Explore topic-wise MCQs in Testing Subject.

This section includes 657 Mcqs, each offering curated multiple-choice questions to sharpen your Testing Subject knowledge and support exam preparation. Choose a topic below to get started.

1.

If \[\int{\sec x\cos ec\,\,x\,\,dx=\log \left| g(x) \right|}+c,\] then what is \[g(x)\] equal to?

A. \[\sin x\cos x\]
B. \[{{\sec }^{2}}x\]
C. \[\tan x\]
D. \[\log \left| \tan x \right|\]
Answer» D. \[\log \left| \tan x \right|\]
2.

The value of the integral \[\int_{-1}^{3}{(\left| x \right|+\left| x-1 \right|)dx}\] is

A. 4
B. 9
C. 2
D. \[\frac{9}{2}\]
Answer» C. 2
3.

\[\left[ \sum\limits_{n=1}^{10}{\int\limits_{-2n-1}^{-2n}{{{\sin }^{27}}xdx}} \right]+\left[ \sum\limits_{n=1}^{10}{\int\limits_{2n}^{2n+1}{{{\sin }^{27}}}xdx} \right]=\]

A. \[{{27}^{2}}\]
B. \[-54\]
C. \[54\]
D. 0
Answer» E.
4.

\[\int{32{{x}^{3}}{{(\log \,\,x)}^{2}}dx}\] is equal to:

A. \[8{{x}^{4}}{{(\log \,\,x)}^{2}}+C\]
B. \[{{x}^{4}}\{8{{(\log \,\,x)}^{2}}-4(\log \,\,x)+1\}+C\]
C. \[{{x}^{4}}\{8{{(\log \,\,x)}^{2}}-4(\log \,\,x)\}+C\]
D. \[{{x}^{3}}\{{{(\log \,\,x)}^{2}}-2\log \,\,x\}+C\]
Answer» C. \[{{x}^{4}}\{8{{(\log \,\,x)}^{2}}-4(\log \,\,x)\}+C\]
5.

\[\int\limits_{0}^{\pi }{xf(\sin \,\,x)dx}\] is equal to

A. \[\pi \int\limits_{0}^{\pi }{f(cos\,\,x)dx}\]
B. \[\pi \int\limits_{0}^{\pi }{f(sin\,\,x)dx}\]
C. \[\frac{\pi }{2}\int\limits_{0}^{\pi /2}{f(sin\,\,x)dx}\]
D. \[\pi \int\limits_{0}^{\pi /2}{f(cos\,\,x)dx}\]
Answer» E.
6.

\[\int{\frac{dx}{\sin x(3+{{\cos }^{2}}x)}}\] is equal to

A. \[\log \left| {{y}^{2}}-1 \right|-{{\tan }^{-1}}y+C\]
B. \[{{\tan }^{-1}}\frac{y}{\sqrt{3}}+C\]
C. \[\log \left| \frac{y-1}{y+1} \right|+C\]
D. \[\frac{1}{4}\log \left| \frac{y-1}{y+1} \right|-\frac{1}{4\sqrt{3}}{{\tan }^{-1}}\frac{y}{\sqrt{3}}+C\]
Answer» E.
7.

If\[\int{{{\log }_{e}}\left( \sqrt{1-x}+\sqrt{1+x} \right)dx}\]\[=x{{\log }_{e}}\left( \sqrt{1-x}+\sqrt{1+x} \right)+g(x)+C\]. Then \[g(x)=\]

A. \[x-{{\sin }^{-1}}x\]
B. \[{{\sin }^{-1}}x-x\]
C. \[x+{{\sin }^{-1}}x\]
D. \[{{\sin }^{-1}}x-{{x}^{2}}\]
Answer» C. \[x+{{\sin }^{-1}}x\]
8.

\[\int{\frac{(1+x){{e}^{x}}}{\cot (x{{e}^{x}})}dx}\] is equal to

A. \[\log \left| \cos (x{{e}^{x}}) \right|+C\]
B. \[\log \left| \cot (x{{e}^{x}}) \right|+C\]
C. \[\log \left| sec(x{{e}^{-x}}) \right|+C\]
D. \[\log \left| sec(x{{e}^{x}}) \right|+C\]
Answer» E.
9.

