Recent questions in 2001

The shadow of a tower standing on a level ground is found to be 60 metres longer when the sun's altitude is $30^{\circ}$ than when it is $45^{\circ}$ . The height of the tower is :

asked Dec 11, 2018 by pady_1

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In a triangle $ABC, \;a^2 \sin 2C + c^2 \sin 2A$ is equal to :

asked Dec 11, 2018 by pady_1

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In a triangle $ABC, \; \frac{a}{b^2-c^2} + \frac{c}{b^2-a^2} = 0$ , then $B$ is equal to :

asked Dec 11, 2018 by pady_1

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In a triangle $ABC$ , $\frac{\cos C + \cos A}{c+a} + \frac{\cos B}{b}$ is equal to :

asked Dec 11, 2018 by pady_1

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$\sin^{-1} \begin{pmatrix}\frac{x}{\sqrt{1-x^2}} \end{pmatrix}$ is equal to :

asked Dec 11, 2018 by pady_1

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$\sec^2 (\tan^{-1}2) + \mathrm{cosec^2} (\cot^{-1} 3)$ is equal to :

asked Dec 11, 2018 by pady_1

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The equation $\sqrt{3}\sin x +\cos x = 4$ , has:

asked Dec 11, 2018 by pady_1

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If $A, B, C, D$ are angles of a cyclic quadrilateral, then $\cos A + \cos B + \cos C + \cos D$ is equal to :

asked Dec 11, 2018 by pady_1

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$\cos^2 ( \frac{\pi}{6} + \theta) - \sin^2 (\frac{\pi}{6} - \theta)$ is equal to :

asked Dec 11, 2018 by pady_1

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If $\mathrm{cosec} \theta = \frac{p+q}{p-q}$ , then $\cot (\frac{\pi}{4} + \frac{\theta}{2} )$ is equal to :

asked Dec 11, 2018 by pady_1

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$\frac{\sin 5 \theta}{\sin \theta}$ is equal to :

asked Dec 11, 2018 by pady_1

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If $\theta = \frac{\pi}{6}$ , then the $10^{th}$ term of $1 + (\cos \theta + i \sin \theta) +(\cos \theta + i \sin \theta)^2+(\cos \theta + i \sin \theta)^3 + ...$ is equal to :

asked Dec 11, 2018 by pady_1

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If $\begin{vmatrix} 1-i & i \\ 1+2i & -i \end{vmatrix} = x + iy$ , then $x$ is equal to :

asked Dec 11, 2018 by pady_1

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If $a$ is a complex number and $b$ is a real number, then the equation $\overline{a} + a + b =0$ represents a:

asked Dec 11, 2018 by pady_1

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If $A = \begin{bmatrix}0 & 2 \\ 3 & - 4 \end{bmatrix}, \; hA = \begin{bmatrix} 0 & 3a \\ 2b & 24 \end{bmatrix}$ , then the value of $h, \;a,\;b$ are respectively :

asked Dec 11, 2018 by pady_1

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A square matrix $[a_{ij}]$ in which $a_{ij} =0$ for $i \neq j$ and $a_{ij}=k$ (constant) for $i=j$ is called a:

asked Dec 11, 2018 by pady_1

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If $A = \begin{bmatrix} -2 & 2 \\ -3 & 2 \end{bmatrix}, \; B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ , then $(B^{-1}A^{-1})^{-1}$ is equal to :

asked Dec 11, 2018 by pady_1

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The bi-quadratic equation, two of whose roots are $1+i, \; 1-\sqrt{2}$ , is :

asked Dec 11, 2018 by pady_1

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$\displaystyle\lim_{x \to 0} \frac{\sin x \sin^{-1} x}{x^2}$ is equal to :

asked Dec 11, 2018 by pady_1

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If $1$ is a multiple root of order 3 for the equation $x^4 - 2x^3 + 2x-1=0$ , then the other root is :

asked Dec 11, 2018 by pady_1

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The roots of the equation $x^3 - 14 x^2 + 56x -64=0$ are in :

asked Dec 11, 2018 by pady_1

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Each of the roots of the equation $x^3 - 6x^2 + 6x-5 =0$ are increased by $h$ . So that the new transformed equation does not contain $x^2$ term, then $h$ is equal to :

asked Dec 11, 2018 by pady_1

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$20^{2-3x^2} = (40\sqrt{5})^{3x^2-2}$ , then $x$ is equal to :

asked Dec 11, 2018 by pady_1

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If $\alpha, \beta$ are the roots of the equation $x^2 + bx + c =0$ and $\alpha + h, \; \beta+h$ are the roots of the equation $x^2 + qx + r =0$ , then $h$ is equal to :

asked Dec 11, 2018 by pady_1

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$|x| <1$ , the coefficient of $x^3$ in the expansion of $\log (1+x+x^2)$ in ascending powers of $x$ , is :

