Class 12 Mathematics 2017 Board Exam

Consider $f:\mathbb{R}-\left\{-\frac{4}{3}\right\}\to \mathbb{R}-\left\{\frac{4}{3}\right\}$ given by $f(x)=\frac{4x+3}{3x+4}$. Show that $f$ is bijective. Find the inverse of $f$ and hence find $f^{-1}(0)$ and $x$ such that $f^{-1}(x)=2$.

OR

Let $A=\mathbb{Q}\times \mathbb{Q}$ and let $\ast$ be a binary operation on $A$ defined by $(a,b)\ast (c,d)=(ac,b+ad)$ for $(a,b),(c,d)\in A$. Determine, whether $\ast$ is commutative and associative. Then, with respect to $\ast$ on $A$

(i) Find the identity element in $A$.

(ii) Find the invertible elements of $A$.

If ${\tan }^{-1}\left(\frac{x-3}{x-4}\right)+{\tan }^{-1}\left(\frac{x+3}{x+4}\right)=\frac{\pi }{4}$, then find the value of $x$.

If for any $2\times 2$ square matrix $A$, $A(\text{adj }A)=\left[\begin{array}{cc}8& 0\\ 0& 8\end{array}\right]$, then write the value of $|A|$.

If $A$ is a skew-symmetric matrix of order 3, then prove that $\det A=0$.

Determine the product

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

and use it to solve the system of equations $x-y+z=4$, $x-2y-2z=9$, $2x+y+3z=1$.

If $A=\left[\begin{array}{ccc}1& 1& 2\\ 3& 2& 4\\ 2& 3& 5\end{array}\right]$, find $A^{-1}$. Hence using $A^{-1}$ solve the system of equations $2x-3y+5z=11$, $3x+2y-4z=-5$, $x+y-2z=-3$.

If $A=\left[\begin{array}{ccc}2& -3& 5\\ 3& 2& -4\\ 1& 1& -2\end{array}\right]$, then find $A^{-1}$ and hence solve the system of linear equations $2x-3y+5z=11$, $3x+2y-4z=-5$ and $x+y-2z=-3$.

Using properties of determinants, prove that

$$\left|\begin{array}{ccc}{a}^2+2a& 2a+1& 1\\ 2a+1& a+2& 1\\ 3& 3& 1\end{array}\right|=(a-1{)}^3$$

OR

Find matrix $A$ such that

$$\left[\begin{array}{cc}2& -1\\ 1& 0\\ -3& 4\end{array}\right]A=\left[\begin{array}{cc}-1& -8\\ 1& -2\\ 9& 22\end{array}\right]$$

Determine the value of '$k$' for which the following function is continuous at $x=3$ :

$$f(x)=\{\begin{array}{ll}\frac{(x+3{)}^2-36}{x-3},& x\ne 3\\ k,& x=3\end{array}$$

Find the value of $c$ in Rolle's theorem for the function $f(x)=x^3-3x$ in $[-\sqrt{3},0]$.