The sum of all real roots of equation $$\left(e^{2x} - 4\right)\left(6e^{2x} - 5e^x + 1\right) = 0$$ is
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The sum of all real roots of equation $$\left(e^{2x} - 4\right)\left(6e^{2x} - 5e^x + 1\right) = 0$$ is
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Let $$x, y > 0$$. If $$x^3 y^2 = 2^{15}$$, then the least value of $$3x + 2y$$ is
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The number of solutions of the equation $$\cos\left(x + \frac{\pi}{3}\right) \cos\left(\frac{\pi}{3} - x\right) = \frac{1}{4}\cos^2 2x, x \in [-3\pi, 3\pi]$$ is:
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Let the area of the triangle with vertices $$A(1, \alpha)$$, $$B(\alpha, 0)$$ and $$C(0, \alpha)$$ be $$4$$ sq. units. If the points $$(\alpha, -\alpha)$$, $$(-\alpha, \alpha)$$ and $$(\alpha^2, \beta)$$ are collinear, then $$\beta$$ is equal to
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A particle is moving in the $$xy$$-plane along a curve $$C$$ passing through the point $$(3, 3)$$. The tangent to the curve $$C$$ at the point $$P$$ meets the $$x$$-axis at $$Q$$. If the $$y$$-axis bisects the segment $$PQ$$, then $$C$$ is a parabola with
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Let the maximum area of the triangle that can be inscribed in the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{4} = 1, a > 2$$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the $$y$$-axis, be $$6\sqrt{3}$$. Then the eccentricity of the ellipse is:
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Consider the following statements:
$$A$$: Rishi is a judge.
$$B$$: Rishi is honest.
$$C$$: Rishi is not arrogant.
The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" is
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Let the system of linear equations
$$x + y + az = 2$$
$$3x + y + z = 4$$
$$x + 2z = 1$$
have a unique solution $$(x^*, y^*, z^*)$$. If $$((a, x^*), (y^*, \alpha)$$ and $$(x^*, -y^*)$$ are collinear points, then the sum of absolute values of all possible values of $$\alpha$$ is:
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Let $$x \times y = x^2 + y^3$$ and $$(x \times 1) \times 1 = x \times (1 \times 1)$$. Then a value of $$2\sin^{-1}\left(\frac{x^4 + x^2 - 2}{x^4 + x^2 + 2}\right)$$ is
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Let $$f(x) = \begin{cases} \frac{\sin(x-|x|)}{x-|x|}, & x \in (-2, -1) \\ \max(2x, 3[|x|]), & |x| < 1 \\ 1, & \text{otherwise} \end{cases}$$
where $$[t]$$ denotes greatest integer $$\leq t$$. If $$m$$ is the number of points where $$f$$ is not continuous and $$n$$ is the number of points where $$f$$ is not differentiable, the ordered pair $$(m, n)$$ is:
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If $$y = \tan^{-1}\left(\sec x^3 - \tan x^3\right), \frac{\pi}{2} < x^3 < \frac{3\pi}{2}$$, then
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The number of distinct real roots of the equation $$x^7 - 7x - 2 = 0$$ is
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Let $$\lambda^*$$ be the largest value of $$\lambda$$ for which the function $$f_\lambda(x) = 4\lambda x^3 - 36\lambda x^2 + 36x + 48$$ is increasing for all $$x \in \mathbb{R}$$. Then $$f_{\lambda^*}(1) + f_{\lambda^*}(-1)$$ is equal to:
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The value of the integral $$\int_{-\pi/2}^{\pi/2} \frac{dx}{(1+e^x)(\sin^6 x + \cos^6 x)}$$ is equal to
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$$\lim_{n \to \infty} \left(\frac{n^2}{(n^2+1)(n+1)} + \frac{n^2}{(n^2+4)(n+2)} + \frac{n^2}{(n^2+9)(n+3)} + \cdots + \frac{n^2}{(n^2+n^2)(n+n)}\right)$$ is equal to
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The slope of normal at any point $$(x, y), x > 0, y > 0$$ on the curve $$y = y(x)$$ is given by $$\frac{x^2}{xy - x^2y^2 - 1}$$. If the curve passes through the point $$(1, 1)$$, then $$e \cdot y(e)$$ is equal to
