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Application of Derivatives

127 Questions
All Exams JEE Mains JEE Advanced TS-EAMCET AP-EAPCET
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 MCQ
IIT-JEE 1987
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
A.
always zero
B.
always negative
C.
always positive
D.
strictly increasing
1987 JEE Advanced MCQ
IIT-JEE 1987
The smallest positive root of the equation, $\tan x - x = 0$ lies in
A.
$\left( {0,{\pi \over 2}} \right)$
B.
$\left( {{\pi \over 2},\pi } \right)$
C.
$\left( {\pi ,{{3\pi } \over 2}} \right)$
D.
$\left( {{{3\pi } \over 2},2\pi } \right)$
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 $(0, -2)$.
1987 JEE Advanced Numerical
IIT-JEE 1987
The set of all $x$ for which $in\left( {1 + x} \right) \le x$ is equal to ..........
1986 JEE Advanced MCQ
IIT-JEE 1986
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
A.
neither a maximum nor a minimum
B.
only one maximum
C.
only one minimum
D.
only one maximum and only one minimum
1986 JEE Advanced MSQ
IIT-JEE 1986
If the line $ax+by+c=0$ is a normal to the curve $xy=1$, then
A.
$a > 0,b > 0$
B.
$a > 0,b < 0$
C.
$a < 0,b > 0$
D.
$a < 0,b < 0$
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.
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
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
A.
it makes a constant angle with the $x$-axis
B.
it passes through the origin
C.
it is at a constant distance from the origin
D.
none of these
1983 JEE Advanced MCQ
IIT-JEE 1983
If $a+b+c=0$, then the quadratic equation $3a{x^2} + 2bx + c = 0$ has
A.
at least one root in $\left[ {0,1} \right]$
B.
one root in $\left[ {2,3} \right]$ and the other in $\left[ {-2,-1} \right]$
C.
imaginary roots
D.
none of these
1983 JEE Advanced MCQ
IIT-JEE 1983
$AB$ is a diameter of a circle and $C$ is any point on the circumference of the circle. Then
A.
the area of $\Delta ABC$ is maximum when it is isosceles
B.
the area of $\Delta ABC$ is minimum when it is isosceles
C.
the perimeter of $\Delta ABC$ is minimum when it is isosceles
D.
none of these
1983 JEE Advanced MCQ
IIT-JEE 1983
If $y = a\,\,In\,x + b{x^2} + x$ has its extreamum values at $x=-1$ and $x=2$, then
A.
$a = 2,b = - 1$
B.
$a = 2,b = - {1 \over 2}$
C.
$a = - 2,b = {1 \over 2}$
D.
none of these
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.
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 ............
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
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 ${\left( {\sqrt {a - c} + \sqrt {b - c} } \right)^2}$.