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.

Section A

Attempt any two questions.

1. (a) If $\vec{a} = (4,0,3)$ and $\vec{b} = (-2,1,5)$, find $|\vec{a}|$, $3\vec{a} + \vec{b}$ and $2\vec{a} + 5\vec{b}$.

   (b) Estimate the value of $\lim_{x \to 0} \frac{(x)(x^2 + 9) - 3}{x^2}$

2. (a) The area of the parabola $y = x^2$ from $(1,1)$ to $(2,4)$ is rotated about the $y$-axis. Find the area of the resulting surface.

   (b) Find the solution of the equation $y^2dy = x^2dx$ that satisfies the initial condition $y(0) = 2$.

   (c) As dry air moves upward, it expands and cools. If the ground temperature is $20°C$ and the temperature at height of 1 km is $10C$, express the temperature $T$(in $°C$) as a function of height $h$(in kilometer), assuming that the linear model is appropriate.

   (b) Draw a graph of the function in part(a). What does the slope represent?

   (c) What is the temperature at a height of $2.5$ km?

Section B

Attempt any eight questions.

4. Integrate $\int_1^2 x^4(x^3 + 1) dx$.

5. Find the Maclaurin series expansion of $f(x) = e^x$ at $x = 0$.

6. Find where the function $f(x) = 3x^4 - 4x^3 - 12x^2 + 5$ is increasing and where it is decreasing.

7. Find $y$ if $x^2 + y^2 = 6xy$.

8. Show that $y = x - 1/x$ is a solution of the differential equation $xy’ + y = 2x$.

9. Sketch the graph and find the domain and range of the function $f(x) = 2x - 1$.

10. Determine whether the series $\sum_{n=1}^{\infty} \frac{n^2}{(5n^2 + 4)}$ converges or diverges.

11. If $f(x,y) = x^2 + x^2y^2 - 2y^2$, find $f_x(2,1)$ and $f_y(2,1)$.

12. Show that the function $f(x) = x^2 + \sqrt{7-x}$ is continuous at $x = 4$.