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 x and y are positive integers, the the solution sets of the inequations xle3,yle2 and 5x+6yle21 are-

Answer»




ANSWER :B
Discussion
2.

Integrate the functions xlogx

Answer»


ANSWER :`(X^(2))/2logx-(x^(2))/4+C`
Discussion
3.

int_(-pi)^(pi) (x Cos x)/(1+Sin^(2)x)dx=

Answer»

1
0
-1
`1/2`

ANSWER :B
Discussion
4.

Find the value of .^(20)C_(0) xx .^(13)C_(10) - .^(20)C_(1) xx .^(12)C_(9) + .^(20)C_(2) xx .^(11)C_(8) - "……" + .^(20)C_(10).

Answer»

Solution :`.^(20)C_(0) xx .^(13)C_(10) - .^(20)C_(1) xx .^(12)C_(9) + .^(20)C_(2)xx .^(11)C_(8) -"…." + .^(20)C_(10)`
`=` Coefficient of `X^(10)` in
`(.^(20)C_(0) - .^(20)C_(1)x + .^(20)C_(2)x^(2) - .^(20)C_(3)x^(3) + "…..")`
`(1+.^(4)C_(1)x + .^(5)C_(2)x^(2)+.^(6)C_(3)x^(3)+".....")`
`=` Coefficientof `x^(10)` in `(1-x)^(20)(1-x)^(-4)`
`=` Coefficient of `x^(10)` in `(1-x)^(16)`
`= .^(16)C_(10)`.
Discussion
5.

If bar(a) and bar(b) any two non-collinear vectors lying in the same plane, then prove that any vector bar(r) coplanar with them can be uniquely expressed as bar(r)=t_(1)bar(a)+t_(2)bar(b), where t_(1)andt_(2) are scalars.

Answer»

Solution :Take any points O in the plane of `bar(a),bar(b)andbar(r)`. Represents the vectors `bar(a),bar(b)andbar(r)` by `bar(OA),bar(OB)andbar(OR)`. Take the points P on `bar(a)` and Q `bar(b)` such that OPRQ is a PARALLELOGRAM.
Now `bar(OP)andbar(OA)` are collinear vectors.
`:.` there exists a non - zero scalar `t_(1)` such that
`bar(OP)=t_(a)*bar(OA)=t_(1)*bar(a)`.
Also `bar(OQ)andbar(OB)` are collinear vectors.
`:.` there exixts a non-zero scalar `t_(2)` such that `bar(OQ)=t_(2)*bar(OB)=t_(2)*bar(b)`.
Now, by parallelogram law of addition of vectors,
`bar(OR)=bar(OP)+bar(OQ)"":.bar(r)=t_(1)bar(a)+t_(2)bar(b)`
Thus `bar(r)` expressed as linear combination `t_(1)bar(a)+t_(2)bar(b)`
Uniqueness:
LET, if possible, `bar(r)=t_(1)^(')bar(a)+t_(2)^(')bar(b)`, where `t_(1)^('),t_(2)^(')` are non-zero scalars. Then
`t_(1)bar(a)+t_(2)bar(b)=t_(1)^(')bar(a)+t_(2)^(')bar(b)`
`:.(t_(1)-t_(1)^('))bar(a)=-(t_(2)-t_(2)^('))bar(b)`. . . . (1)
WE want to show that `t_(1)=t_(1)^(')andt_(2)=t_(2)^(')`.
Suppose `t_(1)!=t_(1)^('),i.e.,t_(1)-t_(1)^(')!=0andt_(2)!=t_(2)^(')!=0`.
Then dividing both sides of (1) by `t_(1)-t_(1)^(')`, we GET,
`bar(a)=-((t_(2)-t_(2)^('))/(t_(1)-t_(1)^(')))bar(b)`
This shows that the VECTOR `bar(a)` is a non-zero scalar MULTIPLE of `bar(b)`.
`:.bar(a)andbar(b)` are collinear vectors.
This is a contradiction, since `bar(a),bar(b)` are given to be non-collinear.
`:.t_(1)=t_(1)^(')`
Similarly, we can show tath `t_(2)=t_(2)^(')`
This shows that `bar(r)` is uniquwly expressed as a linear combination `t_(1)bar(a)+t_(2)bar(b)`.
Discussion
6.

y dx + (x - y^(2))dy = 0

Answer»


ANSWER :`X = (y^(2))/(3) + (C)/(y)`
Discussion
7.

