iCON Education - Differentiation JEE Advanced Test

Let $\mathbb{R}$ denote the set of all real numbers. Consider the polynomial function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by

$ f(x)=\frac{d^{10}}{d x^{10}}\left(\left(x^2-1\right)^{10}\right), \quad \text { for all } x \in \mathbb{R} $

Here $\frac{d^{10}}{d x^{10}}\left(\left(x^2-1\right)^{10}\right)$ is the $10^{\text {th }}$ order derivative of the function $\left(x^2-1\right)^{10}$.

Then which of the following statements is (are) TRUE ?

The coefficient of $x^8$ in the polynomial $f(x)$ is $(-10)\left( \frac{18!}{8!} \right)$

The value of $f(1) + f(-1)$ is equal to $10! \cdot 2^{11}$

The degree of the polynomial $f(x)$ is $10$

The constant term of the polynomial $f(x)$ is $- \left( \frac{10!}{5!} \right)$

Let $S$ be the set of all twice differentiable functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that $\frac{d^2 f}{d x^2}(x)>0$ for all $x \in(-1,1)$. For $f \in S$, let $X_f$ be the number of points $x \in(-1,1)$ for which $f(x)=x$. Then which of the following statements is(are) true?

There exists a function $f \in S$ such that $X_f=0$

For every function $f \in S$, we have $X_f \leq 2$

There exists a function $f \in S$ such that $X_f=2$

There does NOT exist any function $f$ in $S$ such that $X_f=1$

Let $f:\mathbb{R} \to \mathbb{R},\,g:\mathbb{R} \to \mathbb{R}$ and $h:\mathbb{R} \to \mathbb{R}$ be differentiable functions such that $f\left( x \right)= {x^3} + 3x + 2,$ $g\left( {f\left( x \right)} \right) = x$ and $h\left( {g\left( {g\left( x \right)} \right)} \right) = x$ for all $x \in R$. Then

$g'\left( 2 \right) = {1 \over {15}}$

$h'\left( 1 \right) = 666$

$h\left( 0 \right) = 16$

$h\left( {g\left( 3 \right)} \right) = 36$

Let $F:R \to R$ be a thrice differentiable function. Suppose that
$F\left( 1 \right) = 0,F\left( 3 \right) = - 4$ and $F'\left( x \right) < 0$ for all $x \in \left( {{1 \over 2},3} \right).$ Let $f\left( x \right) = xF\left( x \right)$ for all $x \in R.$

The correct statement(s) is (are)

$f'\left( 1 \right) < 0$

$f\left( 2 \right) < 0$

$f'\left( x \right) \ne 0$ for any $x \in \left( {1,3} \right)$

$f'\left( x \right) = 0$ for some $x \in \left( {1,3} \right)$

Let $f:\left[ {0,2} \right] \to R$ be a function which is continuous on $\left[ {0,2} \right]$ and is differentiable on $(0,2)$ with $f(0)=1$. Let
$F\left( x \right) = \int\limits_0^{{x^2}} {f\left( {\sqrt t } \right)dt} $ for $x \in \left[ {0,2} \right]$. If $F'\left( x \right) = f'\left( x \right)$ for all $x \in \left[ {0,2} \right]$, then $F(2)$ equals

Let $g(x) = \log f(x)$, where $f(x)$ is a twice differentiable positive function on (0, $\infty$) such that $f(x + 1) = xf(x)$. Then for N = 1, 2, 3, ..., $g''\left( {N + {1 \over 2}} \right) - g''\left( {{1 \over 2}} \right) = $

$ - 4\left\{ {1 + {1 \over 9} + {1 \over {25}} + ....... + {1 \over {{{\left( {2N - 1} \right)}^2}}}} \right\}$

$4\left\{ {1 + {1 \over 9} + {1 \over {25}} + ....... + {1 \over {{{\left( {2N - 1} \right)}^2}}}} \right\}$

$ - 4\left\{ {1 + {1 \over 9} + {1 \over {25}} + ....... + {1 \over {{{\left( {2N + 1} \right)}^2}}}} \right\}$

$4\left\{ {1 + {1 \over 9} + {1 \over {25}} + ....... + {1 \over {{{\left( {2N + 1} \right)}^2}}}} \right\}$

Which of the following is true?

${\left( {2 + a} \right)^2}f''\left( 1 \right) + {\left( {2 - a} \right)^2}f''\left( { - 1} \right) = 0$

${\left( {2 - a} \right)^2}f''\left( 1 \right) - {\left( {2 + a} \right)^2}f''\left( { - 1} \right) = 0$

$f'\left( 1 \right)f'\left( { - 1} \right) = {\left( {2 - a} \right)^2}$

$f'\left( 1 \right)f'\left( { - 1} \right) = -{\left( {2 + a} \right)^2}$

Let $f$ and $g$ be real valued functions defined on interval $(-1, 1)$ such that $g''(x)$ is continuous, $g\left( 0 \right) \ne 0.$ $g'\left( 0 \right) = 0$, $g''\left( 0 \right) \ne 0$, and $f\left( x \right) = g\left( x \right)\sin x$

