Let $$\alpha$$ and $$\beta$$ be the roots of the equation $$x^{2}+2ax+\left(3a+10\right)=0$$ such that $$\alpha < 1 < \beta$$. Then the set of all possible values of $$a$$ is :
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Let $$\alpha$$ and $$\beta$$ be the roots of the equation $$x^{2}+2ax+\left(3a+10\right)=0$$ such that $$\alpha < 1 < \beta$$. Then the set of all possible values of $$a$$ is :
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Let A= {2, 3, 5, 7, 9}. Let R be the relation on A defined by x R y if and only if $$2x\leq3y$$. Let l be the number of elements in R, and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then l + m is equal to:
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If the system of equations
3x + y + 4z = 3
$$2x+\alpha y-z = -3$$
x+ 2y + z = 4
has no solution, then the value of $$\alpha$$ is equal to :
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If the area of the region $$\left\{\left(x,y\right): 1-2x \leq y \leq4-x^{2}, x\geq 0, y\geq0 \right\}$$ is $$\frac{\alpha}{\beta} , \alpha,\beta \epsilon N$$, gcd $$\left(\alpha,\beta\right)=1$$, then the value of $$\left(\alpha+\beta\right)$$ is
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Let $$f: R\rightarrow R$$ be a twice differentiable function such that $$f''(x) > 0$$ for all $$x\in R$$ and f'(a-1)=0, where a is a real number. Let g(x)= $$f(\tan^{2}x- 2\tan x+a)$$, $$0 < x < \frac{\pi}{2}$$.
Consider the following two statements :
(I) $$\text{g is increasing in } \left(0, \frac{\pi}{4} \right)$$
(II) $$\text{g is deceasing in } \left( \frac{\pi}{4} , \frac{\pi}{2} \right)$$
Then,
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For the matrices $$A=\begin{bmatrix}3 -4 \\1 -1 \end {bmatrix}$$ and $$B=\begin{bmatrix}-29 49 \\-13 18 \end{bmatrix}$$, if $$\left(A^{15} + B \right) \begin{bmatrix}x \\y\end{bmatrix} = \begin{bmatrix}0 \\0 \end{bmatrix}$$, then among the following which one is true ?
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Let one end of a focal chord of the parabola $$y^{2}=16x$$ be (16,16). If $$P\left(\alpha,\beta\right)$$ divides this focal chord internally in the ratio 5 : 2, then the minimum value of $$\alpha+\beta$$ is equal to :
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Let $$A =\left\{x: |x^{2}-10|\leq6 \right\}$$ and $$B= \left\{x:|x-2|>1 \right\}$$. Then
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If the line $$\alpha x+4y=\sqrt{7}$$, where $$\alpha \epsilon R$$, touch the ellipse $$3x^{2}+4y^{2}=1$$ at the point P in the first quadrant, then one of the focal distances of P is:
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Let y = y(x) be the solution of the differential equation $$\sec x \frac{dy}{dx}-2y=2+3\sin x, x\epsilon \left(-\frac{\pi}{2}, \frac{\pi}{2} \right), y(0)=-\frac{7}{4}$$. Then $$y\left(\frac{\pi}{6}\right)$$ is equal to:
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The largest $$n\epsilon N$$, for which $$7^{n}$$ divides 101!, is :
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Let the line $$L_{1}$$ be parallel to the vector $$-3\widehat{i} +2\widehat{j} + 4\widehat{k}$$ and pass through the point (2, 6, 7), and the line $$L_{2}$$ be parallel to the vector $$2\widehat{i} +\widehat{j} + 3\widehat{k}$$ and pass through the point (4, 3, 5). If the line $$L_{3}$$ is parallel to the vector $$-3\widehat{i} +5\widehat{j} + 16\widehat{k}$$ and intersects the lines $$L_{1}$$ and $$L_{2}$$ at the points C and D, respectively, then $$|\overrightarrow{CD}|^2$$ is equal to :
