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Let $$\alpha, \beta$$ be the distinct roots of the equation $$x^2 - (t^2 - 5t + 6)x + 1 = 0, t \in \mathbb{R}$$ and $$a_n = \alpha^n + \beta^n$$. Then the minimum value of $$\frac{a_{2023} + a_{2025}}{a_{2024}}$$ is
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The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is
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Let $$A = \{n \in [100, 700] \cap \mathbb{N} : n$$ is neither a multiple of 3 nor a multiple of 4 $$\}$$. Then the number of elements in $$A$$ is
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Let a variable line of slope $$m > 0$$ passing through the point $$(4, -9)$$ intersect the coordinate axes at the points $$A$$ and $$B$$. The minimum value of the sum of the distances of $$A$$ and $$B$$ from the origin is
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If $$A(3, 1, -1)$$, $$B\left(\frac{5}{3}, \frac{7}{3}, \frac{1}{3}\right)$$, $$C(2, 2, 1)$$ and $$D\left(\frac{10}{3}, \frac{2}{3}, \frac{-1}{3}\right)$$ are the vertices of a quadrilateral $$ABCD$$, then its area is
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A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are $$m$$ and $$n$$, respectively, then $$m + n^2$$ is equal to
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Let $$C$$ be the circle of minimum area touching the parabola $$y = 6 - x^2$$ and the lines $$y = \sqrt{3}|x|$$. Then, which one of the following points lies on the circle $$C$$?
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Let $$f : (-\infty, \infty) - \{0\} \rightarrow \mathbb{R}$$ be a differentiable function such that $$f'(1) = \lim_{a \to \infty} a^2 f\left(\frac{1}{a}\right)$$. Then $$\lim_{a \to \infty} \frac{a(a+1)}{2} \tan^{-1}\left(\frac{1}{a}\right) + a^2 - 2\log_e a$$ is equal to
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The mean and standard deviation of 20 observations are found to be 10 and 2 respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12. The correct standard deviation is
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Let the relations $$R_1$$ and $$R_2$$ on the set $$X = \{1, 2, 3, \ldots, 20\}$$ be given by $$R_1 = \{(x, y) : 2x - 3y = 2\}$$ and $$R_2 = \{(x, y) : -5x + 4y = 0\}$$. If $$M$$ and $$N$$ be the minimum number of elements required to be added in $$R_1$$ and $$R_2$$, respectively, in order to make the relations symmetric, then $$M + N$$ equals
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For $$\alpha, \beta \in \mathbb{R}$$ and a natural number $$n$$, let $$A_r = \begin{vmatrix} r & 1 & \frac{n^2}{2} + \alpha \\ 2r & 2 & n^2 - \beta \\ 3r - 2 & 3 & \frac{n(3n-1)}{2} \end{vmatrix}$$. Then $$\sum_{r=1}^{n} A_r$$ is
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The function $$f: \mathbb{R} \rightarrow \mathbb{R}$$, $$f(x) = \frac{x^2 + 2x - 15}{x^2 - 4x + 9}$$, $$x \in \mathbb{R}$$ is
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If $$f(x) = \begin{cases} x^3 \sin\left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}$$ then
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The interval in which the function $$f(x) = x^x, x > 0$$, is strictly increasing is
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$$\int_0^{\pi/4} \frac{\cos^2 x \sin^2 x}{(\cos^3 x + \sin^3 x)^2} dx$$ is equal to
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Let the area of the region enclosed by the curves $$y = 3x$$, $$2y = 27 - 3x$$ and $$y = 3x - x\sqrt{x}$$ be $$A$$. Then $$10A$$ is equal to
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Let $$y = y(x)$$ be the solution of the differential equation $$(1 + x^2)\frac{dy}{dx} + y = e^{\tan^{-1}x}$$, $$y(1) = 0$$. Then $$y(0)$$ is
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Let $$y = y(x)$$ be the solution of the differential equation $$(2x \log_e x)\frac{dy}{dx} + 2y = \frac{3}{x}\log_e x$$, $$x > 0$$ and $$y(e^{-1}) = 0$$. Then, $$y(e)$$ is equal to
