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Let $$\alpha$$ and $$\beta$$ be the roots of the equation $$x^2 + (2i - 1) = 0$$. Then, the value of $$|\alpha^8 + \beta^8|$$ is equal to
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Let $$\{a_n\}_{n=0}^{\infty}$$ be a sequence such that $$a_0 = a_1 = 0$$ and $$a_{n+2} = 2a_{n+1} - a_n + 1$$ for all $$n \geq 0$$. Then, $$\sum_{n=2}^{\infty} \frac{a_n}{7^n}$$ is equal to
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If the constant term in the expansion of $$\left(3x^3 - 2x^2 + \frac{5}{x^5}\right)^{10}$$ is $$2^k \cdot l$$, where $$l$$ is an odd integer, then the value of $$k$$ is equal to
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The distance between the two points $$A$$ and $$A'$$ which lie on $$y = 2$$ such that both the line segments $$AB$$ and $$A'B$$ (where $$B$$ is the point $$(2, 3)$$) subtend angle $$\frac{\pi}{4}$$ at the origin, is equal to
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Let the tangent to the circle $$C_1 : x^2 + y^2 = 2$$ at the point $$M(-1, 1)$$ intersect the circle $$C_2 : (x-3)^2 + (y-2)^2 = 5$$, at two distinct points $$A$$ and $$B$$. If the tangents to $$C_2$$ at the points $$A$$ and $$B$$ intersect at $$N$$, then the area of the triangle $$ANB$$ is equal to
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Let $$PQ$$ be a focal chord of the parabola $$y^2 = 4x$$ such that it subtends an angle of $$\frac{\pi}{2}$$ at the point $$(3, 0)$$. Let the line segment $$PQ$$ be also a focal chord of the ellipse $$E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, $$a^2 > b^2$$. If $$e$$ is the eccentricity of the ellipse $$E$$, then the value of $$\frac{1}{e^2}$$ is equal to
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Let $$\Delta \in \{\wedge, \vee, \Rightarrow, \Leftrightarrow\}$$ be such that $$(p \wedge q)\Delta((p \vee q) \Rightarrow q)$$ is a tautology. Then $$\Delta$$ is equal to
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Let the mean and the variance of 5 observations $$x_1, x_2, x_3, x_4, x_5$$ be $$\frac{24}{5}$$ and $$\frac{194}{25}$$ respectively. If the mean and variance of the first 4 observations are $$\frac{7}{2}$$ and $$a$$ respectively, then $$(4a + x_5)$$ is equal to
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Let a set $$A = A_1 \cup A_2 \cup \ldots \cup A_k$$, where $$A_i \cap A_j = \phi$$ for $$i \neq j$$; $$1 \leq i, j \leq k$$. Define the relation $$R$$ from $$A$$ to $$A$$ by $$R = \{(x,y) : y \in A_i$$ if and only if $$x \in A_i, 1 \leq i \leq k\}$$. Then, $$R$$ is:
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The probability that a randomly chosen $$2 \times 2$$ matrix with all the entries from the set of first 10 primes, is singular, is equal to
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Let $$A = [a_{ij}]$$ be a square matrix of order 3 such that $$a_{ij} = 2^{j-i}$$, for all $$i, j = 1, 2, 3$$. Then, the matrix $$A^2 + A^3 + \ldots + A^{10}$$ is equal to
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If the system of linear equations
$$2x + y - z = 7$$
$$x - 3y + 2z = 1$$
$$x + 4y + \delta z = k$$, where $$\delta, k \in R$$
has infinitely many solutions, then $$\delta + k$$ is equal to
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The domain of the function $$\cos^{-1}\left(\frac{2\sin^{-1}\left(\frac{1}{4x^2-1}\right)}{\pi}\right)$$ is
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Let $$f : R \to R$$ be a function defined by:
$$f(x) = \begin{cases} \max\{t^3 - 3t\}; & t \leq x, \quad x \leq 2 \\ x^2 + 2x - 6; & 2 < x < 3 \\ [x-3] + 9; & 3 \leq x \leq 5 \\ 2x + 1; & x > 5 \end{cases}$$
Where $$[t]$$ is the greatest integer less than or equal to $$t$$. Let $$m$$ be the number of points where $$f$$ is not differentiable and $$I = \int_{-2}^{2} f(x) dx$$. Then the ordered pair $$(m, I)$$ is equal to
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A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is
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$$\int_0^5 \cos\left(\pi\left(x - \left[\frac{x}{2}\right]\right)\right)dx$$, where $$[t]$$ denotes greatest integer less than or equal to $$t$$, is equal to
