Application of Derivatives · Mathematics · JEE Advanced
MCQ (More than One Correct Answer)
1
Let ℝ denote the set of all real numbers. Let f: ℝ → ℝ be defined by
$f(x) = \begin{cases} \dfrac{6x + \sin x}{2x + \sin x}, & \text{if } x \neq 0, \\ \dfrac{7}{3}, & \text{if } x = 0. \end{cases}$
Then which of the following statements is (are) TRUE?
JEE Advanced 2025 Paper 2 Online
2
Let
$$ \alpha=\sum\limits_{k = 1}^\infty {{{\sin }^{2k}}\left( {{\pi \over 6}} \right)} $$
Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by
$$ g(x)=2^{\alpha x}+2^{\alpha(1-x)} . $$
Then, which of the following statements is/are TRUE ?
$$ \alpha=\sum\limits_{k = 1}^\infty {{{\sin }^{2k}}\left( {{\pi \over 6}} \right)} $$
Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by
$$ g(x)=2^{\alpha x}+2^{\alpha(1-x)} . $$
Then, which of the following statements is/are TRUE ?
JEE Advanced 2022 Paper 2 Online
3
Let f : R $$ \to $$ R be given by
$$f(x) = (x - 1)(x - 2)(x - 5)$$. Define
$$F(x) = \int\limits_0^x {f(t)dt} $$, x > 0
Then which of the following options is/are correct?
$$f(x) = (x - 1)(x - 2)(x - 5)$$. Define
$$F(x) = \int\limits_0^x {f(t)dt} $$, x > 0
Then which of the following options is/are correct?
JEE Advanced 2019 Paper 2 Offline
4
Let, $$f(x) = {{\sin \pi x} \over {{x^2}}}$$, x > 0
Let x1 < x2 < x3 < ... < xn < ... be all the points of local maximum of f and y1 < y2 < y3 < ... < yn < ... be all the points of local minimum of f.
Then which of the following options is/are correct?
Let x1 < x2 < x3 < ... < xn < ... be all the points of local maximum of f and y1 < y2 < y3 < ... < yn < ... be all the points of local minimum of f.
Then which of the following options is/are correct?
JEE Advanced 2019 Paper 2 Offline
5
f : R $$ \to $$ R is a differentiable function such that f'(x) > 2f(x) for all x$$ \in $$R, and f(0) = 1 then
JEE Advanced 2017 Paper 2 Offline
6
If $$f(x) = \left| {\matrix{
{\cos 2x} & {\cos 2x} & {\sin 2x} \cr
{ - \cos x} & {\cos x} & { - \sin x} \cr
{\sin x} & {\sin x} & {\cos x} \cr
} } \right|$$,
then
then
JEE Advanced 2017 Paper 2 Offline
7
Let f: R $$ \to \left( {0,\infty } \right)$$ and g : R $$ \to $$ R be twice differentiable functions such that f'' and g'' are continuous functions on R. Suppose f'$$(2)$$ $$=$$ g$$(2)=0$$, f''$$(2)$$$$ \ne 0$$ and g'$$(2)$$ $$ \ne 0$$. If
$$\mathop {\lim }\limits_{x \to 2} {{f\left( x \right)g\left( x \right)} \over {f'\left( x \right)g'\left( x \right)}} = 1,$$ then
$$\mathop {\lim }\limits_{x \to 2} {{f\left( x \right)g\left( x \right)} \over {f'\left( x \right)g'\left( x \right)}} = 1,$$ then
JEE Advanced 2016 Paper 2 Offline
8
Let $$f, g :$$ $$\left[ { - 1,2} \right] \to R$$ be continuous functions which are twice differentiable on the interval $$(-1, 2)$$. Let the values of f and g at the points $$-1, 0$$ and $$2$$ be as given in the following table:
| X = -1 | X = 0 | X = 2 | |
|---|---|---|---|
| f(x) | 3 | 6 | 0 |
| g(x) | 0 | 1 | -1 |
In each of the intervals $$(-1, 0)$$ and $$(0, 2)$$ the function $$(f-3g)''$$ never vanishes. Then the correct statement(s) is (are)
JEE Advanced 2015 Paper 2 Offline
9
The function $$f(x) = 2\left| x \right| + \left| {x + 2} \right| - \left| {\left| {x + 2} \right| - 2\left| x \right|} \right|$$ has a local minimum or a local maximum at x =
JEE Advanced 2013 Paper 2 Offline
10
A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $$8:15$$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is $$100$$, the resulting box has maximum volume. Then the lengths of the vsides of the rectangular sheet are
JEE Advanced 2013 Paper 1 Offline
11
If $$f\left( x \right) = \int_0^x {{e^{{t^2}}}} \left( {t - 2} \right)\left( {t - 3} \right)dt$$ for all $$x \in \left( {0,\infty } \right),$$ then
IIT-JEE 2012 Paper 2 Offline
12
For the function
$$$f\left( x \right) = x\cos \,{1 \over x},x \ge 1,$$$
IIT-JEE 2009 Paper 2 Offline
13
$$f(x)$$ is cubic polynomial with $$f(2)=18$$ and $$f(1)=-1$$. Also $$f(x)$$ has local maxima at $$x=-1$$ and $$f'(x)$$ has local minima at $$x=0$$, then
IIT-JEE 2006
14
Let $$f\left( x \right) = \left\{ {\matrix{