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The shortest distance between the lines $$\frac{x-3}{2} = \frac{y+15}{-7} = \frac{z-9}{5}$$ and $$\frac{x+1}{2} = \frac{y-1}{1} = \frac{z-9}{-3}$$ is
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A company has two plants $$A$$ and $$B$$ to manufacture motorcycles. 60% motorcycles are manufactured at plant $$A$$ and the remaining are manufactured at plant $$B$$. 80% of the motorcycles manufactured at plant $$A$$ are rated of the standard quality, while 90% of the motorcycles manufactured at plant $$B$$ are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If $$p$$ is the probability that it was manufactured at plant $$B$$, then $$126p$$ is
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Let $$x_1, x_2, x_3, x_4$$ be the solution of the equation $$4x^4 + 8x^3 - 17x^2 - 12x + 9 = 0$$ and $$(4 + x_1^2)(4 + x_2^2)(4 + x_3^2)(4 + x_4^2) = \frac{125}{16}m$$. Then the value of $$m$$ is
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Let the first term of a series be $$T_1 = 6$$ and its $$r^{th}$$ term $$T_r = 3T_{r-1} + 6^r$$, $$r = 2, 3, \ldots, n$$. If the sum of the first $$n$$ terms of this series is $$\frac{1}{5}(n^2 - 12n + 39)(4 \cdot 6^n - 5 \cdot 3^n + 1)$$, then $$n$$ is equal to ______
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If the second, third and fourth terms in the expansion of $$(x + y)^n$$ are 135, 30 and $$\frac{10}{3}$$, respectively, then $$6(n^3 + x^2 + y)$$ is equal to _______
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Let a conic $$C$$ pass through the point $$(4, -2)$$ and $$P(x, y), x \geq 3$$, be any point on $$C$$. Let the slope of the line touching the conic $$C$$ only at a single point $$P$$ be half the slope of the line joining the points $$P$$ and $$(3, -5)$$. If the focal distance of the point $$(7, 1)$$ on $$C$$ is $$d$$, then $$12d$$ equals ______
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Let $$L_1, L_2$$ be the lines passing through the point $$P(0, 1)$$ and touching the parabola $$9x^2 + 12x + 18y - 14 = 0$$. Let $$Q$$ and $$R$$ be the points on the lines $$L_1$$ and $$L_2$$ such that the $$\triangle PQR$$ is an isosceles triangle with base $$QR$$. If the slopes of the lines $$QR$$ are $$m_1$$ and $$m_2$$, then $$16(m_1^2 + m_2^2)$$ is equal to _______
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Let $$\alpha\beta\gamma = 45$$; $$\alpha, \beta, \gamma \in \mathbb{R}$$. If $$x(\alpha, 1, 2) + y(1, \beta, 2) + z(2, 3, \gamma) = (0, 0, 0)$$ for some $$x, y, z \in \mathbb{R}, xyz \neq 0$$, then $$6\alpha + 4\beta + \gamma$$ is equal to _______
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For $$n \in \mathbb{N}$$, if $$\cot^{-1}3 + \cot^{-1}4 + \cot^{-1}5 + \cot^{-1}n = \frac{\pi}{4}$$, then $$n$$ is equal to _____
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Let $$r_k = \frac{\int_0^1 (1-x^7)^k dx}{\int_0^1 (1-x^7)^{k+1} dx}$$, $$k \in \mathbb{N}$$. Then the value of $$\sum_{k=1}^{10} \frac{1}{7(r_k - 1)}$$ is equal to ________
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Let $$\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$$, $$\vec{b} = 3\hat{i} + 4\hat{j} - 5\hat{k}$$ and a vector $$\vec{c}$$ be such that $$\vec{a} \times (\vec{b} + \vec{c}) + \vec{b} \times \vec{c} = \hat{i} + 8\hat{j} + 13\hat{k}$$. If $$\vec{a} \cdot \vec{c} = 13$$, then $$(24 - \vec{b} \cdot \vec{c})$$ is equal to _______
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Let $$P$$ be the point $$(10, -2, -1)$$ and $$Q$$ be the foot of the perpendicular drawn from the point $$R(1, 7, 6)$$ on the line passing through the points $$(2, -5, 11)$$ and $$(-6, 7, -5)$$. Then the length of the line segment $$PQ$$ is equal to ________
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