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A natural number has prime factorization given by $$n = 2^x 3^y 5^z$$, where $$y$$ and $$z$$ are such that $$y + z = 5$$ and $$y^{-1} + z^{-1} = \frac{5}{6}$$, $$y > z$$. Then the number of odd divisors of $$n$$, including 1, is:
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The sum of the series $$\sum_{n=1}^{\infty} \frac{n^2 + 6n + 10}{(2n+1)!}$$ is equal to
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If $$0 < a, b < 1$$, and $$\tan^{-1}a + \tan^{-1}b = \frac{\pi}{4}$$, then the value of $$(a+b) - \left(\frac{a^2 + b^2}{2}\right) + \left(\frac{a^3 + b^3}{3}\right) - \left(\frac{a^4 + b^4}{4}\right) + \ldots$$ is:
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If the locus of the mid-point of the line segment from the point $$(3, 2)$$ to a point on the circle, $$x^2 + y^2 = 1$$ is a circle of radius $$r$$, then $$r$$ is equal to
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Let $$A(1, 4)$$ and $$B(1, -5)$$ be two points. Let $$P$$ be a point on the circle $$(x-1)^2 + (y-1)^2 = 1$$, such that $$(PA)^2 + (PB)^2$$ have maximum value, then the points $$P$$, $$A$$ and $$B$$ lie on
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Let $$f(x)$$ be a differentiable function at $$x = a$$ with $$f'(a) = 2$$ and $$f(a) = 4$$. Then $$\lim_{x \to a} \frac{xf(a) - af(x)}{x - a}$$ equals:
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Let $$F_1(A, B, C) = (A \wedge \sim B) \vee [\sim C \wedge (A \vee B)] \vee \sim A$$ and $$F_2(A, B) = (A \vee B) \vee (B \to \sim A)$$ be two logical expressions. Then:
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Consider the following system of equations:
$$x + 2y - 3z = a$$
$$2x + 6y - 11z = b$$
$$x - 2y + 7z = c$$
where $$a, b$$ and $$c$$ are real constants. Then the system of equations:
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Let $$A = \{1, 2, 3, \ldots, 10\}$$ and $$f : A \to A$$ be defined as
$$f(k) = \begin{cases} k + 1 & \text{if } k \text{ is odd} \\ k & \text{if } k \text{ is even} \end{cases}$$
Then the number of possible functions $$g : A \to A$$ such that $$gof = f$$ is:
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Let $$f(x) = \sin^{-1}x$$ and $$g(x) = \frac{x^2 - x - 2}{2x^2 - x - 6}$$. If $$g(2) = \lim_{x \to 2} g(x)$$, then the domain of the function $$fog$$ is
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Let $$f: R \to R$$ be defined as $$f(x) = \begin{cases} 2\sin\left(-\frac{\pi x}{2}\right), & \text{if } x < -1 \\ |ax^2 + x + b|, & \text{if } -1 \leq x \leq 1 \\ \sin(\pi x), & \text{if } x > 1 \end{cases}$$
If $$f(x)$$ is continuous on $$R$$, then $$a + b$$ equals:
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The triangle of maximum area that can be inscribed in a given circle of radius 'r' is:
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For $$x > 0$$, if $$f(x) = \int_1^x \frac{\log_e t}{(1+t)} dt$$, then $$f(e) + f\left(\frac{1}{e}\right)$$ is equal to:
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Let $$f(x) = \int_0^x e^t f(t)dt + e^x$$ be a differentiable function for all $$x \in R$$. Then $$f(x)$$ equals:
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Let $$A_1$$ be the area of the region bounded by the curves $$y = \sin x$$, $$y = \cos x$$ and $$y$$-axis in the first quadrant. Also, let $$A_2$$ be the area of the region bounded by the curves $$y = \sin x$$, $$y = \cos x$$, $$x$$-axis and $$x = \frac{\pi}{2}$$ in the first quadrant. Then,
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Let slope of the tangent line to a curve at any point $$P(x, y)$$ be given by $$\frac{xy^2 + y}{x}$$. If the curve intersects the line $$x + 2y = 4$$ at $$x = -2$$, then the value of $$y$$, for which the point $$(3, y)$$ lies on the curve, is:
