The number of pairs $$a, b$$ of real numbers, such that whenever $$\alpha$$ is a root of the equation $$x^2 + ax + b = 0$$, $$\alpha^2 - 2$$ is also a root of this equation, is:
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The number of pairs $$a, b$$ of real numbers, such that whenever $$\alpha$$ is a root of the equation $$x^2 + ax + b = 0$$, $$\alpha^2 - 2$$ is also a root of this equation, is:
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Let $$P_1, P_2, \ldots, P_{15}$$ be 15 points on a circle. The number of distinct triangles formed by points $$P_i, P_j, P_k$$ such that $$i + j + k \neq 15$$, is:
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Let $$S_n = 1 \cdot (n-1) + 2 \cdot (n-2) + 3 \cdot (n-3) + \ldots + (n-1) \cdot 1$$, $$n \geq 4$$.
The sum $$\sum_{n=4}^{\infty} \frac{2 S_n}{n!} - \frac{1}{(n-2)!}$$ is equal to:
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Let $$a_1, a_2, \ldots, a_{21}$$ be an A.P. such that $$\sum_{n=1}^{20} \frac{1}{a_n a_{n+1}} = \frac{4}{9}$$. If the sum of this A.P. is 189, then $$a_6 a_{16}$$ is equal to:
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If $$n$$ is the number of solutions of the equation $$2\cos x \cdot 4\sin\frac{\pi}{4} + x\sin\frac{\pi}{4} - x - 1 = 1$$, $$x \in [0, \pi]$$ and $$S$$ is the sum of all these solutions, then the ordered pair $$(n, S)$$ is:
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Consider the parabola with vertex $$\left(\frac{1}{2}, \frac{3}{4}\right)$$ and the directrix $$y = \frac{1}{2}$$. Let P be the point where the parabola meets the line $$x = -\frac{1}{2}$$. If the normal to the parabola at P intersects the parabola again at the point Q, then $$(PQ)^2$$ is equal to:
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Let $$\theta$$ be the acute angle between the tangents to the ellipse $$\frac{x^2}{9} + \frac{y^2}{1} = 1$$ and the circle $$x^2 + y^2 = 3$$ at their point of intersection in the first quadrant. Then $$\tan\theta$$ is equal to:
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Which of the following is equivalent to the Boolean expression $$p \wedge \sim q$$?
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Consider the system of linear equations
$$-x + y + 2z = 0$$
$$3x - ay + 5z = 1$$
$$2x - 2y - az = 7$$
Let $$S_1$$ be the set of all $$a \in R$$ for which the system is inconsistent and $$S_2$$ be the set of all $$a \in R$$ for which the system has infinitely many solutions. If $$nS_1$$ and $$nS_2$$ denote the number of elements in $$S_1$$ and $$S_2$$ respectively, then
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$$\cos^{-1}(\cos(-5)) + \sin^{-1}(\sin(6)) - \tan^{-1}(\tan(12))$$ is equal to:
(The inverse trigonometric functions take the principal values)
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The range of the function $$f(x) = \log_{\sqrt{5}}\left(3 + \cos\frac{3\pi}{4} + x + \cos\frac{\pi}{4} + x + \cos\frac{\pi}{4} - x - \cos\frac{3\pi}{4} - x\right)$$ is:
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The function $$f(x) = x^3 - 6x^2 + ax + b$$ is such that $$f(2) = f(4) = 0$$. Consider two statements:
$$S_1$$: there exists $$x_1, x_2 \in (2, 4)$$, $$x_1 \lt x_2$$, such that $$f'(x_1) = -1$$ and $$f'(x_2) = 0$$.