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Let $$\mathbf{a}$$ and $$\mathbf{b}$$ be two unit vectors such that $$|\mathbf{a} + \mathbf{b}| + 2|\mathbf{a} \times \mathbf{b}| = 2$$. If $$\theta \in (0, \pi)$$ is the angle between $$\hat{a}$$ and $$\hat{b}$$, then among the statements:
$$(S1) : 2|\hat{a} \times \hat{b}| = |\hat{a} - \hat{b}|$$
$$(S2)$$ : The projection of $$\hat{a}$$ on $$(\hat{a} + \hat{b})$$ is $$\frac{1}{2}$$
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If the shortest distance between the lines $$\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{\lambda}$$ and $$\frac{x-2}{1} = \frac{y-4}{4} = \frac{z-5}{\frac{1}{\sqrt{3}}}$$, then the sum of all possible values of $$\lambda$$ is:
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Let the points on the plane $$P$$ be equidistant from the points $$(-4, 2, 1)$$ and $$(2, -2, 3)$$. Then the acute angle between the plane $$P$$ and the plane $$2x + y + 3z = 1$$ is
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A random variable $$X$$ has the following probability distribution:
| $$X$$ | 0 | 1 | 2 | 3 | 4 |
| $$P(X)$$ | $$k$$ | $$2k$$ | $$4k$$ | $$6k$$ | $$8k$$ |
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Let $$S = \{z \in \mathbb{C} : |z - 3| \leq 1$$ and $$z(4 + 3i) + \bar{z}(4 - 3i) \leq 24\}$$. If $$\alpha + i\beta$$ is the point in $$S$$ which is closest to $$4i$$, then $$25(\alpha + \beta)$$ is equal to ______.
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The number of 7-digit numbers which are multiples of 11 and are formed using all the digits $$1, 2, 3, 4, 5, 7$$ and $$9$$ is ______.
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The remainder on dividing $$1 + 3 + 3^2 + 3^3 + \ldots + 3^{2021}$$ by $$50$$ is ______.
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Let a circle $$C : (x - h)^2 + (y - k)^2 = r^2, k > 0$$, touch the $$x$$-axis at $$(1, 0)$$. If the line $$x + y = 0$$ intersects the circle $$C$$ at $$P$$ and $$Q$$ such that the length of the chord $$PQ$$ is $$2$$, then the value of $$h + k + r$$ is equal to ______.
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Let $$P_1$$ be a parabola with vertex $$(3, 2)$$ and focus $$(4, 4)$$ and $$P_2$$ be its mirror image with respect to the line $$x + 2y = 6$$. Then the directrix of $$P_2$$ is $$x + 2y =$$ ______.
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Let the hyperbola $$H : \frac{x^2}{a^2} - y^2 = 1$$ and the ellipse $$E : 3x^2 + 4y^2 = 12$$ be such that the length of latus rectum of $$H$$ is equal to the length of latus rectum of $$E$$. If $$e_H$$ and $$e_E$$ are the eccentricities of $$H$$ and $$E$$ respectively, then the value of $$12(e_H^2 + e_E^2)$$ is equal to ______.
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The sum of all the elements of the set $$\{\alpha \in \{1, 2, \ldots, 100\} : HCF(\alpha, 24) = 1\}$$ is ______.
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Let $$S = \left\{\begin{pmatrix} -1 & a \\ 0 & b \end{pmatrix} ; a, b \in \{1, 2, 3, \ldots 100\}\right\}$$ and let $$T_n = \{A \in S : A^{n(n+1)} = I\}$$. Then the number of elements in $$\bigcap_{n=1}^{100} T_n$$ is ______.
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The area (in sq. units) of the region enclosed between the parabola $$y^2 = 2x$$ and the line $$x + y = 4$$ is ______.
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In an examination, there are $$10$$ true-false type questions. Out of $$10$$, a student can guess the answer of $$4$$ questions correctly with probability $$\frac{3}{4}$$ and the remaining $$6$$ questions correctly with probability $$\frac{1}{4}$$. If the probability that the student guesses the answers of exactly $$8$$ questions correctly out of $$10$$ is $$\frac{27k}{4^{10}}$$, then $$k$$ is equal to ______.
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