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Let circle $$C$$ be the image of $$x^2 + y^2 - 2x + 4y - 4 = 0$$ in the line $$2x - 3y + 5 = 0$$ and $$A$$ be the point on $$C$$ such that $$OA$$ is parallel to the $$x$$-axis and $$A$$ lies on the right hand side of the centre $$O$$ of $$C$$. If $$B(\alpha, \beta)$$, with $$\beta < 4$$, lies on $$C$$ such that the length of the arc $$AB$$ is $$\left(1/6\right)^{\text{th}}$$ of the perimeter of $$C$$, then $$\beta + \sqrt{3}\alpha$$ is equal to
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Let in a $$\triangle ABC$$, the length of the side $$AC$$ be $$6$$, the vertex $$B$$ be $$(1, 2, 3)$$ and the vertices $$A, C$$ lie on the line $$ \frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{-2}. $$ Then the area (in sq. units) of $$\triangle ABC$$ is:
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Let the product of the focal distances of the point $$\left( \sqrt{3}, \frac{1}{2} \right)$$ on the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad (a > b),$$ be $$\frac{7}{4}.$$ Then the absolute difference of the eccentricities of two such ellipses is
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If the system of equations $$\begin{aligned} 2x - y + z &= 4, \\ 5x + \lambda y + 3z &= 12, \\ 100x - 47y + \mu z &= 212 \end{aligned}$$ has infinitely many solutions, then $$\mu - 2\lambda$$ is equal to:
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For some $$n \ne 10,$$ let the coefficients of the 5th, 6th and 7th terms in the binomial expansion of $$(1+x)^{n+4}$$ be in A.P. Then the largest coefficient in the expansion of $$(1+x)^{n+4}$$ is:
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The product of all the rational roots of the equation $$(x^2 - 9x + 11)^2 - (x - 4)(x - 5) = 3$$ is equal to:
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Let the line passing through the points $$(-1,2,1)$$ and parallel to the line $$\frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4}$$ intersect the line $$\frac{x+2}{3}=\frac{y-3}{2}=\frac{z-4}{1}$$ at the point $$P.$$ Then the distance of $$P$$ from the point $$Q(4,-5,1)$$ is:
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$$ \text{Let the lines } 3x-4y-\alpha=0,\; 8x-11y-33=0,\; \text{ and } 2x-3y+\lambda=0$$ be concurrent. If the image of the point $$(1,2)$$ in the line $$2x-3y+\lambda=0$$ $$\text{ is } \left(\frac{57}{13},-\frac{40}{13}\right), \text{ then } |\alpha\lambda| \text{ is equal to:} $$
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$$ \text{If } \alpha \text{ and } \beta \text{ are the roots of the equation } 2z^2-3z-2i=0 ,\; \text{ where } i=\sqrt{-1}, \text{ then } 16\cdot \operatorname{Re}\!\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right) \cdot \operatorname{Im}\!\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right) \text{ is equal to:} $$
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For a statistical data $$x_1,x_2,\ldots,x_{10}$$ of 10 values, a student obtained the mean as 5.5 and $$\sum_{i=1}^{10} x_i^2 = 371. $$ He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8 respectively. The variance of the corrected data is:
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The area of the region $$\left\{(x,y) : x^2 + 4x + 2 \le y \le |x+2| \right\}$$ is equal to:
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Let $$S_n=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\cdots$$ upto $$n$$ terms. If the sum of the first six terms of an A.P. with first term $$-p$$ and common difference $$p$$ is $$\sqrt{2026\, S_{2025}},$$ then the absolute difference between the 20th and 15th terms of the A.P. is:
