BT-102 · Mathematics I/Unsolved PYQ Paper
BT-102 Mathematics-I - Jun 2022 Question Paper
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7 MarksaVerify Rolle's theorem for the function $f(x) = \frac{\sin x}{e^x}$.
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7 MarksbFind the minimum value of $x^2 y z^3$, subject to the condition $2x + y + 3z = a$.
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7 MarksaVerify Cauchy's Mean value theorem for the function $\sin x$ and $\cos x$ in $\left[0, \frac{\pi}{2}\right]$.
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7 MarksbState and prove relationship between Beta and Gamma function.
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7 MarksaChange the order of integration and evaluate $\int_0^{4a} \int_{\frac{x^2}{4a}}^{2\sqrt{ax}} dy\,dx$.
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7 MarksbUsing Double integration, find the volume of the tetrahedron bounded by the coordinate planes and the plane $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$.
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14 MarksLet $W_1$ and $W_2$ be two subspace of a finite dimensional vector space $V(F)$. Then $\dim(W_1 + W_2) = \dim W_1 + \dim W_2 - \dim(W_1 \cap W_2)$.
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14 MarksObtain the Fourier series to represent $$f(x) = \frac{1}{4}(\pi - x^2) \quad \text{in } 0 < x < 2\pi.$$
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7 MarksaShow that $T : V_3(\mathbb{R}) \to V_2(\mathbb{R})$ is defined as $T(a_1, a_2, a_3) = (a_1 - a_2, a_1 - a_3)$ is linear transformation.
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7 MarksbTest the convergence of the series $$\sum_{n=1}^\infty \left(\sqrt{n^4+1} - \sqrt{n^4-1}\right).$$
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7 MarksaVerify Cayley-Hamilton theorem for the matrix $$A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{bmatrix}$$
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7 MarksbExamine the consistency of the system and if consistent, solve the equations $$4x - 2y + 6z = 8, \quad x - y - 3z = -1, \quad 15x - 3y + 9z = 21.$$
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14 MarksDiagonalize the matrix $A = \begin{bmatrix} 8 & -8 & -2 \\ 4 & -3 & -2 \\ 3 & -4 & 1 \end{bmatrix}$.
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