If $$z = x + iy$$, $$xy \neq 0$$, satisfies the equation $$z^2 + i\bar{z} = 0$$, then $$|z^2|$$ is equal to :
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If $$z = x + iy$$, $$xy \neq 0$$, satisfies the equation $$z^2 + i\bar{z} = 0$$, then $$|z^2|$$ is equal to :
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Let $$S_a$$ denote the sum of first $$n$$ terms an arithmetic progression. If $$S_{20} = 790$$ and $$S_{10} = 145$$, then $$S_{15} - S_5$$ is :
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If $$2\sin^3 x + \sin 2x \cos x + 4\sin x - 4 = 0$$ has exactly $$3$$ solutions in the interval $$\left[0, \frac{n\pi}{2}\right]$$, $$n \in \mathbb{N}$$, then the roots of the equation $$x^2 + nx + (n - 3) = 0$$ belong to :
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A line passing through the point $$A(9, 0)$$ makes an angle of $$30°$$ with the positive direction of $$x$$-axis. If this line is rotated about $$A$$ through an angle of $$15°$$ in the clockwise direction, then its equation in the new position is
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If the circles $$(x + 1)^2 + (y + 2)^2 = r^2$$ and $$x^2 + y^2 - 4x - 4y + 4 = 0$$ intersect at exactly two distinct points, then
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The maximum area of a triangle whose one vertex is at $$(0, 0)$$ and the other two vertices lie on the curve $$y = -2x^2 + 54$$ at points $$(x, y)$$ and $$(-x, y)$$ where $$y > 0$$ is :
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If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :
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Let $$f : \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathbb{R}$$ be a differentiable function such that $$f(0) = \frac{1}{2}$$. If $$\lim_{x \to 0} \frac{x \int_0^x f(t) dt}{e^{x^2} - 1} = \alpha$$, then $$8\alpha^2$$ is equal to :
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Let $$M$$ denote the median of the following frequency distribution.
Then $$20M$$ is equal to :
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If $$f(x) = \begin{vmatrix} 2\cos^4 x & 2\sin^4 x & 3 + \sin^2 2x \\ 3 + 2\cos^4 x & 2\sin^4 x & \sin^2 2x \\ 2\cos^4 x & 3 + 2\sin^4 x & \sin^2 2x \end{vmatrix}$$ then $$\frac{1}{5}f'(0)$$ is equal to ________.
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Consider the system of linear equation $$x + y + z = 4\mu$$, $$x + 2y + 2\lambda z = 10\mu$$, $$x + 3y + 4\lambda^2 z = \mu^2 + 15$$, where $$\lambda, \mu \in \mathbb{R}$$. Which one of the following statements is NOT correct?
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If the domain of the function $$f(x) = \cos^{-1}\left(\frac{2 - |x|}{4}\right) + (\log_e(3 - x))^{-1}$$ is $$[-\alpha, \beta) - \{\gamma\}$$, then $$\alpha + \beta + \gamma$$ is equal to :
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Let $$g : \mathbb{R} \rightarrow \mathbb{R}$$ be a non constant twice differentiable such that $$g'\left(\frac{1}{2}\right) = g'\left(\frac{3}{2}\right)$$. If a real valued function $$f$$ is defined as $$f(x) = \frac{1}{2}[g(x) + g(2 - x)]$$, then
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The value of $$\lim_{n \to \infty} \sum_{k=1}^{n} \frac{n^3}{(n^2 + k^2)(n^2 + 3k^2)}$$ is :
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The area (in square units) of the region bounded by the parabola $$y^2 = 4(x - 2)$$ and the line $$y = 2x - 8$$.