{{e^x},} & {0 \le x \le 1} \cr
{2 - {e^{x - 1}},} & {1 < x \le 2} \cr
{x - e,} & {2 < x \le 3} \cr
} } \right.$$ and $$g\left( x \right) = \int\limits_0^x {f\left( t \right)dt,x \in \left[ {1,3} \right]} $$
then $$g(x)$$ has
then $$g(x)$$ has
IIT-JEE 2006
15
The function $$f\left( x \right) = \int\limits_{ - 1}^x {t\left( {{e^t} - 1} \right)\left( {t - 1} \right){{\left( {t - 2} \right)}^3}\,\,\,{{\left( {t - 3} \right)}^5}} $$ $$dt$$ has a local minimum at $$x=$$
IIT-JEE 1999
16
Let $$h\left( x \right) = f\left( x \right) - {\left( {f\left( x \right)} \right)^2} + {\left( {f\left( x \right)} \right)^3}$$ for every real number $$x$$. Then
IIT-JEE 1998
17
If $$f\left( x \right) = \left\{ {\matrix{
{3{x^2} + 12x - 1,} & { - 1 \le x \le 2} \cr
{37 - x} & {2 < x \le 3} \cr
} } \right.$$ then:
IIT-JEE 1993
18
If the line $$ax+by+c=0$$ is a normal to the curve $$xy=1$$, then
IIT-JEE 1986
MCQ (Single Correct Answer)
1
Let $Q$ be the cube with the set of vertices $\left\{\left(x_1, x_2, x_3\right) \in \mathbb{R}^3: x_1, x_2, x_3 \in\{0,1\}\right\}$. Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$. Let $S$ be the set of all four lines containing the main diagonals of the cube $Q$; for instance, the line passing through the vertices $(0,0,0)$ and $(1,1,1)$ is in $S$. For lines $\ell_1$ and $\ell_2$, let $d\left(\ell_1, \ell_2\right)$ denote the shortest distance between them. Then the maximum value of $d\left(\ell_1, \ell_2\right)$, as $\ell_1$ varies over $F$ and $\ell_2$ varies over $S$, is :
JEE Advanced 2023 Paper 1 Online
2
Consider the rectangles lying the region
$$\left\{ {(x,y) \in R \times R:0\, \le \,x\, \le \,{\pi \over 2}} \right.$$ and $$\left. {0\, \le \,y\, \le \,2\sin (2x)} \right\}$$
and having one side on the X-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is
$$\left\{ {(x,y) \in R \times R:0\, \le \,x\, \le \,{\pi \over 2}} \right.$$ and $$\left. {0\, \le \,y\, \le \,2\sin (2x)} \right\}$$
and having one side on the X-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is
JEE Advanced 2020 Paper 1 Offline
3
Which of the following options is the only INCORRECT combination?
JEE Advanced 2017 Paper 1 Offline
4
Which of the following options is the only CORRECT combination?
JEE Advanced 2017 Paper 1 Offline
5
Which of the following options is the only CORRECT combination?
JEE Advanced 2017 Paper 1 Offline
6
The least value of a $$ \in R$$ for which $$4a{x^2} + {1 \over x} \ge 1,$$, for all $$x>0$$. is
JEE Advanced 2016 Paper 1 Offline
7
Let $$f:\left[ {0,1} \right] \to R$$ (the set of all real numbers) be a function. Suppose the function $$f$$ is twice differentiable,
$$f(0) = f(1)=0$$ and satisfies $$f''\left( x \right) - 2f'\left( x \right) + f\left( x \right) \ge .{e^x},x \in \left[ {0,1} \right]$$.
$$f(0) = f(1)=0$$ and satisfies $$f''\left( x \right) - 2f'\left( x \right) + f\left( x \right) \ge .{e^x},x \in \left[ {0,1} \right]$$.
Which of the following is true for $$0 < x < 1?$$
JEE Advanced 2013 Paper 2 Offline
8
Let $$f:\left[ {0,1} \right] \to R$$ (the set of all real numbers) be a function. Suppose the function $$f$$ is twice differentiable,
$$f(0) = f(1)=0$$ and satisfies $$f''\left( x \right) - 2f'\left( x \right) + f\left( x \right) \ge .{e^x},x \in \left[ {0,1} \right]$$.
$$f(0) = f(1)=0$$ and satisfies $$f''\left( x \right) - 2f'\left( x \right) + f\left( x \right) \ge .{e^x},x \in \left[ {0,1} \right]$$.
If the function $${e^{ - x}}f\left( x \right)$$ assumes its minimum in the interval $$\left[ {0,1} \right]$$ at $$x = {1 \over 4}$$, which of the following is true?
JEE Advanced 2013 Paper 2 Offline
9
Let $$f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}$$ for all $$x \in IR$$ and let
$$g\left( x \right) = \int\limits_1^x {\left( {{{2\left( {t - 1} \right)} \over {t + 1}} - In\,t} \right)f\left( t \right)dt} $$ for all $$x \in \left( {1,\,\infty } \right)$$.
$$g\left( x \right) = \int\limits_1^x {\left( {{{2\left( {t - 1} \right)} \over {t + 1}} - In\,t} \right)f\left( t \right)dt} $$ for all $$x \in \left( {1,\,\infty } \right)$$.
Which of the following is true?
