If $$\alpha$$ and $$\beta$$ are roots of the equation $$x^2 + px + \frac{3p}{4} = 0$$, such that $$|\alpha - \beta| = \sqrt{10}$$, then $$p$$ belongs to the set :
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If $$\alpha$$ and $$\beta$$ are roots of the equation $$x^2 + px + \frac{3p}{4} = 0$$, such that $$|\alpha - \beta| = \sqrt{10}$$, then $$p$$ belongs to the set :
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If a complex number $$z$$ satisfies the equation $$x + \sqrt{2}|z + 1| + i = 0$$, then $$|z|$$ is equal to :
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The number of ways in which an examiner can assign 30 marks to 8 questions, giving not less than 2 marks to any question, is :
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Given sum of the first $$n$$ terms of an A.P. is $$2n + 3n^2$$. Another A.P. is formed with the same first term and double of the common difference, the sum of $$n$$ terms of the new A.P. is :
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The sum $$\frac{3}{1^2} + \frac{5}{1^2+2^2} + \frac{7}{1^2+2^2+3^2} + \ldots$$ upto 11-terms is:
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If the 7th term in the binomial expansion of $$\left(\frac{3}{\sqrt[3]{84}} + \sqrt{3}\ln x\right)^9$$, $$x \gt 0$$, is equal to 729, then $$x$$ can be:
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The number of solutions of the equation, $$\sin^{-1}x = 2\tan^{-1}x$$ (in principal values) is :
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Statement-1: The number of common solutions of the trigonometric equations $$2\sin^2\theta - \cos 2\theta = 0$$ and $$2\cos^2\theta - 3\sin\theta = 0$$ in the interval $$[0, 2\pi]$$ is two.
Statement-2: The number of solutions of the equation, $$2\cos^2\theta - 3\sin\theta = 0$$ in the interval $$[0, \pi]$$ is two.
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If the $$x$$-intercept of some line $$L$$ is double as that of the line, $$3x + 4y = 12$$ and the $$y$$-intercept of $$L$$ is half as that of the same line, then the slope of $$L$$ is :
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The acute angle between two lines such that the direction cosines $$l, m, n$$, of each of them satisfy the equations $$l + m + n = 0$$ and $$l^2 + m^2 - n^2 = 0$$ is :
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If a circle C passing through (4, 0) touches the circle $$x^2 + y^2 + 4x - 6y - 12 = 0$$ externally at a point (1, -1), then the radius of the circle C is :
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Statement-1: The line $$x - 2y = 2$$ meets the parabola, $$y^2 + 2x = 0$$ only at the point (-2, -2).
Statement-2: The line $$y = mx - \frac{1}{2m}$$ ($$m \neq 0$$) is tangent to the parabola, $$y^2 = -2x$$ at the point $$\left(-\frac{1}{2m^2}, -\frac{1}{m}\right)$$
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Let the equations of two ellipses be
$$E_1 : \frac{x^2}{3} + \frac{y^2}{2} = 1$$ and $$E_2 : \frac{x^2}{16} + \frac{y^2}{b^2} = 1$$,
If the product of their eccentricities is $$\frac{1}{2}$$, then the length of the minor axis of ellipse $$E_2$$ is :
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The statement $$p \rightarrow (q \rightarrow p)$$ is equivalent to :
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Mean of 5 observations is 7. If four of these observations are 6, 7, 8, 10 and one is missing, then the variance of all the five observations is :
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If two vertices of an equilateral triangle are $$A(-a, 0)$$ and $$B(a, 0)$$, $$a > 0$$, and the third vertex C lies above x-axis then the equation of the circumcircle of $$\triangle ABC$$ is :
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Let $$R = \{(3,3)(5,5),(9,9),(12,12),(5,12),(3,9),(3,12),(3,5)\}$$ be a relation on the set $$A = \{3, 5, 9, 12\}$$. Then, R is :
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If $$p, q, r$$ are 3 real numbers satisfying the matrix equation, $$[p \ q \ r]\begin{bmatrix} 3 & 4 & 1 \\ 3 & 2 & 3 \\ 2 & 0 & 2 \end{bmatrix} = [3 \ 0 \ 1]$$ then $$2p + q - r$$ equals :
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If the system of linear equations :
$$x_1 + 2x_2 + 3x_3 = 6$$
$$x_1 + 3x_2 + 5x_3 = 9$$
$$2x_1 + 5x_2 + ax_3 = b$$
is consistent and has infinite number of solutions, then :
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Let $$f(x) = -1 + |x - 2|$$, and $$g(x) = 1 - |x|$$; then the set of all points where $$f \circ g$$ is discontinuous is :
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For $$a > 0$$, $$t \in \left(0, \frac{\pi}{2}\right)$$, let $$x = \sqrt{a^{\sin^{-1}t}}$$ and $$y = \sqrt{a^{\cos^{-1}t}}$$. Then, $$1 + \left(\frac{dy}{dx}\right)^2$$ equals :
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Statement-1: The function $$x^2(e^x + e^{-x})$$ is increasing for all $$x > 0$$.
Statement-2: The functions $$x^2 e^x$$ and $$x^2 e^{-x}$$ are increasing for all $$x > 0$$ and the sum of two increasing functions in any interval $$(a, b)$$ is an increasing function in $$(a, b)$$.
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The maximum area of a right angled triangle with hypotenuse $$h$$ is :
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If $$\int \frac{x^2 - x + 1}{x^2 + 1}e^{\cot^{-1}x}dx = A(x)e^{\cot^{-1}x} + C$$, then $$A(x)$$ is equal to :
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The integral $$\int_{7\pi/4}^{7\pi/3} \sqrt{\tan^2 x} \ dx$$ is equal to :
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The area of the region (in sq. units), in the first quadrant bounded by the parabola $$y = 9x^2$$ and the lines $$x = 0$$, $$y = 1$$ and $$y = 4$$, is :
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Consider the differential equation :
$$$\frac{dy}{dx} = \frac{y^3}{2(xy^2 - x^2)}$$$
Statement-1: The substitution $$z = y^2$$ transforms the above equation into a first order homogeneous differential equation.
Statement-2: The solution of this differential equation is $$y^2 e^{-y^2/x} = C$$.
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If $$\hat{a}$$, $$\hat{b}$$ and $$\hat{c}$$ are unit vectors satisfying $$\hat{a} - \sqrt{3}\hat{b} + \hat{c} = \vec{0}$$, then the angle between the vectors $$\hat{a}$$ and $$\hat{c}$$ is :
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Let Q be the foot of perpendicular from the origin to the plane $$4x - 3y + z + 13 = 0$$ and R be a point (-1, -6) on the plane. Then length QR is :
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Given two independent events, if the probability that exactly one of them occurs is $$\frac{26}{49}$$ and the probability that none of them occurs is $$\frac{15}{49}$$, then the probability of more probable of the two events is :
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