If $$(\sqrt{3} + i)^{100} = 2^{99}(p + iq)$$, then $$p$$ and $$q$$ are roots of the equation:
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If $$(\sqrt{3} + i)^{100} = 2^{99}(p + iq)$$, then $$p$$ and $$q$$ are roots of the equation:
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A 10 inches long pencil $$AB$$ with mid point $$C$$ and a small eraser $$P$$ are placed on the horizontal top of a table such that $$PC = \sqrt{5}$$ inches and $$\angle PCB = \tan^{-1}(2)$$. The acute angle through which the pencil must be rotated about $$C$$ so that the perpendicular distance between eraser and pencil becomes exactly 1 inch is:
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The value of $$$2\sin\left(\frac{\pi}{8}\right)\sin\left(\frac{2\pi}{8}\right)\sin\left(\frac{3\pi}{8}\right)\sin\left(\frac{5\pi}{8}\right)\sin\left(\frac{6\pi}{8}\right)\sin\left(\frac{7\pi}{8}\right)$$$ is:
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A circle $$C$$ touches the line $$x = 2y$$ at the point $$(2, 1)$$ and intersects the circle $$C_1 : x^2 + y^2 + 2y - 5 = 0$$ at two points $$P$$ and $$Q$$ such that $$PQ$$ is a diameter of $$C_1$$. Then the diameter of $$C$$ is:
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The point $$P\left(-2\sqrt{6}, \sqrt{3}\right)$$ lies on the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ having eccentricity $$\frac{\sqrt{5}}{2}$$. If the tangent and normal at $$P$$ to the hyperbola intersect its conjugate axis at the points $$Q$$ and $$R$$ respectively, then $$QR$$ is equal to:
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The locus of the mid points of the chords of the hyperbola $$x^2 - y^2 = 4$$, which touch the parabola $$y^2 = 8x$$, is:
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$$\lim_{x \to 2}\left(\sum_{n=1}^{9} \frac{x}{n(n+1)x^2 + 2(2n+1)x + 4}\right)$$ is equal to:
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Consider the two statements:
$$(S_1) : (p \rightarrow q) \vee (\sim q \rightarrow p)$$ is a tautology.
$$(S_2) : (p \wedge \sim q) \wedge (\sim p \vee q)$$ is a fallacy.
Then:
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Let $$A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0 \end{bmatrix}$$. Then $$A^{2025} - A^{2020}$$ is equal to:
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Two fair dice are thrown. The numbers on them are taken as $$\lambda$$ and $$\mu$$, and a system of linear equations
$$x + y + z = 5$$
$$x + 2y + 3z = \mu$$
$$x + 3y + \lambda z = 1$$
is constructed. If $$p$$ is the probability that the system has a unique solution and $$q$$ is the probability that the system has no solution, then:
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If $$\sum_{r=1}^{50} \tan^{-1} \frac{1}{2r^2} = p$$, then the value of $$\tan p$$ is:
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The domain of the function $$\operatorname{cosec}^{-1}\left(\frac{1+x}{x}\right)$$ is:
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Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Let $$f(x) = x - [x]$$, $$g(x) = 1 - x + [x]$$, and $$h(x) = \min\{f(x), g(x)\}$$, $$x \in [-2, 2]$$. Then $$h$$ is:
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The local maximum value of the function, $$f(x) = \left(\frac{2}{x}\right)^{x^2}$$, $$x \gt 0$$, is:
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The value of $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(\frac{1 + \sin^2 x}{1 + \pi^{\sin x}}\right) dx$$ is:
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If the value of the integral $$\int_0^5 \frac{x + [x]}{e^{x-[x]}} dx = \alpha e^{-1} + \beta$$, where $$\alpha, \beta \in R$$, $$5\alpha + 6\beta = 0$$, and $$[x]$$ denotes the greatest integer less than or equal to $$x$$; then the value of $$(\alpha + \beta)^2$$ is equal to:
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Let $$y(x)$$ be the solution of the differential equation $$2x^2 dy + (e^y - 2x)dx = 0$$, $$x > 0$$. If $$y(e) = 1$$, then $$y(1)$$ is equal to:
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A hall has a square floor of dimension 10 m $$\times$$ 10 m (see the figure) and vertical walls. If the angle GPH between the diagonals AG and BH is $$\cos^{-1}\frac{1}{5}$$, then the height of the hall (in meters) is:
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Let $$P$$ be the plane passing through the point $$(1, 2, 3)$$ and the line of intersection of the planes $$\vec{r} \cdot (\hat{i} + \hat{j} + 4\hat{k}) = 16$$ and $$\vec{r} \cdot (-\hat{i} + \hat{j} + \hat{k}) = 6$$. Then which of the following points does NOT lie on $$P$$?