If cos y= x cos (a + y), with cos a ne +- 1, prove that (dy)/(dx) = (cos^(2) (a + y))/(sin a)

Answer»


ANSWER :`(COS^(2) (a + y))/(SIN a)`
Discussion
8.

Lt_(n rarr oo)[(1)/(3n+1) + (1)/(3n+2)+…+(1)/(4n)]

Answer»


ANSWER :`log_(E)((4)/(3))`
Discussion
9.

2[(1)/(2x+1)+(1)/(3*(2x+1)^(3))+(1)/(5*(2x+1)^(5))+…..oo]=

Answer»

`log_(e )X`
`log_(e )((1)/(x))`
`log_(e ) ((x+1)/(x))`
`log_(e )((x-1)/(x ))`

ANSWER :C
Discussion
10.

Differential equation y log y dx - x dy = 0

Answer»


ANSWER :`y = E^(CX)`
Discussion
11.

Find the rate of change of the area of a circle per second with respect to its radius r when r = 5 cm.

Answer»


ANSWER :`10 PI CM^(2)//cm`
Discussion
12.

If from a variable point P representingthe complex number z_(1) on the curve |z|=4, two tangentsare drawn to thecurve |z|=2, meeting it at points Q(z_(2)) and R(z_(3)), then which of the following statement(s) is(are) correct?

Answer»

Triangle , PQR is isosceles.
The locus of centroid of triangle PQR is |z|=1.
The circumradium of triangle PQR is 2.
The radiusof circle inscribed in triangle PQR is 1.

Solution :
`:.` From above figure , `cos ( anglePOR)=(OR)/(OP)(2)/(4)=(1)/(2)`
`rArr angle POR=(pi)/(3)=anglePOQ rArr angle OPR= angle OPQ=30^(@)`
`rArr angleQPR=60^(@) " ".....(1)`
ALSO, in `DeltaPQR, PQ=PR "" .....(2)`
`:.` From (1) and (2) , we get
`DeltaPQR` is equilateral `rArr` (A) is INCORRECT.
Also, PQOR are CONCYCLIC and `angleOQP and angleORP = 90^(@)`.
So, circumcentre of `DeltaPQR` passes through O(0,0) and OP is diameter of it.
So, circumcentre of `DeltaPQR` = mid POINT of OP
`=((0+4 cos theta)/(2),(0+4 sin theta)/(2))=(2cos theta, 2 sin theta)`
=centroid of `DeltaPQR` [As, `DeltaPQR` is equilateral.]
`:.` The locus of centroid of `DeltaPQR` is |z|=2 `rArr` (B) isincorrect.
Also, circumradius of `DeltaPQR=(OP)/(2)=(4)/(2)=2rArr` (C) is correct.
As, `r=(R)/(2)=(2)/(2)=1` (As, `DeltaPQR` is equilateral.)
`rArr` radius of circle inscribed in `DeltaPQR` is 1. `rArr` (D) is correct.]
Discussion
13.

Solution of the differential equation (dy)/(dx)=(y(x-y ln y))/(x(xlnx-y)) is

Answer»

`("X LN x + y ln y")/(xy)=C`
`(xlnx-ylny)/(xy)=C`
`(LNX)/(x)+(LNY)/(y)=C`
`(lnx)/(x)-(lny)/(y)=C`

ANSWER :A
Discussion
14.

Principal value of sin^-1 (-frac{ 3}{2}) is :

Answer»

π/3
`-π/3`
`π/6`
`-π/6`

ANSWER :B
Discussion
15.

The number of ways that 5 blue balls of different sizes and 5 red balls of different sizes can be arranged in a row so that no two balls of the same colour come together is :

Answer»

<P>`5! 5!`
`5! ""^(6)P_(5)`
`2xx5! 6!`
`2(5!)^(2)`

ANSWER :D
Discussion
16.

Let ABCD be a rhombus such that angle ABC =2pi //3. Let S be a circle inscribed in te rhombus ABCD. If ength of a side of the rhombus is 1, then PA ^(2) +PB ^(2) + PC ^(2) + PD ^(2) is equal to.

