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Let $$p, q \in \mathbb{R}$$ and $$(1 - \sqrt{3}i)^{200} = 2^{199}(p + iq)$$, $$i = \sqrt{-1}$$. Then $$p + q + q^2$$ and $$p - q + q^2$$ are roots of the equation.
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For three positive integers $$p, q, r$$, $$x^{pq^2} = y^{qr} = z^{p^2r}$$ and $$r = pq + 1$$ such that $$3, 3\log_y x, 3\log_z y, 7\log_x z$$ are in A.P. with common difference $$\frac{1}{2}$$. The $$r - p - q$$ is equal to
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The value of $$\sum_{r=0}^{22} {}^{22}C_r \cdot {}^{23}C_r$$ is
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Let a tangent to the curve $$y^2 = 24x$$ meet the curve $$xy = 2$$ at the points $$A$$ and $$B$$. Then the mid-points of such line segments $$AB$$ lie on a parabola with the
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$$\lim_{t \to 0} \left(1^{\frac{1}{\sin^2 t}} + 2^{\frac{1}{\sin^2 t}} + 3^{\frac{1}{\sin^2 t}} + \cdots + n^{\frac{1}{\sin^2 t}}\right)^{\sin^2 t}$$ is equal to
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The compound statement $$(\sim(P \wedge Q)) \vee ((\sim P) \wedge Q) \Rightarrow ((\sim P) \wedge (\sim Q))$$ is equivalent to
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The relation $$R = \{(a, b): \gcd(a, b) = 1, 2a \neq b, a, b \in \mathbb{Z}\}$$ is:
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If $$A$$ and $$B$$ are two non-zero $$n \times n$$ matrices such that $$A^2 + B = A^2B$$, then
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Let $$N$$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations
$$x + y + z = 1$$
$$2x + Ny + 2z = 2$$
$$3x + 3y + Nz = 3$$
has unique solution is $$\frac{k}{6}$$, then the sum of value of $$k$$ and all possible values of $$N$$ is
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Let $$\alpha$$ be a root of the equation $$(a - c)x^2 + (b - a)x + (c - b) = 0$$, where $$a, b, c$$ are distinct real numbers such that the matrix $$\begin{pmatrix} \alpha^2 & \alpha & 1 \\ 1 & 1 & 1 \\ a & b & c \end{pmatrix}$$ is singular. Then the value of $$\frac{(a - c)^2}{(b - a)(c - b)} + \frac{(b - a)^2}{(a - c)(c - b)} + \frac{(c - b)^2}{(a - c)(b - a)}$$ is
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$$\tan^{-1}\frac{1+\sqrt{3}}{3+\sqrt{3}} + \sec^{-1}\sqrt{\frac{8+4\sqrt{3}}{6+3\sqrt{3}}} = $$
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The equation $$x^2 - 4x + [x] + 3 = x[x]$$, where $$[x]$$ denotes the greatest integer function, has:
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Let $$f(x) = \begin{cases} x^2\sin\frac{1}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases}$$, then at $$x = 0$$
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The area enclosed between the curves $$y^2 + 4x = 4$$ and $$y - 2x = 2$$ is
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Let $$y = y(x)$$ be the solution of the differential equation $$x^3 dy + (xy - 1) dx = 0$$, $$x > 0$$, $$y\left(\frac{1}{2}\right) = 3 - e$$. Then $$y(1)$$ is equal to
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Let $$\vec{u} = \hat{i} - \hat{j} - 2\hat{k}$$, $$\vec{v} = 2\hat{i} + \hat{j} - \hat{k}$$, $$\vec{v} \cdot \vec{w} = 2$$ and $$\vec{v} \times \vec{w} = \vec{u} + \lambda\vec{v}$$, then $$\vec{u} \cdot \vec{w}$$ is equal to
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Let $$PQR$$ be a triangle. The points $$A$$, $$B$$ and $$C$$ are on the sides $$QR$$, $$RP$$ and $$PQ$$ respectively such that $$\frac{QA}{AR} = \frac{RB}{BP} = \frac{PC}{CQ} = \frac{1}{2}$$. Then $$\frac{\text{Area}\triangle PQR}{\text{Area}\triangle ABC}$$ is equal to
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The distance of the point $$(7, -3, -4)$$ from the plane containing the points $$(2, -3, 1)$$, $$(-1, 1, -2)$$ and $$(3, -4, 2)$$ is equal to:
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The distance of the point $$(-1, 9, -16)$$ from the plane $$2x + 3y - z = 5$$ measure parallel to the line $$\frac{x+4}{3} = \frac{2-y}{4} = \frac{z-3}{12}$$ is
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Let $$\Omega$$ be the sample space and $$A \subseteq \Omega$$ be an event. Given below are two statements:
(S1): If $$P(A) = 0$$, then $$A = \phi$$
(S2): If $$P(A) = 1$$, then $$A = \Omega$$
Then
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Let $$\lambda \in \mathbb{R}$$ and let the equation $$E$$ be $$|x|^2 - 2|x| + |\lambda - 3| = 0$$. Then the largest element in the set $$S = \{x + \lambda : x \text{ is an integer solution of } E\}$$ is _____
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A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is
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The number of 9 digit numbers, that can be formed using all the digits of the number 123412341 so that the even digits occupy only even places, is _____
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The 4$$^{th}$$ term of GP is 500 and its common ratio is $$\frac{1}{m}$$, $$m \in \mathbb{N}$$. Let $$S_n$$ denote the sum of the first $$n$$ terms of this GP. If $$S_6 > S_5 + 1$$ and $$S_7 < S_6 + \frac{1}{2}$$, then the number of possible values of $$m$$ is _____
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Suppose $$\sum_{r=0}^{2023} r^2 \cdot {}^{2023}C_r = 2023 \times \alpha \times 2^{2022}$$, then the value of $$\alpha$$ is
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Let a tangent to the curve $$9x^2 + 16y^2 = 144$$ intersect the coordinate axes at the points $$A$$ and $$B$$. Then, the minimum length of the line segment $$AB$$ is _____
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Let $$C$$ be the largest circle centred at $$(2, 0)$$ and inscribed in the ellipse $$\frac{x^2}{36} + \frac{y^2}{16} = 1$$. If $$(1, \alpha)$$ lies on $$C$$, then $$10\alpha^2$$ is equal to _____
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The value of $$\frac{8}{\pi} \int_0^{\pi/2} \frac{\cos x^{2023}}{\sin x^{2023} + \cos x^{2023}} dx$$ is _____.
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The value of $$12 \int_0^3 |x^2 - 3x + 2| dx$$ is _____
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The shortest distance between the lines $$\frac{x-2}{3} = \frac{y+1}{2} = \frac{z-6}{2}$$ and $$\frac{x-6}{3} = \frac{1-y}{2} = \frac{z+8}{0}$$ is equal to _____
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