The value of \[\int_{0}^{{{\sin }^{2}}x}{{{\sin }^{-1}}\sqrt{t}\,\,dt}+\int_{0}^{{{\cos }^{2}}x}{{{\cos }^{-1}}\sqrt{t}dt}\] is

A. \[\pi \]
B. \[\frac{\pi }{2}\]
C. \[\frac{\pi }{4}\]
D. 1
Answer» D. 1
10.

What is the value of\[\int\limits_{0}^{1}{(x-1){{e}^{-x}}dx}\]?

A. 0
B. e
C. \[\frac{1}{e}\]
D. \[\frac{-1}{e}\]
Answer» E.
11.

If \[f(x)=\frac{{{e}^{x}}}{1+{{e}^{x}}},{{I}_{1}}=\int\limits_{f(-a)}^{f(a)}{xg\{x(1-x)\}dx}\] and \[{{I}_{2}}=\int\limits_{f(-a)}^{f(a)}{g\{x(1-x)\}dx,}\] then the value of \[\frac{{{I}_{2}}}{{{I}_{1}}}\] is

A. 1
B. \[-3\]
C. \[-1\]
D. \[2\]
Answer» E.
12.

\[\int{\frac{({{x}^{2}}-1)}{x\sqrt{{{x}^{4}}+3{{x}^{2}}+1}}dx}\] is equal to

A. \[\log \left| x+\frac{1}{x}+\sqrt{{{x}^{2}}+\frac{1}{{{x}^{2}}}+3} \right|+C\]
B. \[\log \left| x-\frac{1}{x}+\sqrt{{{x}^{2}}+\frac{1}{{{x}^{2}}}-3} \right|+C\]
C. \[\log \left| x+\sqrt{{{x}^{2}}+3} \right|+C\]
D. None of these
Answer» B. \[\log \left| x-\frac{1}{x}+\sqrt{{{x}^{2}}+\frac{1}{{{x}^{2}}}-3} \right|+C\]
13.

\[\int{\frac{\sqrt{x}}{1+\sqrt[4]{{{x}^{3}}}}}dx\] is equal to

A. \[\frac{4}{3}\left[ 1+{{x}^{3/4}}+\log (1+{{x}^{3/4}}) \right]+C\]
B. \[\frac{4}{3}\left[ 1+{{x}^{3/4}}-\log (1+{{x}^{3/4}}) \right]+C\]
C. \[\frac{4}{3}\left[ 1-{{x}^{3/4}}+\log (1+{{x}^{3/4}}) \right]+C\]
D. None of these
Answer» C. \[\frac{4}{3}\left[ 1-{{x}^{3/4}}+\log (1+{{x}^{3/4}}) \right]+C\]
14.

The solution of \[\frac{dy}{dx}=\left| x \right|\] is:

A. \[y=\frac{x\left| x \right|}{2}+c\]
B. \[y=\frac{\left| x \right|}{2}+c\]
C. \[y=\frac{{{x}^{2}}}{2}+c\]
D. \[y=\frac{{{x}^{3}}}{2}+c\] Where c is an arbitrary constant
Answer» B. \[y=\frac{\left| x \right|}{2}+c\]
15.

Consider the following statements in respect of the differential equation\[\frac{{{d}^{2}}y}{d{{x}^{2}}}+\cos \left( \frac{dy}{dx} \right)=0\] 1. The degree of the differential equation is not defined. 2. The order of the differential equation is 2. Which of the above statements is/are correct?

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2
Answer» D. Neither 1 nor 2
16.