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Let $$y = y(x)$$ be the solution of the differential equation $$\sec x \, dy + \{2(1 - x)\tan x + x(2 - x)\}dx = 0$$ such that $$y(0) = 2$$. Then $$y(2)$$ is equal to :
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Let $$A(2, 3, 5)$$ and $$C(-3, 4, -2)$$ be opposite vertices of a parallelogram $$ABCD$$ if the diagonal $$\vec{BD} = \hat{i} + 2\hat{j} + 3\hat{k}$$ then the area of the parallelogram is equal to
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Let $$\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$$ and $$\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$$ be two vectors such that $$|\vec{a}| = 1$$; $$\vec{a} \cdot \vec{b} = 2$$ and $$|\vec{b}| = 4$$. If $$\vec{c} = 2(\vec{a} \times \vec{b}) - 3\vec{b}$$, then the angle between $$\vec{b}$$ and $$\vec{c}$$ is equal to :
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Let $$(\alpha, \beta, \gamma)$$ be the foot of perpendicular from the point $$(1, 2, 3)$$ on the line $$\frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3}$$. then $$19(\alpha + \beta + \gamma)$$ is equal to :
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Two integers $$x$$ and $$y$$ are chosen with replacement from the set $$\{0, 1, 2, 3, \ldots, 10\}$$. Then the probability that $$|x - y| > 5$$ is :
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Let $$\alpha, \beta \in \mathbb{R}$$ be roots of equation $$x^2 - 70x + \lambda = 0$$, where $$\frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathbb{Z}$$. If $$\lambda$$ assumes the minimum possible value, then $$\frac{(\sqrt{\alpha - 1} + \sqrt{\beta - 1})(\lambda + 35)}{|\alpha - \beta|}$$ is equal to :
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Let $$\alpha = 1^2 + 4^2 + 8^2 + 13^2 + 19^2 + 26^2 + \ldots$$ upto $$10$$ terms and $$\beta = \sum_{n=1}^{10} n^4$$. If $$4\alpha - \beta = 55k + 40$$, then $$k$$ is equal to _______.
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Number of integral terms in the expansion of $$\left\{7^{(1/2)} + 11^{(1/6)}\right\}^{824}$$ is equal to ______.
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Let the latus rectum of the hyperbola $$\frac{x^2}{9} - \frac{y^2}{b^2} = 1$$ subtend an angle of $$\frac{\pi}{3}$$ at the centre of the hyperbola. If $$b^2$$ is equal to $$\frac{l}{m}(1 + \sqrt{n})$$, where $$l$$ and $$m$$ are co-prime numbers, then $$l^2 + m^2 + n^2$$ is equal to __________.
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A group of $$40$$ students appeared in an examination of $$3$$ subjects - Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects, $$20$$ students passed in Mathematics, $$25$$ students passed in Physics, $$16$$ students passed in Chemistry, at most $$11$$ students passed in both Mathematics and Physics, at most $$15$$ students passed in both Physics and Chemistry, at most $$15$$ students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is _____.
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Let $$A = \{1, 2, 3, \ldots, 7\}$$ and let $$P(A)$$ denote the power set of $$A$$. If the number of functions $$f : A \rightarrow P(A)$$ such that $$a \in f(a), \forall a \in A$$ is $$m^n$$, $$m$$ and $$n \in \mathbb{N}$$ and $$m$$ is least, then $$m + n$$ is equal to ______.
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If the function $$f(x) = \begin{cases} \frac{1}{|x|}, & |x| \geq 2 \\ ax^2 + 2b, & |x| < 2 \end{cases}$$ is differentiable on $$\mathbb{R}$$, then $$48(a + b)$$ is equal to _______.
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The value $$9\int_0^9 \left[\sqrt{\frac{10x}{x+1}}\right] dx$$, where $$[t]$$ denotes the greatest integer less than or equal to $$t$$, is _____.
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Let $$y = y(x)$$ be the solution of the differential equation $$(1 - x^2)dy = \left[xy + (x^3 + 2)\sqrt{3(1 - x^2)}\right]dx$$, $$-1 < x < 1$$, $$y(0) = 0$$. If $$y\left(\frac{1}{2}\right) = \frac{m}{n}$$, $$m$$ and $$n$$ are coprime numbers, then $$m + n$$ is equal to __________.
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If $$d_1$$ is the shortest distance between the lines $$x + 1 = 2y = -12z$$, $$x = y + 2 = 6z - 6$$ and $$d_2$$ is the shortest distance between the lines $$\frac{x-1}{2} = \frac{y+8}{-7} = \frac{z-4}{5}$$, $$\frac{x-1}{2} = \frac{y-2}{1} = \frac{z-6}{-3}$$, then the value of $$\frac{32\sqrt{3} \, d_1}{d_2}$$ is :
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