Mathematics old question papers

Section A Attempt any TWO questions.

(a) Define composition of two functions. Let $f(x)=\sqrt{x}$ and $g(x)=\sqrt{3-x}$. Find $(f\circ f)(x)$, $(g\circ g)(x)$, $(g\circ f)(x)$ and $(f\circ g)(x)$. (b) Evaluate: ${\lim }_{x\to \infty }\sqrt{x}(\sqrt{x}-\sqrt{x-5a})$.

(a) Find the local maximum and local minimum value of the function $f(x)=4x^3-6x^2-9x+17$. (b) Examine the continuity of the function $f(x)=\sqrt{9-x^2}$ on the interval $[-3,3]$.

(a) Using method of integration, find the area enclosed by the circle $x^2+y^2=16$. (b) Evaluate: ${\int }_0^1{x}^3\sqrt{1+2x^4}dx$.

Section B Attempt any EIGHT questions.

Sketch the function $y=x-x^2$ indicating its different characteristics. Also, find the range of function.

Verify Mean value theorem for the function $f(x)=(x-1)(x-2)(x-3)$ in $[1,4]$.

Using L'Hospital's rule, evaluate: ${\lim }_{x\to 1}\left(\frac{x}{x-1}-\frac{1}{\ln x}\right)$.

Find $\frac{d^{2}y}{d{x}^2}$ as a function of $t$ when $x=t-\frac{1}{t}$, $y=t+\frac{1}{t}$.

Find the area of surface swept out by revolving the circle $x^2+y^2=1$ about x-axis.

Examine the convergence of $\sum _{n=1}^{\infty }\frac{2n^2+3n}{\sqrt{5+n^5}}$.

Solve: $x{y}^{\prime }+2y=3$, $y(1)=1$.

Find the Maclaurin series of the function $f(x)=e^x$ and its radius of convergence.

Find all second order partial derivatives when $f(x,y)=x^4-5x^3{y}^2+7x^2{y}^3+4x{y}^5$.