MCQ (Single Correct Answer)

1

A particle is projected vertically upwards and is at a height h after t1 seconds and again after t2 seconds then

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2

The equation of the tangent to the conic $${x^2} - {y^2} - 8x + 2y + 11 = 0$$ at (2, 1) is

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3

A particle is moving in a straight line. At time t, the distance between the particle from its starting point is given by x = t $$-$$ 6t2 + t3. Its acceleration will be zero at

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4

The Rolle's theorem is applicable in the interval $$-$$1 $$\le$$ x $$\le$$ 1 for the function

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5

The distance covered by a particle in t seconds is given by x = 3 + 8t $$-$$ 4t2. After 1 second its velocity will be

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6

If the rate of increase of the radius of a circle is 5 cm/sec., then the rate of increase of its area, when the radius is 20 cm, will be

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7

Angle between y2 = x and x2 = y at the origin is

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8

If the normal to the curve y = f(x) at the point (3, 4) makes an angle 3$$\pi$$/4 with the positive x-axis, then f'(3) is

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9

The equation of normal of $${x^2} + {y^2} - 2x + 4y - 5 = 0$$ at (2, 1) is

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10

The point in the interval [0, 2$$\pi$$], where $$f(x) = {e^x}\sin x$$ has maximum slope, is

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11

The co-ordinates of the point on the curve $$y = {x^2} - 3x + 2$$ where the tangent is perpendicular to the straight line y = x are

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12

The acceleration of a particle starting from rest moving in a straight line with uniform acceleration is 8m/sec2. The time taken by the particle to move the second metre is

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13
Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p(1)=6$ and $p(3)=2$, then $p^{\prime}(0)$ is equal to :
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14

The function $f(x)=2 x^3-3 x^2-12 x+4, x \in \mathbb{R}$ has

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15

Let $\phi(x)=f(x)+f(2 a-x), x \in[0,2 a]$ and $f^{\prime \prime}(x)>0$ for all $x \in[0, a]$. Then $\phi(x)$ is

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16

Let $f$ be a function which is differentiable for all real $x$. If $f(2)=-4$ and $f^{\prime}(x) \geq 6$ for all $x \in[2,4]$, then

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17

If $x=-1$ and $x=2$ are extreme points of $f(x)=\alpha \log |x|+\beta x^2+x,(x \neq 0)$, then

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18

Let $f(\theta)=\left|\begin{array}{ccc}1 & \cos \theta & -1 \\ -\sin \theta & 1 & -\cos \theta \\ -1 & \sin \theta & 1\end{array}\right|$. Suppose $A$ and $B$ are respectively maximum and minimum values of $f(\theta)$.Then $(A,B)$ is equal to

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19

The maximum number of common normals of $y^2=4 a x$ and $x^2=4 b y$ is equal to

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20

$$f(x)=\cos x-1+\frac{x^2}{2!}, x \in \mathbb{R}$$ Then $$\mathrm{f}(x)$$ is

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21

Let $$\mathrm{y}=\mathrm{f}(x)$$ be any curve on the $$\mathrm{X}-\mathrm{Y}$$ plane & $$\mathrm{P}$$ be a point on the curve. Let $$\mathrm{C}$$ be a fixed point not on the curve. The length $$\mathrm{PC}$$ is either a maximum or a minimum, then

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22

If a particle moves in a straight line according to the law $$x=a \sin (\sqrt{\lambda} t+b)$$, then the particle will come to rest at two points whose distance is [symbols have their usual meaning]

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23

Let $$\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$$ be given by $$\mathrm{f}(x)=\left|x^2-1\right|$$, then

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24

Consider the function $$\mathrm{f}(x)=x(x-1)(x-2) \ldots(x-100)$$. Which one of the following is correct?

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25

A missile is fired from the ground level rises x meters vertically upwards in t sec, where $$x = 100t - {{25} \over 2}{t^2}$$. The maximum height reached is

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26

The portion of the tangent to the curve $${x^{{2 \over 3}}} + {y^{{2 \over 3}}} = {a^{{2 \over 3}}},a > 0$$ at any point of it, intercepted between the axes

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27

Given $$f(x) = {e^{\sin x}} + {e^{\cos x}}$$. The global maximum value of $$f(x)$$

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28

A particle moving in a straight line starts from rest and the acceleration at any time t is $$a - k{t^2}$$ where a and k are positive constants. The maximum velocity attained by the particle is

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29
Let f : R $$\to$$ R be such that f(0) = 0 and $$\left| {f'(x)} \right| \le 5$$ for all x. Then f(1) is in
WB JEE 2021
30
Two particles A and B move from rest along a straight line with constant accelerations f and f' respectively. If A takes m sec. more than that of B and describes n units more than that of B in acquiring the same velocity, then
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31
If the tangent at the point P with co-ordinates (h, k) on the curve y2 = 2x3 is perpendicular to the straight line 4x = 3y, then
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32
If the tangent to the curve y2 = x3 at (m2, m3) is also a normal to the curve at (m2, m3), then the value of mM is
WB JEE 2020
33
If the function $$f(x) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1$$ [a > 0] attains its maximum and minimum at p and q respectively such that p2 = q, then a is equal to
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34
Let $$f(x) = {x^{13}} + {x^{11}} + {x^9} + {x^7} + {x^5} + {x^3} + x + 12$$.