IIT-JEE 2012 Paper 2 Offline
10
Let $$f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}$$ for all $$x \in IR$$ and let
$$g\left( x \right) = \int\limits_1^x {\left( {{{2\left( {t - 1} \right)} \over {t + 1}} - In\,t} \right)f\left( t \right)dt} $$ for all $$x \in \left( {1,\,\infty } \right)$$.
$$g\left( x \right) = \int\limits_1^x {\left( {{{2\left( {t - 1} \right)} \over {t + 1}} - In\,t} \right)f\left( t \right)dt} $$ for all $$x \in \left( {1,\,\infty } \right)$$.
Consider the statements:
$$P:$$ There exists some $$x \in R$$ such that $$f\left( x \right) + 2x = 2\left( {1 + {x^2}} \right)$$
$$Q:\,\,$$ There exists some $$x \in R$$ such that $$2\,f\left( x \right) + 1 = 2x\left( {1 + x} \right)$$
Then
IIT-JEE 2012 Paper 2 Offline
11
The total number of local maxima and local minima of the function
$$f(x) = \left\{ {\matrix{
{{{(2 + x)}^3},} & { - 3 < x \le - 1} \cr
{{x^{2/3}},} & { - 1 < x < 2} \cr
} } \right.$$ is
IIT-JEE 2008 Paper 1 Offline
12
The tangent to the curve $$y = {e^x}$$ drawn at the point $$\left( {c,{e^c}} \right)$$ intersects the line joining the points $$\left( {c - 1,{e^{c - 1}}} \right)$$ and $$\left( {c + 1,{e^{c + 1}}} \right)$$
IIT-JEE 2007
13
If a continuous function $$f$$ defined on the real line $$R$$, assumes positive and negative values in $$R$$ then the equation $$f(x)=0$$ has a root in $$R$$. For example, if it is known that a continuous function $$f$$ on $$R$$ is positive at some point and its minimum value is negative then the equation $$f(x)=0$$ has a root in $$R$$.
Consider $$f\left( x \right) = k{e^x} - x$$ for all real $$x$$ where $$k$$ is real constant.
Consider $$f\left( x \right) = k{e^x} - x$$ for all real $$x$$ where $$k$$ is real constant.
The line $$y=x$$ meets $$y = k{e^x}$$ for $$k \le 0$$ at
IIT-JEE 2007
14
If a continuous function $$f$$ defined on the real line $$R$$, assumes positive and negative values in $$R$$ then the equation $$f(x)=0$$ has a root in $$R$$. For example, if it is known that a continuous function $$f$$ on $$R$$ is positive at some point and its minimum value is negative then the equation $$f(x)=0$$ has a root in $$R$$.
Consider $$f\left( x \right) = k{e^x} - x$$ for all real $$x$$ where $$k$$ is real constant.
Consider $$f\left( x \right) = k{e^x} - x$$ for all real $$x$$ where $$k$$ is real constant.
For $$k>0$$, the set of all values of $$k$$ for which $$k{e^x} - x = 0$$ has two distinct roots is
IIT-JEE 2007
15
If a continuous function $$f$$ defined on the real line $$R$$, assumes positive and negative values in $$R$$ then the equation $$f(x)=0$$ has a root in $$R$$. For example, if it is known that a continuous function $$f$$ on $$R$$ is positive at some point and its minimum value is negative then the equation $$f(x)=0$$ has a root in $$R$$.
Consider $$f\left( x \right) = k{e^x} - x$$ for all real $$x$$ where $$k$$ is real constant.
Consider $$f\left( x \right) = k{e^x} - x$$ for all real $$x$$ where $$k$$ is real constant.
The positive value of $$k$$ for which $$k{e^x} - x = 0$$ has only one root is
IIT-JEE 2007
16
If $$P(x)$$ is a polynomial of degree less than or equal to $$2$$ and $$S$$ is the set of all such polynomials so that $$P(0)=0$$, $$P(1)=1$$ and $$P'\left( x \right) > 0\,\,\forall x \in \left[ {0,1} \right],$$ then
IIT-JEE 2005 Screening
17
If $$f\left( x \right) = {x^3} + b{x^2} + cx + d$$ and $$0 < {b^2} < c,$$ then in $$\left( { - \infty ,\infty } \right)$$
IIT-JEE 2004 Screening
18
If $$f\left( x \right) = {x^a}\log x$$ and $$f\left( 0 \right) = 0,$$ then the value of $$\alpha $$ for which Rolle's theorem can be applied in $$\left[ {0,1} \right]$$ is
IIT-JEE 2004 Screening
19
Tangent is drawn to ellipse
$${{{x^2}} \over {27}} + {y^2} = 1\,\,\,at\,\left( {3\sqrt 3 \cos \theta ,\sin \theta } \right)\left( {where\,\,\theta \in \left( {0,\pi /2} \right)} \right)$$.
$${{{x^2}} \over {27}} + {y^2} = 1\,\,\,at\,\left( {3\sqrt 3 \cos \theta ,\sin \theta } \right)\left( {where\,\,\theta \in \left( {0,\pi /2} \right)} \right)$$.
