The total number of 5-digit numbers, formed by using the digits $$1, 2, 3, 5, 6, 7$$ without repetition, which are multiple of $$6$$, is
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The total number of 5-digit numbers, formed by using the digits $$1, 2, 3, 5, 6, 7$$ without repetition, which are multiple of $$6$$, is
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Let $$A_1, A_2, A_3, \ldots$$ be an increasing geometric progression of positive real numbers. If $$A_1 A_3 A_5 A_7 = \frac{1}{1296}$$ and $$A_2 + A_4 = \frac{7}{36}$$, then the value of $$A_6 + A_8 + A_{10}$$ is equal to
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If $$\sum_{k=1}^{31} \left(^{31}C_{k}\right) \left(^{31}C_{k-1} \right) - \sum_{k=1}^{30} \left(^{30}C_{k}\right) \left(^{30}C_{k-1} \right)= \frac{\alpha(60!)}{(30!)(31!)}$$, where $$\alpha \in R$$, then the value of $$16\alpha$$ is equal to
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If the tangents drawn at the point $$O(0,0)$$ and $$P(1+\sqrt{5}, 2)$$ on the circle $$x^2 + y^2 - 2x - 4y = 0$$ intersect at the point $$Q$$, then the area of the triangle $$OPQ$$ is equal to
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Let the eccentricity of the hyperbola $$H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ be $$\sqrt{\frac{5}{2}}$$ and length of its latus rectum be $$6\sqrt{2}$$. If $$y = 2x + c$$ is a tangent to the hyperbola $$H$$, then the value of $$c^2$$ is equal to
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Let $$p, q, r$$ be three logical statements. Consider the compound statements
$$S_1 : ((\sim p) \vee q) \vee ((\sim p) \vee r)$$ and $$S_2 : p \to (q \vee r)$$
Then, which of the following is NOT true?
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Let $$AB$$ and $$PQ$$ be two vertical poles, $$160$$ m apart from each other. Let $$C$$ be the middle point of $$B$$ and $$Q$$, which are feet of these two poles. Let $$\frac{\pi}{8}$$ and $$\theta$$ be the angles of elevation from $$C$$ to $$P$$ and $$A$$, respectively. If the height of pole $$PQ$$ is twice the height of pole $$AB$$, then $$\tan^2\theta$$ is equal to
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Let $$A$$ be a matrix of order $$3 \times 3$$ and $$\det(A) = 2$$. Then $$\det(\det(A) \text{ adj}(5 \text{ adj}(A^3)))$$ is equal to
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If the system of linear equations
$$2x + 3y - z = -2$$
$$x + y + z = 4$$
$$x - y + |\lambda|z = 4\lambda - 4$$ where $$\lambda \in \mathbb{R}$$,
has no solution, then
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Let a function $$f : \mathbb{N} \to \mathbb{N}$$ be defined by
$$f(n) = \begin{cases} 2n, & n = 2, 4, 6, 8, \ldots \\ n-1, & n = 3, 7, 11, 15, \ldots \\ \frac{n+1}{2}, & n = 1, 5, 9, 13, \ldots \end{cases}$$
then, $$f$$ is
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Let $$f : \mathbb{R} \to \mathbb{R}$$ be defined as $$f(x) = \begin{cases} [e^x], & x < 0 \\ ae^x + [x-1], & 0 \leq x < 1 \\ b + [\sin(\pi x)], & 1 \leq x < 2 \\ [e^{-x}] - c, & x \geq 2 \end{cases}$$
where $$a, b, c \in \mathbb{R}$$ and $$[t]$$ denotes greatest integer less than or equal to $$t$$. Then, which of the following statements is true?
