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87d55516-470a-4c57-9713-3f320ed53247 | Let $L_1 : \frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4}$ and $L_2 : \frac{x+1}{2} = \frac{y-2}{3} = \frac{z}{1}$ be two lines. Let $L_3$ be a line passing through the point $(\alpha, \beta, \gamma)$ and be perpendicular to both $L_1$ and $L_2$. If $L_3$ intersects $L_1$, then $|5\alpha - 11\beta - 8\gamma|$ equals: | 3 | null | 20 | 18 | 25 | 16 | MCQ |
f131eebf-78ee-4273-b473-b776fb3280b4 | Let $M$ and $m$ respectively be the maximum and the minimum values of
\[
f(x) = \begin{bmatrix} 1 + \sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1 + \cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1 + 4\sin 4x \end{bmatrix}, x \in \mathbb{R}
\]
Then $M^4 - m^4$ is equal to:
| 1 | null | 1280 | 1295 | 1215 | 1040 | MCQ |
cbc72177-b566-4494-87df-ab52acb84bfc | Let \( ABC \) be a triangle formed by the lines \( 7x - 6y + 3 = 0, x + 2y - 31 = 0 \) and \( 9x - 2y - 19 = 0 \). Let the point \((h, k)\) be the image of the centroid of \( \Delta ABC \) in the line \( 3x + 6y - 53 = 0 \). Then \( h^2 + k^2 + hk \) is equal to: | 2 | null | 47 | 36 | 47 | 40 | MCQ |
68edd102-6049-4277-a9d4-e831373e20f9 | The value of \( \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k^4 + 4k^2 + 11k + 5}{(k+3)!} \right) \) is: | 4 | null | 4/3 | 2 | 7/3 | 5/3 | MCQ |
284e0696-65cb-4558-a145-131420581840 | The least value of \( n \) for which the number of integral terms in the Binomial expansion of \( (\sqrt{7} + \sqrt{11})^n \) is 183, is: | 1 | null | 2184 | 2196 | 2148 | 2172 | MCQ |
7547e409-5d97-4179-9041-5f85e5a66c5e | Let \( y = y(x) \) be the solution of the differential equation
\[
\cos x \left( \log_e (\cos x) \right)^2 dy + (\sin x - 3y \sin x \log_e (\cos x)) dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right).\]
If \( y \left( \frac{\pi}{6} \right) = \frac{-1}{\log_2 2} \), then \( y \left( \frac{\pi}{8} \right) \) is equal to:
| 1 | null | \frac{1}{\log_2 (5) - \log_2 (4)} | \frac{2}{\log_2 (3) - \log_2 (4)} | \frac{1}{\log_2 (4) - \log_2 (3)} | \frac{1}{\log_2 (4)} | MCQ |
d87ae9e7-3636-4854-a874-65795666e575 | Let the line \( x + y = 1 \) meet the circle \( x^2 + y^2 = 4 \) at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to: | 3 | null | \sqrt{14} | 3\sqrt{7} | 2\sqrt{14} | 5\sqrt{7} | MCQ |
dd7bf750-08ca-4eee-8b1b-3aa7880e708f | Let the area of the region $ \{(x, y) : 2y \leq x^2 + 3, \quad y \geq |x|, \quad |y| \leq |x - 1|\} $ be A. Then $ 6A $ is equal to: | 3 | null | $16$ | $12$ | $14$ | $18$ | MCQ |
92a4b8da-f6bb-4de1-b0cb-e42c14b3dcd6 | Let \( S = \{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \} \), where \( A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \). Then \( n(S) \) is equal to ______. | 2 | null | 6 | 2 | 0 | 1 | MCQ |
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