Then the value of $$\theta $$ such that sum of intercepts on axes made by this tangent is minimum, is
IIT-JEE 2003 Screening
20
In $$\left[ {0,1} \right]$$ Languages Mean Value theorem is NOT applicable to
IIT-JEE 2003 Screening
21
The point(s) in the curve $${y^3} + 3{x^2} = 12y$$ where the tangent is vertical, is (are)
IIT-JEE 2002 Screening
22
The length of a longest interval in which the function $$3\,\sin x - 4{\sin ^3}x$$ is increasing, is
IIT-JEE 2002 Screening
23
Let $$f\left( x \right) = \left( {1 + {b^2}} \right){x^2} + 2bx + 1$$ and let $$m(b)$$ be the minimum value of $$f(x)$$. As $$b$$ varies, the range of $$m(b)$$ is
IIT-JEE 2001 Screening
24
The triangle formed by the tangent to the curve $$f\left( x \right) = {x^2} + bx - b$$ at the point $$(1, 1)$$ and the coordinate axex, lies in the first quadrant. If its area is $$2$$, then the value of $$b$$ is
IIT-JEE 2001 Screening
25
If $$f\left( x \right) = x{e^{x\left( {1 - x} \right)}},$$ then $$f(x)$$ is
IIT-JEE 2001 Screening
26
Let $$f\left( x \right) = \int {{e^x}\left( {x - 1} \right)\left( {x - 2} \right)dx.} $$ Then $$f$$ decreases in the interval
IIT-JEE 2000 Screening
27
If the normal to the curve $$y = f\left( x \right)$$ and the point $$(3, 4)$$ makes an angle $${{{3\pi } \over 4}}$$ with the positive $$x$$-axis, then $$f'\left( 3 \right) = $$
IIT-JEE 2000 Screening
28
Let $$f\left( x \right) = \left\{ {\matrix{
{\left| x \right|,} & {for} & {0 < \left| x \right| \le 2} \cr
{1,} & {for} & {x = 0} \cr
} } \right.$$ then at $$x=0$$, $$f$$ has
IIT-JEE 2000 Screening
29
For all $$x \in \left( {0,1} \right)$$
IIT-JEE 2000 Screening
30
Consider the following statements in $$S$$ and $$R$$
$$S:$$ $$\,\,\,$$$ Both $$\sin \,\,x$$ and $$\cos \,\,x$$ are decreasing functions in the interval $$\left( {{\pi \over 2},\pi } \right)$$
$$R:$$$$\,\,\,$$ If a differentiable function decreases in an interval $$(a, b)$$, then its derivative also decreases in $$(a, b)$$.
Which of the following is true ?
$$S:$$ $$\,\,\,$$$ Both $$\sin \,\,x$$ and $$\cos \,\,x$$ are decreasing functions in the interval $$\left( {{\pi \over 2},\pi } \right)$$
$$R:$$$$\,\,\,$$ If a differentiable function decreases in an interval $$(a, b)$$, then its derivative also decreases in $$(a, b)$$.
Which of the following is true ?
IIT-JEE 2000 Screening
31
The function $$f(x)=$$ $${\sin ^4}x + {\cos ^4}x$$ increases if
IIT-JEE 1999
32
If $$f\left( x \right) = {{{x^2} - 1} \over {{x^2} + 1}},$$ for every real number $$x$$, then the minimum value of $$f$$
IIT-JEE 1998
33
The number of values of $$x$$ where the function
$$f\left( x \right) = \cos x + \cos \left( {\sqrt 2 x} \right)$$ attains its maximum is
$$f\left( x \right) = \cos x + \cos \left( {\sqrt 2 x} \right)$$ attains its maximum is
IIT-JEE 1998
34
If $$f\left( x \right) = {x \over {\sin x}}$$ and $$g\left( x \right) = {x \over {\tan x}}$$, where $$0 < x \le 1$$, then in this interval
IIT-JEE 1997
35
The function $$f\left( x \right) = {{in\,\left( {\pi + x} \right)} \over {in\,\left( {e + x} \right)}}$$ is
IIT-JEE 1995 Screening
36
On the interval $$\left[ {0,1} \right]$$ the function $${x^{25}}{\left( {1 - x} \right)^{75}}$$ takes its maximum value at the point
IIT-JEE 1995 Screening
37
The slope of the tangent to a curve $$y = f\left( x \right)$$ at $$\left[ {x,\,f\left( x \right)} \right]$$ is $$2x+1$$. If the curve passes through the point $$\left( {1,2} \right)$$, then the area bounded by the curve, the $$x$$-axis and the line $$x=1$$ is
IIT-JEE 1995 Screening
38
Which one of the following curves cut the parabola $${y^2} = 4ax$$ at right angles?
IIT-JEE 1994
39
The function defined by $$f\left( x \right) = \left( {x + 2} \right){e^{ - x}}$$
IIT-JEE 1994
40
Let $$f$$ and $$g$$ be increasing and decreasing functions, respectively from $$\left[ {0,\infty } \right)$$ to $$\left[ {0,\infty } \right)$$. Let $$h\left( x \right) = f\left( {g\left( x \right)} \right).$$ If $$h\left( 0 \right) = 0,$$ then $$h\left( x \right) - h\left( 1 \right)$$ is
IIT-JEE 1987
41
The smallest positive root of the equation, $$\tan x - x = 0$$ lies in
IIT-JEE 1987
42
Let $$P\left( x \right) = {a_0} + {a_1}{x^2} + {a_2}{x^4} + ...... + {a_n}{x^{2n}}$$ be a polynomial in a real variable $$x$$ with
$$0 < {a_0} < {a_1} < {a_2} < ..... < {a_n}.$$ The function $$P(x)$$ has
$$0 < {a_0} < {a_1} < {a_2} < ..... < {a_n}.$$ The function $$P(x)$$ has
IIT-JEE 1986
43
The normal to the curve $$\,x = a\left( {\cos \theta + \theta \sin \theta } \right)$$, $$y = a\left( {\sin \theta - \theta \cos \theta } \right)$$ at any point $$'\theta '$$ is such that
IIT-JEE 1983
44
If $$y = a\,\,In\,x + b{x^2} + x$$ has its extreamum values at $$x=-1$$ and $$x=2$$, then
IIT-JEE 1983
45
$$AB$$ is a diameter of a circle and $$C$$ is any point on the circumference of the circle. Then
IIT-JEE 1983
46
If $$a+b+c=0$$, then the quadratic equation $$3a{x^2} + 2bx + c = 0$$ has
IIT-JEE 1983
Numerical
1
For each positive integer n, let
$${y_n} = {1 \over n}(n + 1)(n + 2)...{(n + n)^{{1 \over n}}}$$.