Under which one of the following conditions does the solution of \[\frac{dy}{dx}=\frac{ax+b}{cy+d}\] represent a parabola?

A. \[a=0,\text{ }c=0\]
B. \[a=1,\text{ }b=2,\text{ }c\ne 0\]
C. \[a=0,c\ne 0,b\ne 0\]
D. \[a=1,c=1\]
Answer» D. \[a=1,c=1\]
17.

If \[\phi (x)\] is a differentiable function, then the solution of the differential equation\[dy+\{y\phi '(x)-\phi (x)\phi '(x)\}dx=0\] is

A. \[y=\{\phi (x)-1\}+c{{e}^{-\phi (x)}}\]
B. \[y\phi (x)={{\{\phi (x)\}}^{2}}+c\]
C. \[y{{e}^{\phi (x)}}=\phi (x){{e}^{\phi (x)}}+c\]
D. None of these
Answer» B. \[y\phi (x)={{\{\phi (x)\}}^{2}}+c\]
18.

The equation of the curve satisfying \[xdy-ydx=\sqrt{{{x}^{2}}-{{y}^{2}}}\] and \[y(1)=0\] is:

A. \[y={{x}^{2}}\log (\sin \,x)\]
B. \[y=x\sin (log\,x)\]
C. \[{{y}^{2}}=x{{(x-1)}^{2}}\]
D. \[y=2{{x}^{2}}(x-1)\]
Answer» C. \[{{y}^{2}}=x{{(x-1)}^{2}}\]
19.

The differential equation of the curve \[\frac{x}{c-1}+\frac{y}{c+1}=1\] is given by

A. \[\left( \frac{dy}{dx}-1 \right)\left( y+x\frac{dy}{dx} \right)=2\frac{dy}{dx}\]
B. \[\left( \frac{dy}{dx}+1 \right)\left( y-x\frac{dy}{dx} \right)=\frac{dy}{dx}\]
C. \[\left( \frac{dy}{dx}+1 \right)\left( y-x\frac{dy}{dx} \right)=2\frac{dy}{dx}\]
D. None of these
Answer» D. None of these
20.

The population of a country doubles in 40 years. Assuming that the rate of increase is proportional to the number of inhabitants, the number of years in which it would treble itself is

A. 80 years
B. \[80\frac{\log 2}{\log 3}years\]
C. \[40\frac{\log 3}{\log 2}years\]
D. \[40\log 2\log 3\,years\]
Answer» D. \[40\log 2\log 3\,years\]
21.

The marginal cost of manufacturing a certain item is given by\[c'(x)=\frac{dc}{dx}=2+0.15x\]. The total cost function c (x), is

A. \[0.075{{x}^{2}}+2x+100\]
B. \[0.15{{x}^{2}}+3x+30\]
C. \[{{x}^{2}}+100.075x+100\]
D. None of these It is given that c (0) = 100
Answer» B. \[0.15{{x}^{2}}+3x+30\]
22.

What is the degree of the differential equation\[{{\left( \frac{{{d}^{3}}y}{d{{x}^{3}}} \right)}^{2/3}}+4-3\left( \frac{{{d}^{2}}y}{d{{x}^{2}}} \right)+5\left( \frac{dy}{dx} \right)=0\]?

A. 3
B. 2
C. 44257
D. Not defined
Answer» C. 44257
23.

What is the degree of the differential equation\[k\frac{{{d}^{2}}y}{d{{x}^{2}}}={{\left[ 1+{{\left( \frac{dy}{dx} \right)}^{3}} \right]}^{3/2}}\], where k is a constant?

A. 1
B. 2
C. 3
D. 4
Answer» C. 3
24.

Solution of the differential equation\[\frac{dx}{dy}-\frac{x\,\,\log \,\,x}{1+\log \,\,x}=\frac{{{e}^{y}}}{1+\log \,\,x'}\] if \[y(1)=0\], is

A. \[{{x}^{x}}={{e}^{y{{e}^{y}}}}\]
B. \[{{e}^{y}}={{x}^{{{e}^{y}}}}\]
C. \[{{x}^{x}}=y{{e}^{^{y}}}\]
D. None of these
Answer» B. \[{{e}^{y}}={{x}^{{{e}^{y}}}}\]
25.