STATEMENT - 1: $\mathop {\lim }\limits_{x \to 0} \,\,\left[ {g\left( x \right)\cot x - g\left( 0 \right)\cos ec\,x} \right] = f''\left( 0 \right)$ and

STATEMENT - 2: $f'\left( 0 \right) = g\left( 0 \right)$

Statement - 1 is True, Statement - 2 is True; Statement - 2 is a correct explanation for Statement - 1

Statement - 1 is True, Statement - 2 is True; Statement - 2 is NOT a correct explanation for Statement - 1

Statement - 1 is True, Statement -2 is False

Statement - 1 is False, Statement -2 is True

If $f\left( { - 10\sqrt 2 } \right) = 2\sqrt 2 ,$ then $f''\left( { - 10\sqrt 2 } \right) = $

${{4\sqrt 2 } \over {{7^3}{3^2}}}$

$-{{4\sqrt 2 } \over {{7^3}{3^2}}}$

${{4\sqrt 2 } \over {{7^3}3}}$

$-{{4\sqrt 2 } \over {{7^3}3}}$

${{{d^2}x} \over {d{y^2}}}$ equals

${\left( {{{{d^2}y} \over {d{x^2}}}} \right)^{ - 1}}$

$ - {\left( {{{{d^2}y} \over {d{x^2}}}} \right)^{ - 1}}{\left( {{{dy} \over {dx}}} \right)^{ - 3}}$

$\left( {{{{d^2}y} \over {d{x^2}}}} \right){\left( {{{dy} \over {dx}}} \right)^{ - 2}}$

$ - \left( {{{{d^2}y} \over {d{x^2}}}} \right){\left( {{{dy} \over {dx}}} \right)^{ - 3}}$

Let $\,\,\,$$f\left( x \right) = 2 + \cos x$ for all real $X$.

STATEMENT - 1: for eachreal $t$, there exists a point $c$ in $\left[ {t,t + \pi } \right]$ such that $f'\left( c \right) = 0$ because
STATEMENT - 2: $f\left( t \right) = f\left( {t + 2\pi } \right)$ for each real $t$.

Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1

Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

Statement-1 is True, Statement-2 is False

Statement-1 is False, Statement-2 is True.

$\frac{d^{2} x}{d y^{2}}$ equals :

$\left(\frac{d^{2} y}{d x^{2}}\right)^{-1}$

$-\left(\frac{d^{2} y}{d x^{2}}\right)^{-1}\left(\frac{d y}{d x}\right)^{-3}$

$\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d y}{d x}\right)^{-2}$

$-\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d y}{d x}\right)^{-3}$

If $f(x)$ is a twice differentiable function and given that $f\left( 1 \right) = 1;f\left( 2 \right) = 4,f\left( 3 \right) = 9$, then

$f''\left( x \right) = 2$ for $\forall x \in \left( {1,3} \right)$

$f''\left( x \right) = f'\left( x \right) = 5$ for some $x \in \left( {2,3} \right)$

$f''\left( x \right) = 3$ for $\forall x \in \left( {2,3} \right)$

$f''\left( x \right) = 2$ for some $x \in \left( {1,3} \right)$

If $y$ is a function of $x$ and log $(x+y)-2xy=0$, then the value of $y'(0)$ is equal to

Let $f:\left( {0,\infty } \right) \to R$ and $F\left( x \right) = \int\limits_0^x {f\left( t \right)dt.} $ If $F\left( {{x^2}} \right) = {x^2}\left( {1 + x} \right)$, then $f(4)$ equals

If ${x^2} + {y^2} = 1$ then

$yy'' - 2{\left( {y'} \right)^2} + 1 = 0$

$yy'' + {\left( {y'} \right)^2} + 1 = 0$

$yy'' + {\left( {y'} \right)^2} - 1 = 0$

$yy'' + 2{\left( {y'} \right)^2} + 1 = 0$

If $y = {\left( {\sin x} \right)^{\tan x}},$ then ${{dy} \over {dx}}$ is equal to

${\left( {\sin x} \right)^{\tan x}}\left( {1 + {{\sec }^2}x\,\log \,\sin \,x} \right)$

$\tan x{\left( {\sin x} \right)^{\tan x - 1}}.\cos x$

${\left( {\sin x} \right)^{\tan x}}{\sec ^2}x\,\log \,\sin \,x$

$\tan x{\left( {\sin x} \right)^{\tan x - 1}}$

Let $f(x)$ be a quadratic expression which is positive for all the real values of $x$. If $g(x)=f(x)+f''(x)$, then for any real $x$,

$g\left( x \right) \ge 0$

If ${y^2} = P\left( x \right)$, a polynomial of degree $3$, then $2{d \over {dx}}\left( {{y^3}{{{d^2}y} \over {d{x^2}}}} \right)$ equals

$P''\left( x \right) + P\left( x \right)$

$P'\left( x \right)P''\left( x \right)$

$P\left( x \right)P''\left( x \right)$

The derivative of an even function is always an odd function.