[10×2] If , and , , and is the set of real numbers, then find and . Solve: Using determinants, find…

If $f : \mathbb{R} \to \mathbb{R}$, $f(x) = x^3$ and $g : \mathbb{R} \to \mathbb{R}$, $g(x) = 2x^2 + 1$, and $\mathbb{R}$ is the set of real numbers, then find $fog(x)$ and $gof(x)$.

Solve: $\sin(2\tan^{-1}x) = 1$

Using determinants, find the values of $k$, if the area of triangle with vertices $(-2, 0)$, $(0, 4)$ and $(0, k)$ is $4$ square units.

Show that $(A + A')$ is symmetric matrix, if $A = \begin{pmatrix} 2 & 4 \\\\ 3 & 5 \end{pmatrix}$.

$f(x) = \frac{x^2 - 9}{x - 3}$ is not defined at $x = 3$. What value should be assigned to $f(3)$ for continuity of $f(x)$ at $x = 3$?

Prove that the function $f(x) = x^3 - 6x^2 + 12x + 5$ is increasing on $\mathbb{R}$.

Evaluate: $\int \frac{\sec^2 x}{\operatorname{cosec}^2 x}\\,dx$

Using L'Hospital's Rule, evaluate: $\lim_{x \to 0} \frac{8^x - 4^x}{4x}$

Two balls are drawn from an urn containing 3 white, 5 red and 2 black balls, one by one without replacement. What is the probability that at least one ball is red?

If events $A$ and $B$ are independent, such that $P(A) = \frac{3}{5}$, $P(B) = \frac{2}{3}$, find $P(A \cup B)$.