answer»

`(1)/(2)`
1
`(3)/(2)`
2

Answer :C
Discussion
12.

int (logx)/((1+logx)^(2))dx=....

Answer»

`(1)/(1+logx)+C`
`(1)/((1+logx)^(2))+c`
`(X)/(1+logx)+c`
`(1)/((1+logx)^(2))+c`

ANSWER :C
Discussion
13.

Find the second order derivatives of the function log x

Answer»


ANSWER :`-(1)/(x^2)`
Discussion
14.

If f(x)=(x^(5))/(5)+(x^(4))/(4)+x^(3)+(kx^(2))/(2)+x be an increasing function AA x in R, then k^(2) is

Answer»

11
12
13
14

Answer :A
Discussion
15.

Binomial coefficients which are in decreasing order are

Answer»

`""^15C_5, ""^15C_6,""^15C_7`
`""^15C_10, ""^15C_9, ""^15C_8`
`""^15C_6, ""^15C_7, ""^15C_8`
`""615C_7, ""^15C_6, ""^15C_5`

ANSWER :D
Discussion
16.

Let= [1, inftY). Define f :S to S by f(x) = 5^(x(x+1)) Then f^(-1)(x) is equal to

Answer»

`(1/5)^(x(x-1))`
`1/2[ 1+ SQRT(log_(5)(x))]`
`1/2[1+ sqrt(1+4 log_(5)(x))]`
`(1/5)^(x(1-x))`

ANSWER :C
Discussion
17.

If S_(r)= sum_(r=1)^(n)T_(1)=n(n+1)(n+2)(n+3) then sum_(r=1)^(10) 1/(T_(r)) is equal to

Answer»

`55/527`
`58/528`
`59/528`
NONE of these

ANSWER :D
Discussion
18.

Two numbers x and y are chosen at random from {1, 2, 3,…,5n} where n ge 2, show that the probability x^(4) - y^(4) is divisible by 5 is (17 n - 5)/(5(5n - 1)).

Answer»


ANSWER :`(17N - 5)/(5(5N - 1))`
Discussion
19.

For x inR, f(x) is defined as follows : f(x)={{:(x+1,,0lexlt2),(|x-4|,,x ge 2):} Then the solution set of the equation isf(x)^(2)+x=f(x)+x^(2) is

Answer»

`{-1,1}`
`[2,OO)`
`[0,2)`
{0,2}`

ANSWER :(d)
Discussion
20.

If the radius of a sphere , is measured as 7m with an error of 0.02 m then find the approximate error in calculating its volume .

Answer»


ANSWER :`3.92 PIM^(3)`
Discussion
21.

Using elementary transformations find the inverse of each of the following matrices , if it exist. [{:(1,2),(2,-1):}]

Answer»


ANSWER :`[{:((1)/(5),(2)/(5)),((2)/(5),-(1)/(5)):}]`
Discussion
22.

A positive integer n le 5 such that int_(0)^(1) e^(2 x-1) (x-1)^(n) dx= (1)/(4) ((7)/( e) -e) is

Answer»


ANSWER :`3`
Discussion
23.

Evaluate the definite integrals . underset(0)overset(1)int (dx)/(sqrt(3-2x)

Answer»


ANSWER :`SQRT(3)-1`
Discussion
24.

If 2a+3b-5c=0, then ratio in which c divides AB is

Answer»

`3 : 2` internally
`3 : 2` externally
`2 : 3` internally
`2 : 3` externally

Solution :GIVEN, `2a+3b-5c=-0`
`rArr""(2a+3b)/5=c`
`rArr""(2a+3b)/(2+3)=c`
`rArr""(a+3/2b)/(1+3/2)=c"...(1)"`
Let c divides AB in the ratio `lamda:1`
`"Then,"c=(a+lamdab)/(1+lamda)"…(ii)"`
On comparing (i) and (ii), we GET `lamda=3/2`
`:.""` REQUIRED ratio is 3 : 2 internally.
Discussion
25.

Show that the points (2,3,4), (-1,-2,1), (5,8,7) are colinear

Answer»


ANSWER :LET the GIVEN points be A, B and C respectively.
Then the direction ration of AB are -1, -2, -2, -3, 1-4 i.e., -3, -5, -3 and the direction ratios of AB are 5, -2, 8-3, 7-4 i.e., 3, 5, 3
Since `(-3)/3=(-5)/5=(-3)/5`, the direction ratios of AB and AC are proportional. Thus A, B and C are COLLINEAR
Discussion
26.

