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.

If cos^(3)theta+cos^(3)((2pi)/(3)+theta)+cos^(3)((4pi)/(3)+theta)=a cos 3 theta, then a=

Answer»

`(1)/(4)`
`(3)/(4)`
`(5)/(4)`
`(7)/(4)`

ANSWER :B
Discussion
2.

Show that the differential equation 2ye^(x)/(y) dx + ( y - 2xe^(x)/(y)) dy = 0 is homogeneous and find its particular solution, given that, x = 0 when y = 1.

Answer»


ANSWER :`2E^(X)/(y) + LOG |y| = 2`
Discussion
3.

If the vectors _a^rarr=2overset^j-3overset^k and _b^rarr=2overset^j+aoverset^j+6overset^k are parallel, white the value ofalpha.

Answer»

SOLUTION :`|alphaoversetrarra|=2rArr|alpha||OVERSETRARRA|=2rArralpha=_pm(2)/|oversetrarra|`
Discussion
4.

If |{:(x-1, 5x, 7), ( x ^(2) -1, x -1, 8),( 2x, 3x, 0):}|=a x ^(3) + bx ^(2) + cx +d, then c is equal to

Answer»


ANSWER :17
Discussion
5.

Evaluate the following integrals : (i) int(x^(2)-4)/((x^(2)+1)(x^(2)+2)(x^(2)+3))dx (ii) int(1)/((x+1)(x^(2)+1)^(2))dx (iii) int(1)/(x^(3)(x^(2)+1)^(2))dx

Answer»

Solution :`(i)` We have,
`(x^(2)-4)/((x^(2)+1)(x^(2)+2)(x^(2)+3))=(A)/(x^(2)+1)+(B)/(x^(2)+2)+(C)/(x^(2)+3)`
`implies(t-4)/((t+1)(t+2)(t+3))=(A)/(t+1)+(B)/(t+2)+(C)/(t+3)`, for `=x^(2)`
`impliesA=(-5)/(2)`, `B=+6`, `C=(-7)/(2)`
`implies(x^(2)-4)/((x^(2)+1)(x^(2)+2)(x^(2)+3))=(-5)/(2)*(1)/(x^(2)+1)+(6)/(x^(2)+2)-(7)/(2)*(1)/(x^(2)+3)`
`impliesint(x^(2)-4)/((x^(2)+1)(x^(2)+2)(x^(2)+3))dx=(-5)/(2)tan^(-1)x+(6)/(sqrt(2))tan^(-1)((x)/(sqrt(2)))-(7)/(2)*(1)/(sqrt(3))tan^(-1)((x)/(sqrt(3)))+C`
`=(-5)/(2)tan^(-1)x+3sqrt(2)tan^(-1)((x)/(sqrt(2)))-(7)/(2sqrt(3))tan^(-1)((x)/(sqrt(3)))+C`
`(ii)` We have
`(1)/((x+1)(x^(2)+1)^(2))=(A)/(x+1)+(Bx+C)/(x^(2)+1)+(px+Q)/((x^(2)+1)^(2))`
`implies1=A(x^(2)+1)^(2)+(Bx+C)(x^(2)+1)(x+1)+(px+q)(x+1)`.
Equating the coefficients of `x^(4)`, `x^(3)`, `x^(2)`, `x` and constant term from both sides, we GET `A+B=0`, `B+C=0`, `2A+B+C+p=0`, `B+C+p+q=0` and `A+C+q=0`
`impliesA=(1)/(4)`, `B=-(1)/(4)`, `C=(1)/(4)`, `p=-(1)/(2)`, `q=(1)/(2)`
Thus `int(1)/((x+1)(x^(2)+1)^(2))dx=(1)/(4)int(1)/(x+1)dx-(1)/(4)int(x-1)/(x^(2)+1)dx-(1)/(2)int(x-1)/((x^(2)+1)^(2))dx`
`=(1)/(4)ln|x+1|-(1)/(8)int(2x)/(x^(2)+1)dx+(1)/(4)int(1)/(x^(2)+1)dx-(1)/(4)int(2x)/((x^(2)+1)^(2))dx+(1)/(2)(dx)/((x^(2)+1)^(2))`
`=(1)/(4)ln|x+1|-(1)/(8)ln(x^(2)+1)+(1)/(4)tan^(-1)x+(1)/(4)*(1)/((x^(2)+1))+(1)/(2)int(1)/((x^(2)+1)^(2))dx`
Now `int(1)/((x^(2)+1)^(2))dx=int(sec^(2)theta)/(sec^(4)theta)d theta` [Putting `(x=tantheta)` so that `dx=sec^(2)theta d theta`]
`=int(1+cos2theta)/(2)d theta=(1)/(2)[theta+(sin2theta)/(2)]`
`=(1)/(2)(theta+sinthetacostheta)=(1)/(2)[tan^(-1)x+(x)/(x^(2)+1)]`
Hence `int(1)/((x+1)(x^(2)+1)^(2))dx=(1)/(4)ln|x+1|-(1)/(8)ln(x^(2)+1)+(1)/(2)tan^(-1)x+(x+1)/(4(x^(2)+))+C` , `C` is constant of integration.
`(III)` We have,
`int(1)/(x^(3)(x^(2)+1)^(2))dx=(1)/(2)int(2x)/(x^(4)(x^(2)+1)^(2))dx`
Let US put `x^(2)+1=t`
`2xdx=dt`
`x^(2)=t-1`
`=(1)/(2)int(dt)/(t^(2)(t-1)^(2))`
Now `(1)/(t^(2)(t-1)^(2))=(p)/(t-1)+(q)/((t-1)^(2))+(r )/(t)+(s)/(t^(2))`
`implies1=p(t-1)t^(2)+qt^(2)+rt(t-1)^(2)+s(t-1)^(2)`
whence `p=-2`, `q=1`, `r=2`, `s=1`
Thus `intint(1)/(x^(3)(x^(2)+1)^(2))dx=(1)/(2)[int(-2)/(t-1)dt+int(1)/((t-1)^(2))dt+int(2)/(t)dt+int(1)/(t^(2))dt]`
`=-ln|t-1|+ln|t|-(1)/(2(t-1))-(1)/(2t)+C`
`=ln|(x^(2)+1)/(x^(2))|-(1)/(2x^(2))-(1)/(2(x^(2)+1))+C `
Discussion
6.