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The number of real solutions of $$x^7 + 5x^3 + 3x + 1 = 0$$ is equal to ______
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Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Then the value of the integral $$\int_0^1 [-8x^2 + 6x - 1] dx$$ is equal to
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The area of the region $$S = \{(x,y) : y^2 \leq 8x, y \geq \sqrt{2}x, x \geq 1\}$$ is
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Let the solution curve $$y = y(x)$$ of the differential equation, $$\left[\frac{x}{\sqrt{x^2-y^2}} + e^{y/x}\right]x\frac{dy}{dx} = x + \left[\frac{x}{\sqrt{x^2-y^2}} + e^{y/x}\right]y$$ pass through the points $$(1, 0)$$ and $$(2\alpha, \alpha), \alpha > 0$$. Then $$\alpha$$ is equal to
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Let $$y = y(x)$$ be the solution of the differential equation $$x(1 - x^2)\frac{dy}{dx} + (3x^2y - y - 4x^3) = 0, x > 1$$ with $$y(2) = -2$$. Then $$y(3)$$ is equal to
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If two distinct point $$Q$$, $$R$$ lie on the line of intersection of the planes $$-x + 2y - z = 0$$ and $$3x - 5y + 2z = 0$$ and $$PQ = PR = \sqrt{18}$$ where the point $$P$$ is $$(1, -2, 3)$$, then the area of the triangle $$PQR$$ is equal to
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The acute angle between the planes $$P_1$$ and $$P_2$$, when $$P_1$$ and $$P_2$$ are the planes passing through the intersection of the planes $$5x + 8y + 13z - 29 = 0$$ and $$8x - 7y + z - 20 = 0$$ and the points $$(2, 1, 3)$$ and $$(0, 1, 2)$$, respectively, is
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Let the plane $$P : \vec{r} \cdot \vec{a} = d$$ contain the line of intersection of two planes $$\vec{r} \cdot (\hat{i} + 3\hat{j} - \hat{k}) = 6$$ and $$\vec{r} \cdot (-6\hat{i} + 5\hat{j} - \hat{k}) = 7$$. If the plane $$P$$ passes through the point $$(2, 3, \frac{1}{2})$$, then the value of $$\frac{|13\vec{a}|^2}{d^2}$$ is equal to
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The probability, that in a randomly selected 3-digit number at least two digits are odd, is
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The number of real solutions of the equation $$e^{4x} + 4e^{3x} - 58e^{2x} + 4e^x + 1 = 0$$ is ______
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The number of elements in the set $$\{z = a + ib \in C : a, b \in \mathbb{Z}$$ and $$1 < |z - 3 + 2i| < 4\}$$ is ______
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The number of positive integers $$k$$ such that the constant term in the binomial expansion of $$\left(2x^3 + \frac{3}{x^k}\right)^{12}, x \neq 0$$ is $$2^8 \cdot l$$, where $$l$$ is an odd integer, is ______
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A ray of light passing through the point $$P(2, 3)$$ reflects on the $$X$$-axis at point $$A$$ and the reflected ray passes through the point $$Q(5, 4)$$. Let $$R$$ be the point that divides the line segment $$AQ$$ internally into the ratio $$2 : 1$$. Let the co-ordinates of the foot of the perpendicular $$M$$ from $$R$$ on the bisector of the angle $$PAQ$$ be $$(\alpha, \beta)$$. Then, the value of $$7\alpha + 3\beta$$ is equal to ______
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Let the lines $$y + 2x = \sqrt{11} + 7\sqrt{7}$$ and $$2y + x = 2\sqrt{11} + 6\sqrt{7}$$ be normal to a circle $$C : (x-h)^2 + (y-k)^2 = r^2$$. If the line $$\sqrt{11}y - 3x = \frac{5\sqrt{77}}{3} + 11$$ is tangent to the circle $$C$$, then the value of $$(5h - 8k)^2 + 5r^2$$ is equal to ______
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The mean and standard deviation of 15 observations are found to be $$8$$ and $$3$$ respectively. On rechecking it was found that, in the observations, $$20$$ was misread as $$5$$. Then, the correct variance is equal to ______
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Let $$R_1$$ and $$R_2$$ be relations on the set $$\{1, 2, \ldots, 50\}$$ such that $$R_1 = \{(p, p^n) : p$$ is a prime and $$n \geq 0$$ is an integer$$\}$$ and $$R_2 = \{(p, p^n) : p$$ is a prime and $$n = 0$$ or $$1\}$$. Then, the number of elements in $$R_1 - R_2$$ is ______
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Let $$A = \{1, a_1, a_2, \ldots a_{18}, 77\}$$ be a set of integers with $$1 < a_1 < a_2 < \ldots < a_{18} < 77$$. Let the set $$A + A = \{x + y : x, y \in A\}$$ contain exactly $$39$$ elements. Then, the value of $$a_1 + a_2 + \ldots + a_{18}$$ is equal to ______
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Let $$l$$ be a line which is normal to the curve $$y = 2x^2 + x + 2$$ at a point $$P$$ on the curve. If the point $$Q(6, 4)$$ lies on the line $$l$$ and $$O$$ is origin, then the area of the triangle $$OPQ$$ is equal to ______
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If $$\vec{a} = 2\hat{i} + \hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} + 3\hat{j} + \hat{k}$$ and $$\vec{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}$$ are coplanar vectors and $$\vec{a} \cdot \vec{c} = 5, \vec{b} \perp \vec{c}$$, then $$122(c_1 + c_2 + c_3)$$ is equal to ______
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