Evaluate the definite integralint_(-1//sqrt(3))^(1//sqrt(3))((x^(4))/(1-x^(4)))cos^(-1)((2x)/(1+x^(2)))dx.

Answer»


ANSWER :`I=(PI)/(12)[pi+3 LN (2+sqrt(3))-4sqrt(3)]`
Discussion
27.

A player tosses a coin and score one point for every head and two points for every tail that turns up. He plays on until his score reaches or passes n. P_(n) denotes the probability of getting a score of exactly n. The value of P_(n)+(1//2)P_(n-1) is equal to

Answer»

`1//2`
`2//3`
1
None of these

SOLUTION :The scores of n can be reached in the following two MUTUALLY EXCLUSIVE events:
(i) by THROWING a head when the score is `(n-1),`
(ii) by throwing a tail when the score is `(n-2)`
Hence `P_(n)=P_(n-1)xx1/2+P_(n-2)+1/2""[becauseP("head")="(tail")=1//2"]"`
`=1/2[P_(n-1)+P_(n-2)]""(1)`
` impliesP_(0)+1/2P_(n-1)=P_(n-1)+1/2P_(n-2)`
`""("adding"(1//2)P_(n-1)"on both sides")`
`=P_(n-2)+1/2P_(n-3)`
`=P_(2)+1/2P_(1)""(2)`
Now, a score of 1 can be obtained by throwing a head at a single toss. Therefore, `P_(1)=1/2`
And a score of 2 can be obtained by throwing either a tail at a single toss or a head at the first toss as well as second toss. Therefore,
`P_(2)=1/2+((1)/(2)xx(1)/(2))=3/4`
From Eq. (2), we have
`P_(n)+1/2P_(n-1)=3/4+1/2((1)/(2))=1`
`or P_(n)=1-1/2P_(n-1)`
`or P_(n)-2/3=1-1/2P_(n-1)-2/3`
`or P_(n)-2/3=-1/2(P_(n-1)(2)/(3))`
`=(-(1)/(2))^(2)(P_(n-1)-(2)/(3))`
`=(-(1)/(2))^(3)(P_(n-3)-(2)/(3))`
`=(-(1)/(2))^(n-1)(P_(1)-(2)/(3))`
`=(-(1)/(2))^(n-1)((1)/(2)-(2)/(3))`
`=(-(1)/(2))^(n-1)(-(1)/(6))`
`(-(1)/(2))^(n)1/3`
`or P_(n)=2/3+((-1)^(n))/(2^(n))1/3=1/3{2+((-1)^(n))/(2^(n))}`
Now, `P_(100)=2/3+(1)/(3xx2^(101))gt2/3`
and `P_(101)=2/3-(1)/(3xx2^(101))lt2/3`
`P_(101)lt2/3ltP_(100)`
Discussion
28.

A player tosses a coin and score one point for every head and two points for every tail that turns up. He plays on until his score reaches or passes n. P_(n) denotes the probability of getting a score of exactly n. Which of the following is not true ?

Answer»

`P_(100)gt2//3`
`P_(100)lt2//3`
`P_(100),P_(101)gt2//3`
None of these

Solution :The scores of n can be reached in the following two mutually exclusive EVENTS:
(i) by throwing a head when the score is `(n-1),`
(II) by throwing a tail when the score is `(n-2)`
Hence `P_(n)=P_(n-1)xx1/2+P_(n-2)+1/2""[becauseP("head")="(tail")=1//2"]"`
`=1/2[P_(n-1)+P_(n-2)]""(1)`
` impliesP_(0)+1/2P_(n-1)=P_(n-1)+1/2P_(n-2)`
`""("adding"(1//2)P_(n-1)"on both sides")`
`=P_(n-2)+1/2P_(n-3)`
`=P_(2)+1/2P_(1)""(2)`
Now, a score of 1 can be OBTAINED by throwing a head at a single toss. Therefore, `P_(1)=1/2`
And a score of 2 can be obtained by throwing either a tail at a single toss or a head at the first toss as well as second toss. Therefore,
`P_(2)=1/2+((1)/(2)xx(1)/(2))=3/4`
From EQ. (2), we have
`P_(n)+1/2P_(n-1)=3/4+1/2((1)/(2))=1`
`or P_(n)=1-1/2P_(n-1)`
`or P_(n)-2/3=1-1/2P_(n-1)-2/3`
`or P_(n)-2/3=-1/2(P_(n-1)(2)/(3))`
`=(-(1)/(2))^(2)(P_(n-1)-(2)/(3))`
`=(-(1)/(2))^(3)(P_(n-3)-(2)/(3))`
`=(-(1)/(2))^(n-1)(P_(1)-(2)/(3))`
`=(-(1)/(2))^(n-1)((1)/(2)-(2)/(3))`
`=(-(1)/(2))^(n-1)(-(1)/(6))`
`(-(1)/(2))^(n)1/3`
`or P_(n)=2/3+((-1)^(n))/(2^(n))1/3=1/3{2+((-1)^(n))/(2^(n))}`
Now, `P_(100)=2/3+(1)/(3xx2^(101))gt2/3`
and `P_(101)=2/3-(1)/(3xx2^(101))lt2/3`
`P_(101)lt2/3ltP_(100)`
Discussion
29.

