Class 12 Mathematics 2018 Board Exam

If $a\ast b$ denotes the larger of '$a$' and '$b$' and if $a\circ b=(a\ast b)+3$, then write the value of $(5)\circ (10)$, where $\ast$ and $\circ$ are binary operations.

Let $A=\{x\in \mathbb{Z}:0\le x\le 12\}$. Show that

$R=\{(a,b):a,b\in A,|a-b|\text{ is divisible by }4\}$ is an equivalence relation.

Find the set of all elements related to 1. Also write the equivalence class $[2]$.

OR

Show that the function $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=\frac{x}{x^2+1},\forall x\in \mathbb{R}$ is neither one-one nor onto. Also, if $g:\mathbb{R}\to \mathbb{R}$ is defined as $g(x)=2x-1$, find $fog(x)$.

Find the value of ${\tan }^{-1}\sqrt{3}-{\cot }^{-1}(-\sqrt{3})$.

Prove that:

$$3{\sin }^{-1}x={\sin }^{-1}(3x-4x^3),x\in \left[-\frac{1}{2},\frac{1}{2}\right]$$

If the matrix $A=\left[\begin{array}{ccc}0& a& -3\\ 2& 0& -1\\ b& 1& 0\end{array}\right]$ is skew symmetric, find the values of $a$ and $b$.

Given $A=\left[\begin{array}{cc}2& -3\\ -4& 7\end{array}\right]$, compute $A^{-1}$ and show that $2A^{-1}=9I-A$.

If $A=\left[\begin{array}{ccc}1& 1& -2\\ 3& 2& -4\\ 2& -3& 5\end{array}\right]$, find $A^{-1}$. Use it to solve the system of equations

$$2x-3y+5z=11$$
$$3x+2y-4z=-5$$
$$x+y-2z=-3$$

OR

Using elementary row transformations, find the inverse of the matrix

$A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 5& 7\\ -2& -4& -5\end{array}\right]$.

Using properties of determinants, prove that

$$\left|\begin{array}{ccc}1& 1& 1+3x\\ 1+3y& 1& 1\\ 1& 1+3z& 1\end{array}\right|=9(3xyz+xy+yz+zx)$$

Differentiate ${\tan }^{-1}\left(\frac{1+\cos x}{\sin x}\right)$ with respect to $x$.

If $(x^2+y^2{)}^2=xy$, find $\frac{dy}{dx}$.

OR

If $x=a(2\theta -\sin 2\theta )$ and $y=a(1-\cos 2\theta )$, find $\frac{dy}{dx}$ when $\theta =\frac{\pi }{3}$.