Let $$S = \{x \in R : x \geq 0$$ & $$2|\sqrt{x} - 3| + \sqrt{x}(\sqrt{x} - 6) + 6 = 0\}$$. Then S:
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Let $$S = \{x \in R : x \geq 0$$ & $$2|\sqrt{x} - 3| + \sqrt{x}(\sqrt{x} - 6) + 6 = 0\}$$. Then S:
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If $$\alpha, \beta \in C$$ are the distinct roots of the equation $$x^2 - x + 1 = 0$$, then $$\alpha^{101} + \beta^{107}$$ is equal to:
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From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is:
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Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series $$1^2 + 2 \cdot 2^2 + 3^2 + 2 \cdot 4^2 + 5^2 + 2 \cdot 6^2 + \ldots$$ If $$B - 2A = 100\lambda$$, then $$\lambda$$ is equal to:
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Let $$a_1, a_2, a_3, \ldots, a_{49}$$ be in A.P. such that $$\sum_{k=0}^{12} a_{4k+1} = 416$$ and $$a_9 + a_{43} = 66$$. If $$a_1^2 + a_2^2 + \ldots + a_{17}^2 = 140m$$, then m is equal to:
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The sum of the co-efficient of all odd degree terms in the expansion of $$\left(x + \sqrt{x^3 - 1}\right)^5 + \left(x - \sqrt{x^3 - 1}\right)^5$$, $$(x > 1)$$ is:
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If sum of all the solutions of the equation $$8\cos x \cdot \left(\cos\left(\frac{\pi}{6} + x\right) \cdot \cos\left(\frac{\pi}{6} - x\right) - \frac{1}{2}\right) = 1$$ in $$[0, \pi]$$ is $$k\pi$$, then k is equal to:
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A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is:
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If the tangent at (1, 7) to the curve $$x^2 = y - 6$$ touches the circle $$x^2 + y^2 + 16x + 12y + c = 0$$ then the value of c is:
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Tangent and normal are drawn at P(16, 16) on the parabola $$y^2 = 16x$$, which intersect the axis of the parabola at A & B, respectively. If C is the center of the circle through the points P, A & B and $$\angle CPB = \theta$$, then a value of $$\tan \theta$$ is:
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Two sets A and B are as under: $$A = \{(a, b) \in R \times R : |a - 5| < 1$$ and $$|b - 5| < 1\}$$; $$B = \{(a, b) \in R \times R : 4(a - 6)^2 + 9(b - 5)^2 \leq 36\}$$. Then:
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Tangents are drawn to the hyperbola $$4x^2 - y^2 = 36$$ at the points P and Q. If these tangents intersect at the point T(0, 3) then the area (in sq. units) of $$\triangle PTQ$$ is:
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For each $$t \in R$$, let $$[t]$$ be the greatest integer less than or equal to t. Then $$\lim_{x \to 0^+} x\left(\left[\frac{1}{x}\right] + \left[\frac{2}{x}\right] + \ldots + \left[\frac{15}{x}\right]\right)$$
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The Boolean expression $$\sim(p \vee q) \vee (\sim p \wedge q)$$ is equivalent to:
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If $$\sum_{i=1}^{9}(x_i - 5) = 9$$ and $$\sum_{i=1}^{9}(x_i - 5)^2 = 45$$, then the standard deviation of the 9 items $$x_1, x_2, \ldots, x_9$$ is:
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PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR. If the angles of elevation of the top of the tower at P, Q and R are respectively 45$$^\circ$$, 30$$^\circ$$ and 30$$^\circ$$, then the height of the tower (in m) is:
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Let the orthocentre and centroid of a triangle be A(-3, 5) and B(3, 3) respectively. If C is the circumcentre of this triangle, then the radius of the circle having line segment AC as diameter, is:
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If the system of linear equations
$$x + ky + 3z = 0$$
$$3x + ky - 2z = 0$$
$$2x + 4y - 3z = 0$$
has a non-zero solution (x, y, z), then $$\frac{xz}{y^2}$$ is equal to:
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If $$\begin{vmatrix} x-4 & 2x & 2x \\ 2x & x-4 & 2x \\ 2x & 2x & x-4 \end{vmatrix} = (A + Bx)(x - A)^2$$, then the ordered pair (A, B) is equal to:
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Let $$S = \{t \in R : f(x) = |x - \pi| \cdot (e^{|x|} - 1)\sin|x|$$ is not differentiable at $$t\}$$. Then the set S is equal to:
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If the curves $$y^2 = 6x$$, $$9x^2 + by^2 = 16$$ intersect each other at right angles, then the value of b is:
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Let $$f(x) = x^2 + \frac{1}{x^2}$$ and $$g(x) = x - \frac{1}{x}$$, $$x \in R - \{-1, 0, 1\}$$. If $$h(x) = \frac{f(x)}{g(x)}$$, then the local minimum value of h(x) is:
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The integral $$\int \frac{\sin^2 x \cos^2 x}{(\sin^5 x + \cos^3 x \sin^2 x + \sin^3 x \cos^2 x + \cos^5 x)^2} dx$$ is equal to
(where C is the constant of integration).
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The value of $$\int_{-\pi/2}^{\pi/2} \frac{\sin^2 x}{1+2^x} dx$$ is:
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Let $$g(x) = \cos x^2$$, $$f(x) = \sqrt{x}$$, and $$\alpha, \beta (\alpha < \beta)$$ be the roots of the quadratic equation $$18x^2 - 9\pi x + \pi^2 = 0$$. Then the area (in sq. units) bounded by the curve $$y = (gof)(x)$$ and the lines $$x = \alpha$$, $$x = \beta$$ and $$y = 0$$, is:
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Let $$y = y(x)$$ be the solution of the differential equation $$\sin x \frac{dy}{dx} + y \cos x = 4x$$, $$x \in (0, \pi)$$. If $$y\left(\frac{\pi}{2}\right) = 0$$, then $$y\left(\frac{\pi}{6}\right)$$ is equal to:
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Let $$\vec{u}$$ be a vector coplanar with the vectors $$\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$$ and $$\vec{b} = \hat{j} + \hat{k}$$. If $$\vec{u}$$ is perpendicular to $$\vec{a}$$ and $$\vec{u} \cdot \vec{b} = 24$$, then $$|\vec{u}|^2$$ is equal to:
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If $$L_1$$ is the line of intersection of the planes $$2x - 2y + 3z - 2 = 0$$, $$x - y + z + 1 = 0$$ and $$L_2$$ is the line of intersection of the planes $$x + 2y - z - 3 = 0$$, $$3x - y + 2z - 1 = 0$$, then the distance of the origin from the plane, containing the lines $$L_1$$ and $$L_2$$ is:
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The length of the projection of the line segment joining the points (5, -1, 4) and (4, -1, 3) on the plane, $$x + y + z = 7$$ is:
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A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its color is observed and this ball along with two additional balls of the same color are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is:
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