If $$x$$ is a solution of the equation $$\sqrt{2x+1} - \sqrt{2x-1} = 1$$, $$\left(x \geq \frac{1}{2}\right)$$, then $$\sqrt{4x^2 - 1}$$ is equal to:
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If $$x$$ is a solution of the equation $$\sqrt{2x+1} - \sqrt{2x-1} = 1$$, $$\left(x \geq \frac{1}{2}\right)$$, then $$\sqrt{4x^2 - 1}$$ is equal to:
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Let $$z = 1 + ai$$, be a complex number, $$a > 0$$, such that $$z^3$$ is a real number. Then, the sum $$1 + z + z^2 + \ldots + z^{11}$$ is equal to:
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If $$\frac{^{n+2}C_6}{^{n-2}P_2} = 11$$, then $$n$$ satisfies the equation:
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Let $$a_1, a_2, a_3, \ldots a_n, \ldots$$, be in A.P. If $$a_3 + a_7 + a_{11} + a_{15} = 72$$, then the sum of its first 17 terms is equal to:
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The sum $$\sum_{r=1}^{10} (r^2 + 1) \times (r!)$$, is equal to:
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If the coefficients of $$x^{-2}$$ and $$x^{-4}$$, in the expansion of $$\left(x^{1/3} + \frac{1}{2x^{1/3}}\right)^{18}$$, $$(x \gt 0)$$, are $$m$$ and $$n$$ respectively, then $$\frac{m}{n}$$ is equal to
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If $$A > 0$$, $$B > 0$$ and $$A + B = \frac{\pi}{6}$$, then the minimum positive value of $$(\tan A + \tan B)$$ is:
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Let $$P = \{\theta : \sin\theta - \cos\theta = \sqrt{2}\cos\theta\}$$ and $$Q = \{\theta : \sin\theta + \cos\theta = \sqrt{2}\sin\theta\}$$, be two sets. Then
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A straight line through origin $$O$$ meets the lines $$3y = 10 - 4x$$ and $$8x + 6y + 5 = 0$$ at points $$A$$ and $$B$$ respectively. Then, $$O$$ divides the segment $$AB$$ in the ratio
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A ray of light is incident along a line which meets another line $$7x - y + 1 = 0$$ at the point $$(0, 1)$$. The ray is then reflected from this point along the line $$y + 2x = 1$$. Then the equation of the line of incidence of the ray of light is:
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Equation of the tangent to the circle, at the point $$(1, -1)$$, whose center is the point of intersection of the straight lines $$x - y = 1$$ and $$2x + y = 3$$ is:
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$$P$$ and $$Q$$ are two distinct points on the parabola, $$y^2 = 4x$$, with parameters $$t$$ and $$t_1$$, respectively. If the normal at $$P$$ passes through $$Q$$, then the minimum value of $$t_1^2$$, is
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A hyperbola whose transverse axis is along the major axis of the conic $$\frac{x^2}{3} + \frac{y^2}{4} = 4$$ and has vertices at the foci of the conic. If the eccentricity of the hyperbola is $$\frac{3}{2}$$, then which of the following points does not lie on the hyperbola?
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$$\lim_{x \to 0} \frac{(1 - \cos 2x)^2}{2x\tan x - x\tan 2x}$$ is
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The contrapositive of the following statement, "If the side of a square doubles, then its area increases four times", is
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The mean of 5 observations is 5 and their variance is 12.4. If three of the observations are 1, 2 & 6; then the value of the remaining two is:
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The angle of elevation of the top of a vertical tower from a point A, due east of it is 45°. The angle of elevation of the top of the same tower from a point B, due south of A is 30°. If the distance between A and B is $$54\sqrt{2}$$ m, then the height of the tower (in meters), is:
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Let $$A$$, be a $$3 \times 3$$ matrix, such that $$A^2 - 5A + 7I = O$$.
Statement - I: $$A^{-1} = \frac{1}{7}(5I - A)$$.
Statement - II: The polynomial $$A^3 - 2A^2 - 3A + I$$, can be reduced to $$5(A - 4I)$$. Then:
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If $$A = \begin{bmatrix} -4 & -1 \\ 3 & 1 \end{bmatrix}$$, then the determinant of the matrix $$(A^{2016} - 2A^{2015} - A^{2014})$$ is:
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Let $$a, b \in R$$, $$(a \neq 0)$$. If the function $$f$$, defined as
$$f(x) = \begin{cases} \frac{2x^2}{a}, & 0 \leq x \lt 1 \\ a, & 1 \leq x \lt \sqrt{2} \\ \frac{2b^2 - 4b}{x^3}, & \sqrt{2} \leq x \lt 8 \end{cases}$$
is continuous in the interval $$[0, \infty)$$, then an ordered pair $$(a, b)$$ can be
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Let C be a curve given by $$y(x) = 1 + \sqrt{4x - 3}$$, $$x > \frac{3}{4}$$. If $$P$$ is a point on C, such that the tangent at $$P$$ has slope $$\frac{2}{3}$$, then a point through which the normal at $$P$$ passes, is:
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Let $$f(x) = \sin^4 x + \cos^4 x$$. Then, $$f$$ is an increasing function in the interval:
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The integral $$\int \frac{dx}{(1+\sqrt{x})\sqrt{x - x^2}}$$ is equal to
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For $$x \in R$$, $$x \neq 0$$, if $$y(x)$$ is a differentiable function such that $$x\int_1^x y(t)dt = (x+1)\int_1^x ty(t)dt$$, then $$y(x)$$ equals (where C is a constant)
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The value of the integral $$\int_4^{10} \frac{[x^2]}{[x^2 - 28x + 196] + [x^2]}dx$$, where $$[x]$$ denotes the greatest integer less than or equal to $$x$$, is
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The solution of the differential equation $$\frac{dy}{dx} + \frac{y}{2}\sec x = \frac{\tan x}{2y}$$, where $$0 \leq x < \frac{\pi}{2}$$ and $$y(0) = 1$$, is given by
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The number of distinct real values of $$\lambda$$, for which the lines $$\frac{x-1}{1} = \frac{y-2}{2} = \frac{z+3}{\lambda^2}$$ and $$\frac{x-3}{1} = \frac{y-2}{\lambda^2} = \frac{z-1}{2}$$, are coplanar is
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$$ABC$$ is a triangle in a plane with vertices $$A(2, 3, 5)$$, $$B(-1, 3, 2)$$ and $$C(\lambda, 5, \mu)$$. If the median through $$A$$ is equally inclined to the coordinate axes, then the value of $$(\lambda^3 + \mu^3 + 5)$$ is
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Let $$ABC$$ be a triangle whose circumcentre is at $$P$$. If the position vectors of $$A$$, $$B$$, $$C$$ and $$P$$ are $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ and $$\frac{\vec{a}+\vec{b}+\vec{c}}{4}$$ respectively, then the position vector of the orthocentre of this triangle, is:
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An experiment succeeds twice as often as it fails. The probability of at least 5 successes in the six trials of this experiment is
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