The largest $$n \in \mathbb{N}$$ such that $$3^n$$ divides 50! is:
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The largest $$n \in \mathbb{N}$$ such that $$3^n$$ divides 50! is:
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Let one focus of the hyperbola H: $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ be at $$(\sqrt{10}, 0)$$ and the corresponding directrix be $$x = \frac{9}{\sqrt{10}}$$. If $$e$$ and $$l$$ respectively are the eccentricity and the length of the latus rectum of H, then $$9(e^2 + l)$$ is equal to:
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The number of sequences of ten terms, whose terms are either 0 or 1 or 2, that contain exactly five 1s and exactly three 2s, is equal to:
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Let $$f : \mathbb{R} \to \mathbb{R}$$ be a twice differentiable function such that $$(\sin x \cos y)(f(2x+2y) - f(2x-2y)) = (\cos x \sin y)(f(2x+2y) + f(2x-2y))$$, for all $$x, y \in \mathbb{R}$$. If $$f'(0) = \frac{1}{2}$$, then the value of $$24 f''\left(\frac{5\pi}{3}\right)$$ is:
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Let $$A = \begin{bmatrix} \alpha & -1 \\ 6 & \beta \end{bmatrix}$$, $$\alpha > 0$$, such that $$\det(A) = 0$$ and $$\alpha + \beta = 1$$. If I denotes the $$2 \times 2$$ identity matrix, then the matrix $$(1 + A)^8$$ is:
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The term independent of $$x$$ in the expansion of $$\left(\frac{(x+1)}{\left(x^{2/3} + 1 - x^{1/3}\right)} - \frac{(x+1)}{\left(x - x^{1/2}\right)}\right)^{10}$$, $$x > 1$$ is:
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If $$\theta \in [-2\pi, 2\pi]$$, then the number of solutions of $$2\sqrt{2}\cos^2\theta + (2 - \sqrt{6})\cos\theta - \sqrt{3} = 0$$, is equal to:
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Let $$a_1, a_2, a_3, \ldots$$ be in an A.P. such that $$\sum_{k=1}^{12} a_{2k-1} = -\frac{72}{5} a_1$$, $$a_1 \neq 0$$. If $$\sum_{k=1}^{n} a_k = 0$$, then $$n$$ is:
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If the function $$f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$$, where $$a > 0$$, attains its local maximum and local minimum values at $$p$$ and $$q$$, respectively, such that $$p^2 = q$$, then $$f(3)$$ is equal to:
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Let $$z$$ be a complex number such that $$|z| = 1$$. If $$\frac{2 + k^2 z}{k + \bar{z}} = kz$$, $$k \in \mathbb{R}$$, then the maximum distance of $$k + ik^2$$ from the circle $$|z - (1 + 2i)| = 1$$ is:
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If $$\vec{a}$$ is nonzero vector such that its projections on the vectors $$2\hat{i} - \hat{j} + 2\hat{k}$$, $$\hat{i} + 2\hat{j} - 2\hat{k}$$ and $$\hat{k}$$ are equal, then a unit vector along $$\vec{a}$$ is:
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Let A be the set of all functions $$f: \mathbb{Z} \to \mathbb{Z}$$ and R be a relation on A such that $$R = \{(f, g) : f(0) = g(1) \text{ and } f(1) = g(0)\}$$. Then R is:
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For $$\alpha, \beta, \gamma \in \mathbb{R}$$, if $$\lim_{x \to 0} \frac{x^2 \sin \alpha x + (\gamma - 1)e^{x^2}}{\sin 2x - \beta x} = 3$$, then $$\beta + \gamma - \alpha$$ is equal to:
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If the system of linear equations
$$3x + y + \beta z = 3$$
$$2x + \alpha y - z = -3$$
$$x + 2y + z = 4$$
has infinitely many solutions, then the value of $$22\beta - 9\alpha$$ is:
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Let $$P_n = \alpha^n + \beta^n$$, $$n \in \mathbb{N}$$. If $$P_{10} = 123$$, $$P_9 = 76$$, $$P_8 = 47$$ and $$P_1 = 1$$, then the quadratic equation having roots $$\frac{1}{\alpha}$$ and $$\frac{1}{\beta}$$ is:
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If S and S' are the foci of the ellipse $$\frac{x^2}{18} + \frac{y^2}{9} = 1$$ and P be a point on the ellipse, then $$\min(SP \cdot S'P) + \max(SP \cdot S'P)$$ is equal to:
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Let the vertices Q and R of the triangle PQR lie on the line $$\frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3}$$, $$QR = 5$$ and the coordinates of the point P be $$(0, 2, 3)$$. If the area of the triangle PQR is $$\frac{m}{n}$$, then:
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Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of the triangles ABC, ACD and ADB be 5, 6 and 7 square units respectively. Then the area (in square units) of the $$\triangle BCD$$ is equal to:
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Let $$a \in \mathbb{R}$$ and A be a matrix of order $$3 \times 3$$ such that $$\det(A) = -4$$ and $$A + I = \begin{bmatrix} 1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2 \end{bmatrix}$$, where I is the identity matrix of order $$3 \times 3$$. If $$\det((a+1)\operatorname{adj}((a-1)A))$$ is $$2^m 3^n$$, $$m, n \in \{0,1,2,\ldots,20\}$$, then $$m + n$$ is equal to:
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Let the focal chord PQ of the parabola $$y^2 = 4x$$ with the positive x-axis, make an angle of $$60^\circ$$ where P lies in the first quadrant. If the circle, whose one diameter is PS, S being the focus of the parabola, touches the y-axis at the point $$(0, \alpha)$$, then $$5\alpha^2$$ is equal to:
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Let $$[\cdot]$$ denote the greatest integer function. If $$\int_0^{e^3} \left[\frac{1}{e^{x-1}}\right] dx = \alpha - \log_e 2$$, then $$\alpha^3$$ is equal to _____.
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Let $$f : \mathbb{R} \to \mathbb{R}$$ be a thrice differentiable odd function satisfying $$f'(x) \geq 0$$, $$f''(x) = f(x)$$, $$f(0) = 0$$, $$f'(0) = 3$$. Then $$9f(\log_e 3)$$ is equal to _____.
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If the area of the region $$\{(x,y) : |4 - x^2| \leq y \leq x^2, y \leq 4, x \geq 0\}$$ is $$\left(\frac{80\sqrt{2}}{\alpha} - \beta\right)$$, $$\alpha, \beta \in \mathbb{N}$$, then $$\alpha + \beta$$ is equal to _____.
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Three distinct numbers are selected randomly from the set $$\{1, 2, 3, \ldots, 40\}$$. If the probability, that the selected numbers are in an increasing G.P. is $$\frac{m}{n}$$, $$\gcd(m, n) = 1$$, then $$m + n$$ is equal to _____.
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The absolute difference between the squares of the radii of the two circles passing through the point $$(-9, 4)$$ and touching the lines $$x + y = 3$$ and $$x - y = 3$$, is equal to _____.
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