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If the two roots of the equation, $$(a-1)(x^4 + x^2 + 1) + (a+1)(x^2 + x + 1)^2 = 0$$ are real and distinct, then the set of all values of $$a$$ is equal to
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If $$z$$ is a non-real complex number, then the minimum value of $$\frac{Im\ z^5}{(Im\ z)^5}$$ is (Where $$Im\ z$$ = Imaginary part of $$z$$)
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Let $$A = \{x_1, x_2, \ldots, x_7\}$$ and $$B = \{y_1, y_2, y_3\}$$ be two sets containing seven and three distinct elements respectively. Then the total number of functions $$f : A \rightarrow B$$ that are onto, if there exist exactly three elements $$x$$ in $$A$$ such that $$f(x) = y_2$$, is equal to:
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If in a regular polygon the number of diagonals is 54, then the number of sides of this polygon is:
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The sum of the 3$$^{rd}$$ and the 4$$^{th}$$ terms of a G.P. is 60 and the product of its first three terms is 1000. If the first term of this G.P. is positive, then its 7$$^{th}$$ term is:
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If $$\sum_{n=1}^{5}\frac{1}{n(n+1)(n+2)(n+3)} = \frac{k}{3}$$, then $$k$$ is equal to:
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The term independent of $$x$$ in the binomial expansion of $$\left(1 - \frac{1}{x} + 3x^5\right)\left(2x^2 - \frac{1}{x}\right)^8$$ is
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If $$\cos \alpha + \cos \beta = \frac{3}{2}$$ and $$\sin \alpha + \sin \beta = \frac{1}{2}$$ and $$\theta$$ is the arithmetic mean of $$\alpha$$ and $$\beta$$, then $$\sin 2\theta + \cos 2\theta$$ is equal to:
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A straight line $$L$$ through the point $$(3, -2)$$ is inclined at an angle of $$60^\circ$$ to the line $$\sqrt{3}x + y = 1$$. If $$L$$ also intersects the X-axis, then the equation of $$L$$ is:
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If a circle passing through the point $$(-1, 0)$$ touches y-axis at $$(0, 2)$$, then the x-intercept of the circle is
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If the incentre of an equilateral triangle is $$(1, 1)$$ and the equation of its one side is $$3x + 4y + 3 = 0$$, then the equation of the circumcircle of this triangle is:
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If $$PQ$$ be a double ordinate of the parabola, $$y^2 = -4x$$, where $$P$$ lies in the second quadrant. If $$R$$ divides $$PQ$$ in the ratio 2 : 1, then the locus of $$R$$ is
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If the distance between the foci of an ellipse is half the length of its latus rectum, then the eccentricity of the ellipse is:
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Consider the following statements:
P: Suman is brilliant
Q: Suman is rich
R: Suman is honest
The negation of the statement, "Suman is brilliant and dishonest if and only if Suman is rich" can be equivalently expressed as
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Let 10 vertical poles standing at equal distances on a straight line, subtend the same angle of elevation $$\alpha$$ at a point $$O$$ on this line and all the poles are on the same side of $$O$$. If the height of the longest pole is $$h$$ and the distance of the foot of the smallest pole from $$O$$ is $$a$$; then the distance between two consecutive poles, is
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If $$A$$ is a $$3 \times 3$$ matrix such that $$|5 \cdot adj A| = 5$$, then $$|A|$$ is equal to
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If $$\begin{vmatrix} x^2+x & x+1 & x-2 \\ 2x^2+3x-1 & 3x & 3x-3 \\ x^2+2x+3 & 2x-1 & 2x-1 \end{vmatrix} = ax - 12$$, then $$a$$ is equal to:
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Let $$k$$ be a non-zero real number. If $$f(x) = \begin{cases} \frac{(e^x - 1)^2}{\sin\left(\frac{x}{k}\right)\log\left(1 + \frac{x}{4}\right)}, & x \neq 0 \\ 12, & x = 0 \end{cases}$$ is a continuous function at $$x = 0$$, then the value of $$k$$ is
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The equation of a normal to the curve, $$\sin y = x\sin\left(\frac{\pi}{3} + y\right)$$ at $$x = 0$$, is:
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Let $$k$$ and $$K$$ be the minimum and the maximum values of the function $$f(x) = \frac{(1+x)^{0.6}}{1+x^{0.6}}$$ in $$[0, 1]$$, respectively, then the ordered pair $$(k, K)$$ is equal to:
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If $$\int \frac{\log\left(t + \sqrt{1+t^2}\right)}{\sqrt{1+t^2}} dt = \frac{1}{2}(g(t))^2 + c$$, where c is a constant, then $$g(2)$$ is equal to
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Let $$f : R \rightarrow R$$ be a function such that $$f(2-x) = f(2+x)$$ and $$f(4-x) = f(4+x)$$, for all $$x \in R$$ and $$\int_0^2 f(x)dx = 5$$. Then the value of $$\int_{10}^{50} f(x)dx$$ is
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Let $$f : (-1, 1) \rightarrow R$$ be a continuous function. If $$\int_0^{\sin x} f(t) dt = \frac{\sqrt{3}}{2}x$$, then $$f\left(\frac{\sqrt{3}}{2}\right)$$ is equal to:
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The solution of the differential equation $$ydx - (x + 2y^2)dy = 0$$ is $$x = f(y)$$. If $$f(-1) = 1$$, then $$f(1)$$ is equal to
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In a parallelogram $$ABCD$$, $$\left|\overrightarrow{AB}\right| = a$$, $$\left|\overrightarrow{AD}\right| = b$$ and $$\left|\overrightarrow{AC}\right| = c$$. $$\overrightarrow{DB} \cdot \overrightarrow{AB}$$ has the value:
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A plane containing the point $$(3, 2, 0)$$ and the line $$\frac{x-1}{1} = \frac{y-2}{5} = \frac{z-3}{4}$$ also contains the point
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The shortest distance between the z-axis and the line $$x + y + 2z - 3 = 0 = 2x + 3y + 4z - 4$$, is
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If the mean and the variance of a binomial variate $$X$$ are 2 and 1 respectively, then the probability that $$X$$ takes a value greater than or equal to one is:
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If the lengths of the sides of a triangle are decided by the three throws of a single fair die, then the probability that the triangle is of maximum area given that it is an isosceles triangle, is:
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