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The area enclosed by $$y^2 = 8x$$ and $$y = \sqrt{2}x$$ that lies outside the triangle formed by $$y = \sqrt{2}x$$, $$x = 1$$, $$y = 2\sqrt{2}$$, is equal to
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Let the solution curve of the differential equation $$x\frac{dy}{dx} - y = \sqrt{y^2 + 16x^2}$$, $$y(1) = 3$$ be $$y = y(x)$$. Then $$y(2)$$ is equal to
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Let $$\vec{a} = \alpha \hat{i} + 3\hat{j} - \hat{k}$$, $$\vec{b} = 3\hat{i} - \beta \hat{j} + 4\hat{k}$$ and $$\vec{c} = \hat{i} + 2\hat{j} - 2\hat{k}$$ where $$\alpha, \beta \in R$$. If the projection of $$\vec{a}$$ on $$\vec{c}$$ is $$\frac{10}{3}$$ and $$\vec{b} \times \vec{c} = -6\hat{i} + 10\hat{j} + 7\hat{k}$$, then the value of $$\alpha + \beta$$ equal to
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If the mirror image of the point $$(2, 4, 7)$$ in the plane $$3x - y + 4z = 2$$ is $$(a, b, c)$$, the $$2a + b + 2c$$ is equal to
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Let $$S = \{z \in C : |z - 2| \leq 1, z(1+i) + \bar{z}(1-i) \leq 2\}$$. Let $$|z - 4i|$$ attains minimum and maximum values, respectively, at $$z_1 \in S$$ and $$z_2 \in S$$. If $$5(|z_1|^2 + |z_2|^2) = \alpha + \beta\sqrt{5}$$, where $$\alpha$$ and $$\beta$$ are integers, then the value of $$\alpha + \beta$$ is equal to ______.
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Let $$b_1b_2b_3b_4$$ be a 4-element permutation with $$b_i \in \{1, 2, 3, \ldots, 100\}$$ for $$1 \leq i \leq 4$$ and $$b_i \neq b_j$$ for $$i \neq j$$, such that either $$b_1, b_2, b_3$$ are consecutive integers or $$b_2, b_3, b_4$$ are consecutive integers. Then the number of such permutations $$b_1b_2b_3b_4$$ is equal to ______.
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The number of elements in the set $$S = \{\theta \in [-4\pi, 4\pi] : 3\cos^2 2\theta + 6\cos 2\theta - 10\cos^2\theta + 5 = 0\}$$ is ______.
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The number of solutions of the equation $$2\theta - \cos^2\theta + \sqrt{2} = 0$$ in $$R$$ is equal to ______.
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Let $$H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$, $$a > 0$$, $$b > 0$$, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $$4(2\sqrt{2} + \sqrt{14})$$. If the eccentricity $$H$$ is $$\frac{\sqrt{11}}{2}$$, then value of $$a^2 + b^2$$ is equal to ______.
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$$50\tan\left(3\tan^{-1}\left(\frac{1}{2}\right) + 2\cos^{-1}\left(\frac{1}{\sqrt{5}}\right)\right) + 4\sqrt{2}\tan\left(\frac{1}{2}\tan^{-1}(2\sqrt{2})\right)$$ is equal to ______.
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Let $$c, k \in R$$. If $$f(x) = (c+1)x^2 + (1-c^2)x + 2k$$ and $$f(x+y) = f(x) + f(y) - xy$$, for all $$x, y \in R$$, then the value of $$|2(f(1) + f(2) + f(3) + \ldots + f(20))|$$ is equal to ______.
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Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} + \frac{\sqrt{2}y}{2\cos^4x - \cos 2x} = xe^{\tan^{-1}(\sqrt{2}\cot 2x)}$$, $$0 < x < \frac{\pi}{2}$$ with $$y\left(\frac{\pi}{4}\right) = \frac{\pi^2}{32}$$. If $$y\left(\frac{\pi}{3}\right) = \frac{\pi^2}{18}e^{-\tan^{-1}(\alpha)}$$, then the value of $$3\alpha^2$$ is equal to ______.
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Let $$d$$ be the distance between the foot of perpendiculars of the points $$P(1, 2, -1)$$ and $$Q(2, -1, 3)$$ on the plane $$-x + y + z = 1$$. Then $$d^2$$ is equal to ______.
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Let $$P_1 : \vec{r} \cdot (2\hat{i} + \hat{j} - 3\hat{k}) = 4$$ be a plane. Let $$P_2$$ be another plane which passes through the points $$(2, -3, 2)$$, $$(2, -2, -3)$$ and $$(1, -4, 2)$$. If the direction ratios of the line of intersection of $$P_1$$ and $$P_2$$ be $$16, \alpha, \beta$$, then the value of $$\alpha + \beta$$ is equal to ______.
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