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Let $$\alpha$$ and $$\beta$$ be the roots of $$x^2 - 6x - 2 = 0$$. If $$a_n = \alpha^n - \beta^n$$ for $$n \geq 1$$, then the value of $$\dfrac{a_{10} - 2a_8}{3a_9}$$ is:
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If $$\alpha, \beta \in R$$ are such that $$1 - 2i$$ (here $$i^2 = -1$$) is a root of $$z^2 + \alpha z + \beta = 0$$, then $$(\alpha - \beta)$$ is equal to:
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The minimum value of $$f(x) = a^{a^x} + a^{1 - a^x}$$, where $$a, x \in R$$ and $$a > 0$$, is equal to:
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If $$0 < x, y < \pi$$ and $$\cos x + \cos y - \cos(x + y) = \frac{3}{2}$$, then $$\sin x + \cos y$$ is equal to:
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If the curve $$x^2 + 2y^2 = 2$$ intersects the line $$x + y = 1$$ at two points $$P$$ and $$Q$$, then the angle subtended by the line segment $$PQ$$ at the origin is
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A hyperbola passes through the foci of the ellipse $$\frac{x^2}{25} + \frac{y^2}{16} = 1$$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is:
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The contrapositive of the statement "If you will work, you will earn money" is:
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If for the matrix, $$A = \begin{bmatrix} 1 & -\alpha \\ \alpha & \beta \end{bmatrix}$$, $$AA^T = I_2$$, then the value of $$\alpha^4 + \beta^4$$ is:
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Let $$A$$ be a $$3 \times 3$$ matrix with det$$(A) = 4$$. Let $$R_i$$ denote the $$i^{th}$$ row of $$A$$. If a matrix $$B$$ is obtained by performing the operation $$R_2 \to 2R_2 + 5R_3$$ on $$2A$$, then det$$(B)$$ is equal to:
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The following system of linear equations
$$2x + 3y + 2z = 9$$
$$3x + 2y + 2z = 9$$
$$x - y + 4z = 8$$
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$$\operatorname{cosec}\left[2\cot^{-1}(5) + \cos^{-1}\left(\frac{4}{5}\right)\right]$$ is equal to:
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A function $$f(x)$$ is given by $$f(x) = \frac{5^x}{5^x + 5}$$, then the sum of the series $$f\left(\frac{1}{20}\right) + f\left(\frac{2}{20}\right) + f\left(\frac{3}{20}\right) + \ldots + f\left(\frac{39}{20}\right)$$ is equal to:
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Let $$x$$ denote the total number of one-one functions from a set $$A$$ with 3 elements to a set $$B$$ with 5 elements and $$y$$ denote the total number of one-one functions from the set $$A$$ to the set $$A \times B$$. Then:
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The shortest distance between the line $$x - y = 1$$ and the curve $$x^2 = 2y$$ is:
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The integral $$\int \frac{e^{3\log_e 2x} + 5e^{2\log_e 2x}}{e^{4\log_e x} + 5e^{3\log_e x} - 7e^{2\log_e x}} dx$$, $$x > 0$$, is equal to (where $$c$$ is a constant of integration)
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If $$I_n = \int_{\pi/4}^{\pi/2} \cot^n x \, dx$$, then
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$$\lim_{n \to \infty} \left[\frac{1}{n} + \frac{n}{(n+1)^2} + \frac{n}{(n+2)^2} + \ldots + \frac{n}{(2n-1)^2}\right]$$ is equal to
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A plane passes through the points $$A(1, 2, 3)$$, $$B(2, 3, 1)$$ and $$C(2, 4, 2)$$. If $$O$$ is the origin and $$P$$ is $$(2, -1, 1)$$, then the projection of $$\vec{OP}$$ on this plane is of length:
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In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chances of getting a particular chest disorder are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is:
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Let $$A$$ be a set of all 4-digit natural numbers whose exactly one digit is 7. Then the probability that a randomly chosen element of $$A$$ leaves remainder 2 when divided by 5 is:
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The total number of two digit numbers $$'n'$$, such that $$3^n + 7^n$$ is a multiple of 10, is ______.
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If the remainder when $$x$$ is divided by 4 is 3, then the remainder when $$(2020 + x)^{2022}$$ is divided by 8 is ______.
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A line is a common tangent to the circle $$(x - 3)^2 + y^2 = 9$$ and the parabola $$y^2 = 4x$$. If the two points of contact $$(a, b)$$ and $$(c, d)$$ are distinct and lie in the first quadrant, then $$2(a + c)$$ is equal to ______.
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If $$\lim_{x \to 0} \frac{ax - (e^{4x} - 1)}{ax(e^{4x} - 1)}$$ exists and is equal to $$b$$, then the value of $$a - 2b$$ is ______.
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A function $$f$$ is defined on $$[-3, 3]$$ as
$$f(x) = \begin{cases} \min\{|x|, 2 - x^2\}, & -2 \leq x \leq 2 \\ [|x|], & 2 < |x| \leq 3 \end{cases}$$
where $$[x]$$ denotes the greatest integer $$\leq x$$. The number of points, where $$f$$ is not differentiable in $$(-3, 3)$$ is ______.
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If the curves $$x = y^4$$ and $$xy = k$$ cut at right angles, then $$(4k)^6$$ is equal to ______.
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The value of $$\int_{-2}^{2} |3x^2 - 3x - 6| dx$$ is ______
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If the curve, $$y = y(x)$$ represented by the solution of the differential equation $$(2xy^2 - y)dx + x \, dy = 0$$, passes through the intersection of the lines, $$2x - 3y = 1$$ and $$3x + 2y = 8$$, then $$|y(1)|$$ is equal to ______.
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Let $$\vec{a} = \hat{i} + \alpha\hat{j} + 3\hat{k}$$ and $$\vec{b} = 3\hat{i} - \alpha\hat{j} + \hat{k}$$. If the area of the parallelogram whose adjacent sides are represented by the vectors $$\vec{a}$$ and $$\vec{b}$$ is $$8\sqrt{3}$$ square units, then $$\vec{a} \cdot \vec{b}$$ is equal to ______.
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A line $$l$$ passing through origin is perpendicular to the lines
$$l_1: \vec{r} = (3 + t)\hat{i} + (-1 + 2t)\hat{j} + (4 + 2t)\hat{k}$$
$$l_2: \vec{r} = (3 + 2s)\hat{i} + (3 + 2s)\hat{j} + (2 + s)\hat{k}$$
If the co-ordinates of the point in the first octant on $$l_2$$ at a distance of $$\sqrt{17}$$ from the point of intersection of $$l$$ and $$l_1$$ are $$(a, b, c)$$, then $$18(a + b + c)$$ is equal to ______.
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