The equation $$\sqrt{3x^2 + x + 5} = x - 3$$, where x is real, has:
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The equation $$\sqrt{3x^2 + x + 5} = x - 3$$, where x is real, has:
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For all complex numbers z of the form $$1 + i\alpha$$, $$\alpha \in R$$, if $$z^2 = x + iy$$, then:
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Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between themselves exceeds the number of games that the men played with the women by 66, then the number of men who participated in the tournament lies in the interval:
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Let $$f(n) = \left[\frac{1}{3} + \frac{3n}{100}\right]n$$, where $$[n]$$ denotes the greatest integer less than or equal to n. Then $$\sum_{n=1}^{56} f(n)$$ is equal to:
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The number of terms in an A.P. is even, the sum of the odd terms in it is 24 and that of the even terms is 30. If the last term exceeds the first term by $$10\frac{1}{2}$$, then the number of terms in the A.P. is:
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The coefficient of $$x^{1012}$$ in the expansion of $$(1 + x^n + x^{253})^{10}$$, (where $$n \leq 22$$ is any positive integer), is:
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If a line L is perpendicular to the line $$5x - y = 1$$, and the area of the triangle formed by the line L and the coordinate axes is 5 sq units, then the distance of the line L from the line $$x + 5y = 0$$ is:
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The circumcentre of a triangle lies at the origin and its centroid is the midpoint of the line segment joining the points $$(a^2 + 1, a^2 + 1)$$ and $$(2a, -2a)$$, $$a \neq 0$$. Then for any a, the orthocentre of this triangle lies on the line:
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The equation of the circle described on the chord $$3x + y + 5 = 0$$ of the circle $$x^2 + y^2 = 16$$ as the diameter is:
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A chord is drawn through the focus of the parabola $$y^2 = 6x$$ such that its distance from the vertex of this parabola is $$\frac{\sqrt{5}}{2}$$, then its slope can be:
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The tangent at an extremity (in the first quadrant) of the latus rectum of the hyperbola $$\frac{x^2}{4} - \frac{y^2}{5} = 1$$, meets the x-axis and y-axis at A and B, respectively. Then $$OA^2 - OB^2$$, where O is the origin, equals:
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The contrapositive of the statement "if I am not feeling well, then I will go to the doctor" is:
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Let $$\bar{x}$$, M and $$\sigma^2$$ be respectively the mean, mode and variance of n observations $$x_1, x_2, \ldots, x_n$$ and $$d_i = -x_i - a$$, i = 1, 2, ..., n, where a is any number.
Statement I: Variance of d$$_1$$, d$$_2$$, ..., d$$_n$$ is $$\sigma^2$$.
Statement II: Mean and mode of d$$_1$$, d$$_2$$, ..., d$$_n$$ are $$-\bar{x} - a$$ and $$-M - a$$, respectively.
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Let A and B be any two $$3 \times 3$$ matrices. If A is symmetric and B is skew symmetric, then the matrix AB $$-$$ BA is:
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If $$\Delta_r = \begin{vmatrix} r & 2r-1 & 3r-2 \\ \frac{n}{2} & n-1 & a \\ \frac{1}{2}n(n-1) & (n-1)^2 & \frac{1}{2}(n-1)(3n+4) \end{vmatrix}$$, then the value of $$\sum_{r=1}^{n-1} \Delta_r$$:
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The principal value of $$\tan^{-1}\left(\cot\frac{43\pi}{4}\right)$$ is:
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The function $$f(x) = |\sin 4x| + |\cos 2x|$$, is a periodic function with a fundamental period:
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Let $$f : R \to R$$ be defined by $$f(x) = \frac{|x|-1}{|x|+1}$$, then f is:
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If the function $$f(x) = \begin{cases} \frac{\sqrt{2+\cos x}-1}{(\pi-x)^2}, & x \neq \pi \\ k, & x = \pi \end{cases}$$ is continuous at $$x = \pi$$, then k equals:
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Let $$f : R \to R$$ be a function such that $$|f(x)| \leq x^2$$, for all $$x \in R$$. Then, at x = 0, f is:
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If the volume of a spherical ball is increasing at the rate of $$4\pi$$ cc/sec then the rate of increase of its radius (in cm/sec), when the volume is $$288\pi$$ cc is:
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If non-zero real numbers b and c are such that $$\min f(x) > \max g(x)$$, where $$f(x) = x^2 + 2bx + 2c^2$$ and $$g(x) = -x^2 - 2cx + b^2$$, $$(x \in R)$$; then $$\left|\frac{c}{b}\right|$$ lies in the interval:
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If m is a non-zero number and $$\int \frac{x^{5m-1}+2x^{4m-1}}{(x^{2m}+x^m+1)^3} dx = f(x) + c$$, then $$f(x)$$ is equal to:
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Let, the function F be defined as $$F(x) = \int_1^x \frac{e^t}{t} dt$$, $$x > 0$$, then the value of the integral $$\int_1^x \frac{e^t}{t+a} dt$$, where a > 0, is:
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The area of the region (in square units) above the x-axis bounded by the curve $$y = \tan x$$, $$0 \leq x \leq \frac{\pi}{2}$$ and the tangent to the curve at $$x = \frac{\pi}{4}$$ is:
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If $$\frac{dy}{dx} + y\tan x = \sin 2x$$ and $$y(0) = 1$$, then $$y(\pi)$$ is equal to:
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If $$\vec{x} = 3\hat{i} - 6\hat{j} - \hat{k}$$, $$\vec{y} = \hat{i} + 4\hat{j} - 3\hat{k}$$ and $$\vec{z} = 3\hat{i} - 4\hat{j} - 12\hat{k}$$, then the magnitude of the projection of $$\vec{x} \times \vec{y}$$ on $$\vec{z}$$ is:
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If the angle between the line $$2(x+1) = y = z + 4$$ and the plane $$2x - y + \sqrt{\lambda}z + 4 = 0$$ is $$\frac{\pi}{6}$$, then the value of $$\lambda$$ is:
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Equation of the line of the shortest distance between the lines $$\frac{x}{1} = \frac{y}{-1} = \frac{z}{1}$$ and $$\frac{x-1}{0} = \frac{y+1}{-2} = \frac{z}{1}$$ is:
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Let A and E be any two events with positive probabilities.
Statement I: $$P(E/A) \geq P(A/E)P(E)$$.
Statement II: $$P(A/E) \geq P(A \cap E)$$.
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