Find the value of lambda such that line (x-2)/(6) = (y-1)/(lambda) = (z+5)/(-4) is perpendicular to the plane 3x-y-2z=7.

Answer»


SOLUTION :N/A
Discussion
30.

Let A_(n)=sum_(r=1)^(n)cos^(-1).(r )/(n) and B_(n)=sum_(r=0)^(n-1)cos^(-1)((r )/(n))" for "n=1, 2, 3……., then

Answer»

`a_(2010) GT 2010`
`A_(2010) lt 2010`
`B_(2010) gt 2010`
`B_(2010) lt 2010`

Answer :C,D
Discussion
31.

(d)/(dx) (a^(a))= ……(a gt 0)

Answer»

`a^(a) (1 + LOG a)`
0
`a^(a)`
Does not exist

Answer :B
Discussion
32.

Evaluate the integrals in exercise. overset((pi)/(2)) underset(0)int (sinx)/(1+cos^(2)x)dx

Answer»


ANSWER :`(PI)/(4)`
Discussion
33.

Let a hyperbola whose centre is at origin. A line x + y =2 thuches this hyperbola at P (1,1) andintersects the asymtotes at A and B such that AB=6 sqrt2 units. (you can use the oncept that incase of hyperbola portion of tangent intercepted between asymptotes is bisected at the point of constant). Equation of the tangent to the hyperbola at (-1, (7)/(2)) is

Answer»

`5X +2y =2`
`3x + 2y =4`
`3x + 4y =11`
NONE of these

Answer :B
Discussion
34.

Let a hyperbola whose centre is at origin. A line x + y =2 thuches this hyperbola at P (1,1) andintersects the asymtotes at A and B such that AB=6 sqrt2 units. (you can use the oncept that incase of hyperbola portion of tangent intercepted between asymptotes is bisected at the point of constant). Angle subtended by AB at centre of the hyperbola is

Answer»

`SIN ^(-1) "" 4/5`
`sin ^(-1) "" 2/5`
`sin^(-1) "" 3/5`
NONE of these

ANSWER :C
Discussion
35.

Let a hyperbola whose centre is at origin. A line x + y =2 thuches this hyperbola at P (1,1) andintersects the asymtotes at A and B such that AB=6 sqrt2 units. (you can use the oncept that incase of hyperbola portion of tangent intercepted between asymptotes is bisected at the point of constant). Equation of asymptotes are

Answer»

`5XY+ 2x ^(2) + 2Y ^(2) =0`
`3xd ^(2) + 4Y ^(2) + 6xy=0`
`2x^(2) + 2y ^(2) - 5xy =0`
none of these

Answer :A
Discussion
36.

If x^(n)=a_(0)+a_(1)(1+x)+a_(2)(1+x)^(2)+….+a_(n)(1+x)^(n)=b_(0)+b_(1)(1-x)+b_(2)(1-x)^(2)+….+b_(n)(1-x)^(n) then for n=101, (a_(50),b_(50)) equals:

Answer»

`(-""^(101)C_(50), ""^(101)C_(50))`
`(""^(101)C_(50), -""^(101)C_(50))`
`(-""^(101)C_(50), -""^(101)C_(50))`
`(""^(101)C_(50), ""^(101)C_(50))`

ANSWER :A
Discussion
37.

