If $$z \neq 0$$ be a complex number such that $$\left|z - \frac{1}{z}\right| = 2$$, then the maximum value of $$|z|$$ is
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If $$z \neq 0$$ be a complex number such that $$\left|z - \frac{1}{z}\right| = 2$$, then the maximum value of $$|z|$$ is
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Let $$S = \{z = x + iy : |z-1+i| \geq |z|, |z| < 2, |z+i| = |z-1|\}$$. Then the set of all values of x, for which $$w = 2x + iy \in S$$ for some $$y \in \mathbb{R}$$, is
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Let $$\{a_n\}_{n=0}^{\infty}$$ be a sequence such that $$a_0 = a_1 = 0$$ and $$a_{n+2} = 3a_{n+1} - 2a_n + 1$$, $$\forall n \geq 0$$. Then $$a_{25}a_{23} - 2a_{25}a_{22} - 2a_{23}a_{24} + 4a_{22}a_{24}$$ is equal to
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$$\sum_{r=1}^{20}(r^2+1)(r!)$$ is equal to
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The number of elements in the set $$S = \left\{x \in \mathbb{R} : 2\cos\left(\frac{x^2+x}{6}\right) = 4^x + 4^{-x}\right\}$$ is
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Let $$m_1, m_2$$ be the slopes of two adjacent sides of a square of side a such that $$a^2 + 11a + 3(m_1^2 + m_2^2) = 220$$. If one vertex of the square is $$10(\cos\alpha - \sin\alpha, \sin\alpha + \cos\alpha)$$, where $$\alpha \in (0, \frac{\pi}{2})$$ and the equation of one diagonal is $$(\cos\alpha - \sin\alpha)x + (\sin\alpha + \cos\alpha)y = 10$$, then $$72(\sin^4\alpha + \cos^4\alpha) + a^2 - 3a + 13$$ is equal to
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Let $$A(\alpha, -2)$$, $$B(\alpha, 6)$$ and $$C\left(\frac{\alpha}{4}, -2\right)$$ be vertices of a $$\Delta ABC$$. If $$\left(5, \frac{\alpha}{4}\right)$$ is the circumcentre of $$\Delta ABC$$, then which of the following is NOT correct about $$\Delta ABC$$
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The statement $$(p \Rightarrow q) \vee (p \Rightarrow r)$$ is NOT equivalent to:
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Which of the following matrices can NOT be obtained from the matrix $$\begin{pmatrix} -1 & 2 \\ 1 & -1 \end{pmatrix}$$ by a single elementary row operation?
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If the system of equations
$$x + y + z = 6$$
$$2x + 5y + \alpha z = \beta$$
$$x + 2y + 3z = 14$$
has infinitely many solutions, then $$\alpha + \beta$$ is equal to
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The domain of the function $$f(x) = \sin^{-1}\left(\frac{x^2-3x+2}{x^2+2x+7}\right)$$ is
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Let the function $$f(x) = \begin{cases} \frac{\log_e(1+5x) - \log_e(1+\alpha x)}{x} & \text{if } x \neq 0 \\ 10 & \text{if } x = 0 \end{cases}$$ be continuous at $$x = 0$$. Then $$\alpha$$ is equal to
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For $$I(x) = \int \frac{\sec^2 x - 2022}{\sin^{2022} x} dx$$, if $$I\left(\frac{\pi}{4}\right) = 2^{1011}$$, then
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If $$[t]$$ denotes the greatest integer $$\leq t$$, then the value of $$\int_0^1 [2x - |3x^2 - 5x + 2| + 1] dx$$ is
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If the solution curve of the differential equation $$\frac{dy}{dx} = \frac{x+y-2}{x-y}$$ passes through the point (2, 1) and (k+1, 2), k > 0, then
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Let $$y = y(x)$$ be the solution curve of the differential equation $$\frac{dy}{dx} + \frac{2x^2+11x+13}{x^3+6x^2+11x+6}y = \frac{x+3}{x+1}$$, $$x > -1$$, which passes through the point (0, 1). Then $$y(1)$$ is equal to
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If $$\langle 2, 3, 9 \rangle$$, $$\langle 5, 2, 1 \rangle$$, $$\langle 1, \lambda, 8 \rangle$$ and $$\langle \lambda, 2, 3 \rangle$$ are coplanar, then the product of all possible values of $$\lambda$$ is
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Let $$\vec{a}, \vec{b}, \vec{c}$$ be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and $$(\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{c}) + (\vec{b} \times \vec{c}) \cdot (\vec{c} \times \vec{a}) + (\vec{c} \times \vec{a}) \cdot (\vec{a} \times \vec{b}) = 168$$ then $$|\vec{a}| + |\vec{b}| + |\vec{c}|$$ is equal to
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Let Q be the foot of perpendicular drawn from the point P(1, 2, 3) to the plane $$x + 2y + z = 14$$. If R is a point on the plane such that $$\angle PRQ = 60°$$, then the area of $$\Delta PQR$$ is equal to
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Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is
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Let $$\alpha, \beta$$ ($$\alpha > \beta$$) be the roots of the quadratic equation $$x^2 - x - 4 = 0$$. If $$P_n = \alpha^n - \beta^n$$, $$n \in \mathbb{N}$$, then $$\frac{P_{15}P_{16} - P_{14}P_{16} - P_{15}^2 + P_{14}P_{15}}{P_{13}P_{14}}$$ is equal to _____
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The number of natural numbers lying between 1012 and 23421 that can be formed using the digits 2, 3, 4, 5, 6 (repetition of digits is not allowed) and divisible by 55 is _____
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If $$\sum_{k=1}^{10} K^2 (10_{C_{K}})^{2} = 22000L$$, then L is equal to _____
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Let AB be a chord of length 12 of the circle $$(x-2)^2 + (y+1)^2 = \frac{169}{4}$$. If tangents drawn to the circle at points A and B intersect at the point P, then five times the distance of point P from chord AB is equal to _____
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Let $$S = \{(x,y) \in \mathbb{N} \times \mathbb{N} : 9(x-3)^2 + 16(y-4)^2 \leq 144\}$$ and $$T = \{(x,y) \in \mathbb{R} \times \mathbb{R} : (x-7)^2 + (y-4)^2 \leq 36\}$$. Then $$n(S \cap T)$$ is equal to _____
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Let $$X = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$$ and $$A = \begin{pmatrix} -1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1 \end{pmatrix}$$. For $$k \in \mathbb{N}$$, if $$X'A^kX = 33$$, then $$k$$ is equal to
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If $$[t]$$ denotes the greatest integer $$\leq t$$, then number of points, at which the function $$f(x) = 4|2x+3| + 9\left[x + \frac{1}{2}\right] - 12[x+20]$$ is not differentiable in the open interval $$(-20, 20)$$, is _____
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If the tangent to the curve $$y = x^3 - x^2 + x$$ at the point (a, b) is also tangent to the curve $$y = 5x^2 + 2x - 25$$ at the point (2, -1), then $$|2a + 9b|$$ is equal to _____
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Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{a}+\vec{b}|^2 = |\vec{a}|^2 + 2|\vec{b}|^2$$, $$\vec{a} \cdot \vec{b} = 3$$ and $$|\vec{a} \times \vec{b}|^2 = 75$$. Then $$|\vec{a}|^2$$ is equal to _____
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The sum and product of the mean and variance of a binomial distribution are 82.5 and 1350 respectively. Then the number of trials in the binomial distribution is
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