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If vectors $$\vec{a_1} = x\hat{i} - \hat{j} + \hat{k}$$ and $$\vec{a_2} = \hat{i} + y\hat{j} + z\hat{k}$$ are collinear, then a possible unit vector parallel to the vector $$x\hat{i} + y\hat{j} + z\hat{k}$$ is:
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Let $$L$$ be a line obtained from the intersection of two planes $$x + 2y + z = 6$$ and $$y + 2z = 4$$. If point $$P(\alpha, \beta, \gamma)$$ is the foot of perpendicular from $$(3, 2, 1)$$ on $$L$$, then the value of $$21(\alpha + \beta + \gamma)$$ equals:
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If the mirror image of the point $$(1, 3, 5)$$ with respect to the plane $$4x - 5y + 2z = 8$$ is $$(\alpha, \beta, \gamma)$$, then $$5(\alpha + \beta + \gamma)$$ equals:
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A seven digit number is formed using digits 3, 3, 4, 4, 4, 5, 5. The probability, that number so formed is divisible by 2, is
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Let $$\alpha$$ and $$\beta$$ be two real numbers such that $$\alpha + \beta = 1$$ and $$\alpha\beta = -1$$. Let $$p_n = (\alpha)^n + (\beta)^n$$, $$p_{n-1} = 11$$ and $$p_{n+1} = 29$$ for some integer $$n \geq 1$$. Then, the value of $$p_n^2$$ is ______.
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Let $$z$$ be those complex numbers which satisfy $$|z + 5| \leq 4$$ and $$z(1 + i) + \bar{z}(1 - i) \geq -10$$, $$i = \sqrt{-1}$$. If the maximum value of $$|z + 1|^2$$ is $$\alpha + \beta\sqrt{2}$$, then the value of $$(\alpha + \beta)$$ is
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The total number of 4-digit numbers whose greatest common divisor with 18 is 3 is ______.
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If the arithmetic mean and the geometric mean of the $$p^{th}$$ and $$q^{th}$$ terms of the sequence $$-16, 8, -4, 2, \ldots$$ satisfy the equation $$4x^2 - 9x + 5 = 0$$, then $$p + q$$ is equal to ______.
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Let $$L$$ be a common tangent line to the curves $$4x^2 + 9y^2 = 36$$ and $$(2x)^2 + (2y)^2 = 31$$. Then the square of the slope of the line $$L$$ is ______.
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Let $$X_1, X_2, \ldots, X_{18}$$ be eighteen observations such that $$\sum_{i=1}^{18}(X_i - \alpha) = 36$$ and $$\sum_{i=1}^{18}(X_i - \beta)^2 = 90$$, where $$\alpha$$ and $$\beta$$ are distinct real numbers. If the standard deviation of these observations is 1, then the value of $$|\alpha - \beta|$$ is ______.
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If the matrix $$A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 3 & 0 & -1 \end{bmatrix}$$ satisfies the equation $$A^{20} + \alpha A^{19} + \beta A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ for some real numbers $$\alpha$$ and $$\beta$$, then $$\beta - \alpha$$ is equal to ______.
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Let the normals at all the points on a given curve pass through a fixed point $$(a, b)$$. If the curve passes through $$(3, -3)$$ and $$(4, -2\sqrt{2})$$, given that $$a - 2\sqrt{2}b = 3$$, then $$(a^2 + b^2 + ab)$$ is equal to ______.
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Let $$a$$ be an integer such that all the real roots of the polynomial $$2x^5 + 5x^4 + 10x^3 + 10x^2 + 10x + 10$$ lie in the interval $$(a, a+1)$$. Then, $$|a|$$ is equal to ______.
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If $$I_{m,n} = \int_0^1 x^{m-1}(1-x)^{n-1}dx$$, for $$m, n \geq 1$$, and $$\int_0^1 \frac{x^{m-1} + x^{n-1}}{(1+x)^{m+n}} dx = \alpha I_{m,n}$$, $$\alpha \in R$$, then $$\alpha$$ equals ______.
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