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Question 1:

A metal rod was cut into three parts A, B and \( C \), such that the length of \( A \) and \( B \) are in the ratio \( 4: 5 \), while that of \( B \) and \( C \) are in the ratio \( 3: 2 \). If the difference between the lengths of \( A \) and \( C \) is 8 cm, find the length of the metal rod.

136 cm

148 cm

152 cm

164 cm

Question 2:

If \( x=3 t^{2} \) and \( y=4 t^{3} \) the \( \frac{d y}{d x} \) will be equal to

\(\frac{4}{3}\)

\(\frac{4}{3} t\)

\(2 t\)

\(3 t\)

Question 3:

\( 18 \int \frac{1+\cos 4 t}{2} \,dt \)

\( \frac{t}{4}+\frac{\sin 4 t}{8}+C \)

\( t+\frac{\sin 4 t}{8}+C \)

\( \frac{t}{2}+\frac{\sin 4 t}{8}+C \)

\( t+\frac{\sin 4 t}{4}+C \)

Question 4:

The roots of the equation \(3148 x^{2}-13 x-1=0\) are:

\(\frac{-1}{16}, \frac{1}{2}\)

\(\frac{-1}{16}, \frac{1}{3}\)

\(\frac{2}{16}, \frac{1}{5}\)

\(\frac{1}{5}, \frac{2}{2}\)

Question 5:

Plot the graph of given equation \( Y=\sqrt{3} X+5 \)

None of the above

Question 6:

In which of the following options LHS \( \neq \) RHS

\(\frac{6}{7}-\frac{2}{9}=\frac{40}{63}\)

\(\frac{7}{2}+7 \frac{1}{2}=11\)

\(1+\frac{3 / 2}{4 / 2}=\frac{7}{4}\)

\(\frac{p}{q}+\frac{r}{s}=\frac{p r+q s}{q s}\)

Question 7:

If \( x^{3}+3 x y+y^{3}=1 \), then correct option is/are:- (A) \( \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{(1,1)}=-1 \) (B) \( \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{(1,1)}=-2 \) (C) \( \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{(1,0)}=-1 \) (D) \( \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{(1,0)}=-3 \)

A and B

B and D

A and C

B and C

Question 8:

If \( y=\frac{x}{x+1} \) then find \( \frac{dy}{dx} \)

\( \frac{1}{(x+1)^{2}} \)

\( \frac{x}{(1+x)^{2}} \)

\( (1+x)^{2} \)

1

Question 9:

Find the value of \( \frac{\log_{2} 27}{\log_{2} 3} \).

3

2

5

7

Question 10:

x = a \sin t, y = a \cos t. Find \(\frac{dy}{dx}\).

\(\tan t\)

\(\cot t\)

\(-\frac{1}{\cot t}\)

\(-\frac{1}{\tan t}\)

Question 11:

You are given the equation of a curve: \( \frac{x^{2}}{16}+\frac{y^{2}}{4}=1 \). Which of the following correctly represents the graph between \( x \) and \( y \)?

An ellipse centered at origin with major axis along x-axis and x-intercepts at \( \pm 4 \)

An ellipse centered at origin with major axis along y-axis and y-intercepts at \( \pm 4 \)

A parabola opening along x-axis

A circle of radius 4 centered at origin

Question 12:

Find the coordinates of local maxima and local minima of \(Y = x^{3} - 3x^{2} + 4\).

\((2,4),(2,0)\)

\((3,6)(4,0)\)

\((0,4),(2,0)\)

\((6,3)(2,0)\)

Question 13:

2, 4, 6, 8, 10 ..... 80. Find total no of terms and sum of this series.

40,1680

40,1640

30,1840

30,1740

Question 14:

If \(y = 4 \sin\theta \cos\theta\) then find the value of \(y_{\max}\) and the angle at which \(y\) will be maximum.

