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Let $$a, b, c$$ be in arithmetic progression. Let the centroid of the triangle with vertices $$(a, c)$$, $$(2, b)$$ and $$(a, b)$$ be $$\left(\frac{10}{3}, \frac{7}{3}\right)$$. If $$\alpha, \beta$$ are the roots of the equation $$ax^2 + bx + 1 = 0$$, then the value of $$\alpha^2 + \beta^2 - \alpha\beta$$ is:
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If $$n \geq 2$$ is a positive integer, then the sum of the series $${}^{n+1}C_2 + 2({}^2C_2 + {}^3C_2 + {}^4C_2 + \ldots + {}^nC_2)$$ is
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If $$P$$ is a point on the parabola $$y = x^2 + 4$$ which is closest to the straight line $$y = 4x - 1$$, then the co-ordinates of $$P$$ are:
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The negation of the statement $$\sim p \wedge (p \vee q)$$ is:
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For the statements $$p$$ and $$q$$, consider the following compound statements:
$$(a)$$ $$(\sim q \wedge (p \to q)) \to \sim p$$
$$(b)$$ $$((p \vee q) \wedge \sim p) \to q$$
Then which of the following statements is correct?
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The angle of elevation of a jet plane from a point A on the ground is 60°. After a flight of 20 seconds at the speed of 432 km/hour, the angle of elevation changes to 30°. If the jet plane is flying at a constant height, then its height is:
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For the system of linear equations:
$$x - 2y = 1$$, $$x - y + kz = -2$$, $$ky + 4z = 6$$, $$k \in R$$
Consider the following statements:
(A) The system has unique solution if $$k \neq 2, k \neq -2$$.
(B) The system has unique solution if $$k = -2$$.
(C) The system has unique solution if $$k = 2$$.
(D) The system has no-solution if $$k = 2$$.
(E) The system has infinite number of solutions if $$k \neq -2$$.
Which of the following statements are correct?
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Let $$A$$ and $$B$$ be $$3 \times 3$$ real matrices such that $$A$$ is a symmetric matrix and $$B$$ is a skew-symmetric matrix. Then the system of linear equations $$(A^2B^2 - B^2A^2)X = O$$, where $$X$$ is a $$3 \times 1$$ column matrix of unknown variables and $$O$$ is a $$3 \times 1$$ null matrix, has:
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A possible value of $$\tan\left(\frac{1}{4}\sin^{-1}\frac{\sqrt{63}}{8}\right)$$ is:
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For which of the following curves, the line $$x + \sqrt{3}y = 2\sqrt{3}$$ is the tangent at the point $$\left(\frac{3\sqrt{3}}{2}, \frac{1}{2}\right)$$?
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Let $$f : R \to R$$ be defined as
$$f(x) = \begin{cases} -55x, & \text{if } x < -5 \\ 2x^3 - 3x^2 - 120x, & \text{if } -5 \leq x \leq 4 \\ 2x^3 - 3x^2 - 36x - 336, & \text{if } x > 4 \end{cases}$$
Let $$A = \{x \in R : f \text{ is increasing}\}$$. Then $$A$$ is equal to:
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If the curve $$y = ax^2 + bx + c$$, $$x \in R$$, passes through the point (1, 2) and the tangent line to this curve at origin is $$y = x$$, then the possible values of $$a, b, c$$ are:
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The value of the integral, $$\int_1^3 [x^2 - 2x - 2] dx$$, where $$[x]$$ denotes the greatest integer less than or equal to $$x$$, is
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The area of the region: $$R = \{(x, y) : 5x^2 \leq y \leq 2x^2 + 9\}$$ is
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Let $$f$$ be a twice differentiable function defined on $$R$$ such that $$f(0) = 1$$, $$f'(0) = 2$$ and $$f'(x) \neq 0$$ for all $$x \in R$$. If $$\begin{vmatrix} f(x) & f'(x) \\ f'(x) & f''(x) \end{vmatrix} = 0$$, for all $$x \in R$$, then the value of $$f(1)$$ lies in the interval
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If a curve $$y = f(x)$$ passes through the point (1, 2) and satisfies $$x\frac{dy}{dx} + y = bx^4$$, then for what value of $$b$$, $$\int_1^2 f(x)dx = \frac{62}{5}$$?