For x$$ \in $$R, let [x] be the greatest integer less than or equal to x. If $$\mathop {\lim }\limits_{n \to \infty } {y_n} = L$$, then the value of [L] is ..............JEE Advanced 2018 Paper 1 Offline
2
A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $$V$$ $$m{m^3}$$, has a $$2$$ mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness $$2$$ mm and is of radius equal to the outer radius of the container.
If the volume of the material used to make the container is minimum when the inner radius of the container is $$10 $$ mm,
then the value of $${V \over {250\pi }}$$ is
JEE Advanced 2015 Paper 1 Offline
3
The slope of the tangent to the curve $${\left( {y - {x^5}} \right)^2} = x{\left( {1 + {x^2}} \right)^2}$$ at the point $$(1, 3)$$ is
JEE Advanced 2014 Paper 1 Offline
4
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 $$p'(0)$$ is
IIT-JEE 2012 Paper 1 Offline
5
Let $$f:IR \to IR$$ be defined as $$f\left( x \right) = \left| x \right| + \left| {{x^2} - 1} \right|.$$ The total number of points at which $$f$$ attains either a local maximum or a local minimum is
IIT-JEE 2012 Paper 1 Offline
6
Let $$f$$ be a real-valued differentiable function on $$R$$ (the set of all real numbers) such that $$f(1)=1$$. If the $$y$$-intercept of the tangent at any point $$P(x,y)$$ on the curve $$y=f(x)$$ is equal to the cube of the abscissa of $$P$$, then find the value of $$f(-3)$$
IIT-JEE 2010 Paper 1 Offline
7
Let $$f$$ be a function defined on $$R$$ (the set of all real numbers)
such that $$f'\left( x \right) = 2010\left( {x - 2009} \right){\left( {x - 2010} \right)^2}{\left( {x - 2011} \right)^3}{\left( {x - 2012} \right)^4}$$ for all $$x \in $$$$R$$
such that $$f'\left( x \right) = 2010\left( {x - 2009} \right){\left( {x - 2010} \right)^2}{\left( {x - 2011} \right)^3}{\left( {x - 2012} \right)^4}$$ for all $$x \in $$$$R$$
If $$g$$ is a function defined on $$R$$ with values in the interval $$\left( {0,\infty } \right)$$ such that
$$$f\left( x \right) = ln\,\left( {g\left( x \right)} \right),\,\,for\,\,all\,\,x \in R$$$
then the number of points in $$R$$ at which $$g$$ has a local maximum is ___________.
IIT-JEE 2010 Paper 2 Offline
8
Let $$p(x)$$ be a polynomial of degree $$4$$ having extremum at
$$x = 1,2$$ and $$\mathop {\lim }\limits_{x \to 0} \left( {1 + {{p\left( x \right)} \over {{x^2}}}} \right) = 2$$.
$$x = 1,2$$ and $$\mathop {\lim }\limits_{x \to 0} \left( {1 + {{p\left( x \right)} \over {{x^2}}}} \right) = 2$$.
Then the value of $$p (2)$$ is
IIT-JEE 2009 Paper 2 Offline
9
The maximum value of the function $$f(x) = 2{x^3} - 15{x^2} + 36x - 48$$ on the set $$A = \{ x|{x^2} + 20 \le 9x|\} $$ is __________.
IIT-JEE 2009 Paper 2 Offline
Subjective
1
For a twice differentiable function $$f(x),g(x)$$ is defined as $$4\sqrt {65} g\left( x \right) = \left( {f'{{\left( x \right)}^2} + f''\left( x \right)} \right)\,\,f\left( x \right)$$ on $$\,\,\,\left[ {a,\,\,\,e} \right].$$ If for $$a < b < c < d < e,\,f\left( a \right) = 0,f\left( b \right) = 2,f\left( c \right) = - 1,f\left( d \right) = 2,f\left( e \right) = 0$$ then find the minimum number of zeros of $$g(x)$$.
IIT-JEE 2006
2
If $$\left| {f\left( {{x_1}} \right) - f\left( {{x_2}} \right)} \right| < {\left( {{x_1} - {x_2}} \right)^2},$$ for all $${x_1},{x_2} \in R$$. Find the equation of tangent to the cuve $$y = f\left( x \right)$$ at the point $$(1, 2)$$.
IIT-JEE 2005
3
If $$p(x)$$ be a polynomial of degree $$3$$ satisfying $$p(-1)=10, p(1)=-6$$ and $$p(x)$$ has maxima at $$x=-1$$ and $$p'(x)$$ has minima at $$x=1$$. Find the distance between the local maxima and local minima of the curve.