The equation of the curve passing through the point \[\left( 0,\frac{\pi }{4} \right)\] whose differential equation is\[sin\text{ }x\text{ }cos\text{ }y\text{ }dx+cos\text{ }x\text{ }sin\text{ }y\text{ }dy=0\], is

A. \[sec\,\,x\,\,sec\,\,y=\sqrt{2}\]
B. \[cos\,\,x\,\,cos\,\,y=\sqrt{2}\]
C. \[\sec \,\,x=\sqrt{2}\,\,\cos \,\,y\]
D. \[cos\,\,y=\sqrt{2}\,\,\sec \,\,y\]
Answer» B. \[cos\,\,x\,\,cos\,\,y=\sqrt{2}\]
26.

The function \[f(\theta )=\frac{d}{d\theta }\int\limits_{0}^{\theta }{\frac{dx}{1-\cos \theta \,\,\cos x}}\] satisfies the differential equation

A. \[\frac{df}{d\theta }+2f(\theta )cot\theta =0\]
B. \[\frac{df}{d\theta }-2f(\theta )cot\theta =0\]
C. \[\frac{df}{d\theta }+2f(\theta )=0\]
D. \[\frac{df}{d\theta }-2f(\theta )=0\]
Answer» B. \[\frac{df}{d\theta }-2f(\theta )cot\theta =0\]
27.

The differential equation\[\frac{{{d}^{2}}y}{d{{x}^{2}}}+x\frac{dy}{dx}+\sin y+{{x}^{2}}=0\] is of the following type

A. Linear
B. Homogeneous
C. Order two
D. Degree two
Answer» D. Degree two
28.

The solution of the differential equation \[\frac{dy}{dx}+\frac{y}{x}\log \,\,y=\frac{y}{{{x}^{2}}}(\log \,\,{{y}^{2}})\] is

A. \[y=\log ({{x}^{2}}+cx)\]
B. \[\log \,\,y=x\left( c{{x}^{2}}+\frac{1}{2} \right)\]
C. \[x=\log \,\,y\left( c{{x}^{2}}+\frac{1}{2} \right)\]
D. None of these.
Answer» D. None of these.
29.

The degree of the differential equation\[\frac{dy}{dx}-x={{\left( y-x\frac{dy}{dx} \right)}^{-4}}\] is

A. 2
B. 3
C. 4
D. 5
Answer» E.
30.

The general solution the differential equation\[\frac{dy}{dx}-\frac{\tan \,\,y}{1+x}={{(1+x\,\,e)}^{x}}\sec \,\,y\] is

A. \[\sin (1+x)=y({{e}^{x}}+c)\]
B. \[y\sin (1+x)=c{{e}^{x}}\]
C. \[(1+x)\sin \,\,y={{e}^{x}}+c\]
D. \[\sin \,\,y=(1+x)({{e}^{x}}+c)\]
Answer» E.
31.

What is the degree of the differential equation\[y=x\frac{dy}{dx}+{{\left( \frac{dy}{dx} \right)}^{-1}}\]?

A. 1
B. 2
C. -1
D. Degree does not exist.
Answer» C. -1
32.

The degree and order respectively of the differential equation are \[\frac{dy}{dx}=\frac{1}{x+y+1}\].

A. 1, 1
B. 1, 2
C. 2, 1
D. 2, 2
Answer» B. 1, 2
33.

What is the differential equation for\[{{y}^{2}}=4a(x-a)\]?

