The number of real roots of the equation $$x|x| - 5|x + 2| + 6 = 0$$, is
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The number of real roots of the equation $$x|x| - 5|x + 2| + 6 = 0$$, is
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If the set $$\left\{Re\left(\frac{z - \bar{z} + z\bar{z}}{2 - 3z + 5\bar{z}}\right) : z \in \mathbb{C}, \ Re \ z = 3\right\}$$ is equal to the interval $$(\alpha, \beta]$$, then $$24(\beta - \alpha)$$ is equal to
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The total number of three-digit numbers, divisible by 3, which can be formed using the digits 1, 3, 5, 8, if repetition of digits is allowed, is
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Let $$A_1$$ and $$A_2$$ be two arithmetic means and $$G_1$$, $$G_2$$ and $$G_3$$ be three geometric means of two distinct positive numbers. Then $$G_1^4 + G_2^4 + G_3^4 + G_1^2 G_3^2$$ is equal to
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Let $$(a + bx + cx^2)^{10} = \sum_{i=0}^{20} p_i x^i$$, $$a, b, c \in \mathbb{N}$$. If $$p_1 = 20$$ and $$p_2 = 210$$, then $$2(a + b + c)$$ is equal to
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If $$(\alpha, \beta)$$ is the orthocenter of the triangle $$ABC$$ with vertices $$A(3, -7)$$, $$B(-1, 2)$$ and $$C(4, 5)$$, then $$9\alpha - 6\beta + 60$$ is equal to
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The number of common tangents, to the circles $$x^2 + y^2 - 18x - 15y + 131 = 0$$ and $$x^2 + y^2 - 6x - 6y - 7 = 0$$, is
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Negation of $$p \wedge (q \wedge \sim(p \wedge q))$$ is
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The mean and standard deviation of 10 observations are 20 and 8 respectively. Later on, it was observed that one observation was recorded as 50 instead of 40. Then the correct variance is
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Let the determinant of a square matrix $$A$$ of order $$m$$ be $$m - n$$, where $$m$$ and $$n$$ satisfy $$4m + n = 22$$ and $$17m + 4n = 93$$. If $$det(n \ adj(adj(mA))) = 3^a 5^b 6^c$$, then $$a + b + c$$ is equal to
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Let the system of linear equations
$$-x + 2y - 9z = 7$$
$$-x + 3y + 7z = 9$$
$$-2x + y + 5z = 8$$
$$-3x + y + 13z = \lambda$$
has a unique solution $$x = \alpha, y = \beta, z = \gamma$$. Then the distance of the point $$(\alpha, \beta, \gamma)$$ from the plane $$2x - 2y + z = \lambda$$ is
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If the domain of the function $$f(x) = \log_e(4x^2 + 11x + 6) + \sin^{-1}(4x + 3) + \cos^{-1}\left(\frac{10x + 6}{3}\right)$$ is $$(\alpha, \beta]$$, then $$36|\alpha + \beta|$$ is equal to
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Let $$[x]$$ denote the greatest integer function and $$f(x) = \max\{1 + x + [x], 2 + x, x + 2[x]\}$$, $$0 \leq x \leq 2$$, where $$f$$ is not continuous and $$n$$ be the number of points in $$(0, 2)$$, where $$f$$ is not differentiable. Then $$(m + n)^2 + 2$$ is equal to
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If $$\int_0^1 \frac{1}{(5+2x-2x^2)(1+e^{(2-4x)})} dx = \frac{1}{\alpha} \log_e\left(\frac{\alpha+1}{\beta}\right)$$, $$\alpha, \beta > 0$$, then $$\alpha^4 - \beta^4$$ is equal to
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Let $$x = x(y)$$ be the solution of the differential equation $$2(y+2)\log_e(y+2)dx + (x + 4 - 2\log_e(y+2))dy = 0$$, $$y > -1$$ with $$x(e^4 - 2) = 1$$. Then $$x(e^9 - 2)$$ is equal to
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Let $$S$$ be the set of all $$(\lambda, \mu)$$ for which the vectors $$\lambda\hat{i} - \hat{j} + \hat{k}$$, $$\hat{j} + 2\hat{j} + \mu\hat{k}$$ and $$3\hat{i} - 4\hat{j} + 5\hat{k}$$, where $$\lambda - \mu = 5$$, are coplanar, then $$\sum_{(\lambda,\mu) \in S} 80(\lambda^2 + \mu^2)$$ is equal to
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Let $$ABCD$$ be a quadrilateral. If $$E$$ and $$F$$ are the mid points of the diagonals $$AC$$ and $$BD$$ respectively and $$\left(\vec{AB} - \vec{BC}\right) + \left(\vec{AD} - \vec{DC}\right) = k\vec{FE}$$, then $$k$$ is equal to
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Let the foot of perpendicular of the point $$P(3, -2, -9)$$ on the plane passing through the points $$(-1, -2, -3)$$, $$(9, 3, 4)$$, $$(9, -2, 1)$$ be $$Q(\alpha, \beta, \gamma)$$. Then the distance $$Q$$ from the origin is
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Let $$S$$ be the set of all values of $$\lambda$$, for which the shortest distance between the lines $$\frac{x-\lambda}{0} = \frac{y-3}{-4} = \frac{z+6}{1}$$ and $$\frac{x+\lambda}{3} = \frac{y}{-4} = \frac{z-6}{0}$$ is 13. Then $$8|\sum_{\lambda \in S} \lambda|$$ is equal to
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A bag contains 6 white and 4 black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is
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A person forgets his 4-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is 7 and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct code is _____.