IIT-JEE 2005
4
Using Rolle's theorem, prove that there is at least one root
in $$\left( {{{45}^{1/100}},46} \right)$$ of the polynomial
$$P\left( x \right) = 51{x^{101}} - 2323{\left( x \right)^{100}} - 45x + 1035$$.
in $$\left( {{{45}^{1/100}},46} \right)$$ of the polynomial
$$P\left( x \right) = 51{x^{101}} - 2323{\left( x \right)^{100}} - 45x + 1035$$.
IIT-JEE 2004
5
Prove that for $$x \in \left[ {0,{\pi \over 2}} \right],$$ $$\sin x + 2x \ge {{3x\left( {x + 1} \right)} \over \pi }$$. Explain
the identity if any used in the proof.
the identity if any used in the proof.
IIT-JEE 2004
6
Using the relation $$2\left( {1 - \cos x} \right) < {x^2},\,x \ne 0$$ or otherwise,
prove that $$\sin \left( {\tan x} \right) \ge x,\,\forall x \in \left[ {0,{\pi \over 4}} \right]$$
prove that $$\sin \left( {\tan x} \right) \ge x,\,\forall x \in \left[ {0,{\pi \over 4}} \right]$$
IIT-JEE 2003
7
Find a point on the curve $${x^2} + 2{y^2} = 6$$ whose distance from
the line $$x+y=7$$, is minimum.
the line $$x+y=7$$, is minimum.
IIT-JEE 2003
8
If the function $$f:\left[ {0,4} \right] \to R$$ is differentiable then show that
(i)$$\,\,\,\,\,$$ For $$a, b$$$$\,\,$$$$ \in \left( {0,4} \right),{\left( {f\left( 4 \right)} \right)^2} - {\left( {f\left( 0 \right)} \right)^2} = gf'\left( a \right)f\left( b \right)$$
(ii)$$\,\,\,\,\,$$ $$\int\limits_0^4 {f\left( t \right)dt = 2\left[ {\alpha f\left( {{\alpha ^2}} \right) + \beta \left( {{\beta ^2}} \right)} \right]\forall 0 < \alpha ,\beta < 2} $$
(i)$$\,\,\,\,\,$$ For $$a, b$$$$\,\,$$$$ \in \left( {0,4} \right),{\left( {f\left( 4 \right)} \right)^2} - {\left( {f\left( 0 \right)} \right)^2} = gf'\left( a \right)f\left( b \right)$$
(ii)$$\,\,\,\,\,$$ $$\int\limits_0^4 {f\left( t \right)dt = 2\left[ {\alpha f\left( {{\alpha ^2}} \right) + \beta \left( {{\beta ^2}} \right)} \right]\forall 0 < \alpha ,\beta < 2} $$
IIT-JEE 2003
9
If $$P(1)=0$$ and $${{dp\left( x \right)} \over {dx}} > P\left( x \right)$$ for all $$x \ge 1$$ then prove that
$$P(x)>0$$ for all $$x>1$$.
$$P(x)>0$$ for all $$x>1$$.
IIT-JEE 2003
10
Let $$ - 1 \le p \le 1$$. Show that the equation $$4{x^3} - 3x - p = 0$$
has a unique root in the interval $$\left[ {1/2,\,1} \right]$$ and identify it.
has a unique root in the interval $$\left[ {1/2,\,1} \right]$$ and identify it.
IIT-JEE 2001
11
Suppose $$p\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + .......... + {a_n}{x^n}.$$ If
$$\left| {p\left( x \right)} \right| \le \left| {{e^{x - 1}} - 1} \right|$$ for all $$x \ge 0$$, prove that
$$\left| {{a_1} + 2{a_2} + ........ + n{a_n}} \right| \le 1$$.
$$\left| {p\left( x \right)} \right| \le \left| {{e^{x - 1}} - 1} \right|$$ for all $$x \ge 0$$, prove that
$$\left| {{a_1} + 2{a_2} + ........ + n{a_n}} \right| \le 1$$.
IIT-JEE 2000
12
Suppose $$f(x)$$ is a function satisfying the following conditions
(a) $$f(0)=2,f(1)=1$$,
(b) $$f$$has a minimum value at $$x=5/2$$, and
(c) for all $$x$$, $$$f'\left( x \right) = \matrix{ {2ax} & {2ax - 1} & {2ax + b + 1} \cr b & {b + 1} & { - 1} \cr {2\left( {ax + b} \right)} & {2ax + 2b + 1} & {2ax + b} \cr } $$$
where $$a,b$$ are some constants. Determine the constants $$a, b$$ and the function $$f(x)$$.
(a) $$f(0)=2,f(1)=1$$,
(b) $$f$$has a minimum value at $$x=5/2$$, and
(c) for all $$x$$, $$$f'\left( x \right) = \matrix{ {2ax} & {2ax - 1} & {2ax + b + 1} \cr b & {b + 1} & { - 1} \cr {2\left( {ax + b} \right)} & {2ax + 2b + 1} & {2ax + b} \cr } $$$
where $$a,b$$ are some constants. Determine the constants $$a, b$$ and the function $$f(x)$$.
IIT-JEE 1998
13
A curve $$C$$ has the property that if the tangent drawn at any point $$P$$ on $$C$$ meets the co-ordinate axes at $$A$$ and $$B$$, then $$P$$ is the mid-point of $$AB$$. The curve passes through the point $$(1, 1)$$. Determine the equation of the curve.