Prove that ((n^(2))!)/((n!)^(n)) is a natural number for all n in N.

Answer»

Solution :We have to PROVE that `(n^(2))!` is divisible by `(n!)^(n)`.
We know that product of r CONSECUTIVE integers is divisible by r!
Now `(n^(2))!=1xx2xx3xx4xx..XX n^(2)`
`=(1xx2xx3xx..xx n)xx`
`[(n+1)(n+2)xx.. .. Xx(2n)]xx`
`[(2n+1)(2n+2).. .. (3N)]xx`
.. ..
.. ..
`[(n^(2)-(n-1)(n^(2)-n).. .. n^(2)]`
THUS `(n^(2))!` consists n groups of product of n consecutive integers.
Each group is divisible by n!.
So, `(n^(2))!` is divisible by `(n!)^(n)`.
Discussion
7.

Let a !=0, and p(x) be a polynomial of degree greater than 2. If p(x) leaves remainders a and -a when divided respectively by x+a and x-a, then the remainder when p(x) is divided by x^(2)-a^(2) is

Answer»

`X`
`-x`
`-2X`
`2x`

ANSWER :B
Discussion
8.

Ifcos alpha + cos beta + cos gamma = 0= sin alpha + sin beta + sin gammathen show that cos ( alpha + beta) +cos ( beta + gamma) + cos ( gamma + alpha) = 0

Answer»
Discussion
9.

For what value of x the tangent of the curve y = x^(3) - 3x^(2) + 2x -5 is parallel to x + y + 3 = 0

Answer»


ANSWER :`(-1,0)`
Discussion
10.

Consider the family of circles : x^(2)+y^(2)-3x-4y-c_(1)=0, c_(1)inN (i=1,2,3,…,n) Also, let all circles intersects X-axis at integral points only and c_(1)ltc_(2)ltc_(3)ltc_(4)…ltc_(n)A point (x, y) is said to be integral point, if both coordinates x and y are integers.

Answer»


ANSWER :B
Discussion
11.

At which point the curve y=-x^(3)+3x^(2)+2x-27 has maximum slope ? How much ?

Answer»


ANSWER :MAXIMUM SLOPE = 5 at `(1,-23)`
Discussion
12.

Consider the family of circles : x^(2)+y^(2)-3x-4y-c_(1)=0, c_(1)inN (i=1,2,3,…,n) Also, let all circles intersects X-axis at integral points only and c_(1)ltc_(2)ltc_(3)ltc_(4)…ltc_(n)A point (x, y) is said to be integral point, if both coordinates x and y are integers. If circle x^(2)+y^(2)-3x-4y-(c_(2)-c_(1))=0 and circle x^(2)+y^(2)=r^(2) have only one common tangent, then

Answer»

`R=1//2`
tangent PASSES through `(10,0)`
(3, 4) lies outside the circle `x^(2)+y^(2)=r^(2)`
`c_(2)=2r+c_(1)`

Answer :D
Discussion
13.

