Mathematics II TU Board 2023 question paper

Group A

Group B

Attempt any SIX question.(6 × 5 = 30)

1. 2.

   Evaluate the limit $$\underset{x\to \theta }{\lim }\frac{x\cos \theta -\theta \cos x}{x-\theta }.$$

   Answer comingAlso asked in 2026, 2024, 2022

   5
2. 3.

   Find the derivative of $y=e^{3x-1}$ by the definition method.

   Answer coming

   5
3. 4.

   Verify Rolle's theorem for the function $f(x)=\sin x,x\in [0,\pi ]$ and find the point on the curve where the tangent is parallel to the $x$-axis.

   Answer comingAlso asked in 2026

   5
4. 5.

   Find maximum and minimum values of the function $$f(x)=x^3+6x^2+9x-2.$$ Also find the point of inflection, if any.

   Answer comingAlso asked in 2026

   5
5. 6.

   Evaluate the integral a. $\int \frac{2x+5}{\sqrt{x^2+5x}}dx$ b. $\int e^{2x}\log xdx$

   Answer comingAlso asked in 2022

   5
6. 7.

   Evaluate $${\int }_0^2\sqrt{1+x^3}dx$$ by using Simpson's $\frac{1}{3}$rule, taking $n=4$.

   Answer comingAlso asked in 2026

   5
7. 8.

   Solve the differential equation: $$2\frac{dy}{dx}=\frac{2y}{x}+\frac{y^2}{x^2}.$$

   Answer comingAlso asked in 2026

   5

Group C

Attempt any TWO questions(2 × 10 = 20)

1. 9.

   Define pivot element, pivot column, and consistency in a system of equations. Using the simplex method, maximize $$F=5x-3y$$ subject to $$\begin{gathered} 3x+2y\le =6, \\ -x+3y\ge =-4, \\ x\ge 0,y\ge =0. \end{gathered}$$

   Answer coming

   10
2. 10.

   a. Verify Lagrange's mean value theorem for the function $$f(x)=\sqrt{x-1},x\in [1,3].$$ b. Solve the differential equation $$xy\frac{dy}{dx}=x^2+y^2.$$

   Answer coming

   10
3. 11.

   Compute the approximate value of the integrate $$\int \frac{1}{1+x^2}dx$$ 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 2024

   10