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Consider the following reaction sequence.

Chemistry
Hydrocarbons

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For the matrices $ A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} $ and $ B = \begin{bmatrix} -29 & 49 \\ -13 & 18 \end{bmatrix} $, if $ (A^{15} + B) \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0\\ 0 \end{bmatrix} $, then among the following which one is true?}

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What is the equivalent gate for the following digital circuit?

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Foot of perpendicular from origin on a line passing through $(1, 1, 1)$ having direction ratios $\langle 2, 3, 4 \rangle$, is:

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Monascus purpureus is a yeast used commercially in the production of which one of the following?

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A line through $(1, 1, 1)$ and perpendicular to both $\hat{i} + 2\hat{j} + 2\hat{k}$ and $2\hat{i} + 2\hat{j} + \hat{k}$, let $(a, b, c)$ be foot of perpendicular from origin then $34 (a + b + c)$ is:

Mathematics
Geometry and Vectors

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If angular position of a particle is given by $\theta = \frac{t^4}{4} + t^2$. Find angular acceleration at $t = 1s$:

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Sum of $ \frac{1^3}{1} + \frac{1^3 + 2^3}{1 + 3} + \frac{1^3 + 2^3 + 3^3}{1 + 3 + 5} + \dots $ up to 8 terms is:

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If $50 \left( \frac{2x}{1 + 3i} + \frac{y}{1 - 2i} \right) = 31 + 17i$ where $x, y \in R$ & $i = \sqrt{-1}$ then value of $10(x + 3y)$ is ________

Mathematics
Complex Numbers and Quadratic Equations

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Consider two AP's $S_1$ & $S_2$ such that $S_1 : \{ \text{First term} = 1, \text{Common difference} = 5, \text{no. of terms} = 101 \}$
$S_2 : \{ \text{First term} = 9, \text{Common difference} = 7, \text{no. of terms} = 71 \}$
Find the number of common terms in both AP's.

Mathematics
Sequences and Series

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If domain of $f(x) = \sqrt{\log_{0.6} \frac{2x - 5}{x^2 - 4}}$ is $(-\infty, a] \cup \{b\} \cup [c, d) \cup (e, \infty)$ then the value of $(a + b + c + d + e)$ is :

Mathematics
Calculus

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let mid "“ point of sides of $\Delta$ are $(\frac{5}{2}, 3), (\frac{5}{2}, 7) \, \& \, (4, 5)$. If incentre is $(h, k)$ then value of $3h + k$ is:

Mathematics
Triangles

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If $^{36}C_{r+1} = 6 \cdot \frac{^{35}C_r}{k^2 - 3}$, then number of ordered pairs $(r, k)$, where $r, k \in Z$ is ______.

Mathematics
Complex Numbers and Quadratic Equations

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If $\lim_{x \to 2} \frac{\sin(x^3 - 5x^2 + ax + b)}{(\sqrt{x-1}-1) \log_e(x-1)} = m$ (exists finitely), then find the value of $a + b + m$.

Mathematics
limits and derivatives

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If $\sum_{k=1}^{n} a_k = 6n^3$ then evaluate $\sum_{k=1}^{6} \left( \frac{a_{k+1} - a_k}{36} \right)^2$.

Mathematics
limits and derivatives

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Value of definite integral $I = \int_{0}^{2\sqrt{3}} \log_2(x^2 + 4) dx + \int_{2}^{4} \sqrt{2^x - 4} dx$

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Let $K = \sin \frac{\pi}{18} \sin \frac{5\pi}{18} \sin \frac{7\pi}{18}$ then the value of $\sin \left( 10K \frac{\pi}{3} \right)$ is :

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$\alpha, \alpha + 2 \in Z$ and $n \in N$, where $\alpha, \alpha+2$ are roots of equation $x(x + 2) + (x + 1) (x + 3) + ... + (x + n-1) (x + n+1) = 4n$ then $\alpha + n =$

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