Recent questions tagged 1998

The minimum value of $x^2-8x+17, \; x \in R$ is :

asked Dec 19, 2018 by pady_1

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If at any point on the curve $y=f(x)$ the length of the subnormal is constant, then the curve will be a / an :

asked Dec 19, 2018 by pady_1

1 answer

If $f(x) = \begin{vmatrix} 2 \cos x & 1 & 0 \\ x - \frac{\pi}{2} & 2 \cos x & 1 \\ 0 & 1 & 2 \cos x \end{vmatrix}$, then $\frac{df}{dx} $ at $x = \frac{\pi}{2}$ is:

asked Dec 19, 2018 by pady_1

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If $x$ and $y$ are strictly positive such that $x+y=1$, then the minimum value of $x \log x + y \log y$ is :

asked Dec 19, 2018 by pady_1

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If $\log y = \tan^{-1}x$, then $(1+x^2) \frac{d^2y}{dx^2} + (2x-1) \frac{dy}{dx}$ is equal to :

asked Dec 19, 2018 by pady_1

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If the side of an equilateral triangle expands at the rate of $2\; cm/sec$, then the rate of increase of its area when the side is 10 cm is (in sq cm) :

asked Dec 19, 2018 by pady_1

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If $\tan y = \frac{2t}{1-t^2}, \; sin x = \frac{2t}{1+t^2}$, then $\frac{dy}{dx}$ is equal to :

asked Dec 19, 2018 by pady_1

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If $y = \sqrt{\cos 2x}$, then $y \frac{d^2y}{dx^2} + 2y^2$ is equal to :

asked Dec 19, 2018 by pady_1

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If $\begin{align}y = e^{{-x}^2} \int_0^x e^{t^2} dt \end{align}$, then $\frac{dy}{dx} + 2xy$ is equal to :

asked Dec 19, 2018 by pady_1

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$\displaystyle\lim_{x \to 0} \frac{\sin ax}{\sin bx} $ is equal to :

asked Dec 19, 2018 by pady_1

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$\displaystyle\lim_{x \to \infty} \begin{vmatrix} \frac{a^{1/x} + b^{1/x} + c^{1/x} }{3} \end{vmatrix}^x$, where $a, b, c$ are real and non-zero is equal to :

asked Dec 19, 2018 by pady_1

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

asked Dec 19, 2018 by pady_1

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If $f : R \to R$ is continuous such that $f(x+y) = f(x) + f(y), \; \; \forall \; \; x, y \in R$ and $f(1) = 2$, then $f(100)$ is equal to :

asked Dec 19, 2018 by pady_1

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$\displaystyle\lim_{x \to 0} \log \begin{vmatrix} \frac{\log (1+x)}{x} \end{vmatrix}$ is equal to :

asked Dec 19, 2018 by pady_1

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The equaion of the line common to the pairs of lines <br> $(p^2 - q^2)x^2 + (q^2 - r^2) xy + (r^2 - p^2) y^2 = 0$ and $(f-m)x^2 + (m-n)xy + (n-f)y^2 = 0$ is :

asked Dec 19, 2018 by pady_1

0 answers

If $a, h, b$ are in A.P., then the triangular area formed by the pair of lines $ax^2 + 2hxy + by^2=0$ and the line $x-y=-2$ (in square units) is :

asked Dec 19, 2018 by pady_1

1 answer

If the pair of lines given by $(x^2+y^2) \sin \alpha = (x \cos \alpha - y \sin \alpha)^2$ are perpendicular to each other, then $\alpha$ is equal to :

asked Dec 19, 2018 by pady_1

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The angle betwen the pair of lines $2(x+2)^2 + 3(x+2)(y+2) - 2(y-2)^2 = 0$ is :

asked Dec 19, 2018 by pady_1

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The equation of the pair of lines through $(1, -1)$ and perpendicular to the pair of lines $x^2 - xy - 2y^2 = 0$ is :

asked Dec 19, 2018 by pady_1

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The perpendicular distance from $(1,2)$ to the straight line $12x + 5y=7$, is :

asked Dec 19, 2018 by pady_1

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Let $a$ and $b$ be non-zero real numbers such that $a \neq b$. Then the equation of the line passing through the origin and the point of intersection of $\frac{x}{a} + \frac{y}{b} = 1$ and $\frac{x}{b} + \frac{y}{a} = 1$ is :

asked Dec 19, 2018 by pady_1

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$k$ is non-zero constant. If $k = \frac{a+b}{ab}$, then straight line $\frac{x}{a} + \frac{y}{b} = 1$ passes through the point :

asked Dec 19, 2018 by pady_1

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The angle at which the axes are to be rotated to remove the $xy$ term in the equation $x^2 + 2\sqrt{3}xy = y^2$ is :

asked Dec 19, 2018 by pady_1

0 answers

$A = (-9, 0)$ and $B = (-1, 0)$ are two points. If $P=(x,y)$ is a point such that $3PB = PA$, then the locus of $P$ is :