IIT-JEE 1998
14
Let $$a+b=4$$, where $$a<2,$$ and let $$g(x)$$ be a differentiable function.
If $${{dg} \over {dx}} > 0$$ for all $$x$$, prove that $$\int_0^a {g\left( x \right)dx + \int_0^b {g\left( x \right)dx} } $$
increases as $$(b-a)$$ increases.
IIT-JEE 1997
15
Let $$f\left( x \right) = \left\{ {\matrix{
{x{e^{ax}},\,\,\,\,\,\,\,x \le 0} \cr
{x + a{x^2} - {x^3},\,x > 0} \cr
} } \right.$$
Where a is a positive constant. Find the interval in which $$f'(x)$$ is increasing.
IIT-JEE 1996
16
A curve $$y=f(x)$$ passes through the point $$P(1, 1)$$. The normal to the curve at $$P$$ is $$a(y-1)+(x-1)=0$$. If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, determine the equation of the curve. Also obtain the area bounded by the $$y$$-axis, the curve and the normal to the curve at $$P$$.
IIT-JEE 1996
17
Determine the points of maxima and minima of the function
$$f\left( x \right) = {1 \over 8}\ell n\,x - bx + {x^2},x > 0,$$ where $$b \ge 0$$ is a constant.
$$f\left( x \right) = {1 \over 8}\ell n\,x - bx + {x^2},x > 0,$$ where $$b \ge 0$$ is a constant.
IIT-JEE 1996
18
Let $$(h, k)$$ be a fixed point, where $$h > 0,k > 0.$$. A straight line passing through this point cuts the possitive direction of the coordinate axes at the points $$P$$ and $$Q$$. Find the minimum area of the triangle $$OPQ$$, $$O$$ being the origin.
IIT-JEE 1995
19
The curve $$y = a{x^3} + b{x^2} + cx + 5$$, touches the $$x$$-axis at $$P(-2, 0)$$ and cuts the $$y$$ axis at a point $$Q$$, where its gradient is $$3$$. Find $$a, b, c$$.
IIT-JEE 1994
20
The circle $${x^2} + {y^2} = 1$$ cuts the $$x$$-axis at $$P$$ and $$Q$$. Another circle with centre at $$Q$$ and variable radius intersects the first circle at $$R$$ above the $$x$$-axis and the line segment $$PQ$$ at $$S$$. Find the maximum area of the triangle $$QSR$$.
IIT-JEE 1994
21
Let $$f\left( x \right) = \left\{ {\matrix{
{ - {x^3} + {{\left( {{b^3} - {b^2} + b - 1} \right)} \over {\left( {{b^2} + 3b + 2} \right)}},} & {0 \le x < 1} \cr
{2x - 3} & {1 \le x \le 3} \cr
} } \right.$$
Find all possible real values of $$b$$ such that $$f(x)$$ has the smallest value at $$x=1$$.
IIT-JEE 1993
22
Find the equation of the normal to the curve
$$y = {\left( {1 + x} \right)^y} + {\sin ^{ - 1}}\left( {{{\sin }^2}x} \right)$$ at $$x=0$$
$$y = {\left( {1 + x} \right)^y} + {\sin ^{ - 1}}\left( {{{\sin }^2}x} \right)$$ at $$x=0$$
IIT-JEE 1993
23
In this questions there are entries in columns $$I$$ and $$II$$. Each entry in column $$I$$ is related to exactly one entry in column $$II$$. Write the correct letter from column $$II$$ against the entry number in column $$I$$ in your answer book.
Let the functions defined in column $$I$$ have domain $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$
$$\,\,\,\,$$Column $$I$$
(A) $$x + \sin x$$
(B) $$\sec x$$
$$\,\,\,\,$$Column $$II$$
(p) increasing
(q) decreasing
(r) neither increasing nor decreasing
IIT-JEE 1992
24
A cubic $$f(x)$$ vanishes at $$x=2$$ and has relative minimum / maximum at $$x=-1$$ and $$x = {1 \over 3}$$ if $$\int\limits_{ - 1}^1 {f\,\,dx = {{14} \over 3}} $$, find the cubic $$f(x)$$.
IIT-JEE 1992
25
What normal to the curve $$y = {x^2}$$ forms the shortest chord?
IIT-JEE 1992
26
A window of perimeter $$P$$ (including the base of the arch) is in the form of a rectangle surmounded by a semi circle. The semi-circular portion is fitted with coloured glass while the rectangular part is fitted with clear glass transmits three times as such light per square meter as the coloured glass does.
What is the ratio for the sides of the rectangle so that the window transmits the maximum light ?
IIT-JEE 1991
27
Show that $$2\sin x + \tan x \ge 3x$$ where $$0 \le x < {\pi \over 2}$$.
IIT-JEE 1990
28
A point $$P$$ is given on the circumference of a circle of radius $$r$$. Chord $$QR$$ is parallel to the tangent at $$P$$. Determine the maximum possible area of the triangle $$PQR$$.
IIT-JEE 1990
29
Find all maxima and minima of the function
$$$y = x{\left( {x - 1} \right)^2},0 \le x \le 2$$$
Also determine the area bounded by the curve $$y = x{\left( {x - 1} \right)^2}$$,
the $$y$$-axis and the line $$y-2$$.
Also determine the area bounded by the curve $$y = x{\left( {x - 1} \right)^2}$$,
the $$y$$-axis and the line $$y-2$$.