\(2,45^{\circ}\)

\(4,45^{\circ}\)

\(2,90^{\circ}\)

\(4,90^{\circ}\)

Question 15:

Match the matrix

Point	Slope
A	(p) Zero
B	(q) Negative
C	(r) Maximum +ve
D	(s) Positive

A-(r), B-(s), C-(q), D-(p)

A-(r), B-(s), C-(p), D-(q)

A-(s), B-(r), C-(p), D-(q)

A-(s), B-(r), C-(q), D-(p)

Question 16:

If \( V=x^{2} y+y^{4} z+z^{3} x \) then which of the following option is correct.

\( \left(\frac{\partial V}{\partial x}\right)=2 x y+4 y^{3} z+z^{3} \)

\( \left(\frac{\partial V}{\partial y}\right)=x^{2}+4 y^{3} z+3 x \)

\( \left(\frac{\partial V}{\partial z}\right)=y^{4}+3 z^{2} x \)

\( \left(\frac{\partial V}{\partial x}\right)=2 x y+4 y^{3} z+3 z^{2} \)

Question 17:

In the given figure, each box represents a function machine. A function machine illustrates what it does with the input.
Which of the following statements is correct?

\(z=(2 x+3)^{2}\)

\(z=2(x+3)\)

\(z=\sqrt{2 x+3}\)

\(z=\sqrt{2(x+3)}\)

Question 18:

The distance between points ( \( \mathbf{a}+\mathbf{b}, \mathbf{b}+\mathbf{c} \) ) and \( (a-b, c-b) \) is :-

\( 2 \sqrt{a^{2}+b^{2}} \)

\( 2 \sqrt{b^{2}+c^{2}} \)

\( 2 \sqrt{2} \mathrm{~b} \)

\( \sqrt{a^{2}-c^{2}} \)

Question 19:

Determine the average value of \( \mathbf{y}=\mathbf{2 x}+\mathbf{3} \) in the interval \( 0 \leq x \leq 1 \).

1

5

3

4

Question 20:

A metallic disc is being heated. Its area at any time \( t \) is given by \( A = 5t^{2} + 4t + 8 \). Calculate rate of increase in area at \( t = 3\ \mathrm{s} \).

\(30 \mathrm{~m}^{2} / \mathrm{s}\)

\(34 \mathrm{~m}^{2} / \mathrm{s}\)

\(28 \mathrm{~m}^{2} / \mathrm{s}\)

\(20 \mathrm{~m}^{2} / \mathrm{s}\)

Question 21:

Find the max value of \(33 \sin x + 56 \cos x\)

65

-65

55

-55

Question 22:

If \( \sin \theta = \frac{\sqrt{2}}{\sqrt{3}} \) and \( \theta \) lies in the first quadrant, the value of \( \tan \theta \) is:

\( \sqrt{2} \)

\( \frac{1}{\sqrt{2}} \)

\( \frac{\sqrt{3}}{\sqrt{2}} \)

\( \frac{1}{2} \)

Question 23:

Find θ for which sin θ = cos θ, if 180 < θ < 360

135°

315°

225°

150°

Question 24:

The length of hypotenuse of a right angle triangle exceeds the length of its base by 2 cm and exceeds twice the length of altitude by 1 cm. Find length of each side of the triangle.

\(6,8,10\)

\(7,24,25\)

\(8,15,17\)

\(7,40,41\)

Question 25:

Correct graph of \( y=-(x+2)^{2} \) is

(1)

(2)

(3)

(4)

Question 26:

If \(x y = c^{2}\), then what is \(\frac{dy}{dx}\)?