A. \[yy'-2xyy'+{{y}^{2}}=0\]
B. \[yy'(yy'+2x)+{{y}^{2}}=0\]
C. \[yy'(yy'-2x)+{{y}^{2}}=0\]
D. \[yy'-2xyy'+y=0\]
Answer» D. \[yy'-2xyy'+y=0\]
34.

A curve is such that the portion of the x-axis cut off between the origin and the tangent at a point is twice the abscissa and which passes through the point (1, 2). The equation of the curve is

A. \[xy=1\]
B. \[xy=2\]
C. \[xy=3\]
D. None of these
Answer» C. \[xy=3\]
35.

If for the differential equation \[y'=\frac{y}{x}+\phi \left( \frac{x}{y} \right),\] the general solution is \[y=\frac{x}{\log \left| Cx \right|},\] then \[\phi (x/y)\] is given by

A. \[-{{x}^{2}}/{{y}^{2}}\]
B. \[-{{y}^{2}}/{{x}^{2}}\]
C. \[{{x}^{2}}/{{y}^{2}}\]
D. \[-{{y}^{2}}/{{x}^{2}}\]
Answer» E.
36.

An integrating factor of the differential equation \[\sin x\frac{dy}{dx}+2y\cos x=1\] is

A. \[{{\sin }^{2}}x\]
B. \[\frac{2}{\sin x}\]
C. \[\log \left| \sin \,\,x \right|\]
D. \[\frac{1}{{{\sin }^{2}}x}\]
Answer» B. \[\frac{2}{\sin x}\]
37.

If \[y=y(x)\] and \[\frac{2+\sin x}{1+y}\left( \frac{dy}{dx} \right)=-\cos \,x,y(0)=1,\]then \[y\left( \frac{\pi }{2} \right)\] equals

A. 44256
B. 44257
C. -0.333333333333333
D. 1
Answer» B. 44257
38.

\[y=2\,cos\text{ }x+3\,sin\text{ }x\] satisfies which of the following differential equations? 1. \[\frac{{{d}^{2}}y}{d{{x}^{2}}}+y=0\] 2. \[{{\left( \frac{dy}{dx} \right)}^{2}}+\frac{dy}{dx}=0\] Select the correct answer using the code given below.

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2
Answer» B. 2 only
39.

If \[x\,dy=y\,dx+{{y}^{2}}dy,y>0\] and\[y\text{(1})=1\], then what is \[y(-3)\] equal to?

A. 3 only
B. -1 only
C. Both -1 and 3
D. Neither -1 nor 3
Answer» B. -1 only
40.

If \[y={{e}^{4x}}+2{{e}^{-x}}\] satisfies the relation \[\frac{{{d}^{3}}y}{d{{x}^{3}}}+A\frac{dy}{dx}+By=0,\] then values of A and B respectively are:

A. -13, 14
B. -13, -12
C. -13, 12
D. 12, -13
Answer» C. -13, 12
41.

The solution of the differential equation\[\frac{dy}{dx}+\frac{2yx}{1+{{x}^{2}}}=\frac{1}{{{(1+{{x}^{2}})}^{2}}}\] is:

A. \[y(1+{{x}^{2}})=c+{{\tan }^{-1}}x\]
B. \[\frac{y}{1+{{x}^{2}}}=c+{{\tan }^{-1}}x\]
C. \[y\log (1+{{x}^{2}})=c+{{\tan }^{-1}}x\]
D. \[y(1+{{x}^{2}})=c+{{\sin }^{-1}}x\]
Answer» B. \[\frac{y}{1+{{x}^{2}}}=c+{{\tan }^{-1}}x\]
42.

The solution to of the differential equation\[(x+1)\frac{dy}{dx}-y={{e}^{3x}}{{(x+1)}^{2}}\] is

A. \[y=(x+1){{e}^{3x}}+c\]
B. \[3y=(x+1)+{{e}^{3x}}+c\]
C. \[\frac{3y}{x+1}={{e}^{3x}}+c\]
D. \[y{{e}^{-3x}}=3(x+1)+c\]
Answer» D. \[y{{e}^{-3x}}=3(x+1)+c\]
43.