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For a triangle ABC, let $$\overrightarrow{p} = \overrightarrow{BC}, \overrightarrow{q}= \overrightarrow{CA}$$ and $$\overrightarrow{r} = \overrightarrow{BA}$$. If $$|\overrightarrow{p}| = 2\sqrt{3}, |\overrightarrow{q}|=2$$ and $$\cos\theta = \frac{1}{\sqrt{3}}$$ where $$\theta$$ is the angle between $$\overrightarrow{p}$$ and $$\overrightarrow{q}$$, then $$|\overrightarrow{p} \times \left(\overrightarrow{q}-\overrightarrow{3r}\right)|^2 +3|\overrightarrow{r}|^2$$ is equal to :
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The positive integer n, for which the solutions of the equation x(x + 2) + (x + 2)(x + 4) + .... + (x + 2n - 2)(x + 2n) = $$\dfrac{8n}{3}$$ are two consecutive even integers, is:
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Let $$y^{2}=12x$$ be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that $$\angle OPA =90^\circ$$. Then the locus of the centroid of such triangles OPA is:
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Let $$f(x) = x^{3}+ x^{2}f'(1)+2xf''(2)+f'''(3)$$, $$x\epsilon R$$. Then the value of f'(5) is :
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A random varaible X takes values 0,1,2,3 with probabilities $$\frac{2a+1}{30},\frac{8a-1}{30},\frac{4a+1}{30}$$, b respectively, where $$a,b \epsilon R$$. let $$\mu$$ and $$\sigma$$ respectively be the mean and standard deviation of X such that $$\sigma^{2}+\mu^{2}=2$$. Then $$\frac{a}{b}$$ is equal to :
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Let z be the complex number satisfying $$|z-5|\leq 3$$ and having maximum positive principal argument. Then $$34|\frac{5z-12}{5iz+16}|^2$$ is equal to :
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Let the line L pass through the point ( - 3, 5, 2) and make equal angles with the positive coordinate axes. If the distance of L from the point ( - 2, r, 1) is $$\sqrt{\frac{14}{3}}$$, then the sum of all possible values of r is:
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Let $$a_{1},\dfrac{a_{2}}{2},\dfrac{a_{3}}{2^{2}},....,\dfrac{a_{10}}{2^{9}}$$ be a G.P. of common ratio $$\dfrac{1}{\sqrt{2}}$$. If $$a_{1}+a_{2}+....+a_{10}=62$$, then $$a_{1}$$ is equal to:
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If $$\left(\frac{1}{{}^{15}C_0} + \frac{1}{{}^{15}C_1}\right) \left(\frac{1}{{}^{15}C_1} + \frac{1}{{}^{15}C_2}\right) \cdots \left(\frac{1}{{}^{15}C_{12}} + \frac{1}{{}^{15}C_{13}}\right) = \frac{\alpha^{13}}{{}^{14}C_0 {}^{14}C_1 \cdots {}^{14}C_{12}}$$ then $$30\alpha$$ is equal to __________
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If $$\displaystyle \int_{0}^{1} 4\cot^{-1}(1 - 2x + 4x^2)\,dx = a\tan^{-1}(2) - b\log_e(5)$$, where $$a,b\epsilon N$$ then (2a + b} is equal to _________
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Let [·] denote the greatest integer function and $$f(x) = \lim_{n \to \infty} \frac{1}{n^3} \sum_{k=1}^{n} \left[\frac{k^2}{3^x}\right]$$. Then $$12 \sum_{j=1}^{\infty} f(j)$$ is equal to _______
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If P is a point on the circle $$x^{2}+y^{2}=4$$, Q is a point on the straight line 5x + y + 2 = 0 and x- y + 1 = 0 is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such points P is __________
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Let the maximum value of $$\left(\sin^{-1}x\right)^2+\left(\cos^{-1}x\right)^2$$ for $$x\epsilon \left[-\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}\right]$$ be $$\frac{m}{n}\pi^{2}$$, where gcd
(m, n) = l. Then m + n is equal to ____________
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