Answer»


ANSWER :`2.75`
Discussion
17.

Prove that the lines x = py + q, z = ry + s and x = p'y + q', z = r'y + s' are perpendicular if pp' + rr' + 1 = 0.

Answer»


ANSWER :`THEREFORE` pp'+rr'+1=0
Discussion
18.

If (m_(i),1/(m_(i))),m_(i)gt0,i=1,2,3,4 are four distinct points on a circle, show that m_(1)m_(2)m_(3)m_(4)=1.

Answer»


ANSWER :1
Discussion
19.

If alpha and beta are the roots of x^(2)-2x+4=0, then the value of alpha^(6)+beta^(6) is

Answer»

32
64
128
356

Answer :C
Discussion
20.

If linear function f(x) and g(x) satisfy int[(3x-1) cos x + (1-2x)sinx]dx = f(x) cos + g(x) sin x + C, then

Answer»

`F(X) = 3(x-1)`
`f(x) = 3x-5`
`G(x) = 3(x-1)`
`g(x) = 3+x`

Answer :C
Discussion
21.

On a toss of two dice, A throws a total of 5. Then the probability that he will throw another 5 before he throws 7 is

Answer»

`(2)/(45)`
`(2)/(5)`
`(1)/(81)`
`(1)/(9)`

Answer :B
Discussion
22.

If f : N to R is defined by f(1) = -1 and f(n+1)=3f(n)+2 for n gt 1, then f is

Answer»

one - one
onto
a CONSTANT FUNCTION
`f(n)gt 0` for `n gt 1`

ANSWER :C
Discussion
23.

Prove that (tan^(-1)(1)/(e ))^(2)+(2e)/(sqrt(e^(2)+1)

Answer»

Solution :we have to PROVE
`TAN^(-1)(1)/(C )sqrt(e^(2)+1)lt(tan^(-1)e)^(2)+(32)/(e^(2)+1)`
or `tan^(-1)(1)/(e )^(2)+(2)/sqrt(1)(e )^(2)+1lt(tan^(-1)e)^(2)+(2)/sqrt(e^(2)+1)`
Now f(X) =`(2tan^(-1)x)/(1+x^(2))-(2x)/(x^(2)+1)`
It is difficult to study the sign of f(x)
so we let g(x) =`tn^(-1)x-(x)/sqrt(x^(2)+1)`
g(x) is andincreasing function
for `xgt 0,g(x)gtg(0)`
`g(x)gt0`
`f(x)gt0`
f(x) is increasing function
Since `(1)/(e )lte`, we have `f(1/e)ltf(x)`
Discussion
24.

The mid-point of the chord x+2y +4 =0on the hyperbola3x^(2) -4y^(2) =12is

Answer»

`(1,2) `
`( 2,3) `
`( 2,-3) `
`(2,-1)`

ANSWER :C
Discussion
25.

The statement, 'r is sufficient for s', is also expressed as.

Answer»

s if R
r only if s'
r IMPLIES s'
all of these

Answer :D
Discussion
26.

C_0+3. C_1+5. C_2 +…...(2n+1).C_n=

Answer»

`(4n-10) 2^N + n+ 6`
`(2N-5) 2^n + n + 6`
`(2n -5 ) 2^(n-1)+n+6`
`(3n-4)2^(n-1) + 1`

Answer :B
Discussion
27.

Choose the correct answer int(10x^(9)+10^(x)log_(e)10dx)/(x^(10)+10^(x)) equals

Answer»

`10^(x)-x^(10)+C`
`10^(x)+x^(10)+C`
`(10^(x)-x^(10))^(-1)+C`
`LOG(10^(x)+x^(10))+C`

Answer :D
Discussion
28.

PN is the ordinate of any point P on the hyperbolax^(2)//a^(2) -y^(2)//b^(2) =1. If Q divides AP in the retioa^(2) ,b^(2)then NQ is

Answer»

<P>PERPENDICULAR to A'P
Parallel to A'P
perpendicular to OP
NONE

ANSWER :A
Discussion
29.