Then
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35
In open interval $$\left( {0,\,{\pi \over 2}} \right)$$
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36
Consider the curve $$y = b{e^{ - x/a}}$$, where a and b are non-zero real numbers. Then
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37
If the radius of a spherical balloon increases by 0.1%, then its volume increases approximately by
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38
The law of motion of a body moving along a straight line is x = $${1 \over 2}$$ vt. x being its distance from a fixed point on the line at time t and v is its velocity there. Then
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39
A ladder 20 ft long leans against a vertical wall. The top end slides downwards at the rate of 2 ft per second. The rate at which the lower end moves on a horizontal floor when it is 12 ft from the wall is
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40
The normal to the curve $$y = {x^2} - x + 1$$, drawn at the points with the abscissa $${x_1} = 0$$, $${x_2} = - 1$$ and $${x_3} = {5 \over 2}$$
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41
Let, $$F(x) = {e^x},G(x) = {e^{ - x}}$$ and $$H(x) = G(F(x))$$, where x is a real variable. Then, $${{dH} \over {dx}}$$ at x = 0 is
WB JEE 2017
42
The chord of the curve $$y = {x^2} + 2ax + b$$, joining the points where x = $$\alpha$$ and x = $$\beta$$, is parallel to the tangent to the curve at abscissa x is equal to
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43
The value of K in order that f(x) = sin x $$-$$ cos x $$-$$ kx + 5 decreases for all positive real values of x is given by
WB JEE 2017
44
Time period T of a simple pendulum of length l is given by $$T = 2\pi \sqrt {{l \over g}} $$. If the length is increased by 2%, then an approximate change in the time period is
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Subjective

MCQ (More than One Correct Answer)

1

Let $f(x)=x^3, x \in[-1,1]$. Then which of the following are correct?

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2

The acceleration f $$\mathrm{ft} / \mathrm{sec}^2$$ of a particle after a time $$\mathrm{t}$$ sec starting from rest is given by $$\mathrm{f}=6-\sqrt{1.2 \mathrm{t}}$$. Then the maximum velocity $$\mathrm{v}$$ and time $$\mathrm{T}$$ to attend this velocity are

WB JEE 2024
3

A balloon starting from rest is ascending from ground with uniform acceleration of 4 ft/sec$$^2$$. At the end of 5 sec, a stone is dropped from it. If T be the time to reach the stone to the ground and H be the height of the balloon when the stone reaches the ground, then

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4

If $$f(x) = 3\root 3 \of {{x^2}} - {x^2}$$, then

WB JEE 2023
5

From a balloon rising vertically with uniform velocity v ft/sec a piece of stone is let go. The height of the balloon above the ground when the stone reaches the ground after 4 sec is [g = 32 ft/sec2]

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6
The greatest and least value of $$f(x) = {\tan ^{ - 1}} - {1 \over 2}\,ln \,x\,on\,\left[ {{1 \over {\sqrt 3 }},\sqrt 3 } \right]$$ are
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7
A particle is projected vertically upwards. If it has to stay above the ground for 12 sec, then
WB JEE 2020
8
Tangent is drawn at any point P(x, y) on a curve, which passes through (1, 1). The tangent cuts X-axis and Y-axis at A and B respectively. If AP : BP = 3 : 1, then
WB JEE 2020
9
Two particles A and B move from rest along a straight line with constant accelerations f and h, respectively. If A takes m seconds more than B and describes n units more than that of B acquiring the same speed, then
WB JEE 2019
10
A particle is in motion along a curve 12y = x3. The rate of change of its ordinate exceeds that of abscissa in
WB JEE 2018
11
Let $$f(x) = \cos \left( {{\pi \over x}} \right),x \ne 0$$, then assuming k as an integer,
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12
If the line ax + by + c = 0, ab $$ \ne $$ 0, is a tangent to the curve xy = 1 $$-$$ 2x, then
WB JEE 2017
13
Two particles move in the same straight line starting at the same moment from the same point in the same direction. The first moves with constant velocity u and the second starts from rest with constant acceleration f. Then,
WB JEE 2017
14
If f(x) is a function such that f'(x) = (x $$-$$ 1)2(4 $$-$$ x), then
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