asked Dec 11, 2018 by pady_1

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$\frac{2}{2!} + \frac{2+4}{3!} + \frac{2+4+6}{4!}$ + ...$ is equal to:

asked Dec 11, 2018 by pady_1

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If $\frac{x-4}{x^2-5x-2k} = \frac{2}{x-2} - \frac{1}{x+k'}$ then $k$ is equal to :

asked Dec 11, 2018 by pady_1

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If $(1+x)^n = C_0 + C_1x + C_2x^2 + ....+ C_nx^n$ , then $C_0+2C_1 + 3C_2 + ....+(n+1)C_n$ is equal to :

asked Dec 11, 2018 by pady_1

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The co-efficient of $x^4$ in the expansion of $\frac{(1-3x)^2}{(1-2x)}$ is equal to :

asked Dec 11, 2018 by pady_1

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$1+\frac{1}{4} + \frac{1.3}{4.8} + \frac{1.3.5}{4.8.12} + ...$ is equal to :

asked Dec 11, 2018 by pady_1

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Using the digits $0, 2, 4, 6, 8$ not more than once in any number, the number of 5 digited numbers that can be formed, is :

asked Dec 11, 2018 by pady_1

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The number of ways in which 5 boys and 4 girls sit around a circular tables. So that no two girls sit together is :

asked Dec 11, 2018 by pady_1

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If $y = A \cos nx + B \sin nx$ , then $y_2 = n^2y$ is equal to :

asked Dec 11, 2018 by pady_1

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If $2^3 + 4^3 + 6^3 + ....+(2n)^3 = hn^2 (n+1)^2$ , then $h$ is equal to :

asked Dec 11, 2018 by pady_1

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If $x =\log_{0.1} 0.001, \; y = \log_9 81$ , then $\sqrt{x -2\sqrt{y}}$ is equal to :

asked Dec 11, 2018 by pady_1

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$\frac{\sqrt{8 +\sqrt{28} } + \sqrt{8 - \sqrt{28}}}{\sqrt{8 +\sqrt{28} } - \sqrt{8 - \sqrt{28}}}$ is equal to :

asked Dec 11, 2018 by pady_1

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Two functions $f : R \to R, \; g: R \to R$ are defined as follows $f(x) = \begin{cases} 0 , & \quad \text{x } \text{ is rational}\\ 1, & \quad \text{x} \text{ is irrational} \end{cases}$ $g(x) = \begin{cases} -1 , & \quad \text{x } \text{ is rational}\\ 0, & \quad \text{x} \text{ is irrational} \end{cases}$ Then $(fog) (\pi) + (gof)(e)$ is equal to :

asked Dec 11, 2018 by pady_1

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Let $f : R \to R$ is defined by $f(x) = \begin{cases} x+2, & x \leq -1 \\ x^2 , & -1 < x <1 \\ 2-x, & x \geq 1 \end{cases}$ Then the value of $f(-1.75) + f(0.5) + f(1.5)$ is :

asked Dec 11, 2018 by pady_1

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Let $Z$ denote the set of integers define $f : Z \to Z$ by $f(x) = \begin{cases} \frac{x}{2}, & \quad \text{x} \text{ is even}\\ 0, & \quad \text{x} \text{ is odd} \end{cases}$ then $f$ is :

asked Dec 11, 2018 by pady_1

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$f(x) = (20 - x^4)^{1/4}$ for $0 < x < \sqrt{5}$ , then $f(f(1/2))$ is equal to :

asked Dec 11, 2018 by pady_1

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If $\tan \theta + \cot \theta = 2$ , then $\sin \theta$ is equal to :

asked Dec 11, 2018 by pady_1

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The solution of $\frac{dy}{dx} +y =e^x$ is :

asked Dec 11, 2018 by pady_1

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The solution of $x^2 +y^2 \frac{dy}{dx}=4$ is :

asked Dec 11, 2018 by pady_1

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The family of curves in which the sub-tangent at any point to any curve is double the abscissa is given by :

asked Dec 11, 2018 by pady_1

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The solution of $x \; dx + y \;dy = x^2y\; dy - xy^2\;dx$ is :

asked Dec 11, 2018 by pady_1

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Using the Trapezoidal rule, the approximate value of $\begin{align} \int_1^4 y \;dx \end{align}$

asked Dec 11, 2018 by pady_1

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$\begin{align} \int_{-1}^1 (ax^3 +bx ) dx =0 \end{align}$ for :

asked Dec 11, 2018 by pady_1

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$\begin{align} \int_0^{\pi/2} \sin^8x \cos^2 x \;dx \end{align}$ is equal to :

asked Dec 11, 2018 by pady_1

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$\begin{align} \int \frac{dx}{a^2 \sin^2x +b^2 \cos^2 x} \end{align}$ is equal to :

asked Dec 11, 2018 by pady_1

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$\begin{align} \int (x+1)^2 e^x \;dx \end{align}$ is equal to :

asked Dec 11, 2018 by pady_1

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