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Let the complex number $$z = x + iy$$ be such that $$\frac{2z - 3i}{2z + i}$$ is purely imaginary. If $$x + y^2 = 0$$, then $$y^4 + y^2 - y$$ is equal to
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Let the first term a and the common ratio r of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to
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If the coefficient of $$x^7$$ in $$\left(ax - \frac{1}{bx^2}\right)^{13}$$ and the coefficient of $$x^{-5}$$ in $$\left(ax + \frac{1}{bx^2}\right)^{13}$$ are equal, then $$a^4 b^4$$ is equal to:
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$$96 \cos\frac{\pi}{33} \cos\frac{2\pi}{33} \cos\frac{4\pi}{33} \cos\frac{8\pi}{33} \cos\frac{16\pi}{33}$$ is equal to
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A line segment $$AB$$ of length $$\lambda$$ moves such that the points $$A$$ and $$B$$ remain on the periphery of a circle of radius $$\lambda$$. Then the locus of the point, that divides the line segment $$AB$$ in the ratio 2:3, is a circle of radius
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Let the ellipse $$E: x^2 + 9y^2 = 9$$ intersect the positive $$x$$- and $$y$$-axes at the points $$A$$ and $$B$$ respectively. Let the major axis of $$E$$ be a diameter of the circle $$C$$. Let the line passing through $$A$$ and $$B$$ meet the circle $$C$$ at the point $$P$$. If the area of the triangle with vertices $$A$$, $$P$$ and the origin $$O$$ is $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime, then $$m - n$$ is equal to
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The negation of the statement $$p \vee q \wedge q \vee \sim r$$ is
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If $$A$$ is a $$3 \times 3$$ matrix and $$|A| = 2$$, then $$|3 \text{ adj}(|3A| \cdot A^2)|$$ is equal to
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For the system of linear equations
$$2x - y + 3z = 5$$
$$3x + 2y - z = 7$$
$$4x + 5y + \alpha z = \beta$$,
which of the following is NOT correct?
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If $$f(x) = \frac{\tan^{-1} x + \log_e 123}{x \log_e 1234 - \tan^{-1} x}$$, $$x > 0$$, then the least value of $$f(f(x)) + f\left(f\left(\frac{4}{x}\right)\right)$$ is
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A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm$$^2$$) is equal to
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If $$Ix = \int e^{\sin^2 x} \sin 2x \cdot \sin^2 x \, dx$$ and $$I(0) = 1$$, then $$I\left(\frac{\pi}{3}\right)$$ is equal to
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Let $$f$$ be a differentiable function such that $$x^2 f(x) - x = 4\int_0^x tf(t) \, dt$$, $$f(1) = \frac{2}{3}$$. Then $$18f(3)$$ is equal to
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The slope of tangent at any point $$(x, y)$$ on a curve $$y = y(x)$$ is $$\frac{x^2 + y^2}{2xy}$$, $$x > 0$$. If $$y(2) = 0$$, then a value of $$y(8)$$ is
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An arc $$PQ$$ of a circle subtends a right angle at its centre $$O$$. The mid point of the arc $$PQ$$ is $$R$$. If $$\overrightarrow{OP} = \vec{u}$$, $$\overrightarrow{OR} = \vec{v}$$ and $$\overrightarrow{OQ} = \alpha\vec{u} + \beta\vec{v}$$, then $$\alpha$$, $$\beta^2$$, are the roots of the equation
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Let $$O$$ be the origin and the position vector of the point $$P$$ be $$-\hat{i} - 2\hat{j} + 3\hat{k}$$. If the position vectors of the points $$A$$, $$B$$ and $$C$$ are $$-2\hat{i} + \hat{j} - 3\hat{k}$$, $$2\hat{i} + 4\hat{j} - 2\hat{k}$$ and $$-4\hat{i} + 2\hat{j} - \hat{k}$$ respectively, then the projection of the vector $$\overrightarrow{OP}$$ on a vector perpendicular to the vectors $$\overrightarrow{AB}$$ and $$\overrightarrow{AC}$$ is
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Let two vertices of a triangle $$ABC$$ be $$(2, 4, 6)$$ and $$(0, -2, -5)$$, and its centroid be $$(2, 1, -1)$$. If the image of the third vertex in the plane $$x + 2y + 4z = 11$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha\beta + \beta\gamma + \gamma\alpha$$ is equal to
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The shortest distance between the lines $$\frac{x+2}{1} = \frac{y}{-2} = \frac{z-5}{2}$$ and $$\frac{x-4}{1} = \frac{y-1}{2} = \frac{z+3}{0}$$ is
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Let $$P$$ be the point of intersection of the line $$\frac{x+3}{3} = \frac{y+2}{1} = \frac{1-z}{2}$$ and the plane $$x + y + z = 2$$. If the distance of the point $$P$$ from the plane $$3x - 4y + 12z = 32$$ is $$q$$, then $$q$$ and $$2q$$ are the roots of the equation
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Let $$N$$ denote the sum of the numbers obtained when two dice are rolled. If the probability that $$2^N < N!$$ is $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime, $$4m - 3n$$ is equal to
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Let $$a$$, $$b$$, $$c$$ be the three distinct positive real numbers such that $$2a^{\log_e a} = bc^{\log_e b}$$ and $$b^{\log_e 2} = a^{\log_e c}$$. Then $$6a + 5bc$$ is equal to _______.
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The number of permutations, of the digits 1, 2, 3, ..., 7 without repetition, which neither contain the string 153 nor the string 2467, is _______.
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Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total numbers of persons, who participated in the tournament, is _______.
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The sum of all those terms, of the arithmetic progression 3, 8, 13, ..., 373, which are not divisible by 3, is equal to _______.
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The coefficient of $$x^7$$ in $$(1 - x + 2x^3)^{10}$$ is _______.
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Let a common tangent to the curves $$y^2 = 4x$$ and $$(x-4)^{2}+y^{2}=16$$, touch the curves at the points $$P$$ and $$Q$$. Then $$PQ^2$$ is equal to _______.
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If the mean of the frequency distribution
| Class | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
| Frequency | 2 | 3 | $$x$$ | 5 | 4 |
is 28, then its variance is _______.
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The number of elements in the set $$\{n \in \mathbb{Z}: |n^2 - 10n + 19| < 6\}$$ is _______.
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Let $$f: [-2, 2] \to \mathbb{R}$$ be defined by $$f(x) = \begin{cases} x[x], & -2 < x < 0 \\ (x - 1)[x], & 0 \leq x \leq 2 \end{cases}$$ where $$[x]$$ denotes the greatest integer function. If $$m$$ and $$n$$ respectively are the number of points in $$(-2, 2)$$ at which $$y = |f(x)|$$ is not continuous and not differentiable, then $$m + n$$ is equal to _______.
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Let $$y = px$$ be the parabola passing through the points $$(-1, 0)$$, $$(0, 0)$$, $$(1, 0)$$ and $$(1, 0)$$. If the area of the region $$\{(x, y): (x+1)^2 + (y-1)^2 \leq 1, y \leq px\}$$ is $$A$$, then $$12\pi - 4A$$ is equal to _______.
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