If a, b are positive integers, define a^(**)b = alpha where ab -= alpha (modulo 7), with this ** operation, the inverse of 3 in group G = { 1, 2, 3, 4, 5, 6}

Answer»

3
1
5
4

Answer :C
Discussion
14.

Show that the plane 2x-y-2z-4=0 touches the shpere x^2+y^2+z^2+2x-6y+1=0.

Answer»

Solution :Equation of the sphere concentric to the GIVEN sphere and passing through ORIGIN is `x^2+y^2+z^2-4x-6y+8z=0`.
Discussion
15.

If alpha = underset(x rarr 0)Lt (x.2^(x) -x)/(1 - cos x) and beta = underset(x rarr 0)(Lt) (x.2^(x) -x)/(sqrt(1 + x^(2)) -sqrt(1-x^(2))), then

Answer»

`ALPHA = beta`
`alpha = 2beta`
`alpha = (beta)/(2)`
`alpha = 3 beta`

ANSWER :B
Discussion
16.

Evaluate int_(0)^(1)x^(6) sin^(-1) x dx

Answer»


ANSWER :`(PI)/(14)-(16)/(245)`
Discussion
17.

Let Sn = n^2 + 20n + 12 where n is a positive integer. What is the sum of all possible values of n for which S_(n)is a perfect square ?

Answer»


ANSWER :16
Discussion
18.

Leta_k =k(^(10)C _k ) , b_k =(10 = k ),^(10 ) C _k and A_k =[{:( a_k ,0),( 0, b_k) :}] , "If" A = underset(k=1) overset (theta ) sum A_k = [{:( a,0),( 0,b) :}]then find the value of(ab)/( 2^(theta )-1) .

Answer»


ANSWER :`51100`
Discussion
19.

If x = (729 + 6(2)(243) + 15(4)(81) + 20(8)(27) + 15(16)(9) + 6(32)3 + 64)/(1 + 4(4) + 6(16) + 4(64) + 256) then sqrt(x)-1/sqrt(x) =

Answer»

0.2
4.8
1.02
5.2

Answer :B
Discussion
20.

Compute the 5!+6!

Answer»

SOLUTION :`5!+6!=5!+6*5!`
`=5!(1+6)=120*7=840`
Discussion
21.

Find the co-ordinates of the foot of the perpendicular drawn from the origin. 5y + 8 = 0

Answer»


ANSWER :The DIRECTION cosines of the normal to the plane are 0, -1, 0 and the distance of the plane from the ORIGIN is `8/5` (see example 1 (d) above)
`therefore` The co-ordinates of the FOOT of the PERPENDICUALR `= (0, (-8)/5, 0)`
Discussion
22.

Find the co-ordinates of the foot of the perpendicular drawn from the origin. x + y + z = 1

Answer»


Answer :The DIRECTION cosines of the normal to the plane are `(1)/sqrt3, (1)/sqrt3, (1)/sqrt3,` and the distance of the plane from the origin `= (1)/sqrt3` (SEE example 1(B) above)
`therefore` The FOOT of the PERPENDICULAR drawn from the origin `= (1/3, 1/3, 1/3)`
Discussion
23.

Find the order and degree of the D.E. : (1) (d^(2)y)/(dx^(2)) + x(dy)/(dx0 + y = 2sin x

Answer»


ANSWER :order=2 , DEGREE =1
Discussion
24.

Three cards are drawn from the pack of 52 playing cards. Find the probability distribution of numbers of kings.

Answer»


ANSWER :`(##KPK_AIO_MAT_XII_P2_C13_E01_052_A01##)`
Discussion
25.

Find the ratio of the curves into which the circle x^(2) + y^(2)= 64 is divided by the curve y^(2)=12x

Answer»


ANSWER :`4PI + SQRT3 : 8pi - sqrt3`
Discussion
26.

Prove that the following functions do not have maxima or minima :f(x)=e^(x)

Answer»


ANSWER :no MAXIMA or MINIMA.
Discussion
27.

If a(p + q)^(2) + 2 bpq + c = 0 and a(p +r)^(2) + 2bpr + c = 0, then |q - r| equals

Answer»

`(2)/(|a|)sqrt((2a + B) bp^(2) - ac)`
`(2)/(|a|) sqrt(p^(2) - 4AC)`
`p^(2) +(C)/(a)`
NONE of these

Answer :A
Discussion
28.

