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Application of Derivatives Syllabus Reduced

126 Questions
All Exams JEE Mains JEE Advanced
1992 JEE Advanced Numerical
IIT-JEE 1992
What normal to the curve $y = {x^2}$ forms the shortest chord?
1992 JEE Advanced Numerical
IIT-JEE 1992
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

1992 JEE Advanced Numerical
IIT-JEE 1992
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 $f(x)$.
1991 JEE Advanced Numerical
IIT-JEE 1991
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 ?

1990 JEE Advanced Numerical
IIT-JEE 1990
Show that $2\sin x + \tan x \ge 3x$ where $0 \le x < {\pi \over 2}$.
1990 JEE Advanced Numerical
IIT-JEE 1990
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 $PQR$.
1989 JEE Advanced Numerical
IIT-JEE 1989
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 $y-2$.
1988 JEE Advanced Numerical
IIT-JEE 1988
Investigate for maxima and minimum the function $$f\left( x \right) = \int\limits_1^x {\left[ {2\left( {t - 1} \right){{\left( {t - 2} \right)}^3} + 3{{\left( {t - 1} \right)}^2}{{\left( {t - 2} \right)}^2}} \right]} dt$$
1987 JEE Advanced Numerical
IIT-JEE 1987
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 $(0, -2)$.
1985 JEE Advanced Numerical
IIT-JEE 1985
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 $x+2y=0$.
1985 JEE Advanced Numerical
IIT-JEE 1985
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.
1983 JEE Advanced Numerical
IIT-JEE 1983
Show that $1+x$ $In\left( {x + \sqrt {{x^2} + 1} } \right) \ge \sqrt {1 + {x^2}} $ for all $x \ge 0$
1983 JEE Advanced Numerical
IIT-JEE 1983
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.
1982 JEE Advanced Numerical
IIT-JEE 1982
If $a{x^2} + {b \over x} \ge c$ for all positive $x$ where $a>0$ and $b>0$ show that $27a{b^2} \ge 4{c^3}$.
1982 JEE Advanced Numerical
IIT-JEE 1982
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 $f'(c)=2g'(c)$.
1981 JEE Advanced Numerical
IIT-JEE 1981
Use the function $f\left( x \right) = {x^{1/x}},x > 0$. to determine the bigger of the two numbers ${e^\pi }$ and ${\pi ^e}$
1981 JEE Advanced Numerical
IIT-JEE 1981
Let $x$ and $y$ be two real variables such that $x>0$ and $xy=1$. Find the minimum value of $x+y$.
1981 JEE Advanced Numerical
IIT-JEE 1981
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 $\left[ {0,1} \right]$.
1979 JEE Advanced Numerical
IIT-JEE 1979
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 ${\left( {\sqrt {a - c} + \sqrt {b - c} } \right)^2}$.
1994 JEE Advanced Numerical
IIT-JEE 1994
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 ..........
1994 JEE Advanced Numerical
IIT-JEE 1994
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 $V=$ .................
1987 JEE Advanced Numerical
IIT-JEE 1987
The set of all $x$ for which $in\left( {1 + x} \right) \le x$ is equal to ..........
1983 JEE Advanced Numerical
IIT-JEE 1983
The larger of $\cos \left( {In\,\,\theta } \right)$ and $In $ $\left( {\cos \,\,\theta } \right)$ If ${e^{ - \pi /2}} < \theta < {\pi \over 2}$ is ..................
1983 JEE Advanced Numerical
IIT-JEE 1983
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 ............
1984 JEE Advanced MCQ
IIT-JEE 1984
For $0 < a < x,$ the minimum value of the function $lo{g_a}x + {\log _x}a$ is $2$.
A.
TRUE
B.
FALSE
1983 JEE Advanced MCQ
IIT-JEE 1983
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.
A.
TRUE
B.
FALSE