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Let $$\overrightarrow{a}=-\widehat{i}+2\widehat{j}+2\widehat{k},\overrightarrow{b}=8\widehat{i}+7\widehat{j}-3\widehat{k} \text { and } \overrightarrow{c}$$ be a vector such that $$\overrightarrow{a}\times\overrightarrow{c}=\overrightarrow{b}$$. If $$\overrightarrow{c}\cdot(\widehat{i}+\widehat{j}+\widehat{k})=4$$, then $$\mid\overrightarrow{a}+\overrightarrow{c}\mid^{2}$$ is equal to :
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Let PQ and MN be two straight lines touching the circle $$x^{2}+y^{2}-4x-6y-3=0$$ at the points A and B respectively. Let O be the centre of the circle and $$\angle AOB=\pi/3$$. Then the locus of the point of intersection of the lines PQ and MN is:
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The number of relations, defined on the set {a, b, c, d}, which are both reflexive and symmetric, is equal to:
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If $$x^{2}+x+1=0$$, then the value of $$\left(x+\frac{1}{x}\right)^{4}+\left(x^{2}+\frac{1}{x^{2}}\right)^{4}+\left(x^{3}+\frac{1}{x^{3}}\right)^{4}+...+\left(x^{25}+\frac{1}{x^{25}}\right)^{4}$$ is:
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Let $$f: R \rightarrow (0, \infty)$$ be a twice differentiable function such that f(3) = 18, f'(3) = 0 and f" (3) = 4. Then $$\lim_{x \rightarrow 1}\left(\log_{a}\left(\frac{f(2+x)}{f(3)}\right)^{\frac{18}{(x-1)^{2}}}\right)$$ ls equal to :
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The value of $$\operatorname{cosec}10°-\sqrt{3}\sec10°$$ is equal to :
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If the coefficient of x in the expansion of $$(ax^{2}+bx+c)(1-2x)^{26}$$. is - 56 and the coefficients of $$x^{2}\text{ and }x^{3}$$ are both zero, then a + b + c is equal to:
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Let a point A lie between the parallel lines $$L_{1}\text{ and }L_{2}$$ such that its distances from $$L_{1}\text{ and }L_{2}$$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC, where the points B and C lie on the lines $$L_{1}\text{ and }L_{2}$$, respectively, is:
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Let O be the vertex of the parabola $$x^{2}=4y$$ and Q be any point on it. Let the locus of the point P, which divides the line segment OQ internally in the ratio 2: 3 be the conic C. Then the equation of the chord of C, which is bisected at the point (1, 2), is:
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Let $$\overrightarrow{c} \text{ and } \overrightarrow{d}$$ be vectors such that $$\mid\overrightarrow{c}+\overrightarrow{d}\mid=\sqrt{29}$$ and $$\overrightarrow{c}\times( 2\widehat{i}+3\widehat{j}+4\widehat{k})=(2\widehat{i}+3\widehat{j}+4\widehat{k})\times\overrightarrow{d}$$. If $$\lambda_{1}, \lambda_{2}( \lambda_{1}> \lambda_{2})$$ are the possible values of $$(\overrightarrow{c}+\overrightarrow{d})\cdot(-7\widehat{i}+2\widehat{j}+3\overrightarrow{k})$$, then the equation $$K^{2}x^{2}+(K^{2}-5K+\lambda_{1})xy+\left(3K+\frac{\lambda_{2}}{2} \right)y^{2}-8x+12y+\lambda_{2}=0$$ represents a circle, for K equal to :
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Let $$a_{1},a_{2},a_{3},...$$ be a G.P. of increasing positive terms such that $$a_{2}.a_{3}.a_{4}=64\text{ and }a_{1}+a_{3}+a_{5}=\frac{813}{7}.\text{ Then }a_{3}+a_{5}+a_{7}$$ is equal to :
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The sum of all the roots of the equation $$(x-1)^2-5\mid x-1\mid+\ 6=0$$ is:
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Let the mean and variance of 7 observations 2, 4, 10, x, 12, 14, y, x > y, be 8 and 16 respectively. Two numbers are chosen from {1, 2, 3, x - 4,y,5} one after an other without replacement, then the probability, that the smaller number among the two chosen numbers is less than 4, is :
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Let the foci of a hyperbola coincide with the foci of the ellipse $$\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$$. If the eccentricity of the hyperbola is 5, then the length of its latus rectum is :
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If the domain of the function $$f(x)=\cos^{-1}\left(\frac{2x-5}{11-3x}\right)+\sin^{-1}(2x^{2}-3x+1)$$ is the interval $$[\alpha, \beta]$$, then $$\alpha+2\beta$$ is equal to:
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The area of the region, inside the ellipse $$x^{2}+4y^{2}=4$$ and outside the region bounded by the curves y=|x|-1 and y=1-|x|, is:
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Let y=y(x) be the solution curve of the differential equation $$(1+x^{2})dy+(y-\tan^{-1}x)dx=0,y(0)=1$$. Then the value of y (1) is :
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Let $$(\alpha,\beta,\gamma)$$ be the co-ordinates of the foot of the perpendicular drawn from the point (5, 4, 2) on the line $$\overrightarrow{r}=(-\widehat{i}+3\widehat{j}+\widehat{k})+\lambda(2\widehat{i}+3\widehat{j}-\widehat{k}).$$ Then the length of the projection of the vector $$\alpha\widehat{i}+\beta\widehat{j}+\gamma\widehat{k}$$ on the vector $$6\widehat{i}+2\widehat{j}+3\widehat{k}$$ is:
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The number of strictly increasing functions f from the set {1, 2, 3, 4, 5, 6} to the set {1, 2, 3, ... , 9} such that $$f(i)\neq i \text{ for }1\leq i\leq 6$$, is equal to:
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The value of $$\int_{\frac{-\pi}{6}}^{\frac{\pi}{6}}\left(\frac{\pi+4x^{11}}{1-\sin(|x|+\frac{\pi}{6})}\right)dx$$ is equal to :
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$$6\int_{0}^{\pi} |(\sin3x+\sin2x+\sin x)|dx$$ is equal to _________.
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Let $$a_{1}=1$$ and for $$n\geq1,a_{n+1}=\frac{1}{2}a_{n}+\frac{n^{2}-2n-1}{n^{2}(n+1)^2}$$. Then $$\left|\sum_{ n=1}^{ \infty}\left( a_n - \frac{2}{n^2}\right)\right|$$ is equal to ______.
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Let S= {(m, n) :m, n $$\epsilon$$ {1, 2, 3, .... , 50}}. lf the number of elements (m, n) in S such that $$6^m+9^n$$ is a multiple of 5 is p and the number of elements (m, n) in S such that m + n is a square of a prime number is q, then p +q is equal to ________.
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Let $$f:R\rightarrow R$$ be a twice differentiable function such that the quadratic equation $$f(x)m^{2}-2 f'(x)m+ f''(x)=0$$ in m, has two equal roots for every $$x \epsilon R$$. If $$ f(0)=1,f'(0)=2$$, and $$(\alpha,\beta)$$ is the largest interval in which the function $$f(\log_{e}{x-x})$$ is increasing, then $$\alpha+\beta$$ is equal to ________.
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For some $$\alpha,\beta\epsilon R$$, let $$A=\begin{bmatrix}\alpha & 2 \\ 1 & 2 \end{bmatrix}\text{ and }B=\begin{bmatrix}1 & 1 \\1 & \beta \end{bmatrix}$$ be such that $$A^{2}-4A+2I=B^2-3B+I=0$$. Then $$(det(adj(A^3-B^3)))^2$$ is equal to _______.
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