Let $$\alpha$$ be a root of the equation $$1 + x^2 + x^4 = 0$$. Then the value of $$\alpha^{1011} + \alpha^{2022} - \alpha^{3033}$$ is equal to:
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Let $$\alpha$$ be a root of the equation $$1 + x^2 + x^4 = 0$$. Then the value of $$\alpha^{1011} + \alpha^{2022} - \alpha^{3033}$$ is equal to:
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Let $$(z)$$ represent the principal argument of the complex number $$z$$. The, $$|z| = 3$$ and $$\arg(z-1) - \arg(z+1) = \frac{\pi}{4}$$ intersect:
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The sum of the infinite series $$1 + \frac{5}{6} + \frac{12}{6^2} + \frac{22}{6^3} + \frac{35}{6^4} + \frac{51}{6^5} + \frac{70}{6^6} + \ldots$$ is equal to:
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Let $$n \geq 5$$ be an integer. If $$9^n - 8n - 1 = 64\alpha$$ and $$6^n - 5n - 1 = 25\beta$$, then $$\alpha - \beta$$ is equal to:
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The distance of the origin from the centroid of the triangle whose two sides have the equations $$x - 2y + 1 = 0$$ and $$2x - y - 1 = 0$$ and whose orthocenter is $$\left(\frac{7}{3}, \frac{7}{3}\right)$$ is:
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Let a triangle $$ABC$$ be inscribed in the circle $$x^2 - \sqrt{2}(x+y) + y^2 = 0$$ such that $$\angle BAC = \frac{\pi}{2}$$. If the length of side $$AB$$ is $$\sqrt{2}$$, then the area of the $$\triangle ABC$$ is equal to:
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Let $$P : y^2 = 4ax$$, $$a > 0$$ be a parabola with focus $$S$$. Let the tangents to the parabola $$P$$ make an angle of $$\frac{\pi}{4}$$ with the line $$y = 3x + 5$$ touch the parabola $$P$$ at $$A$$ and $$B$$. Then the value of $$a$$ for which $$A$$, $$B$$ and $$S$$ are collinear is:
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The value of $$\lim_{x \to 1} \frac{(x^2-1)\sin^2(\pi x)}{x^4-2x^3+2x-1}$$ is equal to:
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Negation of the Boolean statement $$(p \vee q) \Rightarrow ((\sim r) \vee p)$$ is equivalent to:
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The number of values of $$a \in N$$ such that the variance of 3, 7, 12, $$a$$, 43 $$- a$$ is a natural number is:
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From the base of a pole of height 20 meter, the angle of elevation of the top of a tower is 60°. The pole subtends an angle 30° at the top of the tower. Then the height of the tower is:
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Let $$A = \begin{pmatrix} 2 & -1 \\ 0 & 2 \end{pmatrix}$$. If $$B = I - {}^5C_1(\text{adj } A) + {}^5C_2(\text{adj } A)^2 - \ldots - {}^5C_5(\text{adj } A)^5$$, then the sum of all elements of the matrix $$B$$ is:
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Let $$f : R \to R$$ be a function defined by $$f(x) = (x-3)^{n_1}(x-5)^{n_2}$$, $$n_1, n_2 \in N$$. The, which of the following is NOT true?
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Let $$f$$ be a real valued continuous function on $$[0, 1]$$ and $$f(x) = x + \int_0^1 (x-t)f(t)dt$$. Then which of the following points $$(x, y)$$ lies on the curve $$y = f(x)$$?
