Functions - Calculus - Mathematics Previous Year Questions

Find the range of value of \(t\) for which

$$2 \sin t=\frac{1-2 x+5 x^{2}}{3 x^{2}-2 x-1}, t \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$

If \(f''(x)=-f(x)\) and \(g(x)=f'(x)\) and \(\mathrm{F}(x)=\left(f\left(\frac{x}{2}\right)\right)^{2}+\left(g\left(\frac{x}{2}\right)\right)^{2}\) and given that \(\mathrm{F}(5)=5\), then \(\mathrm{F}(10)\) is equal to :

If the function \(f(x) = {x^3} + {e^{x/2}}\) and \(g(x) = {f^{ - 1}}(x)\), then the value of \(g'(1)\) is _________.

Let $f, g$ and $h$ be real valued functions defined on the interval $[0,1]$ by

$f(x)=e^{x^2}+e^{-x^2}$,

$g(x)=x e^{x^2}+e^{-x^2}$

and $h(x)=x^2 e^{x^2}+e^{-x^2}$.

If $a, b$ and $c$ denote, respectively, the absolute maximum of $f, g$ and $h$ on $[0,1]$, then :

Consider the polynomial
$$f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.$$
Let \(s\) be the sum of all distinct real roots of \(f(x)\) and let \(t = \left| s \right|.\)

The real numbers lies in the interval

Consider the polynomial
$$f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.$$
Let \(s\) be the sum of all distinct real roots of \(f(x)\) and let \(t = \left| s \right|.\)

The function\(f'(x)\) is

Let \(f:(0,1) \to R\) be defined by \(f(x) = {{b - x} \over {1 - bx}}\), where b is a constant such that \(0 < b < 1\). Then

Let f(x) = x² and g(x) = sin x for all x \(\in\) R. Then the set of all x satisfying \((f \circ g \circ g \circ f)(x) = (g \circ g \circ f)(x)\), where \((f \circ g)(x) = f(g(x))\), is

Match the statements given in Column I with the intervals/union of intervals given in Column II :

The function \(f:[0,3] \to [1,29]\), defined by \(f(x) = 2{x^3} - 15{x^2} + 36x + 1\), is

Let \(f:( - 1,1) \to R\) be such that \(f(\cos 4\theta ) = {2 \over {2 - {{\sec }^2}\theta }}\) for \(\theta \in \left( {0,{\pi \over 4}} \right) \cup \left( {{\pi \over 4},{\pi \over 2}} \right)\). Then the value(s) of \(f\left( {{1 \over 3}} \right)\) is(are)

$${f_1}\left( x \right) = \left\{ {\matrix{ \[{\left| x \right|} & {if\,x < 0,} \cr\] {{e^x}} & {if\,x \ge 0;} \cr } } \right.$$

f₂(x) = x² ;

and

$${f_4}\left( x \right) = \left\{ {\matrix{ \[{{f_2}\left( {{f_1}\left( x \right)} \right)} & {if\,x < 0,} \cr\] \[{{f_2}\left( {{f_1}\left( x \right)} \right) - 1} & {if\,x \ge 0;} \cr\] } } \right.$$

Let \(f(x) = \sin \left( {{\pi \over 6}\sin \left( {{\pi \over 2}\sin x} \right)} \right)\) for all \(x \in R\) and g(x) = \({{\pi \over 2}\sin x}\) for all x\(\in\)R. Let \((f \circ g)(x)\) denote f(g(x)) and \((g \circ f)(x)\) denote g(f(x)). Then which of the following is/are true?