$$S_2$$: there exists $$x_3, x_4 \in (2, 4)$$, $$x_3 \lt x_4$$, such that $$f$$ is decreasing in $$(2, x_4)$$, increasing in $$(x_4, 4)$$ and $$2f'(x_3) = \sqrt{3}f(x_4)$$. Then
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Let $$f : R \rightarrow R$$ be a continuous function. Then $$\lim_{x \to \pi/4} \frac{\frac{\pi}{4}\int_2^{\sec^2 x} f(x) dx}{x^2 - \frac{\pi^2}{16}}$$ is equal to:
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Let $$I_{n,m} = \int_0^{1/2} \frac{x^n}{x^m-1} dx$$, $$\forall n > m$$ and $$n, m \in N$$. Consider a matrix $$A = a_{ij_{3 \times 3}}$$ where $$a_{ij} = \begin{cases} I_{6+i,3} - I_{i+3,3}, & i \leq j \\ 0, & i > j \end{cases}$$. Then adj $$A^{-1}$$ is:
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The function $$f(x)$$, that satisfies the condition $$f(x) = x + \int_0^{\pi/2} \sin x \cos y f(y) dy$$, is:
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The area, enclosed by the curves $$y = \sin x + \cos x$$ and $$y = |\cos x - \sin x|$$ and the lines $$x = 0$$, $$x = \frac{\pi}{2}$$, is:
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If $$y = y(x)$$ is the solution curve of the differential equation $$x^2 dy + (y - \frac{1}{x}) dx = 0$$; $$x > 0$$ and $$y(1) = 1$$, then $$y\left(\frac{1}{2}\right)$$ is equal to:
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The distance of line $$3y - 2z - 1 = 0 = 3x - z + 4$$ from the point $$(2, -1, 6)$$ is:
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Let the acute angle bisector of the two planes $$x - 2y - 2z + 1 = 0$$ and $$2x - 3y - 6z + 1 = 0$$ be the plane P. Then which of the following points lies on P?
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Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common is:
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If for the complex numbers $$z$$ satisfying $$|z - 2 - 2i| \leq 1$$, the maximum value of $$|3iz + 6|$$ is attained at $$a + ib$$, then $$a + b$$ is equal to _________.
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All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two R appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word FARMER in this list is _________.
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If the sum of the coefficients in the expansion of $$(x + y)^n$$ is 4096, then the greatest coefficient in the expansion is _________.
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Let the points of intersections of the lines $$x - y + 1 = 0$$, $$x - 2y + 3 = 0$$ and $$2x - 5y + 11 = 0$$ are the mid points of the sides of a triangle ABC. Then the area of the triangle ABC is _________.
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A man starts walking from the point $$P(-3, 4)$$, touches the x-axis at $$R$$, and then turns to reach at the point $$Q(0, 2)$$. The man is walking at a constant speed. If the man reaches the point Q in the minimum time, then $$50[(PR)^2 + (RQ)^2]$$ is equal to _________.
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Let $$f(x) = x^6 + 2x^4 + x^3 + 2x + 3$$, $$x \in R$$. Then the natural number $$n$$ for which $$\lim_{x \to 1} \frac{x^n f(1) - f(x)}{x - 1} = 44$$ is _________.
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Let $$f(x)$$ be a polynomial of degree 3 such that $$f(k) = -\frac{2}{k}$$ for $$k = 2, 3, 4, 5$$. Then the value of $$52 - 10 f(10)$$ is equal to _________.
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Let $$[t]$$ denote the greatest integer $$\leq t$$. The number of points where the function $$f(x) = [x]|x^2 - 1| + \sin\frac{\pi}{[x]+3} - [x+1]$$, $$x \in (-2, 2)$$ is not continuous is _________.
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Let $$\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k}$$ and $$\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$$. Let a vector $$\vec{v}$$ be in the plane containing $$\vec{a}$$ and $$\vec{b}$$. If $$\vec{v}$$ is perpendicular to the vector $$3\hat{i} + 2\hat{j} - \hat{k}$$ and its projection on $$\vec{a}$$ is 19 units, then $$|2\vec{v}|^2$$ is equal to _________.
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Let $$X$$ be a random variable with distribution.
If the mean of $$X$$ is 2.3 and variance of $$X$$ is $$\sigma^2$$, then $$100\sigma^2$$ is equal to _________.
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