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$$ \text{Let } f:\mathbb{R}\setminus\{0\}\to\mathbb{R} \text{ be a function such that } f(x)-6f\!\left(\frac{1}{x}\right)=\frac{35}{3x}-\frac{5}{2}. \text{ If } \lim_{x\to 0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta, \; \alpha,\beta\in\mathbb{R}, \text{ then } \alpha+2\beta \text{ is equal to:} $$
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$$ \text{If } I(m,n)=\int_{0}^{1} x^{m-1}(1-x)^{\,n-1}\,dx,\quad m,n>0, \text{ then } I(9,14)+I(10,13) \text{ is:} $$
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A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8, and B wins if he throws a sum of 8 before A throws a sum of 5. The probability that A wins if A makes the first throw, is:
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Let $$f(x)=\dfrac{2^{x+2}+16}{2^{2x+1}+2^{x+4}+32}$$. Then the value of $$8\left(f\!\left(\dfrac{1}{15}\right)+f\!\left(\dfrac{2}{15}\right)+\cdots+f\!\left(\dfrac{59}{15}\right)\right)$$ is equal to:
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Let $$y=y(x)$$ be the solution of the differential equation $$\left(xy-5x^2\sqrt{1+x^2}\right)dx+(1+x^2)dy=0, \quad y(0)=0.$$ Then $$y(\sqrt{3})$$ is equal to:
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$$\lim_{x\to 0}\cosec x \left( \sqrt{2\cos^2 x+3\cos x} - \sqrt{\cos^2 x+\sin x+4} \right)$$ is:
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Consider the region $$R=\{(x,y): x \le y \le 9-\tfrac{11}{3}x^2,\; x\ge 0\}.$$ The area of the largest rectangle of sides parallel to the coordinate axes and inscribed in $$R$$ is:
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Let $$\vec a=\hat{i}+2\hat{j}+3\hat{k}, b=3\hat{i}+\hat{j}-\hat{k} $$ and be three vectors such that $$c$$ is coplanar with $$ \vec a$$ and $$\vec b$$. If $$\vec c $$ is perpendicular to $$\vec b$$ and $$\vec a\cdot \vec c=5,$$ then $$|\vec c|$$ is equal to:
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Let $$ S=\{p_1,p_2,....,p_{10}\} $$ be the set of first ten prime numbers. Let $$ A=S\cup P, $$ where $$ P $$ is the set of all possible products of distinct elements of $$ S. $$ Then the number of all ordered pairs $$ (x,y),\; x\in S,\; y\in A, $$ such that $$ x $$ divides $$ y,$$ is: $$ \underline{ \hspace{2cm} } $$
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$$ \text{If for some } \alpha,\beta;\; \alpha\le\beta,\; \alpha+\beta=8$$ and $$\sec^2(\tan^{-1}\alpha)+\cosec^2(\cot^{-1}\beta)=36,$$ $$\alpha^2+\beta^2$$ is:_______
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$$ \text{Let } A \text{ be a } 3\times 3 \text{ matrix such that } X^TAX=0 \text{ for all nonzero } 3\times1 \text{ matrices } X=\begin{bmatrix}x\\y\\z\end{bmatrix}. \text{ If } A\begin{bmatrix}1\\1\\1\end{bmatrix} = \begin{bmatrix}1\\4\\-5\end{bmatrix}, \; A\begin{bmatrix}1\\2\\1\end{bmatrix} = \begin{bmatrix}0\\4\\-8\end{bmatrix}, \text{ and } \det(\operatorname{adj}(2(A+I)))=2^\alpha 3^\beta 5^\gamma, \; \alpha,\beta,\gamma\in\mathbb{N}, \text{ then } \alpha^2+\beta^2+\gamma^2 \text{ is:} \underline{\hspace{2cm}}$$
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Let $$f$$ be a differentiable function such that $$2(x+2)^2f(x)-3(x+2)^2 = 10\int_0^x (t+2)f(t)\,dt,\quad x\ge0.$$ Then $$f(2)$$ is equal to: $$\underline{\hspace{1cm}}$$
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The number of 3-digit numbers that are divisible by 2 and 3, but not divisible by 4 and 9, is_______
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