The real number $$k$$ for which the equation, $$2x^3 + 3x + k = 0$$ has two distinct real roots in $$[0, 1]$$ belongs to
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The real number $$k$$ for which the equation, $$2x^3 + 3x + k = 0$$ has two distinct real roots in $$[0, 1]$$ belongs to
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If the equations $$x^2 + 2x + 3 = 0$$ and $$ax^2 + bx + c = 0$$, $$a, b, c \in R$$, have a common root, then $$a : b : c$$ is:
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If $$z$$ is a complex number of unit modulus and argument $$\theta$$, then $$\arg\left(\frac{1+z}{1+\bar{z}}\right)$$ can be equal to (given $$z \neq -1$$)
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Let $$T_n$$ be the number of all possible triangles formed by joining vertices of an $$n$$-sided regular polygon. If $$T_{n+1} - T_n = 10$$, then the value of $$n$$ is :
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If $$x$$, $$y$$, $$z$$ are positive numbers in A.P. and $$\tan^{-1}x$$, $$\tan^{-1}y$$ and $$\tan^{-1}z$$ are also in A.P., then which of the following is correct.
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The sum of first 20 terms of the sequence 0.7, 0.77, 0.777, ......, is :
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The term independent of $$x$$ in the expansion of $$\left(\frac{x+1}{x^{2/3} - x^{1/3} + 1} - \frac{x-1}{x - x^{1/2}}\right)^{10}$$ is
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The expression $$\frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A}$$ can be written as :
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A ray of light along $$x + \sqrt{3}y = \sqrt{3}$$ gets reflected upon reaching X-axis, the equation of the reflected ray is
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The $$x$$-coordinate of the incentre of the triangle that has the coordinates of midpoints of its sides as (0, 1), (1, 1) and (1, 0) is
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The circle passing through (1, -2) and touching the axis of $$x$$ at (3, 0) also passes through the point
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Given : A circle, $$2x^2 + 2y^2 = 5$$ and a parabola, $$y^2 = 4\sqrt{5}x$$.
Statement - I : An equation of a common tangent to these curves is $$y = x + \sqrt{5}$$.
Statement - II : If the line, $$y = mx + \frac{\sqrt{5}}{m}$$ $$(m \neq 0)$$ is their common tangent, then $$m$$ satisfies $$m^4 - 3m^2 + 2 = 0$$.
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The equation of the circle passing through the foci of the ellipse $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$, and having centre at (0, 3) is
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The value of $$\lim_{x \to 0} \frac{(1 - \cos 2x)(3 + \cos x)}{x \tan 4x}$$ is equal to
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Consider :
Statement - I : $$(p \wedge \sim q) \wedge (\sim p \wedge q)$$ is a fallacy.
Statement - II : $$(p \rightarrow q) \leftrightarrow (\sim q \rightarrow \sim p)$$ is a tautology.
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All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given?
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$$ABCD$$ is a trapezium such that $$AB$$ and $$CD$$ are parallel and $$BC \perp CD$$. If $$\angle ADB = \theta$$, $$BC = p$$ and $$CD = q$$, then $$AB$$ is equal to
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Let $$A$$ and $$B$$ be two sets containing 2 elements and 4 elements respectively. The number of subsets of $$A \times B$$ having 3 or more elements is :
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If $$P = \begin{bmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix}$$ is the adjoint of a $$3 \times 3$$ matrix $$A$$ and $$|A| = 4$$, then $$\alpha$$ is equal to
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The number of values of $$k$$, for which the system of equations :
$$(k+1)x + 8y = 4k$$
$$kx + (k+3)y = 3k - 1$$
has no solution, is :
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If $$y = \sec(\tan^{-1}x)$$, then $$\frac{dy}{dx}$$ at $$x = 1$$ is equal to
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The intercepts on the $$x$$-axis made by tangents to the curve, $$y = \int_0^x |t| \ dt$$, $$x \in R$$, which are parallel to the line $$y = 2x$$, are equal to
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If $$\int f(x)dx = \psi(x)$$, then $$\int x^5 f(x^3)dx$$, is equal to
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The area (in square units) bounded by the curves $$y = \sqrt{x}$$, $$2y - x + 3 = 0$$, X-axis and lying in the first quadrant is
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At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production P w.r.t. additional number of workers $$x$$ is given by $$\frac{dP}{dx} = 100 - 12\sqrt{x}$$. If the firm employs 25 more workers, then the new level of production of items is
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If the vectors $$\vec{AB} = 3\hat{i} + 4\hat{k}$$ and $$\vec{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$$ are the sides of a triangle $$ABC$$, then the length of the median through $$A$$ is:
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If the lines $$\frac{x-2}{1} = \frac{y-3}{1} = \frac{z-4}{-k}$$ and $$\frac{x-1}{k} = \frac{y-4}{2} = \frac{z-5}{1}$$ are coplanar, then $$k$$ can have
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Distance between two parallel planes $$2x + y + 2z = 8$$ and $$4x + 2y + 4z + 5 = 0$$ is
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A multiple choice examination has 5 questions. Each question has three alternative answers out of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is :
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