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If $$\int_0^2 \left(\sqrt{2x} - \sqrt{2x - x^2}\right)dx = \int_0^1 \left(1 - \sqrt{1-y^2}-\frac{y^2}{2}\right)dy + \int_1^2 \left(2 - \frac{y^2}{2}\right)dy + I$$, then $$I$$ equal to
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If $$y = y(x)$$ is the solution of the differential equation $$(1+e^{2x})\frac{dy}{dx} + 2(1+y^2)e^x = 0$$ and $$y(0) = 0$$, then $$6\left(y'(0) + \left(y\left(\log_e \sqrt{3}\right)\right)^2\right)$$ is equal to:
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Let $$A$$, $$B$$, $$C$$ be three points whose position vectors respectively are:
$$\vec{a} = \hat{i} + 4\hat{j} + 3\hat{k}$$
$$\vec{b} = 2\hat{i} + \alpha\hat{j} + 4\hat{k}$$, $$\alpha \in R$$
$$\vec{c} = 3\hat{i} - 2\hat{j} + 5\hat{k}$$
If $$\alpha$$ is the smallest positive integer for which $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ are non-collinear, then the length of the median, through $$A$$, of $$\triangle ABC$$ is:
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Let $$\frac{x-2}{3} = \frac{y+1}{-2} = \frac{z+3}{-1}$$ lie on the plane $$px - qy + z = 5$$, for some $$p, q \in R$$. The shortest distance of the plane from the origin is:
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Let $$Q$$ be the mirror image of the point $$P(1, 2, 1)$$ with respect to the plane $$x + 2y + 2z = 16$$. Let $$T$$ be a plane passing through the point $$Q$$ and contains the line $$\vec{r} = -\hat{k} + \lambda(\hat{i} + \hat{j} + 2\hat{k})$$, $$\lambda \in R$$. Then, which of the following points lies on $$T$$?
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The probability that a relation $$R$$ from $$\{x, y\}$$ to $$\{x, y\}$$ is both symmetric and transitive, is equal to:
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The total number of four digit numbers such that each of the first three digits is divisible by the last digit, is equal to ______.
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Let 3, 6, 9, 12, ... upto 78 terms and 5, 9, 13, 17, ... upto 59 terms be two series. Then, the sum of the terms common to both the series is equal to ______.
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Let the coefficients of $$x^{-1}$$ and $$x^{-3}$$ in the expansion of $$\left(2x^{1/5} - \frac{1}{x^{1/5}}\right)^{15}$$, $$x > 0$$, be $$m$$ and $$n$$ respectively. If $$r$$ is a positive integer such $$mn^2 = {}^{15}C_r \cdot 2^r$$, then the value of $$r$$ is equal to ______.
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The number of solutions of the equation $$\sin x = \cos^2 x$$ in the interval $$(0, 10)$$ is ______.
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Let $$M = \begin{bmatrix} 0 & -\alpha \\ \alpha & 0 \end{bmatrix}$$, where $$\alpha$$ is a non-zero real number and $$N = \sum_{k=1}^{49} M^{2k}$$. If $$(I - M^2)N = -2I$$, then the positive integral value of $$\alpha$$ is ______.
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Let $$f(x)$$ and $$g(x)$$ be two real polynomials of degree 2 and 1 respectively. If $$f(g(x)) = 8x^2 - 2x$$, and $$g(f(x)) = 4x^2 + 6x + 1$$, then the value of $$f(2) + g(2)$$ is ______.
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Let $$f$$ and $$g$$ be twice differentiable even functions on $$(-2, 2)$$ such that $$f\left(\frac{1}{4}\right) = 0$$, $$f\left(\frac{1}{2}\right) = 0$$, $$f(1) = 1$$ and $$g\left(\frac{3}{4}\right) = 0$$, $$g(1) = 2$$. Then, the minimum number of solutions of $$f(x)g''(x) + f'(x)g'(x) = 0$$ in $$(-2, 2)$$ is equal to ______.
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For real numbers $$a$$, $$b$$ ($$a > b > 0$$), let
Area $$\{(x, y) : x^2 + y^2 \leq a^2$$ and $$\frac{x^2}{a^2} + \frac{y^2}{b^2} \geq 1\} = 30\pi$$
and
Area $$\{(x, y) : x^2 + y^2 \geq b^2$$ and $$\frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1\} = 18\pi$$
Then the value of $$(a-b)^2$$ is equal to ______.
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Let $$y = y(x)$$, $$x > 1$$, be the solution of the differential equation $$(x-1)\frac{dy}{dx} + 2xy = \frac{1}{x-1}$$, with $$y(2) = \frac{1+e^4}{2e^4}$$. If $$y(3) = \frac{e^{\alpha}+1}{\beta e^{\alpha}}$$, then the value of $$\alpha + \beta$$ is equal to ______.
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Let $$\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$$, $$\vec{b} = \hat{i} + \hat{j} + \hat{k}$$ and $$\vec{c}$$ be a vector such that $$\vec{a} \times (\vec{b}+ \vec{c}) = \vec{0}$$, then the value of $$3(\vec{c} \cdot \vec{a})$$ is equal to ______.
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