If the sum of the squares of the reciprocals of the roots $$\alpha$$ and $$\beta$$ of the equation $$3x^2 + \lambda x - 1 = 0$$ is $$15$$, then $$6(\alpha^3 + \beta^3)^2$$ is equal to
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If the sum of the squares of the reciprocals of the roots $$\alpha$$ and $$\beta$$ of the equation $$3x^2 + \lambda x - 1 = 0$$ is $$15$$, then $$6(\alpha^3 + \beta^3)^2$$ is equal to
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Let $$A = \{z \in \mathbb{C} : 1 \leqslant |z - (1+i)| \leqslant 2\}$$ and $$B = \{z \in A : |z - (1-i)| = 1\}$$. Then, $$B$$
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If $$\{a_i\}_{i=1}^{n}$$, where $$n$$ is an even integer, is an arithmetic progression with common difference $$1$$, and $$\sum_{i=1}^{n} a_i = 192$$, $$\sum_{i=1}^{n/2} a_{2i} = 120$$, then $$n$$ is equal to
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The remainder when $$3^{2022}$$ is divided by $$5$$ is
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Let $$S = \left\{\theta \in [-\pi, \pi] - \left\{\pm \frac{\pi}{2}\right\} : \sin\theta \tan\theta + \tan\theta = \sin 2\theta\right\}$$. If $$T = \sum_{\theta \in S} \cos 2\theta$$, then $$T + n(S)$$ is equal to
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Let $$x^2 + y^2 + Ax + By + C = 0$$ be a circle passing through $$(0, 6)$$ and touching the parabola $$y = x^2$$ at $$(2, 4)$$. Then $$A + C$$ is equal to ______
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Let $$\lambda x - 2y = \mu$$ be a tangent to the hyperbola $$a^2x^2 - y^2 = b^2$$. Then $$\left(\frac{\lambda}{a}\right)^2 - \left(\frac{\mu}{b}\right)^2$$ is equal to
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The number of choices for $$\Delta \in \{\wedge, \vee, \Rightarrow, \Leftrightarrow\}$$, such that $$(p \Delta q) \Rightarrow ((p \Delta \sim q) \vee ((\sim p) \Delta q))$$ is a tautology, is
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Let $$S = \{\sqrt{n} : 1 \leqslant n \leqslant 50$$ and $$n$$ is odd$$\}$$. Let $$a \in S$$ and $$A = \begin{bmatrix} 1 & 0 & a \\ -1 & 1 & 0 \\ -a & 0 & 1 \end{bmatrix}$$. If $$\sum_{a \in S} \det(\text{adj } A) = 100\lambda$$, then $$\lambda$$ is equal to
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The number of values of $$\alpha$$ for which the system of equations
$$x + y + z = \alpha$$
$$\alpha x + 2\alpha y + 3z = -1$$
$$x + 3\alpha y + 5z = 4$$
is inconsistent, is
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The set of all values of $$k$$ for which $$(\tan^{-1}x)^3 + (\cot^{-1}x)^3 = k\pi^3, x \in R$$, is the interval
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The domain of $$f(x) = \frac{\cos^{-1}\left(\frac{x^2 - 5x + 6}{x^2 - 9}\right)}{\log(x^2 - 3x + 2)}$$ is
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For the function $$f(x) = 4\log_e(x-1) - 2x^2 + 4x + 5, x > 1$$, which one of the following is NOT correct?
