Section A: Calculus

Q1. Integrate $\int x e^x dx$.

Ans: Using integration by parts: $\int u dv = uv – \int v du$

Let $u=x, dv=e^x dx \implies du=dx, v=e^x$.

$$ = x e^x – \int e^x dx $$

$$ = x e^x – e^x + C $$

$$ = e^x(x-1) + C $$

Q2. Evaluate $\lim_{x \to 0} \frac{\sin 5x}{x}$.

Ans: Multiply and divide by 5.

$$ 5 \times \lim_{x \to 0} \frac{\sin 5x}{5x} = 5(1) = 5 $$

Section B: Vectors

Q3. Find projection of $\vec{a} = \hat{i} + \hat{j}$ on $\vec{b} = \hat{i} – \hat{j}$.

Ans: Projection $p = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$.

$\vec{a} \cdot \vec{b} = (1)(1) + (1)(-1) = 1 – 1 = 0$.

So, projection is 0.

Q4. Find angle between vectors.

Ans: $\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}$.