The number of ways of selecting two numbers $$a$$ and $$b$$, $$a \in \{2, 4, 6, \ldots, 100\}$$ and $$b \in \{1, 3, 5, \ldots, 99\}$$ such that $$2$$ is the remainder when $$a + b$$ is divided by $$23$$ is
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The number of ways of selecting two numbers $$a$$ and $$b$$, $$a \in \{2, 4, 6, \ldots, 100\}$$ and $$b \in \{1, 3, 5, \ldots, 99\}$$ such that $$2$$ is the remainder when $$a + b$$ is divided by $$23$$ is
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Let $$a, b, c > 1$$, $$a^3, b^3$$ and $$c^3$$ be in A.P. and $$\log_a b$$, $$\log_c a$$ and $$\log_b c$$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is $$\frac{a+4b+c}{3}$$ and the common difference is $$\frac{a-8b+c}{10}$$ is $$-444$$, then $$abc$$ is equal to
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Let $$x = \left(8\sqrt{3} + 13\right)^{13}$$ and $$y = \left(7\sqrt{2} + 9\right)^{9}$$. If $$[t]$$ denotes the greatest integer $$\leq t$$, then
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The parabolas: $$ax^2 + 2bx + cy = 0$$ and $$d^2 + 2ex + fy = 0$$ intersect on the line $$y = 1$$. If $$a, b, c, d, e, f$$ are positive real numbers and $$a, b, c$$ are in G.P., then
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Let $$A$$ be a point on the $$x$$-axis. Common tangents are drawn from $$A$$ to the curves $$x^2 + y^2 = 8$$ and $$y^2 = 16x$$. If one of these tangents touches the two curves at $$Q$$ and $$R$$, then $$(QR)^2$$ is equal to
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Let $$f, g$$ and $$h$$ be the real valued functions defined on $$\mathbb{R}$$ as
$$f(x) = \begin{cases} \frac{x}{|x|}, & x \neq 0 \\ 1, & x = 0 \end{cases}$$, $$g(x) = \begin{cases} \frac{\sin(x+1)}{(x+1)}, & x \neq -1 \\ 1, & x = -1 \end{cases}$$ and $$h(x) = 2[x] - f(x)$$, where $$[x]$$ is the greatest integer $$\leq x$$. Then the value of $$\lim_{x \to 1} g(h(x-1))$$ is
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Consider the following statements:
$$P$$: I have fever
$$Q$$: I will not take medicine
$$R$$: I will take rest
The statement "If I have fever, then I will take medicine and I will take rest" is equivalent to:
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Let $$S$$ be the set of all values of $$a_1$$ for which the mean deviation about the mean of $$100$$ consecutive positive integers $$a_1, a_2, a_3, \ldots, a_{100}$$ is $$25$$. Then $$S$$ is
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If $$P$$ is a $$3 \times 3$$ real matrix such that $$P^T = aP + (a-1)I$$, where $$a > 1$$, then
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For $$\alpha, \beta \in \mathbb{R}$$, suppose the system of linear equations
$$x - y + z = 5$$
$$2x + 2y + \alpha z = 8$$
$$3x - y + 4z = \beta$$
has infinitely many solutions. Then $$\alpha$$ and $$\beta$$ are the roots of
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Let $$a_1 = 1, a_2, a_3, a_4, \ldots$$ be consecutive natural numbers. Then $$\tan^{-1}\left(\frac{1}{1+a_1a_2}\right) + \tan^{-1}\left(\frac{1}{1+a_2a_3}\right) + \ldots + \tan^{-1}\left(\frac{1}{1+a_{2021}a_{2022}}\right)$$ is equal to
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The range of the function $$f(x) = \sqrt{3-x} + \sqrt{2+x}$$ is
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If the functions $$f(x) = \frac{x^3}{3} + 2bx + \frac{ax^2}{2}$$ and $$g(x) = \frac{x^3}{3} + ax + bx^2$$, $$a \neq 2b$$ have a common extreme point, then $$a + 2b + 7$$ is equal to
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$$\lim_{n \to \infty} \frac{3}{n}\left\{4 + \left(2 + \frac{1}{n}\right)^2 + \left(2 + \frac{2}{n}\right)^2 + \ldots + \left(3 - \frac{1}{n}\right)^2\right\}$$ is equal to
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Let $$q$$ be the maximum integral value of $$p$$ in $$[0, 10]$$ for which the roots of the equation $$x^2 - px + \frac{5}{4}p = 0$$ are rational. Then the area of the region $$\{(x,y) : 0 \leq y \leq (x-q)^2, 0 \leq x \leq q\}$$ is
