Full Marks: 80
Pass Marks: 32
Time: 3 Hrs.

Attempt any THREE questions

1. (a) If a function is defined by

   $f(x) = \begin{cases} 1+x & \text{if } x \leq -1 \ x^2 & \text{if } x > -1 \end{cases}$

   evaluate $f(-3)$, $f(-1)$ and $f(0)$ and sketch the graph.

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

2. (a) Sketch the curve $y = x^2 + 1$ with the guidelines of sketching.

   (b) If $z = xy^2 + y^3$, $x = \sin t$, $y = \cos t$, find $dz/dt$ at $t = 0$

3. (a) Estimate the area between the curve $y = x^2$ and the lines $x = 0$ and $x = 1$, using rectangle method, with four sub intervals.

   (b) A particle moves a line so that its velocity $v$ at time $t$ is

   (1) Find the displacement of the particle during the time period $1 \leq t \leq 4$ (2) Find the distance travelled during this time period.

4. (a) Define initial value problem. Solve:

   $y” + y’ - 6y = 0$, $y(0) = 1$, $y’(0) = 0$

   (b) Find the Taylor’s series expansion for $\cos x$ at $x = 0$.

Group B

Attempt any TEN questions

5. Dry air is moving upward. If the ground temperature is $20°$ and the temperature at a height of $2km$ is $10°c$, express the temperature $T$ in $°c$ as a function of the height $h$(in km), assuming that a linear model is appropriate. (b) Draw the graph of the function and find the slope. Hence, give the meaning of slope. (c) What is the temperature at a height of $2km$?

6. Find the equation of the tangent at $(1,3)$ to the curve $y = 2x^2 + 1$.

7. State Rolle’s theorem and verify the theorem for $f(x) = x^2 - 9$, $x \in [-3,3]$

8. Starting with $x_1 = 1$, find the third approximate $x_3$ to the root of the equation $x^3 - x - 5 = 0$

9. Show the integral coverages

   $\int_2^{\infty} dx/x - 1$

10. Use Trapezoidal rule to approximate the integral $\int_1^4 dx/x$ with $n = 5$.

11. Find the derivative of $r(t) = (t^2)i - te^{2t}k + \sin(2t)k$ and find the unit tangent vector at $t = 0$

12. What is sequence? Is the sequence

    $a_n = \frac{n}{\sqrt{5+n}}$

    convergent?

13. Find the angle between the vectors $\vec{a} = (2, 2, -1)$ and $\vec{b} = (1, 3, 2)$

14. Find the partial derivative $f_{xx}$ and $f_{yy}$ of $f(x,y) = x^2 + x^2y^2 - y^2 + xy$ at $(1,2)$.

15. Evaluate

    $\int_0^2 \int_0^{x^2} x^2y , dxdy$