Full Marks: 80 + 20
Pass Marks: 32 + 8
Time: 3 Hours

Candidates are required to give their answers in their own words as far as practicable. The figures in the margin indicate full marks.

Group A

Attempt any three questions

1. (a) A function is defined by $f(x) = {x+2 \text{ if } x<0, 1-x \text{ if } x>0}$. Conclude $f(-1)$, $f(3)$ and sketch graph.

   (b) Prove that $\lim_{x \to 0} \frac{|x|}{x}$ does not exist.

2. Estimate the area between the curve $y^2 = x$ and the lines $x = 0$ and $x = 2$.

3. (a) Find the Maclaurin series for $e^x$ and prove that it represents $e^x$ for all $x$.

   (b) Define Initial Value Problem. Solve that value problem of $y” + 5y = 1$, $y(0) = 2$.

   (c) Find the volume of a sphere of radius $r$.

4. (a) For What values of $x$ does the series $\sum_{n=1}^{\infty} \frac{(x-3)^n}{2^n}$ converge?

   (b) Calculate $\int_R \int f(x, y)dA$ for $f(x,y) = 100 - 6x^2y$ and $R: 0 \leq x \leq 2, -1 \leq y \leq 1$.

Group B

Attempt any ten questions

5. If $f(x) = \sqrt{x}$ and $g(x) = \sqrt{3-x}$, find $gof$ and $gog$.

6. Use Continuity to evaluate the limit, $\lim_{x \to 4} \left(\frac{5+\sqrt{x}}{\sqrt{5+x}}\right)$.

7. Verify Mean Value Theorem by $f(x) = x^3 - 3x + 3$ for $[-1, 2]$.

8. Sketch the curve $y = x^3 + x$.

9. Determine whether the integral $\int_1^{\infty} \left(\frac{1}{x}\right) dx$ is convergent or divergent.

10. Find the length of the arc of the semi cubical parabola $y^2 = x^3$ between the points $(1,1)$ and $(4,8)$.

11. Find the solution of $y”+6y’+9=0$, $y(0)=2$, $y’(0)=1$.

12. Test the convergence of the series $\sum_{n=1}^{\infty} \left(\frac{n^n}{n!}\right)$.

13. Define cross product of two vectors. If $\vec{a} = \vec{i} + 3\vec{j} + 4\vec{k}$ and $\vec{b} = 2\vec{i} + 7\vec{j} - 5\vec{k}$ find the vector $\vec{b} \times \vec{a}$ and $\vec{a} \times \vec{b}$.

14. Define limit of a function. Find limit $\lim_{x \to \infty}(x-\sqrt{x})$.

15. Find the extremes values of $f(x, y) = y^2 - x^2$.