LetF(x)=(1+sin(pi/(2k))(1+sin(k-1)pi/(2k))(1+sin(2k+1)pi/(2k))(1+sin(3k-1)pi/(2k)). The value ofF(1)+F(2)+F(3) is equal to

Answer»

`(3)/(16)`
`(1)/(4)`
`(5)/(16)`
`(7)/(16)`

Answer :D
Discussion
30.

If A and B are coefficients of x^(n) in the expansion of (1+x)^(2n) and (1+x)^(2n-1) respectively, then find the value of (A)/(B).

Answer»

1
2
3
4

Answer :B
Discussion
31.

The value of f(0) so that f(x)=(sin x)/(x) is continuous at x=0 is

Answer»

INCREASING in `(0,pi//2)`
decreasing in `(0,pi//2)`
STATIONARY at `x=pi//2`
none

Answer :B
Discussion
32.

If b gta, then the equation (x -a)(x - b) - 1 = 0has :

Answer»

both roots in [a,b]
both roots in `[- infty , b)`
both roots in `(b, + infty)`
one ROOT in `(-infty, a )` and other in `(b, + infty)` .

ANSWER :D
Discussion
33.

If A = [{:(3,1),(-1,2):}] and A^(2) - 5A= kI then k=.........

Answer»

`5`
`7`
`-7`
`-5`

ANSWER :C
Discussion
34.

if lim_(xto(pi)/(4))((tan((pi)/(4)-x))/((pi)/(4)-x))=

Answer»

0
1
`pi/4`
not defined

Answer :B
Discussion
35.

The sequence a_(n) = sin (( pi )/( 4) + n pi ) for n = 1,2,3,"…….." is :

Answer»

an ARITHMETIC progression
a GEOMETRIC progression
neither an arithmetic nor a geometric progression
both an arithmetic and a geometric progression

Answer :B
Discussion
36.

The monthly sales for the first 11 months of the year of a certain salesman were Rs. 12,000 but due to his illness during the last month the average sales for the whole year came down to Rs 11,375 . The value of the sale during the last month was

Answer»

RS 4,500
Rs 6,000
Rs 10,000
Rs 8,000

Answer :A
Discussion
37.

Sum of coefficients of x^(2r) , r = 1,2,3…. in (1+x)^n is

Answer»

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

ANSWER :A
Discussion
38.

Let m be a vector of magnitude sqrt(3) and perpendicular to the vectors hati+hatj and hatj-hatk let n be another vector of magnitude 2sqrt(6) and perpendicular to the vectors 2hat I -hatj and hat j+2 hatk The areain sq units of the trianglefromed with m and n side is

Answer»

`SQRT(2)`
`sqrt(6)`
`2sqrt(3)`
`3sqrt(2)`

ANSWER :D
Discussion
39.

The solution of (e^(x) + 1) y dy + (y+ 1) dx = 0 is

Answer»

`E^(X+y) = C(y+1)(e^(x +1))`
`e^(x+1) = c(y+1)(e^(x) - 1)`
`e^(x+y) = c(y-1)(e^(x) + 1)`
`e^(x+y) = c(y-1)(e^(x) - 1)`

ANSWER :A
Discussion
40.

Equation of the hyperbola of eccentricity 3 and the distance between whose foci is 24 is

Answer»

`X^(2)-8y^(2)=128`
`x^(2)-16Y^(2)=128`
`16X^(2)-y^(2)=128`
`8x^(2)-y^(2)=128`

ANSWER :D
Discussion
41.

If L = underset( n rarr oo)lim (1+3^(-1))(1+3^(-2))+(1+3^(-4))+(1+3^(-8))…(1+3^(-2^(n))), then

Answer»

`L =(2)/(3)`
`L =(3)/(2)`
`L=1`
`L=2`

ANSWER :B
Discussion
42.

If with reference to the right handed system of mutually perpendicular unit vectors hati,hatj and hatk,vec(alpha)=3hati-hatj,vec(beta)=2hati+hatj-3hatk, then express vec(beta) in the form vec(beta)=vec(beta)_(1)+vec(beta)_(2), where vec(beta)_(1) is parallel to vec(alpha) and vec(beta)_(2) is perpendicular to vec(alpha).

Answer»


ANSWER :`vec(beta)_(1)=(3)/(2)hati-(1)/(2)HATJ` and `vec(beta)_(2)=(1)/(2)hati+(3)/(2)hatj-3hatk`
Discussion
43.

