The minimum value of the sum of the squares of the roots of $$x^2 + (3-a)x = 2(a-1)$$ is
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The minimum value of the sum of the squares of the roots of $$x^2 + (3-a)x = 2(a-1)$$ is
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If $$z = x + iy$$ satisfies $$|z| - 2 = 0$$ and $$|z-i| -| z+5i| = 0$$, then
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$$\displaystyle\sum_{\substack{i,j=0 \\ i \neq j}}^{n}$$ $$^n C_{i}$$ $$^n C_{j}$$ is equal to
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Let the abscissae of the two points $$P$$ and $$Q$$ on a circle be the roots of $$x^2 - 4x - 6 = 0$$ and the ordinates of $$P$$ and $$Q$$ be the roots of $$y^2 + 2y - 7 = 0$$. If $$PQ$$ is a diameter of the circle $$x^2 + y^2 + 2ax + 2by + c = 0$$, then the value of $$a + b - c$$ is
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The equation of a common tangent to the parabolas $$y = x^2$$ and $$y = -(x-2)^2$$ is
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The acute angle between the pair of tangents drawn to the ellipse $$2x^2 + 3y^2 = 5$$ from the point $$(1, 3)$$ is
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If the line $$x - 1 = 0$$ is a directrix of the hyperbola $$kx^2 - y^2 = 6$$, then the hyperbola passes through the point
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Let $$\beta = \displaystyle\lim_{x \to 0} \dfrac{\alpha x - (e^{3x} - 1)}{\alpha x(e^{3x} - 1)}$$ for some $$\alpha \in \mathbb{R}$$. Then the value of $$\alpha + \beta$$ is:
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Negation of the Boolean expression $$p \leftrightarrow (q \rightarrow p)$$ is
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Let $$A = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$$ and $$B = \begin{pmatrix} 9^2 & -10^2 & 11^2 \\ 12^2 & 13^2 & -14^2 \\ -15^2 & 16^2 & 17^2 \end{pmatrix}$$, then the value of $$A'BA$$ is
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If $$0 < x < \dfrac{1}{\sqrt{2}}$$ and $$\dfrac{\sin^{-1}x}{\alpha} = \dfrac{\cos^{-1}x}{\beta}$$, then a value of $$\sin\dfrac{2\pi\alpha}{\alpha + \beta}$$ is
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The value of $$\log_e 2 \cdot \dfrac{d}{dx}(\log_{\cos x} \cosec x)$$ at $$x = \dfrac{\pi}{4}$$ is
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Let $$P$$ and $$Q$$ be any points on the curves $$(x-1)^2 + (y+1)^2 = 1$$ and $$y = x^2$$, respectively. The distance between $$P$$ and $$Q$$ is minimum for some value of the abscissa of $$P$$ in the interval
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If the maximum value of $$a$$, for which the function $$f_a(x) = \tan^{-1}(2x) - 3ax + 7$$ is non-decreasing in $$\left(-\dfrac{\pi}{6}, \dfrac{\pi}{6}\right)$$, is $$\bar{a}$$, then $$f_{\bar{a}}\left(\dfrac{\pi}{8}\right)$$ is equal to
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The integral $$\displaystyle\int \dfrac{1 - \dfrac{1}{\sqrt{3}}(\cos x - \sin x)}{1 + \dfrac{2}{\sqrt{3}}\sin 2x} dx$$ is equal to
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$$\displaystyle\int_0^{20\pi} (|\sin x| + |\cos x|)^2 dx$$ is equal to:
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The area bounded by the curves $$y = |x^2 - 1|$$ and $$y = 1$$ is
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Let the solution curve $$y = f(x)$$ of the differential equation $$\dfrac{dy}{dx} + \dfrac{xy}{x^2 - 1} = \dfrac{x^4 + 2x}{\sqrt{1-x^2}}$$, $$x \in (-1, 1)$$ pass through the origin. Then $$\displaystyle\int_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f(x) dx$$ is equal to
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A vector $$\vec{a}$$ is parallel to the line of intersection of the plane determined by the vectors $$\hat{i}$$, $$\hat{i} + \hat{j}$$ and the plane determined by the vectors $$\hat{i} - \hat{j}$$, $$\hat{i} + \hat{k}$$. The obtuse angle between $$\vec{a}$$ and the vector $$\vec{b} = \hat{i} - 2\hat{j} + 2\hat{k}$$ is
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Let $$X$$ be a binomially distributed random variable with mean $$4$$ and variance $$\dfrac{4}{3}$$. Then $$54 P(X \le 2)$$ is equal to
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Numbers are to be formed between 1000 and 3000, which are divisible by 4, using the digits 1, 2, 3, 4, 5 and 6 without repetition of digits. Then the total number of such numbers is ______.
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If $$\displaystyle\sum_{k=1}^{10} \dfrac{k}{k^4 + k^2 + 1} = \dfrac{m}{n}$$, where $$m$$ and $$n$$ are co-prime, then $$m + n$$ is equal to ______.
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Different A.P.'s are constructed with the first term 100, the last term 199, and integral common differences. The sum of the common differences of all such A.P.'s having at least 3 terms and at most 33 terms is ______.
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If the sum of solutions of the system of equations $$2\sin^2\theta - \cos 2\theta = 0$$ and $$2\cos^2\theta + 3\sin\theta = 0$$ in the interval $$[0, 2\pi]$$ is $$k\pi$$, then $$k$$ is equal to ______.
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The mean and standard deviation of 40 observations are 30 and 5 respectively. It was noticed that two of these observations 12 and 10 were wrongly recorded. If $$\sigma$$ is the standard deviation of the data after omitting the two wrong observations from the data, then $$38\sigma^2$$ is equal to ______.
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Let 𝐴 = {1, 2, 3, 4, 5, 6, 7} and 𝐵 = {3, 6, 7, 9}. Then the number of elements in the set $$\{C \subseteq A : C \cap B \neq \phi\}$$ is
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The number of matrices $$A=\begin{bmatrix}a & b \\c & d \end{bmatrix}$$, where $$𝑎, 𝑏, 𝑐, d ∈ -1, 0, 1, 2, 3, … … , 10,$$ such that $$A=A^{-1}$$, is______.
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Suppose $$𝑦 = 𝑦𝑥$$ be the solution curve to the differential equation $$\frac{dy}{dx}-y=2-e^{-x}$$ such that $$\lim_{x \rightarrow \infty} yx$$ If $$𝑎$$ and $$𝑏$$ are respectively the $$𝑥 -$$ and $$𝑦 -$$ intercept of the tangent to the curve at $$𝑥 = 0$$, then the value of $$𝑎 - 4𝑏$$ is equal to _______.
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The largest value of $$𝑎$$, for which the perpendicular distance of the plane containing the lines $$\vec{r} = \hat{i} + \hat{j} + \lambda( \hat{i} + a \hat{j} - \hat{k} )\quad \text{and} \quad \vec{r} = \hat{i} + \hat{j} + \mu (\hat{i} + \hat{j} - a \hat{k})$$ from the point 2, 1, 4 is $$\sqrt{3}$$, is
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The plane passing through the line $$L:l 𝑥 - 𝑦 + 3(1 - 𝑙 )𝑧 = 1, 𝑥 + 2𝑦 - 𝑧 = 2$$ and perpendicular to the plane $$3𝑥 + 2𝑦 + 𝑧 = 6$$ is $$3𝑥 - 8𝑦 + 7𝑧 = 4$$. If $$\theta$$ is the acute angle between the line $$𝐿$$ and the 𝑦-axis, then $$415 \cos^{2}\theta $$ is equal to_______.
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