If $$x^2 + 9y^2 - 4x + 3 = 0$$, $$x, y \in R$$, then $$x$$ and $$y$$ respectively lie in the intervals
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If $$x^2 + 9y^2 - 4x + 3 = 0$$, $$x, y \in R$$, then $$x$$ and $$y$$ respectively lie in the intervals
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If $$S = \left\{z \in C : \frac{z-i}{z+2i} \in R\right\}$$, then
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If for $$x, y \in R$$, $$x > 0$$, $$y = \log_{10} x + \log_{10} x^{1/3} + \log_{10} x^{1/9} + \ldots$$ upto $$\infty$$ terms and $$\frac{2+4+6+\ldots+2y}{3+6+9+\ldots+3y} = \frac{4}{\log_{10} x}$$, then the ordered pair $$(x, y)$$ is equal to
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If $$0 < x < 1$$, then $$\frac{3}{2}x^2 + \frac{5}{3}x^3 + \frac{7}{4}x^4 + \ldots$$, is equal to
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$$\sum_{k=0}^{20} \left({}^{20}C_k\right)^2$$ is equal to
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Let $$A$$ be a fixed point $$(0, 6)$$ and $$B$$ be a moving point $$(2t, 0)$$. Let $$M$$ be the mid-point of $$AB$$ and the perpendicular bisector of $$AB$$ meets the y-axis at $$C$$. The locus of the mid-point $$P$$ of MC is
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A tangent and a normal are drawn at the point $$P(2, -4)$$ on the parabola $$y^2 = 8x$$, which meet the directrix of the parabola at the points $$A$$ and $$B$$ respectively. If $$Q(a, b)$$ is a point such that $$AQBP$$ is a square, then $$2a + b$$ is equal to
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If $$\alpha, \beta$$ are the distinct roots of $$x^2 + bx + c = 0$$, then $$\lim_{x \to \beta} \frac{e^{2(x^2+bx+c)} - 1 - 2(x^2+bx+c)}{(x-\beta)^2}$$ is equal to
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The statement $$(p \wedge (p \rightarrow q) \wedge (q \rightarrow r)) \rightarrow r$$ is
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Let $$\frac{\sin A}{\sin B} = \frac{\sin(A-C)}{\sin(C-B)}$$, where $$A, B, C$$ are angles of a triangle $$ABC$$. If the lengths of the sides opposite these angles are $$a, b, c$$ respectively, then
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If the matrix $$A = \begin{bmatrix} 0 & 2 \\ K & -1 \end{bmatrix}$$ satisfies $$A(A^3 + 3I) = 2I$$, then the value of $$K$$ is
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If $$(\sin^{-1} x)^2 - (\cos^{-1} x)^2 = a$$; $$0 < x < 1$$, $$a \neq 0$$, then the value of $$2x^2 - 1$$ is
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A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is
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If $$U_n = \left(1 + \frac{1}{n^2}\right)\left(1 + \frac{2^2}{n^2}\right)^2 \cdots \left(1 + \frac{n^2}{n^2}\right)^n$$, then $$\lim_{n \to \infty} (U_n)^{\frac{-4}{n^2}}$$ is equal to
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$$\int_6^{16} \frac{\log_e x^2}{\log_e x^2 + \log_e(x^2 - 44x + 484)} dx$$ is equal to
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Let us consider a curve, $$y = f(x)$$ passing through the point $$(-2, 2)$$ and the slope of the tangent to the curve at any point $$(x, f(x))$$ is given by $$f(x) + xf'(x) = x^2$$. Then
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Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} = 2(y + 2\sin x - 5)x - 2\cos x$$ such that $$y(0) = 7$$. Then $$y(\pi)$$ is equal to
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The distance of the point $$(1, -2, 3)$$ from the plane $$x - y + z = 5$$ measured parallel to a line, whose direction ratios are $$2, 3, -6$$, is
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Equation of a plane at a distance $$\sqrt{\frac{2}{21}}$$ units from the origin, which contains the line of intersection of the planes $$x - y - z - 1 = 0$$ and $$2x + y - 3z + 4 = 0$$, is
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When a certain biased die is rolled, a particular face occurs with probability $$\frac{1}{6} - x$$ and its opposite face occurs with probability $$\frac{1}{6} + x$$. All other faces occur with probability $$\frac{1}{6}$$. Note that opposite faces sum to 7 in any die. If $$0 < x < \frac{1}{6}$$, and the probability of obtaining total sum = 7, when such a die is rolled twice, is $$\frac{13}{96}$$, then the value of $$x$$ is
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If $$A = \{x \in R : |x-2| > 1\}$$, $$B = \{x \in R : \sqrt{x^2 - 3} > 1\}$$, $$C = \{x \in R : |x-4| \geq 2\}$$ and $$Z$$ is the set of all integers, then the number of subsets of the set $$(A \cap B \cap C)^c \cap Z$$ is _________.
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A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55, is _________.
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Let the equation $$x^2 + y^2 + px + (1-p)y + 5 = 0$$ represent circles of varying radius $$r \in (0, 5]$$. Then the number of elements in the set $$S = \{q : q = p^2$$ and $$q$$ is an integer$$\}$$ is _________.
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If the minimum area of the triangle formed by a tangent to the ellipse $$\frac{x^2}{b^2} + \frac{y^2}{4a^2} = 1$$ and the co-ordinate axis is $$kab$$, then $$k$$ is equal to _________.
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Let $$n$$ be an odd natural number such that the variance of 1, 2, 3, 4, ..., n is 14. Then $$n$$ is equal to _________.
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If the system of linear equations
$$2x + y - z = 3$$
$$x - y - z = \alpha$$
$$3x + 3y + \beta z = 3$$
has infinitely many solutions, then $$|\alpha + \beta - \alpha\beta|$$ is equal to _________.
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If $$y^{1/4} + y^{-1/4} = 2x$$, and $$(x^2 - 1)\frac{d^2y}{dx^2} + \alpha x\frac{dy}{dx} + \beta y = 0$$, then $$|\alpha - \beta|$$ is equal to _________.
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The number of distinct real roots of the equation $$3x^4 + 4x^3 - 12x^2 + 4 = 0$$ is _________.
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If $$\int \frac{dx}{(x^2+x+1)^2} = a\tan^{-1}\left(\frac{2x+1}{\sqrt{3}}\right) + b\left(\frac{2x+1}{x^2+x+1}\right) + C$$,$$x > 0$$ where $$C$$ is the constant of integration, then the value of $$9\left(\sqrt{3}a + b\right)$$ is equal to _________.
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Let $$\vec{a} = \hat{i} + 5\hat{j} + \alpha\hat{k}$$, $$\vec{b} = \hat{i} + 3\hat{j} + \beta\hat{k}$$ and $$\vec{c} = -\hat{i} + 2\hat{j} - 3\hat{k}$$ be three vectors such that, $$|\vec{b} \times \vec{c}| = 5\sqrt{3}$$ and $$\vec{a}$$ is perpendicular to $$\vec{b}$$. Then the greatest amongst the values of $$|\vec{a}|^2$$ is _________.
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