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If the tangent at the point $$(x_1, y_1)$$ on the curve $$y = x^3 + 3x^2 + 5$$ passes through the origin, then $$(x_1, y_1)$$ does NOT lie on the curve
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The sum of absolute maximum and absolute minimum values of the function $$f(x) = |2x^2 + 3x - 2| + \sin x \cos x$$ in the interval $$[0, 1]$$ is
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The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is $$3$$ units and after $$5$$ seconds, it becomes $$7$$ units, then its radius after $$9$$ seconds is
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If $$x = x(y)$$ is the solution of the differential equation $$y\frac{dx}{dy} = 2x + y^3(y+1)e^y, x(1) = 0$$; then $$x(e)$$ is equal to
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Let $$\hat{a}, \hat{b}$$ be unit vectors. If $$\vec{c}$$ be a vector such that the angle between $$\hat{a}$$ and $$\vec{c}$$ is $$\frac{\pi}{12}$$, and $$\hat{b} = \vec{c} + 2(\vec{c} \times \hat{a})$$, then $$|6\vec{c}|^2$$ is equal to:
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Bag $$A$$ contains $$2$$ white, $$1$$ black and $$3$$ red balls and bag $$B$$ contains $$3$$ black, $$2$$ red and $$n$$ white balls. One bag is chosen at random and $$2$$ balls drawn from it at random are found to be $$1$$ red and $$1$$ black. If the probability that both balls come from Bag $$A$$ is $$\frac{6}{11}$$, then $$n$$ is equal to
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If a random variable $$X$$ follows the Binomial distribution $$B(33, p)$$ such that $$3P(X = 0) = P(X = 1)$$, then the value of $$\frac{P(X = 15)}{P(X = 18)} - \frac{P(X = 16)}{P(X = 17)}$$ is equal to
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In an examination, there are $$5$$ multiple choice questions with $$3$$ choices, out of which exactly one is correct. There are $$3$$ marks for each correct answer, $$-2$$ marks for each wrong answer and $$0$$ mark if the question is not attempted. Then, the number of ways a student appearing in the examination gets $$5$$ marks is ______
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Let $$A\left(\frac{3}{\sqrt{a}}, \sqrt{a}\right), a > 0$$, be a fixed point in the $$xy$$-plane. The image of $$A$$ in $$y$$-axis be $$B$$ and the image of $$B$$ in $$x$$-axis be $$C$$. If $$D(3\cos\theta, a\sin\theta)$$, is a point in the fourth quadrant such that the maximum area of $$\triangle ACD$$ is $$12$$ square units, then $$a$$ is equal to ______
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If two tangents drawn from a point $$(\alpha, \beta)$$ lying on the ellipse $$25x^2 + 4y^2 = 1$$ to the parabola $$y^2 = 4x$$ are such that the slope of one tangent is four times the other, then the value of $$(10\alpha + 5)^2 + (16\beta^2 + 50)^2$$ equals ______
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The number of one-one functions $$f : \{a, b, c, d\} \to \{0, 1, 2, \ldots, 10\}$$ such that $$2f(a) - f(b) + 3f(c) + f(d) = 0$$ is ______
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The number of points where the function
$$f(x) = \begin{cases} |2x^2 - 3x - 7| & \text{if } x \leqslant -1 \\ [4x^2 - 1] & \text{if } -1 < x < 1 \\ |x+1| + |x-2| & \text{if } x \geqslant 1 \end{cases}$$
where $$[t]$$ denotes the greatest integer $$\leqslant t$$, is discontinuous is ______
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If $$f(\theta) = \sin\theta + \int_{-\pi/2}^{\pi/2} (\sin\theta + t\cos\theta) \cdot f(t)\,dt$$, then $$\left|\int_0^{\pi/2} f(\theta)\,d\theta\right|$$ is ______
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Let $$\underset{0 \leqslant x \leqslant 2}{\text{Max}}\left\{\frac{9-x^2}{5-x}\right\} = \alpha$$ and $$\underset{0 \leqslant x \leqslant 2}{\text{Min}}\left\{\frac{9-x^2}{5-x}\right\} = \beta$$. If $$\int_{\beta - 8/3}^{2\alpha - 1} \text{Max}\left\{\frac{9-x^2}{5-x}, x\right\}dx = \alpha_1 + \alpha_2 \log_e\left(\frac{8}{15}\right)$$, then $$\alpha_1 + \alpha_2$$ is equal to ______
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Let $$S$$ be the region bounded by the curves $$y = x^3$$ and $$y^2 = x$$. The curve $$y = 2|x|$$ divides $$S$$ into two regions of areas $$R_1$$ and $$R_2$$. If $$ |R_1, R_2 | = R_2$$, then $$\frac{R_2}{R_1}$$ is equal to ______
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Let a line having direction ratios $$1, -4, 2$$ intersect the lines $$\frac{x-7}{3} = \frac{y-1}{-1} = \frac{z+2}{1}$$ and $$\frac{x}{2} = \frac{y-7}{3} = \frac{z}{1}$$ at the points $$A$$ and $$B$$. Then $$(AB)^2$$ is equal to ______
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If the shortest distance between the lines $$\vec{r} = (-\hat{i} + 3\hat{k}) + \lambda(\hat{i} - a\hat{j})$$ and $$\vec{r} = (-\hat{j} + 2\hat{k}) + \mu(\hat{i} - \hat{j} + \hat{k})$$ is $$\sqrt{\frac{2}{3}}$$, then the integral value of $$a$$ is equal to ______
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