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

Group A

Attempt any THREE Questions

1. (a) If $f(x) = \sqrt{x}$ and $g(x) = \sqrt{3-x}$ then find $fog$ and its domain and range.

   (b) A rectangular storage container with an open top has a volume of $20m^3$. The length of its base is twice its width. Material for the base costs Rs 10 per square meter material for the sides costs Rs 4 per square meter. Express the cost of materials as a function of the width of the base.

2. (a) Using rectangular, estimate the area under the parabola $y = x^2$ from 0 to 1.

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

   $v = t^2 - t + 6$

   a. Find the displacement of the particle during the time period $1 \leq t \leq 4$.

   b. Find the distance travelled during this time period.

3. (a) Find the area of the region bounded by $y = x^2$ and $y = 2x - x^2$

   (b) Using trapezoidal rule, approximate $\int_1^2 \frac{1}{x} dx$ with $n = 5$

4. (a) Solve $y’ = x^2/y^2$, $y(0) = 2$

   (b) Solve the initial value problem: $y” + y’ - 6y = 0$, $y(0) = 1$, $y’(0) = 0$

Group B

Attempt any TEN Questions

5. Recent studies indicates that the average surface temperature of the earth has been rising rapidly. Some scientists have modeled the temperature by the linear function $T = 0.03t + 8.50$, where $T$ is temperature in degree centigrade and $t$ represents years since 1900.

   a. What do the slope and $T$-intercept represent?

   b. Use the equation to predict the average global surface temperature in 2100

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

7. State Rolle’s theorem and verify the Rolle’s theorem for $f(x) = x^2 - 3x + 2$ in $[0, 3]$

8. Use Newton’s method to find $^6\sqrt{2}$ correct five decimal places.

9. Find the derivatives of $r(t) = (1 + t^2)i - te^{-t} j + \sin 2tk$ and find the unit tangent vector at $t=0$.

10. Find the volume of the solid obtained by rotating about the $y$-axis the region between $y = x$ and $y = x^2$.

11. Solve: $y’ + 2xy - 1 = 0$

12. What is sequence? Is the sequence $a_n = \frac{n}{\sqrt{5+n}}$ convergent?

13. Find a vector perpendicular to the plane that passes through the points $p(1, 4, 6)$, $Q(2, 5, -1)$ and $R(-1, 1)$

14. Find the partial derivative of $f(x, y) = x^2 + 2x^3y^2 - 3y^2 + x + y$ at $(1, 2)$

15. Find the local maximum and minimum values, saddle points of $f(x,y) = x^4 + y^4 - 4xy + 1$