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A fair die is tossed until six is obtained on it. Let $$X$$ be the number of required tosses, then the conditional probability $$P(X \geq 5 \mid X \gt 2)$$ is:
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Let $$\lambda \neq 0$$ be in $$R$$. If $$\alpha$$ and $$\beta$$ are the roots of the equation $$x^2 - x + 2\lambda = 0$$, and $$\alpha$$ and $$\gamma$$ are the roots of the equation $$3x^2 - 10x + 27\lambda = 0$$, then $$\frac{\beta\gamma}{\lambda}$$ is equal to _________
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The least positive integer $$n$$ such that $$\frac{(2i)^n}{(1-i)^{n-2}}$$, $$i = \sqrt{-1}$$, is a positive integer, is _________
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The sum of all 3-digit numbers less than or equal to 500, that are formed without using the digit 1 and they all are multiple of 11, is _________
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Let $$a_1, a_2, \ldots, a_{10}$$ be an A.P. with common difference $$-3$$ and $$b_1, b_2, \ldots, b_{10}$$ be a G.P. with common ratio 2. Let $$c_k = a_k + b_k$$, $$k = 1, 2, \ldots, 10$$. If $$c_2 = 12$$ and $$c_3 = 13$$, then $$\sum_{k=1}^{10} c_k$$ is equal to _________
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Let $$\binom{n}{k}$$ denote $$^nC_k$$ and $$\left[\frac{n}{k}\right] = \begin{cases} \binom{n}{k}, & \text{if } 0 \leq k \leq n \\ 0, & \text{otherwise} \end{cases}$$. If $$A_k = \sum_{i=0}^{9} \binom{9}{i} \left[\binom{12}{12-k+i}\right] + \sum_{i=0}^{8} \binom{8}{i} \left[\binom{13}{13-k+i}\right]$$ and $$A_4 - A_3 = 190p$$, then $$p$$ is equal to _________
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Let the mean and variance of four numbers 3, 7, $$x$$ and $$y$$ ($$x > y$$) be 5 and 10 respectively. Then the mean of four numbers 3 + 2x, 7 + 2y, $$x + y$$ and $$x - y$$ is _________
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Let $$A$$ be a $$3 \times 3$$ real matrix. If $$\det(2\text{Adj}(2\text{Adj}(\text{Adj}(2A)))) = 2^{41}$$, then the value of $$\det(A^2)$$ equals _________
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Let $$a$$ and $$b$$ respectively be the points of local maximum and local minimum of the function $$f(x) = 2x^3 - 3x^2 - 12x$$. If $$A$$ is the total area of the region bounded by $$y = f(x)$$, the $$x$$-axis and the lines $$x = a$$ and $$x = b$$, then 4A is equal to _________
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If the projection of the vector $$\hat{i} + 2\hat{j} + \hat{k}$$ on the sum of the two vectors $$2\hat{i} + 4\hat{j} - 5\hat{k}$$ and $$-\lambda\hat{i} + 2\hat{j} + 3\hat{k}$$ is 1, then $$\lambda$$ is equal to _________
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Let $$Q$$ be the foot of the perpendicular from the point $$P(7, -2, 13)$$ on the plane containing the lines $$\frac{x+1}{6} = \frac{y-1}{7} = \frac{z-3}{8}$$ and $$\frac{x-1}{3} = \frac{y-2}{5} = \frac{z-3}{7}$$. Then $$(PQ)^2$$ is equal to _________
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