IIT-JEE 1989
30
Investigate for maxima and minimum the function
$$$f\left( x \right) = \int\limits_1^x {\left[ {2\left( {t - 1} \right){{\left( {t - 2} \right)}^3} + 3{{\left( {t - 1} \right)}^2}{{\left( {t - 2} \right)}^2}} \right]} dt$$$
IIT-JEE 1988
31
Find the point on the curve $$\,\,\,4{x^2} + {a^2}{y^2} = 4{a^2},\,\,\,4 < {a^2} < 8$$
that is farthest from the point $$(0, -2)$$.
that is farthest from the point $$(0, -2)$$.
IIT-JEE 1987
32
Let $$f\left( x \right) = {\sin ^3}x + \lambda {\sin ^2}x, - {\pi \over 2} < x < {\pi \over 2}.$$ Find the intervals in which $$\lambda $$ should lie in order that $$f(x)$$ has exactly one minimum and exactly one maximum.
IIT-JEE 1985
33
Find all the tangents to the curve
$$y = \cos \left( {x + y} \right),\,\, - 2\pi \le x \le 2\pi ,$$ that are parallel to the line $$x+2y=0$$.
$$y = \cos \left( {x + y} \right),\,\, - 2\pi \le x \le 2\pi ,$$ that are parallel to the line $$x+2y=0$$.
IIT-JEE 1985
34
Show that $$1+x$$ $$In\left( {x + \sqrt {{x^2} + 1} } \right) \ge \sqrt {1 + {x^2}} $$ for all $$x \ge 0$$
IIT-JEE 1983
35
Find the coordinates of the point on the curve $$y = {x \over {1 + {x^2}}}$$
where the tangent to the curve has the greatest slope.
where the tangent to the curve has the greatest slope.
IIT-JEE 1983
36
If $$f(x)$$ and $$g(x)$$ are differentiable function for $$0 \le x \le 1$$ such that $$f(0)=2$$, $$g(0)=0$$, $$f(1)=6$$; $$g(1)=2$$, then show that there exist $$c$$ satisfying $$0 < c < 1$$ and $$f'(c)=2g'(c)$$.
IIT-JEE 1982
37
If $$a{x^2} + {b \over x} \ge c$$ for all positive $$x$$ where $$a>0$$ and $$b>0$$ show that $$27a{b^2} \ge 4{c^3}$$.
IIT-JEE 1982
38
For all $$x$$ in $$\left[ {0,1} \right]$$, let the second derivative $$f''(x)$$ of a function $$f(x)$$ exist and satisfy $$\left| {f''\left( x \right)} \right| < 1.$$ If $$f(0)=f(1)$$, then show that $$\left| {f\left( x \right)} \right| < 1$$ for all $$x$$ in $$\left[ {0,1} \right]$$.
IIT-JEE 1981
39
Let $$x$$ and $$y$$ be two real variables such that $$x>0$$ and $$xy=1$$. Find the minimum value of $$x+y$$.
IIT-JEE 1981
40
Use the function $$f\left( x \right) = {x^{1/x}},x > 0$$. to determine the bigger of the two numbers $${e^\pi }$$ and $${\pi ^e}$$
IIT-JEE 1981
41
Prove that the minimum value of $${{\left( {a + x} \right)\left( {b + x} \right)} \over {\left( {c + x} \right)}},$$
$$a,b > c,x > - c$$ is $${\left( {\sqrt {a - c} + \sqrt {b - c} } \right)^2}$$.
$$a,b > c,x > - c$$ is $${\left( {\sqrt {a - c} + \sqrt {b - c} } \right)^2}$$.
IIT-JEE 1979
Fill in the Blanks
1
Let $$P$$ be a variable point on the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ with foci $${F_1}$$ and $${F_2}$$. If $$A$$ is the area of the triangle $$P{F_1}{F_2}$$ then the maximum value of $$A$$ is ..........
IIT-JEE 1994
2
Let $$C$$ be the curve $${y^3} - 3xy + 2 = 0$$. If $$H$$ is the set of points on the curve $$C$$ where the tangent is horizontal and $$V$$ is the set of the point on the curve $$C$$ where the tangent is vertical then $$H=$$.............. and $$V=$$ .................
IIT-JEE 1994
3
The set of all $$x$$ for which $$in\left( {1 + x} \right) \le x$$ is equal to ..........
IIT-JEE 1987
4
The larger of $$\cos \left( {In\,\,\theta } \right)$$ and $$In $$ $$\left( {\cos \,\,\theta } \right)$$ If $${e^{ - \pi /2}} < \theta < {\pi \over 2}$$ is ..................
IIT-JEE 1983
5
The function $$y = 2{x^2} - In\,\left| x \right|$$ is monotonically increasing for values of $$x\left( {x \ne 0} \right)$$ satisfying the inequalities ......... and monotonically decreasing for values of $$x$$ satisfying the inequalities ............
IIT-JEE 1983
True or False
1
For $$0 < a < x,$$ the minimum value of the function $$lo{g_a}x + {\log _x}a$$ is $$2$$.
IIT-JEE 1984
2
If $$x-r$$ is a factor of the polynomial $$f\left( x \right) = {a_n}{x^4} + ..... + {a_0},$$ repeated $$m$$ times $$\left( {1 < m \le n} \right)$$, then $$r$$ is a root of $$\left( x \right) = 0$$ repeated $$m$$ times.
IIT-JEE 1983