The number of real solutions of the equation, $$x^2 - |x| - 12 = 0$$ is:
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The number of real solutions of the equation, $$x^2 - |x| - 12 = 0$$ is:
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The sum of all those terms which are rational numbers in the expansion of $$\left(2^{\frac{1}{3}} + 3^{\frac{1}{4}}\right)^{12}$$ is:
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If the greatest value of the term independent of $$x$$ in the expansion of $$\left(x \sin \alpha + a\frac{\cos \alpha}{x}\right)^{10}$$ is $$\frac{10!}{(5!)^2}$$, then the value of $$a$$ is equal to:
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The lowest integer which is greater than $$\left(1 + \frac{1}{10^{100}}\right)^{10^{100}}$$ is
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If $$^nP_r = ^nP_{r+1}$$ and $$^nC_r = ^nC_{r-1}$$, then the value of $$r$$ is equal to:
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The value of $$\cot \frac{\pi}{24}$$ is:
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The number of distinct real roots of $$\begin{vmatrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{vmatrix} = 0$$ in the interval $$-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$$ is:
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Let the equation of the pair of lines, $$y = px$$ and $$y = qx$$, can be written as $$(y - px)(y - qx) = 0$$. Then the equation of the pair of the angle bisectors of the lines $$x^2 - 4xy - 5y^2 = 0$$ is:
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If a tangent to the ellipse $$x^2 + 4y^2 = 4$$ meets the tangents at the extremities of its major axis at $$B$$ and $$C$$, then the circle with $$BC$$ as diameter passes through the point.
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Consider the statement "The match will be played only if the weather is good and ground is not wet". Select the correct negation from the following:
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The first of the two samples in a group has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15.6 and standard deviation $$\sqrt{13.44}$$, then the standard deviation of the second sample is:
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If $$P = \begin{bmatrix} 1 & 0 \\ \frac{1}{2} & 1 \end{bmatrix}$$, then $$P^{50}$$ is:
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If $$[x]$$ be the greatest integer less than or equal to $$x$$, then $$\sum_{n=8}^{100} \left[\frac{(-1)^n n}{2}\right]$$ is equal to:
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Consider function $$f : A \rightarrow B$$ and $$g : B \rightarrow C$$ $$(A, B, C \subseteq R)$$ such that $$(gof)^{-1}$$ exists, then:
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If $$f(x) = \begin{cases} \int_0^x (5 + |1 - t|) \, dt, & x > 2 \\ 5x + 1, & x \leq 2 \end{cases}$$, then
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The value of the integral $$\int_{-1}^{1} \log\left(x + \sqrt{x^2 + 1}\right) dx$$ is:
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Let $$y = y(x)$$ be the solution of the differential equation $$x \, dy = (y + x^3 \cos x) \, dx$$ with $$y(\pi) = 0$$, then $$y\left(\frac{\pi}{2}\right)$$ is equal to:
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Let $$a, b$$ and $$c$$ be distinct positive numbers. If the vectors $$a\hat{i} + a\hat{j} + c\hat{k}$$, $$\hat{i} + \hat{k}$$ and $$c\hat{i} + c\hat{j} + b\hat{k}$$ are co-planar, then $$c$$ is equal to:
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If $$|\vec{a}| = 2$$, $$|\vec{b}| = 5$$ and $$|\vec{a} \times \vec{b}| = 8$$, then $$|\vec{a} \cdot \vec{b}|$$ is equal to:
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Let $$X$$ be a random variable such that the probability function of a distribution is given by $$P(X = 0) = \frac{1}{2}$$, $$P(X = j) = \frac{1}{3^j}$$ $$(j = 1, 2, 3, \ldots, \infty)$$. Then the mean of the distribution and $$P(X$$ is positive and even) respectively, are:
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If $$a + b + c = 1$$, $$ab + bc + ca = 2$$ and $$abc = 3$$, then the value of $$a^4 + b^4 + c^4$$ is equal to:
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The equation of a circle is $$\text{Re}(z^2) + 2(\text{Im}(z))^2 + 2\text{Re}(z) = 0$$, where $$z = x + iy$$. A line which passes through the centre of the given circle and the vertex of the parabola, $$x^2 - 6x - y + 13 = 0$$, has $$y$$-intercept equal to _________.
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Let $$n \in \mathbf{N}$$ and $$[x]$$ denote the greatest integer less than or equal to $$x$$. If the sum of $$(n + 1)$$ terms of $$^nC_0, 3 \cdot ^nC_1, 5 \cdot ^nC_2, 7 \cdot ^nC_3, \ldots$$ is equal to $$2^{100} \cdot 101$$, then $$2\left[\frac{n-1}{2}\right]$$ is equal to
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If the co-efficient of $$x^7$$ and $$x^8$$ in the expansion of $$\left(2 + \frac{x}{3}\right)^n$$ are equal, then the value of $$n$$ is equal to:
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Consider the function $$f(x) = \frac{P(x)}{\sin(x - 2)}$$, $$x \neq 2$$, and $$f(x) = 7$$, $$x = 2$$ where $$P(x)$$ is a polynomial such that $$P''(x)$$ is always a constant and $$P(3) = 9$$. If $$f(x)$$ is continuous at $$x = 2$$, then $$P(5)$$ is equal to _________.
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If a rectangle is inscribed in an equilateral triangle of side length $$2\sqrt{2}$$ as shown in the figure, then the square of the largest area of such a rectangle is _________.
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Let a curve $$y = f(x)$$ pass through the point $$\left(2, (\log_e 2)^2\right)$$ and have slope $$\frac{2y}{x \log_e x}$$ for all positive real values of $$x$$. Then the value of $$f(e)$$ is equal to _________.
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If $$\vec{a}$$ and $$\vec{b}$$ are unit vectors and $$\left(\vec{a} + 3\vec{b}\right)$$ is perpendicular to $$\left(7\vec{a} - 5\vec{b}\right)$$ and $$\left(\vec{a} - 4\vec{b}\right)$$ is perpendicular to $$\left(7\vec{a} - 2\vec{b}\right)$$, then the angle between $$\vec{a}$$ and $$\vec{b}$$ (in degrees) is _________.
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If the lines $$\frac{x - k}{1} = \frac{y - 2}{2} = \frac{z - 3}{3}$$ and $$\frac{x + 1}{3} = \frac{y + 2}{2} = \frac{z + 3}{1}$$ are co-planar, then the value of $$k$$ is _________.
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A fair coin is tossed $$n-$$ times such that the probability of getting at least one head is at least 0.9. Then the minimum value of $$n$$ is _________.
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