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\[\int \left( e^{x \log a} + e^{a \log x} \right) dx \text{ is equal to (where } a > 1)\]
\[1.\ \frac{a^x}{\log a} + \frac{x^{a+1}}{a+1} + C : C \text{ is an arbitrary constant}\]\[2.\ (\log a)\, a^x + \frac{x^{a+1}}{a+1} + C : C \text{ is an arbitrary constant}\]\[3.\ \frac{a^x}{a+1} + \frac{x^{a+1}}{\log a} + C : C \text{ is an arbitrary constant}\]\[4.\ (a+1)a^x + (\log a)x^{a+1} + C : C \text{ is an arbitrary constant}\]
\[\text{If } A=\begin{bmatrix}-1 & 2 & 3x \\2y & 4 & -1 \\6 & -1 & 0\end{bmatrix}\text{ is a symmetric matrix, then the value of } 2x-y \text{ is:}\]
\[\text{1. 0}\]
\[\text{2. 1}\]
\[\text{3. 3}\]
\[\text{4. -2}\]
\[\text{If } xy = e^{(x-y)}, \text{ then } \frac{dy}{dx} \text{ is equal to:}\]
1.\[\dfrac{e^{x-y} + y}{x + e^{x-y}}\]
2.\[\dfrac{e^{x-y} - y}{x + e^{x-y}}\]
3.\[\dfrac{e^{x-y} - y}{x - e^{x-y}}\]
4.\[\dfrac{e^{x-y} + y}{x - e^{x-y}}\]
\[\text{The feasible region of a LPP is bounded. The corresponding objective function is } Z = 6x - 7y.\]\[\text{Then the objective function attains:}\]
\[1.\; \text{Only maximum in the feasible region} \]\[2.\; \text{Only minimum in the feasible region} \]\[3.\; \text{both maximum and minimum in the feasible region} \]\[4.\; \text{either maximum or minimum but not both in the feasible region}\]
\[\text{If } A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}, \text{ then } A^2 - 5A \text{ is equal to (where I is identity matrix of order 2)}\]
\[1.\; 14I \]\[2.\; 7I \]\[3.\; -7I \]\[4.\; -5I\]
\[\text{If } A = \begin{bmatrix} 1 & 2 \\ 4 & -3 \end{bmatrix}\text{ and } f(x) = 2x^2 - 4x + 5,\ \text{then } f(A) \text{ is equal to}\]
\[1.\ \begin{bmatrix} 19 & -16 \\ -32 & 51 \end{bmatrix}\]
\[2.\ \begin{bmatrix} 19 & -32 \\ -16 & 51 \end{bmatrix}\]
\[3.\ \begin{bmatrix} 19 & -11 \\ -27 & 51 \end{bmatrix}\]
\[4.\ \begin{bmatrix} -19 & 16 \\ 32 & -51 \end{bmatrix}\]
\[\text{The function } f(x) = x^2 - x + 1 \text{ is}\]
\[1.\ \text{Increasing on } \left(-\frac{1}{2}, 1\right) \text{ and decreasing on } \left(0, \frac{1}{2}\right)\]\[2.\ \text{Increasing on } \left(\frac{1}{2}, \infty\right) \text{ and decreasing on } \left(-\infty, \frac{1}{2}\right)\]\[3.\ \text{Increasing on } \left(-\infty, \frac{1}{2}\right] \text{ and decreasing on } \left[\frac{1}{2}, \infty\right)\]\[4.\ \text{Increasing on } (-\infty, 1) \text{ and decreasing on } (1, \infty)\]
\[\text{The area (in sq. units) of the bigger portion of region enclosed by the curves }4x^{2} + 9y^{2} = 36 \text{ and } 2x + 3y = 6 \text{ is}\]
\[1.\dfrac{3}{2}(\pi - 2) \]
\[2.\dfrac{3}{2}(3\pi + 2) \]
\[3. \ (6\pi)\]
\[4.\ (3\pi + 3)\]
\[\int \frac{dx}{\left(1 + 5 \sin^2 x \right)} \text{ is equal to}\]
\[1.\ \sqrt{6}\cot^{-1}(\sqrt{6}\cot x) + C : C \text{ is an arbitrary constant}\]\[2.\ \sqrt{6}\cot^{-1}(\sqrt{6}\tan x) + C : C \text{ is an arbitrary constant}\]\[3.\ \frac{1}{\sqrt{6}}\tan^{-1}(\sqrt{6}\cot x) + C : C \text{ is an arbitrary constant}\]\[4.\ \frac{1}{\sqrt{6}}\tan^{-1}(\sqrt{6}\tan x) + C : C \text{ is an arbitrary constant}\]
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