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If the sum of the series $$\left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{2^2} - \frac{1}{2 \cdot 3} + \frac{1}{3^2}\right) + \left(\frac{1}{2^3} - \frac{1}{2^2 \cdot 3} + \frac{1}{2 \cdot 3^2} - \frac{1}{3^3}\right) + \left(\frac{1}{2^4} - \frac{1}{2^3 \cdot 3} + \frac{1}{2^2 \cdot 3^2} - \frac{1}{2 \cdot 3^3} + \frac{1}{3^4}\right) + \ldots$$ is $$\frac{\alpha}{\beta}$$, where $$\alpha$$ and $$\beta$$ are co-prime, then $$\alpha + 3\beta$$ is equal to _____.
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Consider the triangles with vertices $$A(2, 1)$$, $$B(0, 0)$$ and $$C(t, 4)$$, $$t = [0, 4]$$. If the maximum and the minimum perimeters of such triangles are obtained at $$t = \alpha$$ and $$t = \beta$$ respectively, then $$6\alpha + 21\beta$$ is equal to _____.
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Let an ellipse with centre $$(1, 0)$$ and latus rectum of length $$\frac{1}{2}$$ have its major axis along x-axis. If its minor axis subtends an angle $$60°$$ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to _____.
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The number of elements in the set $$\{n \in \mathbb{N} : 10 \leq n \leq 100$$ and $$3^n - 3$$ is a multiple of $$7\}$$ is _____.
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Let $$A = \{1, 2, 3, 4\}$$ and $$R$$ be a relation on the set $$A \times A$$ defined by $$R = \{((a, b), (c, d)) : 2a + 3b = 4c + 5d\}$$. Then the number of elements in $$R$$ is _____.
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Let $$f(x) = \int \frac{dx}{(3+4x^2)\sqrt{4-3x^2}}$$, $$|x| < \frac{2}{\sqrt{3}}$$. If $$f(0) = 0$$ and $$f(1) = \frac{1}{\alpha\beta}\tan^{-1}\left(\frac{\alpha}{\beta}\right)$$, $$\alpha, \beta > 0$$, then $$\alpha^2 + \beta^2$$ is equal to _____.
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If the area bounded by the curve $$2y^2 = 3x$$, lines $$x + y = 3$$, $$y = 0$$ and outside the circle $$(x-3)^2 + y^2 = 2$$ is A, then $$4(\pi + 4A)$$ is equal to _____.
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If the line $$x = y = z$$ intersects the line $$x\sin A + y\sin B + z\sin C - 18 = 0 = x\sin 2A + y\sin 2B + z\sin 2C - 9$$, where $$A$$, $$B$$, $$C$$ are the angles of a triangle $$ABC$$, then $$80\left(\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}\right)$$ is equal to _____.
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Let the plane $$P$$ contain the line $$2x + y - z - 3 = 0 = 5x - 3y + 4z + 9$$ and be parallel to the line $$\frac{x+2}{2} = \frac{3-y}{-4} = \frac{z-7}{5}$$. Then the distance of the point $$A(8, -1, -19)$$ from the plane $$P$$ measured parallel to the line $$\frac{x}{-3} = \frac{y-5}{4} = \frac{z-2}{12}$$ is equal to _____.
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