\(\frac{x}{y}\)

\(\frac{y}{x}\)

\(-\frac{x}{y}\)

\(-\frac{y}{x}\)

Question 27:

If \( y=\sin^{3} x - 3\sec^{2} x \), then \( \frac{dy}{dx} \) at \( x=\frac{\pi}{3} \) is

\( \frac{9-96 \sqrt{3}}{4} \)

\( \frac{9-86 \sqrt{3}}{4} \)

\( \frac{9-76 \sqrt{3}}{4} \)

None of these

Question 28:

\( \displaystyle \int_{a}^{b} 2 \frac{dx}{x} = \)

\( \log_{e} b - \log_{e} a \)

\( 2 \log_{e}(b-a) \)

\( \log_{e}\bigl(\tfrac{b^{2}}{a^{2}}\bigr) \)

\( 2 \log_{e}\bigl(\tfrac{a}{b}\bigr) \)

Question 29:

Find area between the curve \(y = x^{3}\) and the x-axis from \(x = -1\) to \(x = 2\).

47

31

\(\frac{15}{4}\)

\(\frac{7}{4}\)

Question 30:

Match the following:

Trigonometric Function		Maxima and Minima values
(1)	y = 3 sin θ + 4 sin θ	(A)	yₘₐₓ = +7, yₘᵢₙ = +3
(2)	y = 4 sin(5 θ)	(B)	yₘₐₓ = +5, yₘᵢₙ = −5
(3)	y = 5 − 2 sin θ	(C)	yₘₐₓ = +4, yₘᵢₙ = −4
(4)	y = 6 sin θ + 8 cos θ	(D)	yₘₐₓ = +10, yₘᵢₙ = −10

1-B; 2-C; 3-A; 4-D

1-A; 2-B; 3-C; 4-D

1-C; 2-D; 3-A; 4-B

1-B; 2-A; 3-C; 4-D

Question 31:

Find sum of the infinite series
\[
1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\frac{1}{32}+\cdots
\]

\(\tfrac{1}{2}\)

\(\tfrac{2}{3}\)

2

\(\tfrac{3}{2}\)

Question 32:

Calculate \( \sqrt{0.99} \)

0.885

0.995

0.985

0.975

Question 33:

Find the value of \( \tan 75^{\circ} \)?

\( \frac{\sqrt{3}+1}{\sqrt{3}-1} \)

\( \frac{\sqrt{3}-1}{\sqrt{3}+1} \)

\( \frac{2}{\sqrt{3}+1} \)

\( \frac{\sqrt{3}-1}{2} \)

Question 34:

Find the value of \( \cos 75^{\circ} \)?

\( \frac{\sqrt{3}-1}{2 \sqrt{2}} \)

\( \frac{\sqrt{3}+1}{2 \sqrt{2}} \)

\( \frac{2}{\sqrt{3}+1} \)

\( \frac{\sqrt{3}-1}{2} \)

Question 35:

\( \int\left(\frac{\sqrt{x}}{2}+\frac{2}{\sqrt{x}}\right) d x \)

\( \frac{x^{3 / 2}}{3}+2 x^{1 / 2}+C \)

\( \frac{x \sqrt{x}}{3}+4 \sqrt{x} \)

\( \frac{x^{3 / 2}}{3}+2 x^{1 / 2} \)

\( \frac{x \sqrt{x}}{3}+4 \sqrt{x}+C \)

Question 36:

A circular arc is length of \( \pi \mathrm{cm} \). find angle subtended by it at the center?

\( 30^{\circ} \)

\( 90^{\circ} \)

\( 60^{\circ} \)

\( 45^{\circ} \)

Question 37:

Evaluate value of \( \cos 74^{\circ} \)

\( \frac{3}{4} \)

\( \frac{3}{5} \)

\( \frac{3}{25} \)

\( \frac{7}{25} \)

Question 38:

Find the volume of a segment of height \( h \) of a sphere of radius \( R \).

V=\( \frac{1}{3} \pi h^{3}(2 R-h) \)

V=\( \frac{1}{3} \pi h^{2}(3 R-h) \)

V=\( \frac{1}{2} \pi h^{2}(2 R-h) \)

V=\( \pi h^{2}(3 R-h) \)

Question 39:

Find \( \frac{d q}{d t} \) if \( q=q_{0}\left(1-e^{-t / \tau}\right) \)

\( \frac{d q}{d t}=\frac{q_{0} e^{-t / \tau}}{\tau} \)

\( \frac{d q}{d t}=\frac{q_{0} e^{+t / \tau}}{\tau} \)

\( \frac{d q}{d t}=\frac{e^{-t / \tau}}{q \cdot \tau} \)

\( \frac{\mathrm{dq}}{\mathrm{dt}}=\frac{\tau \mathrm{e}^{-\mathrm{t} / \tau}}{\mathrm{q}_{0}} \)

Question 40:

If \( \frac{x}{y}=\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}} \) then find \( \frac{x+y}{x-y} \).

