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If three distinct numbers $$a$$, $$b$$, $$c$$ are in G.P. and the equations $$ax^{2} + 2bx + c = 0$$ and $$dx^{2} + 2ex + f = 0$$ have a common root, then which one of the following statements is correct?
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The number of integral values of $$m$$ for which the equation, $$(1 + m^{2})x^{2} - 2(1 + 3m)x + 1 + 8m = 0$$ has no real root, is:
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If $$z = \frac{\sqrt{3}}{2} + \frac{i}{2} (i = \sqrt{-1})$$, then $$(1 + iz + z^{5} + iz^{8})^{9}$$ is equal to:
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The number of four-digit numbers strictly greater than 4321 that can be formed using the digits 0, 1, 2, 3, 4, 5 (repetition of digits is allowed) is:
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The sum $$\sum_{k=1}^{20} k \cdot \frac{1}{2^k}$$ is equal to:
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If the fourth term in the binomial expansion of $$\left(\sqrt{x^{\frac{1}{1+\log_{10}x}}} + x^{\frac{1}{12}}\right)^{6}$$ is equal to 200, and $$x > 1$$, then the value of $$x$$ is:
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Suppose that the points $$(h, k)$$, $$(1, 2)$$ and $$(-3, 4)$$ lie on the line $$L_1$$. If a line $$L_2$$ passing through the points $$h, k$$ and $$(4, 3)$$ is perpendicular to $$L_1$$, then $$\frac{k}{h}$$ equals:
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The tangent and the normal lines at the point $$(\sqrt{3}, 1)$$ to the circle $$x^{2} + y^{2} = 4$$ and the x-axis form a triangle. The area of this triangle (in square units) is:
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The tangent to the parabola $$y^{2} = 4x$$ at the point where it intersects the circle $$x^{2} + y^{2} = 5$$ in the first quadrant, passes through the point:
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In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at $$(0, 5\sqrt{3})$$, then the length of its latus rectum is:
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If the eccentricity of the standard hyperbola passing through the point (4, 6) is 2, then the equation of the tangent to the hyperbola at (4, 6) is:
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Let $$f: R \rightarrow R$$ be a differentiable function satisfying $$f'(3) + f'(2) = 0$$. Then $$\lim_{x \to 0} \frac{1 + f(3 + x) - f(3)}{1 + f(2 - x) - f(2)}^{\frac{1}{x}}$$ is equal to:
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Which one of the following statements is not a tautology?
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A student scores the following marks in five tests: 45, 54, 41, 57, 43. His score is not known for the sixth test. If the mean score is 48 in the six tests, then the standard deviation of the marks in six tests is:
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Two vertical poles of height, 20 m and 80 m stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, from this horizontal plane is:
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If the lengths of the sides of a triangle are in A.P and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is:
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Let the numbers 2, b, c be in an A.P. and $$A = \begin{pmatrix} 1 & 1 & 1 \\ 2 & b & c \\ 4 & b^{2} & c^{2} \end{pmatrix}$$. If $$\det(A) \in [2, 16]$$, then $$c$$ lies in the interval:
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If the system of linear equations
$$x - 2y + kz = 1$$
$$2x + y + z = 2$$
$$3x - y - kz = 3$$
has a solution $$(x, y, z), z \neq 0$$, then $$(x, y)$$ lies on the straight line whose equation is:
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Let $$f(x) = a^x$$ ($$a > 0$$) be written as $$f(x) = f_1(x) + f_2(x)$$, where $$f_1(x)$$ is an even function and $$f_2(x)$$ is an odd function. Then $$f_1(x + y) + f_1(x - y)$$ equals:
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Let $$f: [-1, 3] \rightarrow R$$ be defined as
$$f(x) = \begin{cases} x + x, & -1 \le x < 1 \\ x + x, & 1 \le x < 2 \\ x + x, & 2 \le x \le 3 \end{cases}$$
where [t] denotes the greatest integer less than or equal to t. Then, f is discontinuous at:
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If $$f(1) = 1, f'(1) = 3$$, then the derivative of $$f(f(f(x))) + (f(x))^{2}$$ at $$x = 1$$ is:
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The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is:
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Given that the slope of the tangent to a curve $$y = y(x)$$ at any point $$(x, y)$$ is $$\frac{2y}{x^{2}}$$. If the curve passes through the centre of the circle $$x^{2} + y^{2} - 2x - 2y = 0$$, then its equation is:
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If $$\int \frac{dx}{x^3(1 + x^6)^{2/3}} = xf(x)(1 + x^6)^{1/3} + C$$, where C is a constant of integration, then the function $$f(x)$$ is equal to:
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Let $$f(x) = \int_0^x g(t) \, dt$$, where $$g$$ is a non-zero even function. If $$f(x + 5) = g(x)$$, then $$\int_0^x f(t) \, dt$$ equals:
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Let $$S(\alpha) = \{(x, y): y^{2} \le x, 0 \le x \le \alpha\}$$ and $$A(\alpha)$$ is area of the region $$S(\alpha)$$. If for a $$\lambda$$, $$0 < \lambda < 4$$, $$A(\lambda):A(4) = 2:5$$, then $$\lambda$$ equals:
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Let $$\vec{a} = 3\hat{i} + 2\hat{j} + x\hat{k}$$ and $$\vec{b} = \hat{i} - \hat{j} + \hat{k}$$, for some real $$x$$. Then the condition for $$\vec{a} \times \vec{b} = r$$ to follow is:
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The vector equation of the plane through the line of intersection of the planes $$x + y + z = 1$$ and $$2x + 3y + 4z = 5$$ which is perpendicular to the plane $$x - y + z = 0$$ is:
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If a point $$R(4, y, z)$$ lies on the line segment joining the points $$P(2, -3, 4)$$ and $$Q(8, 0, 10)$$, then the distance of R from the origin is:
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The minimum number of times one has to toss a fair coin so that the probability of observing at least one head is at least 90% is:
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