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The solution of the differential equation $$\frac{dy}{dx} = -\left(\frac{x^2+3y^2}{3x^2+y^2}\right)$$, $$y(1) = 0$$ is
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Let $$\lambda \in \mathbb{R}$$, $$\vec{a} = \lambda\hat{i} + 2\hat{j} - 3\hat{k}$$, $$\vec{b} = \hat{i} - \lambda\hat{j} + 2\hat{k}$$. If $$\left((\vec{a}+\vec{b}) \times (\vec{a} \times \vec{b})\right) \times (\vec{a}-\vec{b}) = 8\hat{i} - 40\hat{j} - 24\hat{k}$$ then $$\left|\lambda(\vec{a}+\vec{b}) \times (\vec{a}-\vec{b})\right|^2$$ is equal to
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Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors. Let $$|\vec{a}| = 1$$, $$|\vec{b}| = 4$$ and $$\vec{a} \cdot \vec{b} = 2$$. If $$\vec{c} = (2\vec{a} \times \vec{b}) - 3\vec{b}$$, then the value of $$\vec{b} \cdot \vec{c}$$ is
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A vector $$\vec{v}$$ in the first octant is inclined to the $$x$$ axis at $$60°$$, to the $$y$$-axis at $$45°$$ and to the $$z$$-axis at an acute angle. If a plane passing through the points $$(\sqrt{2}, -1, 1)$$ and $$(a, b, c)$$, is normal to $$\vec{v}$$, then
If a plane passes through the points $$(-1, k, 0)$$, $$(2, k, -1)$$, $$(1, 1, 2)$$ and is parallel to the line $$\frac{x-1}{1} = \frac{2y+1}{2} = \frac{z+1}{-1}$$, then the value of $$\frac{k^2+1}{(k-1)(k-2)}$$ is
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If the value of real number $$\alpha \gt 0$$ for which $$x^2 - 5\alpha x + 1 = 0$$ and $$x^2 - \alpha x - 5 = 0$$ have a common real roots is $$\frac{3}{\sqrt{2\beta}}$$ then $$\beta$$ is equal to ______.
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The number of seven digits odd numbers, that can be formed using all the seven digits 1, 2, 2, 2, 3, 3, 5 is ______.
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The $$8^{th}$$ common term of the series
$$S_1 = 3 + 7 + 11 + 15 + 19 + \ldots$$
$$S_2 = 1 + 6 + 11 + 16 + 21 + \ldots$$ is
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$$50^{th}$$ root of a number $$x$$ is $$12$$ and $$50^{th}$$ root of another number $$y$$ is $$18$$. Then the remainder obtained on dividing $$(x + y)$$ by $$25$$ is ______.
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Let $$P(a_1, b_1)$$ and $$Q(a_2, b_2)$$ be two distinct points on a circle with center $$C(\sqrt{2}, \sqrt{3})$$. Let $$O$$ be the origin and $$OC$$ be perpendicular to both $$CP$$ and $$CQ$$. If the area of the triangle $$OCP$$ is $$\frac{\sqrt{35}}{2}$$, then $$a_1^2 + a_2^2 + b_1^2 + b_2^2$$ is equal to ______.
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Let $$A = \{1, 2, 3, 5, 8, 9\}$$. Then the number of possible functions $$f : A \to A$$ such that $$f(m \cdot n) = f(m) \cdot f(n)$$ for every $$m, n \in A$$ with $$m \cdot n \in A$$ is equal to
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If $$\int \sqrt{\sec 2x - 1} dx = \alpha \log_e \left|\cos 2x + \beta + \sqrt{\cos 2x\left(1 + \cos\frac{1}{\beta}x\right)}\right|$$ + constant, then $$\beta - \alpha$$ is equal to
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Let $$A$$ be the area of the region $$\{(x,y) : y \geq x^2, y \geq (1-x)^2, y \leq 2x(1-x)\}$$. Then $$540A$$ is equal to
Let a line $$L$$ pass through the point $$P(2, 3, 1)$$ and be parallel to the line $$x + 3y - 2z - 2 = 0 = x - y + 2z$$. If the distance of $$L$$ from the point $$(5, 3, 8)$$ is $$\alpha$$, then $$3\alpha^2$$ is equal to ______.
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A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is $$p$$. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colours is $$q$$. If $$p : q = m : n$$, where $$m$$ and $$n$$ are co-prime, then $$m + n$$ is equal to
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