\( \sqrt{\frac{a}{b}} \)

\( \sqrt{\frac{b}{a}} \)

\( \frac{a}{b} \)

\( \frac{b}{a} \)

Question 41:

If \( f(x)=x^3+3x^2-9x+10 \), which of the following is true?

One local minima & one local maxima

Only one local minima

Only one local maxima

No local extrema

Question 42:

If radius of circle is given by \(r=2t+1\), then \(\frac{d(\text{Area})}{dt}= ?\)

\(t+2\)

\(\pi(4t+1)\)

\(2t+1\)

\(4\pi(2t+1)\)

Question 43:

If \( y=x^{2}-5x+6 \), then \( y-x \) graph is

N/A

Question 44:

Find the \(15^{\text{th}}\) term of the sequence \(20,15,10\ldots\.\).

-45

-50

-55

0

Question 45:

Find \(f'(x)\) if \(f(x)=\frac{1}{\sqrt[3]{x^{2}+x+1}}\)

\(-\frac{1}{2}\left(\mathrm{x}^{2}+\mathrm{x}+1\right)^{-4/3}(2\mathrm{x}+1)\)

\(-\frac{1}{3}\left(x^{2}+x+2\right)^{-4/3}(2x+1)\)

\(-\frac{1}{3}\left(x^{2}+x+1\right)^{-4/3}(2x+1)\)

\(-\frac{1}{2}\left(\mathrm{x}^{2}+\mathrm{x}+2\right)^{-4/3}(2\mathrm{x}+1)\)

Question 46:

Find the derivative of the function \( g(t)=\left(\frac{t-2}{2 t+1}\right)^{9} \)

\( \frac{45(t-2)^{8}}{(2 t+1)^{10}} \)

\( \frac{25(t-2)^{8}}{(2 t+1)^{10}} \)

\( \frac{35(t-1)^{8}}{(2 t+1)^{10}} \)

\( \frac{45(t-1)^{9}}{(2 t+1)^{10}} \)

Question 47:

Differentiate \( y = e^{\sin x} \)

\( e^{\cos x} \sin x \)

\( e^{\cos x} \cos x \)

\( e^{\sin x} \sin x \)

\( e^{\sin x} \cos x \)

Question 48:

Find the points on the curve \( y = x^{4} - 6 x^{2} + 4 \) where the tangent line is horizontal.

(0,2),(\sqrt{3},-5) and (-\sqrt{3},-5)

(0,4),(\sqrt{3},-5) and (-\sqrt{3},-5)

(0,2),(\sqrt{3},-\sqrt{5}) and (-\sqrt{3},-\sqrt{5})

(0,4),(\sqrt{3},-\sqrt{5}) and (-\sqrt{3},-\sqrt{5})

Question 49:

Find an equation of the tangent line to the curve \( y = e x / (1 + x^{2}) \) at the point (1, e/2).

\( y=\frac{1}{e} \)

\( y=e \)

\( y=\frac{e}{2} \)

\( y=\frac{2}{e} \)

Question 50:

The value of \( \tan 18^{\circ} \cdot \tan 36^{\circ} \cdot \tan 54^{\circ} \cdot \tan 72^{\circ} \tan 60^{\circ} \) is

\( \frac{1}{\sqrt{3}} \)

\( \sqrt{3} \)

1

Zero

बिंदु	ढाल
A	(p) शून्य
B	(q) ऋणात्मक
C	(r) अधिकतम +ve
D	(s) धनात्मक

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