For anyinteger n ge 1, underset(k=1) overset(n) sumK ( K+2)=

Answer»

`(N(n+1)(n+2))/(6)`
`(n(n+1)(2n+7))/(6)`
`(n(n+1)(2n+1))/(6)`
`(n(n-1)(2n+8))/(6)`

ANSWER :B
Discussion
38.

Evalute the following integrals int (1)/(x^(2) + 6x + 10)dx

Answer»


ANSWER :`TAN^(-1)(x+3)+C`
Discussion
39.

intdx/(1+sinx)

Answer»

SOLUTION :`I=intdx/(1+sinx)=INT(1-sinx)/cos^2xdx`
=`intsec^2x-secxtanxdx`
=tanx-secx+c
Discussion
40.

If f : IR to IRis defined by{:f(x)={("x - 1,","for "x le 1),(2-x^(2)",","for "1 lt x le 3),("X - 10,","for "3 lt x lt 5),("2x,","for " x ge 5):}thenthe set points of discontinuity of f is

Answer»

LR - {1,5}
{1,3,5}
{1,5}
IR - {1,3,5}

ANSWER :C
Discussion
41.

A line of fixed length (a + b) moves so that its ends are always on two fixed perpendicular straight lines. Prove that a marked point on the line which divides this line into portions of length 'a' and 'b' describes an ellipse and also find the eccentricity of the ellipse when a = 8, b=12.

Answer»


ANSWER :`(SQRT(5))/(3)`
Discussion
42.

Solve the following D.E's (dy)/(dx) = (2x+y+3)/(2y+x+1)

Answer»


ANSWER :`(x+y+(4)/(3))(x-y+2)^(3) = C`
Discussion
43.

If the vertices of a angles are A (1 , 4, 2) , B (-2 , 1 ,2) , C (2 , 3, -4)then findangleA,angleB,angleC .

Answer»


ANSWER :`COS^(-1)(SQRT(19/28))`
Discussion
44.

If the line x+y+1=0 and y=2x+5=0 are tangents to a parabola whose focus at (1,2) then the equation of normal to the parabola through (29/17, 14/17) is

Answer»

`2x+4y-1=0`
`y-3x+2=0`
`4x+y-5=0`
NONE of these

Solution :Normal through GIVEN point be the AXIS of PARABOLA.
`y-2=-4(x-1)`
`4x+y=6`
Discussion
45.

The volume (in cubic unit ) of the tetrahedron with edges hati+hatj+hatk, hati-hatj+hatk and hati+2hatj-hatk, is

Answer»

4
`(2)/(3)`
`(1)/(6)`
`(1)/(3)`

Solution :Volume of TETRAHEDRON `=1/6[(AB, AC, AD)]`
`=1/6 |(1,1,1),(1,-1,1),(1,2,-1)|`
`=1/6 [-1+2+3]=2/3` cu UNIT
Discussion
46.

The set of all x satisying the inequility (4x-1)/(3x+1) ge 1is

Answer»

`(-OO , -1/3) UU [1/4 , oo)`
`(-oo, -2/3) uu [2,oo)`
`(-oo, -1/3) uu [3,oo)`
`(-oo, -2/3) uu [4 , oo)`

Answer :C
Discussion
47.

int (1)/(1 - x^(2)) " ln " ((1 + x)/(1 -x))dx =

Answer»

`(1)/(2) "In" ((1 + X)/(1 - x))^(2) + C`
`(1)/(4) [ "In" ((1 + x)/(1 - x)) ]^(2) + C `
`(1)/(3) "In" ((1 + x)/(1 - x)) + C `
`(1)/(2) [ "In" ((1+ x)/(1 - x) ) ]^(3) ` + C

Answer :B
Discussion
48.

The set of all real values of x for which the real valued function f(x)= (1+(1)/(x))^(x) is defined is

Answer»

`(0,OO)`
R - {0}
`(-oo,-1) cup (0,oo) `
R - {0,-1}

Answer :C
Discussion
49.

If f(x) = sqrt(x) and g(x)=(1)/(sqrt(x)) for xin [3,12] then the value of c in (3,12) for which (f'(c))/(g'(c))=(f(12)-f(3))/(g(12)-g(3))hold is

Answer»

`7.5`
`4.8`
6
9

Answer :C
Discussion
50.

{:(" "Lt),(n rarr oo):} 1/n sum_(r=1)^(n)sec^(2).(rpi)/(4n)=

Answer»

A) 1
B) `2/pi`
C) `pi/4`
D) `4/pi`

ANSWER :D
Discussion