The number of real roots of the equation $$5 + 2^x -1 = 2^x \cdot 2^x - 2$$ is:
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The number of real roots of the equation $$5 + 2^x -1 = 2^x \cdot 2^x - 2$$ is:
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If z and $$\omega$$ are two complex numbers such that $$z\omega = 1$$ and $$\arg(z) - \arg(\omega) = \frac{\pi}{2}$$, then:
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Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is:
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The sum $$1 + \frac{1^3 + 2^3}{1 + 2} + \frac{1^3 + 2^3 + 3^3}{1 + 2 + 3} + \ldots + \frac{1^3 + 2^3 + 3^3 + \ldots + 15^3}{1 + 2 + 3 + \ldots + 15} - \frac{1}{2}(1 + 2 + 3 + \ldots + 15)$$ is equal to
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Let $$a_1, a_2, a_3, \ldots$$ be an A.P. with $$a_6 = 2$$. Then, the common difference of this A.P., which maximise the product $$a_1 \cdot a_4 \cdot a_5$$, is:
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Let a, b and c be in G.P. with common ratio r, where $$a \neq 0$$ and $$0 \lt r \leq \frac{1}{2}$$. If 3a, 7b and 15c are the first three terms of an A.P., then the 4$$^{th}$$ term of this A.P. is:
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The smallest natural number n, such that the coefficient of $$x$$ in the expansion of $$\left(x^2 + \frac{1}{x^3}\right)^n$$ is $$^nC_{23}$$, is
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Lines are drawn parallel to the line $$4x - 3y + 2 = 0$$, at a distance $$\frac{3}{5}$$ units from the origin. Then which one of the following points lies on any of these lines?
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The locus of the centres of the circles, which touch the circle, $$x^2 + y^2 = 1$$ externally, also touch the y-axis and lie in the first quadrant, is:
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If the line $$ax + y = c$$, touches both the curves $$x^2 + y^2 = 1$$ and $$y^2 = 4\sqrt{2}x$$, then c is equal to:
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The tangent and normal to the ellipse $$3x^2 + 5y^2 = 32$$ at the point P(2, 2) meet the x-axis at Q and R, respectively. Then the area (in sq. units) of the triangle PQR is:
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If $$5x + 9 = 0$$ is the directrix of the hyperbola $$16x^2 - 9y^2 = 144$$, then its corresponding focus is:
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If $$\lim_{x \to 1} \frac{x^2 - ax + b}{x - 1} = 5$$, then a + b is equal to:
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The negation of the Boolean expression $$\sim s \vee (\sim r \wedge s)$$ is equivalent to
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If both the mean and the standard deviation of 50 observations $$x_1, x_2, \ldots, x_{50}$$ are equal to 16, then the mean of $$(x_1 - 4)^2, (x_2 - 4)^2, \ldots, (x_{50} - 4)^2$$ is
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The angles A, B & C of a ΔABC are in A.P. and a:b = 1:$$\sqrt{3}$$. If c = 4 cm, then the area (in sq. cm) of this triangle is:
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The sum of the real roots of the equation $$\begin{vmatrix} x & -6 & -1 \\ 2 & -3x & x-3 \\ -3 & 2x & x+2 \end{vmatrix} = 0$$, is equal to:
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Let $$\lambda$$ be a real number for which the system of linear equations
$$x + y + z = 6$$,
$$4x + \lambda y - \lambda z = \lambda - 2$$ and
$$3x + 2y - 4z = -5$$
has infinitely many solutions. Then $$\lambda$$ is a root of the quadratic equation:
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If $$\cos^{-1}x - \cos^{-1}\frac{y}{2} = \alpha$$, where $$-1 \leq x \leq 1$$, $$-2 \leq y \leq 2$$, $$x \leq \frac{y}{2}$$, then for all x, y, $$4x^2 - 4xy\cos\alpha + y^2$$ is equal to:
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Let $$f(x) = \log_e \sin x$$, $$0 < x < \pi$$ and $$g(x) = \sin^{-1}(e^{-x})$$, $$(x \geq 0)$$. If $$\alpha$$ is a positive real number such that$$a=f\circ g'(\alpha)$$ and $$b = f \circ g(\alpha)$$, then
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If the tangent to the curve $$y = \frac{x}{x^2 - 3}$$, $$x \in R$$, $$x \neq \pm\sqrt{3}$$, at a point $$(\alpha, \beta) \neq (0, 0)$$ on it is parallel to the line $$2x + 6y - 11 = 0$$, then:
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A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm$$^3$$/min. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the ice decreases, is:
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If $$\int x^5 e^{-x^2} dx = g(x)e^{-x^2} + c$$, where c is a constant of integration, then g(-1) is equal to
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The integral $$\int_{\pi/6}^{\pi/3} \sec^{2/3}x \cdot \text{cosec}^{4/3}x \, dx$$ is equal to
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The area (in sq. units) of the region bounded by the curves $$y = 2^x$$ and $$y = x + 1$$, in the first quadrant is
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Let $$y = yx$$ be the solution of the differential equation, $$\frac{dy}{dx} + y\tan x = 2x + x^2\tan x$$, $$x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$, such that $$y(0) = 1$$. Then
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The distance of the point having position vector $$-\hat{i} + 2\hat{j} + 6\hat{k}$$ from the straight line passing through the point (2, 3, -4) and parallel to the vector, $$6\hat{i} + 3\hat{j} - 4\hat{k}$$ is
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If the plane $$2x - y + 2z + 3 = 0$$ has the distances $$\frac{1}{3}$$ and $$\frac{2}{3}$$ units from the planes $$4x - 2y + 4z + \lambda = 0$$ and $$2x - y + 2z + \mu = 0$$, respectively, then the maximum value of $$\lambda + \mu$$ is equal to:
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A perpendicular is drawn from a point on the line $$\frac{x-1}{2} = \frac{y+1}{-1} = \frac{z}{1}$$ to the plane $$x + y + z = 3$$ such that the foot of the perpendicular Q also lies on the plane $$x - y + z = 3$$. Then the coordinates of Q are
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Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than 99% is:
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