Let $$z$$ be a complex number such that $$\left|\frac{z-2i}{z+i}\right| = 2$$, $$z \neq -i$$. Then $$z$$ lies on the circle of radius 2 and centre
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Let $$z$$ be a complex number such that $$\left|\frac{z-2i}{z+i}\right| = 2$$, $$z \neq -i$$. Then $$z$$ lies on the circle of radius 2 and centre
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The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is
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Let $$f(x) = 2x^n + \lambda$$, $$\lambda \in \mathbb{R}$$, $$n \in \mathbb{N}$$, and $$f(4) = 133$$, $$f(5) = 255$$. Then the sum of all the positive integer divisors of $$(f(3) - f(2))$$ is
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$$\sum_{k=0}^{6} {}^{51-k}C_3$$ is equal to
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The equations of two sides of a variable triangle are $$x = 0$$ and $$y = 3$$, and its third side is a tangent to the parabola $$y^2 = 6x$$. The locus of its circumcentre is:
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Let $$\triangle, \nabla \in \{\wedge, \vee\}$$ be such that $$(p \to q) \triangle (p \nabla q)$$ is a tautology. Then
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Let $$A, B, C$$ be $$3 \times 3$$ matrices such that $$A$$ is symmetric and $$B$$ and $$C$$ are skew-symmetric. Consider the statements
(S1) $$A^{13}B^{26} - B^{26}A^{13}$$ is symmetric
(S2) $$A^{26}C^{13} - C^{13}A^{26}$$ is symmetric
Then,
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Let $$A = \begin{bmatrix} \frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{-3}{\sqrt{10}} & \frac{1}{\sqrt{10}} \end{bmatrix}$$ and $$B = \begin{bmatrix} 1 & -i \\ 0 & 1 \end{bmatrix}$$, where $$i = \sqrt{-1}$$. If $$M = A^T BA$$, then the inverse of the matrix $$AM^{2023}A^T$$ is
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Let $$f : \mathbb{R} \to \mathbb{R}$$ be a function defined by $$f(x) = \log_{\sqrt{m}} \{\sqrt{2}(\sin x - \cos x) + m - 2\}$$, for some $$m$$, such that the range of $$f$$ is $$[0, 2]$$. Then the value of $$m$$ is _____.
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The number of functions $$f : \{1, 2, 3, 4\} \to \{a \in \mathbb{Z} : |a| \leq 8\}$$ satisfying $$f(n) + \frac{1}{n}f(n+1) = 1$$, $$\forall n \in \{1, 2, 3\}$$ is
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If the function $$f(x) = \begin{cases} (1+|\cos x|)^{\frac{\lambda}{|\cos x|}}, & 0 < x < \frac{\pi}{2} \\ \mu, & x = \frac{\pi}{2} \\ e^{\frac{\cot 6x}{\cot 4x}}, & \frac{\pi}{2} < x < \pi \end{cases}$$ is continuous at $$x = \frac{\pi}{2}$$, then $$9\lambda + 6\log_e \mu + \mu^6 - e^{6\lambda}$$ is equal to
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Let the function $$f(x) = 2x^3 + (2p-7)x^2 + 3(2p-9)x - 6$$ have a maxima for some value of $$x < 0$$ and a minima for some value of $$x > 0$$. Then, the set of all values of $$p$$ is
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The integral $$16\int_1^2 \frac{dx}{x^3(x^2+2)^2}$$ is equal to
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Let T and C respectively, be the transverse and conjugate axes of the hyperbola $$16x^2 - y^2 + 64x + 4y + 44 = 0$$. Then the area of the region above the parabola $$x^2 = y + 4$$, below the transverse axis T and on the right of the conjugate axis C is:
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Let $$y = y(t)$$ be a solution of the differential equation $$\frac{dy}{dt} + \alpha y = \gamma e^{-\beta t}$$ Where, $$\alpha > 0, \beta > 0$$ and $$\gamma > 0$$. Then $$\lim_{t \to \infty} y(t)$$
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If the four points, whose position vectors are $$3\hat{i} - 4\hat{j} + 2\hat{k}$$, $$\hat{i} + 2\hat{j} - \hat{k}$$, $$-2\hat{i} - \hat{j} + 3\hat{k}$$ and $$5\hat{i} - 2\alpha\hat{j} + 4\hat{k}$$ are coplanar, then $$\alpha$$ is equal to
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Let $$\vec{a} = -\hat{i} - \hat{j} + \hat{k}$$, $$\vec{a} \cdot \vec{b} = 1$$ and $$\vec{a} \times \vec{b} = \hat{i} - \hat{j}$$. Then $$\vec{a} - 6\vec{b}$$ is equal to
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The foot of perpendicular of the point $$(2, 0, 5)$$ on the line $$\frac{x+1}{2} = \frac{y-1}{5} = \frac{z+1}{-1}$$ is $$(\alpha, \beta, \gamma)$$. Then, which of the following is NOT correct?
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Let $$a \in R$$ and let $$\alpha, \beta$$ be the roots of the equation $$x^2 + 60^{1/4}x + a = 0$$. If $$\alpha^4 + \beta^4 = -30$$, then the product of all possible values of $$a$$ is _____.
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Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 orange, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer 5 fruits to Anil is _____.
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For the two positive numbers $$a, b$$, if $$a, b$$ and $$\frac{1}{18}$$ are in a geometric progression, while $$\frac{1}{a}$$, 10 and $$\frac{1}{b}$$ are in an arithmetic progression, then, $$16a + 12b$$ is equal to _____.
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The remainder when $$(2023)^{2023}$$ is divided by 35 is
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If $$m$$ and $$n$$ respectively are the numbers of positive and negative value of $$\theta$$ in the interval $$[-\pi, \pi]$$ that satisfy the equation $$\cos 2\theta \cos \frac{\theta}{2} = \cos 3\theta \cos \frac{9\theta}{2}$$, then $$mn$$ is equal to _____.
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A triangle is formed by X-axis, Y-axis and the line $$3x + 4y = 60$$. Then the number of points $$P(a, b)$$ which lie strictly inside the triangle, where $$a$$ is an integer and $$b$$ is a multiple of $$a$$, is _____.
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Points $$P(-3, 2)$$, $$Q(9, 10)$$ and $$R(a, 4)$$ lie on a circle $$C$$ with $$PR$$ as its diameter. The tangents to $$C$$ at the points $$Q$$ and $$R$$ intersect at the point $$S$$. If $$S$$ lies on the line $$2x - ky = 1$$, then $$k$$ is equal to _____.
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If $$\int_{1/3}^{3} |\log_e x| dx = \frac{m}{n} \log_e\left(\frac{n^2}{e}\right)$$, where $$m$$ and $$n$$ are coprime natural numbers, then $$m^2 + n^2 - 5$$ is equal to _____.
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The shortest distance between the lines $$x + 1 = 2y = -12z$$ and $$x = y + 2 = 6z - 6$$ is
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If the shortest distance between the line joining the points $$(1, 2, 3)$$ and $$(2, 3, 4)$$, and the line $$\frac{x-1}{2} = \frac{y+1}{-1} = \frac{z-2}{0}$$ is $$\alpha$$, then $$28\alpha^2$$ is equal to _____.
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Let N be the sum of the numbers appeared when two fair dice are rolled and let the probability that $$N - 2, \sqrt{3N}, N + 2$$ are in geometric progression be $$\frac{k}{48}$$. Then the value of $$k$$ is
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25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer then a non-smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $$\frac{k}{10}$$. Then the value of $$k$$ is _____.
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