The solution of \[(y+x+5)dy=(y-x+1)dx\] is

A. \[\log ({{(y+3)}^{2}}+{{(x+2)}^{2}})+{{\tan }^{-1}}\frac{y+3}{y+2}+C\]
B. \[\log ({{(y+3)}^{2}}+{{(x+2)}^{2}})+{{\tan }^{-1}}\frac{y-3}{y-2}=C\]
C. \[\log ({{(y+3)}^{2}}+{{(x+2)}^{2}})+2{{\tan }^{-1}}\frac{y+3}{y+2}=C\]
D. \[\log ({{(y+3)}^{2}}+{{(x+2)}^{2}})-2{{\tan }^{-1}}\frac{y+3}{y+2}=C\]
Answer» D. \[\log ({{(y+3)}^{2}}+{{(x+2)}^{2}})-2{{\tan }^{-1}}\frac{y+3}{y+2}=C\]
44.

The general solution of the differential equation \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=\cos \,\,nx\] is

A. \[{{n}^{2}}y+\cos \,\,nx={{n}^{2}}(Cx+D)\]
B. \[{{n}^{2}}y-sin\,\,nx={{n}^{2}}(-Cx+D)\]
C. \[{{n}^{2}}y+\cos \,\,nx=\frac{Cx+D}{{{n}^{2}}}\]
D. None of these. [Where C and D are arbitrary constants]
Answer» B. \[{{n}^{2}}y-sin\,\,nx={{n}^{2}}(-Cx+D)\]
45.

If \[y={{(x+\sqrt{1+{{x}^{2}}})}^{n}},\] then \[(1+{{x}^{2}})\frac{{{d}^{2}}y}{d{{x}^{2}}}+x\frac{dy}{dx}\] is

A. \[{{n}^{2}}y\]
B. \[-{{n}^{2}}y\]
C. \[-y\]
D. \[2{{x}^{2}}y\]
Answer» B. \[-{{n}^{2}}y\]
46.

The solution of the differential equation\[\frac{dy}{dx}+\frac{y}{x}\log y=\frac{y}{{{x}^{2}}}{{(\log \,\,y)}^{2}}\] is

A. \[y=\log ({{x}^{2}}+cx)\]
B. \[\log \,\,y=x\left( c{{x}^{2}}+\frac{1}{2} \right)\]
C. \[x=\log \,\,y\left( c{{x}^{2}}+\frac{1}{2} \right)\]
D. None of these
Answer» D. None of these
47.

The solution of the equation \[\frac{dy}{dx}=\sqrt{\frac{1-{{y}^{2}}}{1-{{x}^{2}}}}\] is

A. \[{{\sin }^{-1}}y-{{\sin }^{-1}}x=c\]
B. \[{{\sin }^{-1}}y{{\sin }^{-1}}x=c\]
C. \[{{\sin }^{-1}}(xy)=2\]
D. None of these
Answer» B. \[{{\sin }^{-1}}y{{\sin }^{-1}}x=c\]
48.

If \[{{y}^{2}}=p(x)\] is a polynomial of degree 3, then what is \[2\frac{d}{dx}\left[ {{y}^{3}}\frac{{{d}^{2}}y}{d{{x}^{2}}} \right]\] equal to?

A. p'(x)p"'(x)
B. p"(x)p'"(x)
C. p(x)p"'(x)
D. A constant
Answer» D. A constant
49.

The order and degree of the differential equation of parabolas having vertex at the origin and focus at (a, 0) where a > 0, are respectively

A. 1, 1
B. 2, 1
C. 1, 2
D. 2, 2
Answer» B. 2, 1
50.

What is the order of the differential equation\[\frac{dx}{dy}+\int{y\,dx={{x}^{3}}}\]?

A. 1
B. 2
C. 3
D. Cannot be determined
Answer» C. 3