If alpha, beta are natural numbers, then int_(0)^(pi//2)cos^(alpha)x cos beta x dx equals, provided alpha=beta

Answer»

`(PI)/(2^(ALPHA+1))`
`(pi)/(2^(alpha))`
`(pi)/(2^(alpha-1))`
NONE of these

Answer :A
Discussion
44.

The point of intersection of the lines (x-4)/(1)=(y+3)/(-4)=(z+1)/(4) and (x+1)/(2)=(y+1)/(3)=(z+10)/(8) is

Answer»

(5,7,6)
(5, -7, 6)
(5, 6, 7)
(-5, 6, 7)

ANSWER :B
Discussion
45.

If X is the number of tails in three tosses of a coin, determine the standard deviation of X.

Answer»


ANSWER :`(SQRT(3))/(2)`
Discussion
46.

The locus of the points of intersection of tangents to y^2 = 4ax which intercept a constant length d on the directrix is:

Answer»

`(y^2-4ax)(x+a)^2=d^2x^2`
`(x^2-4ay)(x+a)^2=dx^2`
`(y^2-x)(x+a)^2=x^2`
NONE of these

Answer :A
Discussion
47.

int sin (log x)dx=

Answer»

X [SIN (LOG x ) + cos (log x ) ] + C
x [ sin (log x ) - cos (log x ) ] +c
`(x)/(2)` [ sin (log x ) + cos (log x ) ] +c
`(x)/(2)` [ sin (log x ) - cos (log x ) ]+ c

ANSWER :D
Discussion
48.

If b_(1)b_(2)=2(c_(1)+c_(2)) and b_(1), b_(2), c_(1), c_(2) are all real numbers, then at least one of the equations x^(2)+b_(1)x+c_(1)=0 and x^(2)+b_(2)x+c_(2)=0 has

Answer»

real roots
purely imaginary roots
roots of the FORM `a +ib (a, b in R, AB ne 0)`
RATIONAL roots

Answer :a
Discussion
49.

The relative housing cost for a US city is defined to be the ratio ("average housing cost for the city")/("national average housing cost") expressed as a percent. The scatterplot above shows the relative housing cost and the population density for several large US cities in the year 2005. The line of best fit is also shown and has equation y = 0.0125x + 61. Which of the following best explains how the number 61 in the equation relates to the scatterplot?

Answer»

In 2005, the lowest housing cost in the United States was about $61 per month.
In 2005, the lowest housing cost in the United States was about 61% of the highest housing cost
In 2005, even in cities with low population densities, housing costs were never below 61% of the national average.
In 2005, even in cities with low population densities, housing costs were LIKELY at least 61% of the national average.

Solution :To interpret what the number 61 in the equation of the line of best fit represents, one must first understand what the data in the scatterplot represent. Each of the points in the scatterplot represents a large US city, graphed according to its population density (along the horizontal axis) and its relative housing cost (along the vertical axis). The line of best fit for this datarepresents the expected relative housing cost for a certain population density, based on the data points in the graph.thus, one might say, on average, a city of population density xis expected to have a relative housing cost of y%, where y = 0.0125x + 61. The number 61 in the equation represents the y-intercept of the line of best fit, in that when the population density, x, is 0, there is an expected relative housing cost of 61%. This might not make the best sense within the context of the problem, in that when the population density is 0, the population is 0, so there probably wouldn’t be any housing costs. However, it COULD be interpreted that for cities with low population densities, housing costs were likely around orabove 61% (since below 61% would be for cities with negative population densities, which is impossible).
Choice A is incorrect because it interprets the values of the vertical axis as dollars and not PERCENTAGES. Choice B is incorrect because thelowest housing cost is about 61% of the national average, not 61% of the highest housing cost. Choice C is incorrect because one cannot absolutely assert that no city with a low population density had housing costs below 61% of the national average, as the MODEL shows that it is unlikely, but not impossible.
Discussion
50.

Draw the graph of the following function. -f(x)

Answer»


ANSWER :`-F(X)` is SHOWN as
`(##ARH_AMA_DIF_CAL_C04_E02_004_A01##)`
Discussion