Mathematics II TU Board 2022 question paper

Group A

Group B

Attempt any SIX question.(6 × 5 = 30)

1. 2.

   "A function $f(x)$ is defined as $$f(x)=\{\begin{array}{ll}2x+3& \text{for }-\frac{3}{2}\le x<0\\ 3-2x& \text{for }0\le x\le \frac{3}{2}\\ -3-2x& \text{for }x>\frac{3}{2}\end{array}$$ show that $f(x)$ is continuous at $x=0$ and discontinuous at $x=\frac{3}{2}$."

   Answer comingAlso asked in 2026, 2024

   5
2. 3.

   Find $\frac{dy}{dx}$ when $x=a(t+\sin t)$, $y=a(1-\cos t)$.

   Answer comingAlso asked in 2026

   5
3. 4.

   State L'Hospital's Rule. Use it to evaluate: $$\underset{x\to 0}{\lim }\frac{x{e}^x-In(1+x)}{x^2}$$

   Answer coming

   5
4. 5.

   State Rolle's Theorem. Verify Rolle's Theorem for the function $f(x)=x^2-9$ in the interval $-3\le x\le 3$.

   Answer comingAlso asked in 2024

   5
5. 6.

   Evaluate: a) ${\int }_0^2\frac{dx}{x^2-25}$ b) $\int \frac{2x+5}{\sqrt{x^2}+5x}dx$

   Answer comingAlso asked in 2023

   5
6. 7.

   Solve: $\sin x\frac{dy}{dx}+(\cos x)y=-\sin x\cos x$.

   Answer comingAlso asked in 2026, 2024, 2023

   5
7. 8.

   Solve the following system by Gauss-Seidel method: $$\begin{array}{rl}-4x+y-z& =-8\\ 3x+6y+2z& =1\\ x-y+3z& =2\end{array}$$

   Answer coming

   5

Group C

Attempt any TWO questions(2 × 10 = 20)

1. 9.

   Using simplex method, find optimal solution of $Z=7x1+5x2$ Subject to $x1+2x2\le 6$ $4x1+3x2\le 6$ $x1\ge 0,x2\ge 0$.

   Answer coming

   10
2. 10.

   What do you mean by stationary points and inflection points? Using derivatives, find two numbers whose sum is 20 and sum of whose squares is minimum.

   Answer coming

   10
3. 11.

   Using Simpson's $\frac{1}{3}$ rule evaluate ${\int }_0^1\frac{1}{1+x}dx$ with 3 points of intervals. Find error of approximation. How many points are to be considered to make the approximation value within 10-5?

   Answer coming

   10