Mathematics II TU Board 2024 question paper

Group A

Group B

Attempt any SIX question.(6 × 5 = 30)

1. 2.

   A function $f(x)$ is defined as$$f(x)=\{\begin{array}{ll}2x+3& \text{for }x<1\\ 4& \text{for }x=1\\ 6x-1& \text{for }x>1\end{array}$$ Is the function $f(x)$ continuous at $x=1$? If not, how can you make it continuous at $x=1$?

   Answer comingAlso asked in 2026, 2022

   5
2. 3.

   Find the derivative of $e^{-2x}$ using the first principle. $$-\sum e^{-\frac{x^2}{2}}.$$

   Answer comingAlso asked in 2026

   5
3. 4.

   Evaluate $$\underset{x\to \frac{\pi }{4}}{\lim }\frac{\sin x-\cos x}{x-\frac{\pi }{4}}$$

   Answer comingAlso asked in 2026, 2023, 2022

   5
4. 5.

   Find area of the region enclosed by the parabola $y=2x^2$ and line $y=x$. Evaluate a) $${\int }_0^2\frac{dx}{\sqrt{x^2+4^2}}$$ b) $$\int x^2{e}^{x}dx$$

   Answer coming

   5
5. 6.

   State Mean Value Theorem. Give its geometrical meaning. Verify the Mean Value Theorem for the function $f(x)=x^3+x^2-6x$ in the interval $[-1,4]$.

   Answer comingAlso asked in 2022

   5
6. 7.

   Examine the consistency of the system. Solve it by using Gauss elimination method. $$\{\begin{array}{l}3x+y+z=5\\ 3x-4y+z=-2\\ 3x+y-3z=-1\end{array}$$

   Answer coming

   5
7. 8.

   Using simplex method, find the optimal solution of the following linear programming problem. Minimize $Z=10x+15y$. Subject to $$\{\begin{array}{l}x+y\ge 8\\ 5x+3y\ge 30\\ x\ge 0,y\ge 0\end{array}$$

   Answer coming

   5

Group C

Attempt any TWO questions(2 × 10 = 20)

1. 9.

   Solve the following: a) $\frac{dy}{dx}-3y=x^2$ b) $xdy-ydx=x^{3}ydy$

   Answer coming

   10
2. 10.

   a) Use Simpson's $\frac{1}{3}$ Rule to evaluate $${\int }_0^1\frac{1}{1+x^2}dx$$ taking $n=4$. Also find the error. b) A man who has 130 m of fencing material wishes to enclose a rectangular garden. Find the maximum area he can enclose.

   Answer coming

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3. 11.

   Compute the approximate value of the integral $$\int \frac{1}{1+x^2}dx$$ by using the composite trapezoidal rule with three points, and compare the result with the actual value. Determine the error formula and numerically verify an upper bound on it.

   Answer comingAlso asked in 2023

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