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Let $$f(x)$$ be a differentiable function defined on $$[0, 2]$$ such that $$f'(x) = f'(2 - x)$$ for all $$x \in (0, 2)$$, $$f(0) = 1$$ and $$f(2) = e^2$$. Then the value of $$\int_0^2 f(x)dx$$ is
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The vector equation of the plane passing through the intersection of the planes $$\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1$$ and $$\vec{r} \cdot (\hat{i} - 2\hat{j}) = -2$$, and the point (1, 0, 2) is:
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Let $$a, b \in R$$. If the mirror image of the point $$P(a, 6, 9)$$ with respect to the line $$\frac{x - 3}{7} = \frac{y - 2}{5} = \frac{z - 1}{-9}$$ is $$(20, b, -a - 9)$$, then $$|a + b|$$ is equal to:
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The probability that two randomly selected subsets of the set $$\{1, 2, 3, 4, 5\}$$ have exactly two elements in their intersection, is:
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The number of the real roots of the equation $$(x + 1)^2 + |x - 5| = \frac{27}{4}$$ is ______.
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Let $$i = \sqrt{-1}$$. If $$\frac{(-1 + i\sqrt{3})^{21}}{(1 - i)^{24}} + \frac{(1 + i\sqrt{3})^{21}}{(1 + i)^{24}} = k$$, and $$n = [|k|]$$ be the greatest integral part of $$|k|$$. Then $$\sum_{j=0}^{n+5} (j + 5)^2 - \sum_{j=0}^{n+5} (j + 5)$$ is equal to ______.
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The students $$S_1, S_2, \ldots, S_{10}$$ are to be divided into 3 groups $$A$$, $$B$$ and $$C$$ such that each group has at least one student and the group $$C$$ has at most 3 students. Then the total number of possibilities of forming such groups is ______.
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The sum of first four terms of a geometric progression (G.P.) is $$\frac{65}{12}$$ and the sum of their respective reciprocals is $$\frac{65}{18}$$. If the product of first three terms of the G.P. is 1, and the third term is $$\alpha$$, then $$2\alpha$$ is ______.
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For integers $$n$$ and $$r$$, let $$\binom{n}{r} = \begin{cases} {}^nC_r, & \text{if } n \geq r \geq 0 \\ 0, & \text{otherwise} \end{cases}$$. The maximum value of $$k$$ for which the sum $$\sum_{i=0}^{k} \binom{10}{i}\binom{15}{k-i} + \sum_{i=0}^{k+1} \binom{12}{i}\binom{13}{k+1-i}$$ is maximum, is equal to ______.
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Let a point $$P$$ be such that its distance from the point (5, 0) is thrice the distance of $$P$$ from the point (-5, 0). If the locus of the point $$P$$ is a circle of radius $$r$$, then $$4r^2$$ (in the nearest integer) is equal to ______.
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If the variance of 10 natural numbers 1, 1, 1, ..., 1, $$k$$ is less than 10, then the maximum possible value of $$k$$ is ______.
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If $$a + \alpha = 1, b + \beta = 2$$ and $$af(x) + \alpha f\left(\frac{1}{x}\right) = bx + \frac{\beta}{x}, x \neq 0$$, then the value of the expression $$\frac{f(x) + f\left(\frac{1}{x}\right)}{x + \frac{1}{x}}$$ is ______.
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If the area of the triangle formed by the $$x$$-axis, the normal and the tangent to the circle $$(x - 2)^2 + (y - 3)^2 = 25$$ at the point (5, 7) is $$A$$, then $$24A$$ is equal to ______.
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Let $$\lambda$$ be an integer. If the shortest distance between the lines $$x - \lambda = 2y - 1 = -2z$$ and $$x = y + 2\lambda = z - \lambda$$ is $$\frac{\sqrt{7}}{2\sqrt{2}}$$, then the value of $$|\lambda|$$ is ______.
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