If $$z_1, z_2$$ are two distinct complex numbers such that $$\left|\frac{z_1 - 2z_2}{\frac{1}{2} - z_1\bar{z}_2}\right| = 2$$, then
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If $$z_1, z_2$$ are two distinct complex numbers such that $$\left|\frac{z_1 - 2z_2}{\frac{1}{2} - z_1\bar{z}_2}\right| = 2$$, then
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Let $$0 \leq r \leq n$$. If $$^{n+1}C_{r+1} : ^{n}C_{r} : ^{n-1}C_{r-1} = 55 : 35 : 21$$, then $$2n + 5r$$ is equal to:
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If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $$315^{th}$$ position in this arrangement is :
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Let $$ABC$$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $$ABC$$ and the same process is repeated infinitely many times. If $$P$$ is the sum of perimeters and $$Q$$ is be the sum of areas of all the triangles formed in this process, then :
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A software company sets up $$m$$ number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $$m$$ is equal to:
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If $$P(6, 1)$$ be the orthocentre of the triangle whose vertices are $$A(5, -2)$$, $$B(8, 3)$$ and $$C(h, k)$$, then the point $$C$$ lies on the circle:
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If the locus of the point, whose distances from the point $$(2, 1)$$ and $$(1, 3)$$ are in the ratio $$5 : 4$$, is $$ax^2 + by^2 + cxy + dx + ey + 170 = 0$$, then the value of $$a^2 + 2b + 3c + 4d + e$$ is equal to :
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$$\lim_{n \to \infty} \frac{(1^2 - 1)(n-1) + (2^2 - 2)(n-2) + \cdots + ((n-1)^2 - (n-1)) \cdot 1}{(1^3 + 2^3 + \cdots + n^3) - (1^2 + 2^2 + \cdots + n^2)}$$ is equal to :
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Let $$A = \{1, 2, 3, 4, 5\}$$. Let $$R$$ be a relation on $$A$$ defined by $$xRy$$ if and only if $$4x \leq 5y$$. Let $$m$$ be the number of elements in $$R$$ and $$n$$ be the minimum number of elements from $$A \times A$$ that are required to be added to $$R$$ to make it a symmetric relation. Then $$m + n$$ is equal to :
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If $$A$$ is a square matrix of order 3 such that $$\det(A) = 3$$ and $$\det(\text{adj}(-4 \text{adj}(-3 \text{adj}(3 \text{adj}((2A)^{-1}))))) = 2^m 3^n$$, then $$m + 2n$$ is equal to :
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Let $$f(x) = \frac{1}{7 - \sin 5x}$$ be a function defined on $$\mathbb{R}$$. Then the range of the function $$f(x)$$ is equal to :
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Suppose for a differentiable function $$h$$, $$h(0) = 0$$, $$h(1) = 1$$ and $$h'(0) = h'(1) = 2$$. If $$g(x) = h(e^x)e^{h(x)}$$, then $$g'(0)$$ is equal to:
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If the function $$f(x) = \left(\frac{1}{x}\right)^{2x}; x \gt 0$$ attains the maximum value at $$x = \frac{1}{e}$$ then :
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If $$\int \frac{1}{a^2 \sin^2 x + b^2 \cos^2 x} dx = \frac{1}{12} \tan^{-1}(3 \tan x) +$$ constant, then the maximum value of $$a \sin x + b \cos x$$, is :
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If the area of the region $$\{(x, y) : \frac{a}{x^2} \leq y \leq \frac{1}{x}, 1 \leq x \leq 2, 0 < a < 1\}$$ is $$(\log_e 2) - \frac{1}{7}$$ then the value of $$7a - 3$$ is equal to:
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Suppose the solution of the differential equation $$\frac{dy}{dx} = \frac{(2+\alpha)x - \beta y + 2}{\beta x - 2\alpha y - (\beta\gamma - 4\alpha)}$$ represents a circle passing through origin. Then the radius of this circle is :
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Let $$\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$$, $$\vec{b} = ((\vec{a} \times (\hat{i} + \hat{j})) \times \hat{i}) \times \hat{i}$$. Then the square of the projection of $$\vec{a}$$ on $$\vec{b}$$ is :