A glass vessel is so constructed that when the depth of the liquid in is x cm , the volume of the liquid is (3x^(2)-(x^(3))/(3)) cm^(3).Liquid is poured itno the vessel at such a the rate of 2 cm//sec, Tthen the liquid is being poured in the vessel at tjhe rate of

Answer»

`81 cc//sec`.
`18 cc//sec`.
`9 cc//sec`.
`8 cc//sec`.

Answer :B
Discussion
29.

P is a variable point on the ellipse x^2/a^2+y^2/b^2=1 with foci F_1 and F_2. IF A is the area of the triangle PF_1,F_2, then the maximum value of A is

Answer»

`E/(AB)`
`(ae)/B`
`(ae)/b`
`(ab)/e`

ANSWER :C
Discussion
30.

Evaluate the following definite integrals as limit of sums. int_(0)^(5)(x+1)dx

Answer»


ANSWER :`35/2`
Discussion
31.

When phenol is reacted with CHCI, and NaOH followed by acidification, salicyldehyde is obtained. Which of the following species are involved in the above mentioned reaction as intermediate?

Answer»




All of these

Answer :A
Discussion
32.

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are hat i + 2 hat j - hat k and - hat i + hat j + hat k respectively, in the ratio 2 : 1 (i) internally (ii) externally

Answer»


Answer :(i) `-(1)/(3)HATI+(4)/(3)HATJ+(1)/(3)HATK`
(II) `-3hati+3hatk`
Discussion
33.

Equation of the curve passing through the point (4,3) and having slope = y//2 at a point (x,y) on it is

Answer»

`LOG ((y)/(3)) = (X)/(2) - 2`
`log ((y)/(3)) = (x)/(2) + 2`
`y = 3e^(x-2)`
`y^(2) = x + 5`

ANSWER :A
Discussion
34.

Ifabne0andthe sum ofcoefficientsofx ^ 7 andx ^ 4inthe expansionof((x ^ 2 ) /(a)-(b)/(x) ) ^ (11)is0 , then

Answer»

`a=B `
`a+ b= 0 `
`ab =- 1 `
`a b = 1 `

Solution :Consider` T_(r + 1 ) `termintheexpansion
`therefore T _(r + 1 )=""^(11)c_r ((X^ 2 )/(a)) ^(11 -r )(( - b ) /( x ) ) ^r`
`=""^(11) c _r ((1 )/(a)) ^( 11 - r)( - b) ^rx ^( 22 - 2 r) .x^( - r) `
`=""^(11)c _ r((1 ) /(a)) ^(11 - r )( - b ) ^rx ^(22 -3r ) `
if `x ^(22 - 3r)=x ^7`
`rArr22 - 3r=7`
`rArr3r=15`
`thereforer=5`
If`x^( 22-3r )=x ^ 4`
`rArr22 -3r= 4 `
`3r =18 `
`r=6 `
`therefore`co -efficientof` T_ 6`+ Co-efficientof`T _ 7=0 `
`rArr""^(11)c _ 6 (( 1 ) /(a)) ^ 5(- b ) ^( 6)+""^(11)c _ 5 (( 1 ) /(a)) ^6(- b ) ^( 5)=0 `
`rArr ""^(11)c _5[ (b^6 )/(a ^5 )-(b^5 ) /(a^6 ) ] =0""[ because""^(11)c _ 5 =""^(11)c _6] `
`rArr(b ^ 6 ) /( a ^5 )=(b ^ 5 ) /(a ^6 ) `
` rArr(ab) ^6 =(ab) ^ 5 `
` rArr(ab) ^5 [ab -1 ] =0 `
` rArrab - 1=0[becauseab ne 0] `
`ab = 1`
Discussion
35.

There are n-1 red balls, n green balls and n+1 blue balls in a bag. The number of ways of choosing two balls from the bag that have different colours is 299. What is the value of n?

Answer»


ANSWER :10
Discussion
36.

Let f (x) = (x ^(3) -4)/((x-1)^(3)) AA x ne 1, g (x)== (x ^(4) -2x ^(2))/(4) AA x in R, h (x) (x ^(3) +4)/((x+1)^(3)) AA x ne -1,

Answer»


Answer :ATO Q,R; BtoR,S; C TOQ,R,S; DtoP,R,T`
Discussion
37.