asked Dec 19, 2018 by pady_1

0 answers

If $1, \omega, \omega^2$ are the cube roots of unity, then $(1- \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5$ is equal to

asked Dec 19, 2018 by pady_1

0 answers

The value of $\begin{pmatrix}\frac{1-\sqrt{3}i }{2}\end{pmatrix}^{36} + \begin{pmatrix}\frac{-1 -\sqrt{3} i }{2}\end{pmatrix}^{36} $ is equal to :

asked Dec 19, 2018 by pady_1

0 answers

If $1, \omega, \omega^2$ are the cube roots of unity and $a, b$ are real and $x = a + b$, $y = a \omega^2 + b \omega$, $z = a \omega + b \omega^2$, then $x^2 + y^2 + z^2 $ is equal to :

asked Dec 19, 2018 by pady_1

0 answers

If $P$ represent $z= x + iy$ in the argand plane and $|z-1|^2 + |z+1|^2 = 4$, then the locus of $P$ is :

asked Dec 19, 2018 by pady_1

0 answers

If the angles of depression of the upper and lower ends of a lamp post the top of a hill of height $h$ meters are $\alpha$ and $\beta$ respectively, then the height of the lamp post is equal to :

asked Dec 19, 2018 by pady_1

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If the area of the triangle ABC is $a^2 - (b-c)^2$, then $\tan \frac{A}{2}$ is equal to :

asked Dec 19, 2018 by pady_1

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In a triangle $ABC$, if $\angle{C} = 60^{\circ}$, then $\frac{a}{b+c} + \frac{b}{c+a}$ is equal to :

asked Dec 19, 2018 by pady_1

0 answers

A square, non-singular matrix A satisfies $A^2 - A + 2I=0$, then $A^{-1}$ is equal to :

asked Dec 19, 2018 by pady_1

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The matrix $\begin{bmatrix} 1 & 0 & 1 \\ 2 & 1 & 0 \\ 3 & 1 & 1 \end{bmatrix}$ is :

asked Dec 19, 2018 by pady_1

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$\begin{vmatrix} x & 1 & y+z \\ y & 1 & z+x \\ z & 1 & x+y \end{vmatrix}$ is equal to :

asked Dec 19, 2018 by pady_1

0 answers

If the sum of the squares of the roots of $x^2 + px - 3=0$ is $10$, then the value of $p$ is equal to :

asked Dec 19, 2018 by pady_1

0 answers

If $\tan^{-1}x + \tan^{-1}y + \tan^{-1}z = \frac{\pi}{2}$, then $1-xy-yz - zx$ is equal to :

asked Dec 19, 2018 by pady_1

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If $x = \tan^{-1}y$, then $\log_e(\frac{1+y}{1-y})$, then :

asked Dec 19, 2018 by pady_1

0 answers

If $\sqrt{\sin x} + \cos x = 0$, then $\sin x$ is equal to :

asked Dec 19, 2018 by pady_1

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If $A + B + C = 0$, then $\sin^2A + \sin^2B + \sin^2C$ is equal to :

asked Dec 19, 2018 by pady_1

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Maximum and minimum values of $\sin^2(120^{\circ} + \theta) + \sin^2(120^{\circ} - \theta)$ are respectively :

asked Dec 19, 2018 by pady_1

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The sum of the series $1 + 3x + 5x^2 + 7x^3+ ....+(2n-1)x^{n-1} + ...$ is:

asked Dec 19, 2018 by pady_1

0 answers

If $\frac{(x+1)^2}{x^3 + x} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1}$, then $\sin^{-1} (A/C)$ is equal to :

asked Dec 19, 2018 by pady_1

0 answers

If $C_n$ is the coefficient of $x^n$ in the expansion of $(1+x)^n$, then $C_1 + 2C_2 + 3C_3 + ...+nC_n$ is equal to :

asked Dec 19, 2018 by pady_1

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The term independent of $x$ in the expansion of $(2x^2 - \frac{3}{x^3})^{15}$, is :

asked Dec 19, 2018 by pady_1

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If $a = 1 + \log_x yz, \; b = 1+ \log_y zx, \; c = 1 + \log_z xy$, then $ab + bc + ca$ is equal to :

asked Dec 19, 2018 by pady_1

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If $x = 2\sqrt{2} + \sqrt{7}$, then $x+\frac{1}{x}$ is equal to :

asked Dec 19, 2018 by pady_1

0 answers

A group consists of 6 men and 3 women. A committee is to be formed with 5 people choosing 3 men and 2 women. The number of different committees that can be formed, as :

asked Dec 19, 2018 by pady_1

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$\begin{vmatrix} a & b& c \\ b & c &a \\ c & a & b \end{vmatrix} = 0$ then :

asked Dec 19, 2018 by pady_1

0 answers

$\tan^{-1} x + \cot^{-1} (x+1)$ is equal to :

asked Dec 19, 2018 by pady_1

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Domain of $f(x) = \frac{1}{6} \sqrt{\log_{10} (5x-x^2) }$ is :

asked Dec 19, 2018 by pady_1

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