Maths Previous Year Question Papers 2019-20-21-22-24 PDF

## GUJARAT TECHNOLOGICAL UNIVERSITY ### BE - SEMESTER-I & II (NEW) EXAMINATION - **SUMMER 2024** **Subject Code:** 3110014 **Subject Name:** Mathematics - 1 **Date:** 09-07-2024 **Time:** 02:30 PM TO 05:30 PM **Total Marks:** 70 **Instructions:** 1. Attempt all questions. 2. Make suitable assump...

## GUJARAT TECHNOLOGICAL UNIVERSITY ### BE - SEMESTER-I & II (NEW) EXAMINATION - **SUMMER 2024** **Subject Code:** 3110014 **Subject Name:** Mathematics - 1 **Date:** 09-07-2024 **Time:** 02:30 PM TO 05:30 PM **Total Marks:** 70 **Instructions:** 1. Attempt all questions. 2. Make suitable assumptions wherever necessary. 3. Figures to the right indicate full marks. 4. Simple and non-programmable scientific calculators are allowed. **Q.1** **(a)** Using L' Hospital's rule, evaluate $lim_{x\to 1} \frac{x - x \ln x}{1 + \log x -x}$. **(b)** Define Beta function and evaluate $\int_0^1 x(1-x) dx$. **(c)** Find the Fourier series of $f(x) = \begin{cases} \pi + x & -\pi < x <0 \\ \pi - x & 0 < x < \pi \end{cases}$. **Q.2** **(a)** Show that the sequence {$u_n$}, where $u_n = \frac{\sin n}{n}$ converges to zero. **(b)** Express $f(x) =2x^3 + 3x^2 -8x + 7$ in terms of (x-2). **(c)** Find the area of the surface of revolution of a quadrant of a circular arc as obtained by revolving it about a tangent at one of its ends. **(c)** Find the length of the loop of the curve $9ay^2 = (x-2a)(x-5a)^2$. **Q.3** **(a)** Evaluate $\int_0^1 \frac{1}{(1+v)(1 + tan^{-1} v)} dv$. **(b)** Test the convergence of the series $\sum_{n=1}^{\infty} \dfrac{2n + 1}{n^2 n^2 (n+1)^2}$. **(c)** If $\theta = u^2 + v^2$ then find n so that $\frac{1}{r} \frac{\partial \theta}{\partial r} = 2 \frac{\partial \theta}{\partial t}$. **Q.3** **(a)** Check the convergence of $\int_1^{\infty} \frac{1}{x^3} dx$. **(b)** Evaluate $\int_0^5 (3 - x^2) dx$. **(c)** Find the Fourier series of $f(x) = \frac{1}{2} (x - \pi)$ in the interval (0,2π). Hence, deduce that $\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + ….$. **Q.4** **(a)** Prove that $tan^{-1} x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + ... $. **(b)** Find the Fourier sine series of f(x) = $e^x$ in 0 < x < π. **(c)** Find the extreme values of the function f(x, y) = x² + y³ -3x - 12y + 20. **Q.4** **(a)** Find the directional derivative of $f(x, y, z) = x^2yz + 4xz^2$ at (1, -2, -1) in the direction of 2i - j - 2k. **(b)** Solve the following system by Gauss - Jordan method: -2y + 3z = 1 3x + 6y - 3z = 2 6x + 6y + 3z = 5 **(c)** Change the order of integration and evaluate $\int_0^1 \int_0^{\sqrt{1 -x²}} \frac{cos x}{√(1-x²) √(1-x²-y²)} dx dy$. **Q.5** **(a)** Evaluate $\int_0^1 \int _0^2 (x²+ 3y²) dx dy$. **(b)** Apply Cayley – Hamilton theorem to A = $\begin{bmatrix} 1 & 2 \\ 2 & -1 \end{bmatrix}$ and deduce that A⁴ = 625I. **(c)** Evaluate $\int_0^2 \int_0^{\sqrt{2x-x²}} \int_0^{\sqrt{2√2x-x²-y²}} xyz dx dy dz$ by changing to polar coordinates. **Q.5** **(a)** Evaluate $\int_0^1 \int_0^1 \int_0^1 xyz dx dy dz$. **(b)** Using Gauss Jordan method, find inverse of $A = \begin{bmatrix} 2 & 3 & 4 \\ 4 & 3 & 1 \\ 1 & 2 & 4 \end{bmatrix}$. **(c)** Find the eigen values and eigen vectors of the matrix $A = \begin{bmatrix} 4 & 6 & 6 \\ 1 & 3 & 6 \\ -1 & -4 & -3 \end{bmatrix}$.