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Let $$\vec{a} = 6\hat{i} + \hat{j} - \hat{k}$$ and $$\vec{b} = \hat{i} + \hat{j}$$. If $$\vec{c}$$ is a vector such that $$|\vec{c}| \geq 6$$, $$\vec{a} \cdot \vec{c} = 6|\vec{c}|$$, $$|\vec{c} - \vec{a}| = 2\sqrt{2}$$ and the angle between $$\vec{a} \times \vec{b}$$ and $$\vec{c}$$ is $$60°$$, then $$|(\vec{a} \times \vec{b}) \times \vec{c}|$$ is equal to:
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Let $$P(\alpha, \beta, \gamma)$$ be the image of the point $$Q(3, -3, 1)$$ in the line $$\frac{x-0}{1} = \frac{y-3}{1} = \frac{z-1}{-1}$$ and $$R$$ be the point $$(2, 5, -1)$$. If the area of the triangle $$PQR$$ is $$\lambda$$ and $$\lambda^2 = 14K$$, then $$K$$ is equal to :
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If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is:
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Let $$\alpha, \beta$$ be roots of $$x^2 + \sqrt{2}x - 8 = 0$$. If $$U_n = \alpha^n + \beta^n$$, then $$\frac{U_{10} + \sqrt{2}U_9}{2U_8}$$ is equal to ___________
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If $$S(x) = (1+x) + 2(1+x)^2 + 3(1+x)^3 + \cdots + 60(1+x)^{60}$$, $$x \neq 0$$, and $$(60)^2 S(60) = a(b)^b + b$$, where $$a, b \in N$$, then $$(a + b)$$ equal to ___________
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The length of the latus rectum and directrices of a hyperbola with eccentricity $$e$$ are 9 and $$x = \pm \frac{4}{\sqrt{13}}$$, respectively. Let the line $$y - \sqrt{3}x + \sqrt{3} = 0$$ touch this hyperbola at $$(x_0, y_0)$$. If $$m$$ is the product of the focal distances of the point $$(x_0, y_0)$$, then $$4e^2 + m$$ is equal to ___________
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In a triangle $$ABC$$, $$BC = 7$$, $$AC = 8$$, $$AB = \alpha \in \mathbb{N}$$ and $$\cos A = \frac{2}{3}$$. If $$49\cos(3C) + 42 = \frac{m}{n}$$, where $$\gcd(m, n) = 1$$, then $$m + n$$ is equal to ___________
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If the system of equations $$2x + 7y + \lambda z = 3$$, $$3x + 2y + 5z = 4$$, $$x + \mu y + 32z = -1$$ has infinitely many solutions, then $$(\lambda - \mu)$$ is equal to ___________
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Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Let $$f : [0, \infty) \rightarrow \mathbb{R}$$ be a function defined by $$f(x) = \left[\frac{x}{2} + 3\right] - [\sqrt{x}]$$. Let $$S$$ be the set of all points in the interval $$[0, 8]$$ at which $$f$$ is not continuous. Then $$\sum_{a \in S} a$$ is equal to ___________
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Let $$[t]$$ denote the largest integer less than or equal to $$t$$. If $$\int_0^3 \left([x^2] + \left[\frac{x^2}{2}\right]\right) dx = a + b\sqrt{2} - \sqrt{3} - \sqrt{5} + c\sqrt{6} - \sqrt{7}$$, where $$a, b, c \in \mathbb{Z}$$, then $$a + b + c$$ is equal to ___________
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If the solution $$y(x)$$ of the given differential equation $$(e^y + 1)\cos x \, dx + e^y \sin x \, dy = 0$$ passes through the point $$\left(\frac{\pi}{2}, 0\right)$$, then the value of $$e^{y\left(\frac{\pi}{6}\right)}$$ is equal to ___________
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If the shortest distance between the lines $$\frac{x - \lambda}{3} = \frac{y - 2}{-1} = \frac{z - 1}{1}$$ and $$\frac{x + 2}{-3} = \frac{y + 5}{2} = \frac{z - 4}{4}$$ is $$\frac{44}{\sqrt{30}}$$, then the largest possible value of $$|\lambda|$$ is equal to ___________
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From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable $$X$$ denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of $$X$$ is $$\frac{m}{n}$$, where $$\gcd(m, n) = 1$$, then $$n - m$$ is equal to ___________
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