Two circle S_1=0,S_2=0 of equal radius 'r' intersect such that one circle passes through the centre of the other circle.Another circle (S=0) touches S_1=0 internally and S_2=0 externallyand also touches the line joining the centres of the two circles S_1=0 and S_2=0 .Then radius of S=0 is

Answer»

`(sqrt3r)/2`
`(sqrt3r)/4`
`(sqrt3r)/8`
none of these

Answer :B
Discussion
38.

int_(0)^(oo) (x^(3))/((1+x^(2))^(9//2))dx=

Answer»

`1/35`
`3/35`
`2/35`
`4/35`

ANSWER :C
Discussion
39.

The value of a for which the sum of the squares of the roots of the equation x^(2)-(a-2)x-a-1=0 assume the least value is

Answer»

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

ANSWER :B
Discussion
40.

If the distance between (2, 3) and (-5, 2) is equal to the distance between (x, 2) and (1,3), then the values of x are

Answer»

-6, 8
6, 8
-8, 6
-7, 7

Answer :A
Discussion
41.

Find x

Answer»

`49^(@)`
`52^(@)`
`79^(@)`
`101^(@)`

ANSWER :D
Discussion
42.

Write relations in tabular form and determine their type for R={(x,y) : x "divides" 2- y } "on" A ={1,2,3,4,5}

Answer»

Solution :R`={(x,y):x DIVIDES 2-y } on A `
={1,2,3,4,5}
`={(x,y):2-y "is a MULTIPLE of" x }`
`={(1,1),(1,2),(1,3),(1,4),(1,5),(2,2),(2,4),(3,2),(3,5),(4,2),(5,2)}`
R is NEITHER REFLEXIVE nor SYMMETRIC nor transitive.
Discussion
43.

If g is the acceleration due to gravity on the earth's surface the change in the potential energy of an object of mass m raised from the surface of the eacrth to height to the radius R of the earth is

Answer»

`mgR//2`
2mgR
mgR
#REF!

Solution :`u_(G)=-(GM_(1)M_(2))/(R)`
`RARR""u_(f)-u_(i)=-(GMm)/(2R)=(GMm)/(R)`
`Deltau=(GMm)/(2R)=(GMm)/(R)""{g=(GM)/(R^(2))}`
Discussion
44.

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

Answer»


ANSWER :`(X^(2))/(2)+(1)/(2)log|x+1|+(3)/(2)log|x-1|+c`
Discussion
45.

The focal distance of the point (4,2) on the parabola x^(2)=8y is

Answer»

`SQRT2`
`2SQRT2`
`3sqrt2`
`4sqrt2`

ANSWER :B
Discussion
46.

If P(A) = 0.3, P(B) = 0.4, P(C) = 0.8, P(A nn B) = 0.08, P(A nn C) = 0.28, P(A nn B nn C) = 0.09, P(A uu B uu C) ge 0.75 then show P(B nn C) lies in [0.23, 0.48].

Answer»


ANSWER :`RARR 0.23 LE P(B nn C) le 0.48`
Discussion
47.

If 5^(40) is divided by 11, then remainder is

Answer»

2
3
5
1

Answer :D
Discussion
48.

If the order of a differential equation (d^2y)/(dx^2)-2 ((dy)/(dx))^3 +sin ((dy)/(dx)) +y =0 is l and the degree of the differential equation (1+(d^2y)/(dx^2))^(2/3)=[2-((dy)/(dx))^3]^(3/2)is m, then the differential equation corresponding to the family of curves y= Ax^l +Be ^(mx), where A and B are arbitrary constants is

Answer»

`(4x^2-2x)y^('') +(16x^2-2)y^'+(32x=8)y=0`
`(2x^2-x)y^('')+(8x^2-2)y^'+(16x-4)y=0`
`(2x^2-4t)y^('')-(8x^2-1)y^'+(16x-4)y=0`
`(4x^2-2x)y^('')+(8x^2-1)y^'+(16x-4)y=0`

ANSWER :C
Discussion
49.

The relation R on the set A = {1,2,3}defined as R = {(1,1),(1,2),(2,1),(3,3)} is reflexive , symmetric and transitive.

Answer»


SOLUTION :N/A
Discussion
50.

If the equation of the circle cutting the circlex^(2) + y^(2) + 2x - 4y + 8 = 0 orthogonally and coxal with the circlesx^(2) + y^(2) + 6x + 4y - 12 = 0 , x^(2)4x - 6y - 12 = 0" is " x^(2) + y^(2) + 2ax + 2by + c = 0then the ascending order of a,b,c is

Answer»

a,B,C
b,c,a
b,a,c
a,c,b

Answer :C::D
Discussion