If $$m$$ is chosen in the quadratic equation $$(m^2 + 1)x^2 - 3x + (m^2 + 1)^2 = 0$$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is:
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If $$m$$ is chosen in the quadratic equation $$(m^2 + 1)x^2 - 3x + (m^2 + 1)^2 = 0$$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is:
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Let $$z \in C$$ be such that $$|z| \lt 1$$. If $$\omega = \frac{5 + 3z}{5(1 - z)}$$, then:
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The sum of the series $$1 + 2 \times 3 + 3 \times 5 + 4 \times 7 + \ldots$$ upto 11th term is:
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Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square, whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is:
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If the sum and product of the first three terms in an A.P. are 33 and 1155, respectively, then a value of its 11th term is:
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If some three consecutive coefficients in the binomial expansion of $$(x + 1)^n$$ in powers of $$x$$ are in the ratio 2 : 15 : 70, then the average of these three coefficients is:
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The value of $$\sin 10° \sin 30° \sin 50° \sin 70°$$ is:
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If the two lines $$x + (a-1)y = 1$$ and $$2x + a^2y = 1$$, $$(a \in R - \{0, 1\})$$ are perpendicular, then the distance of their point of intersection from the origin is:
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A rectangle is inscribed in a circle with a diameter lying along the line $$3y = x + 7$$. If the two adjacent vertices of the rectangle are $$(-8, 5)$$ and $$(6, 5)$$, then the area of the rectangle (in sq. units) is:
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The common tangent to the circles $$x^2 + y^2 = 4$$ and $$x^2 + y^2 + 6x + 8y - 24 = 0$$ also passes through the point:
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The area (in sq. units) of the smaller of the two circles that touch the parabola, $$y^2 = 4x$$ at the point $$(1, 2)$$ and the x-axis is:
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If the tangent to the parabola $$y^2 = x$$ at a point $$(\alpha, \beta)$$, $$(\beta > 0)$$ is also a tangent to the ellipse, $$x^2 + 2y^2 = 1$$, then $$\alpha$$ is equal to:
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If $$f(x) = [x] - \left[\frac{x}{4}\right]$$, $$x \in R$$, where $$[x]$$ denotes the greatest integer function, then:
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If $$p \Rightarrow (q \lor r)$$ is False, then the truth values of p, q, r are respectively, (where T is True and F is False)
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The mean and the median of the following ten numbers in increasing order 10, 22, 26, 29, 34, x, 42, 67, 70, y are 42 and 35 respectively, then $$\frac{y}{x}$$ is equal to:
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Two poles standing on a horizontal ground are of heights 5 m and 10 m respectively. The line joining their tops makes an angle of 15° with the ground. Then the distance (in m) between the poles, is:
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The total number of matrices $$A = \begin{pmatrix} 0 & 2y & 1 \\ 2x & y & -1 \\ 2x & -y & 1 \end{pmatrix}$$, $$(x, y \in R, x \neq y)$$ for which $$A^TA = 3I_3$$ is:
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If the system of equations $$2x + 3y - z = 0$$, $$x + ky - 2z = 0$$ and $$2x - y + z = 0$$ has a non-trivial solution $$(x, y, z)$$, then $$\frac{x}{y} + \frac{y}{z} + \frac{z}{x} + k$$ is equal to:
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The domain of the definition of the function $$f(x) = \frac{1}{4 - x^2} + \log_{10}(x^3 - x)$$ is:
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If the function $$f(x) = \begin{cases} a|\pi - x| + 1, & x \le 5 \\ b|x - \pi| + 3, & x \gt 5 \end{cases}$$ is continuous at $$x = 5$$, then the value of $$a - b$$ is:
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A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is $$\tan^{-1}\left(\frac{1}{2}\right)$$. Water is poured into it at a constant rate of 5 cubic m/min. Then the rate (in m/min), at which the level of water is rising at the instant when the depth of water in the tank is 10 m; is:
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If $$\int e^{\sec x}(\sec x \tan x f(x) + (\sec x \tan x + \sec^2 x))dx = e^{\sec x}f(x) + C$$, then a possible choice of $$f(x)$$ is:
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The value of the integral $$\int_0^1 x\cot^{-1}(1 - x^2 + x^4) dx$$ is:
If $$f: R \rightarrow R$$ is a differentiable function and $$f(2) = 6$$, then $$\lim_{x \to 2} \int_6^{f(x)} \frac{2t \, dt}{(x - 2)}$$ is:
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The area (in sq. units) of the region $$A = \left\{(x, y) : \frac{y^2}{2} \le x \le y + 4\right\}$$ is:
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If $$\cos x \frac{dy}{dx} - y\sin x = 6x$$, $$(0 < x < \frac{\pi}{2})$$ and $$y\left(\frac{\pi}{3}\right) = 0$$, then $$y\left(\frac{\pi}{6}\right)$$ is equal to:
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If a unit vector $$\vec{a}$$ makes angles $$\frac{\pi}{3}$$ with $$\hat{i}$$, $$\frac{\pi}{4}$$ with $$\hat{j}$$ and $$\theta \in (0, \pi)$$ with $$\hat{k}$$, then a value of $$\theta$$ is:
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The vertices B and C of a $$\triangle ABC$$ lie on the line, $$\frac{x + 2}{3} = \frac{y - 1}{0} = \frac{z}{4}$$ such that $$BC = 5$$ units. Then the area (in sq. units) of this triangle, given the point $$A(1, -1, 2)$$, is:
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Let P be the plane, which contains the line of intersection of the planes, $$x + y + z - 6 = 0$$ and $$2x + 3y + z + 5 = 0$$ and it is perpendicular to the xy-plane. Then the distance of the point (0, 0, 256) from P is equal to:
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Two newspapers A and B are published in a city. It is known that 25% of the city population reads A and 20% reads B while 8% reads both A and B. Further, 30% of those who read A but not B look into advertisements and 40% of those who read B but not A look into advertisements, while 50% of those who read both A and B look into advertisements. Then the percentage of the population who look into advertisements is:
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