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If $3^x=4^{x-1}$, then $x=$

A horse runs along a circle with a speed of 20 km/hr. A lantern is at the centre of the circle. A fence is along the tangent to the circle at the point at which the horse starts. The speed with which the shadow of the horse move along the fence at the moment when it covers $\frac{1}{8}$ of the circle in km/hr is

If $I_1=\underset{0}{\overset{\pi }{\int }}\frac{\cos x}{(x+2{)}^2}dx,$ $I_2=\underset{0}{\overset{\pi /2}{\int }}\frac{\cos x\sin x}{(x+1)}dx$ and $\lambda I_2=\frac{2}{\mu +\pi }+k-\gamma I_1,$ then

If radii of director circles of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are $2r$ and $r$ respectively and $e_e$ and $e_h$ be the eccentricities of the ellipse and the hyperbola respectively then

If the polar coordinates of a point are $(6,\frac{\pi }{3})$, then the Cartesian coordinates are

The general solutions for the equation $\cos 4x=\frac{\sqrt{5}+1}{4}$ is

If $\left[\begin{array}{cccc}1& 0& 0& 0\\ 2& 3& 0& 0\\ 4& 5& 6& 0\\ 7& 8& 9& 10\end{array}\right]$, then A is

If x satisfies $\left(x-1\right|+\left(x-2\right|+\left(x-3\right|\ge 6$, then the values of x lie in the range

The equation of the circle which passes through the points (2, 3) and (4, 5) and the centre lies on the straight line y - 4x + 3 = 0, is

The letters of the word "RANDOM" are arranged in all possible ways. The number of arrangements in which there are $2$ letters between 'R' and 'D' is

A straight line through the vertex P of a traingle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then

The value of ${\sin }^{2}52{\frac{1}{2}}^{\circ }-{\sin }^{2}22{\frac{1}{2}}^{\circ }$ is

If the distance between a focus and corresponding directrix of an ellipse be 8 and the eccentricity be $\frac{1}{2}$ , then length of the minor axis is

If $a,b,c$ are three distinct positive real numbers, then the number of real root(s) of $a{x}^2+2b|x|+c=0$ is

A matrix ‘B’ is singular if

Let P(6,3) be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.

If the normal at the point P intersects the x-axis at (9,0), then the eccentricity of hyperbola is,

The condition that the straight line $lx+my=n$ may be a normal to the hyperbola $b^2{x}^2-a^2{y}^2=a^2{b}^2$ is given by

If $2x=y^{1/5}+y^{-1/5}$ then $(x^2-1)\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}=$

If $\left|\frac{2}{x-5}\right|>1,$ then $x$ belongs to the interval

Equation of a line in the plane $\pi :2x-y+z-4=0$ which is perpendicular to the line $l$ whose equation is $\frac{x-2}{1}=\frac{y-2}{-1}=\frac{z-3}{-2}$ and which passes through the point of intersection of $l$ and $\pi$ is

If $2+5+8+11+\cdots \text{upto}n\text{terms}=610$, then the value of $n$ is

If one root is $n$ times the other for the equation $a{x}^2+bx+c=0,$ then

The differential equation of all straight lines passing through the point (1,-1)is

[MP PET 1994]

What is the condition for a strictly increasing differentiable function?

The general solution(s) of the equation $\sec 4\theta -\sec 2\theta =2$ can be

The product of all roots of the equation $(x^2-5x+7{)}^2-(x-2\left)\right(x-3)=1$ is

If $\alpha +\beta +\gamma =0,$ ${\alpha }^3+{\beta }^3+{\gamma }^3=12$ and ${\alpha }^5+{\beta }^5+{\gamma }^5=40,$ then the value of ${\alpha }^4+{\beta }^4+{\gamma }^4$ is equal to
(correct answer + 1, wrong answer - 0.25)

Let $A=\{x\in R:x^2-|x|-2=0\}$ and $B=\{\alpha +\beta ,\alpha \beta \}$ where $\alpha ,\beta$ are real roots of the quadratic equation $x^2+|x|-2=0.$ If $(a,b)\in A\times B,$ then the quadratic equation whose roots are $a,b$ is

$\sqrt{3}{x}^2-\sqrt{2x}+3\sqrt{3}=0$

The roots of the equation $2x^4+x^3-11x^2+x+2=0$ is/are

If $\alpha ,\beta ,\gamma$ are the roots of $x^3+2x^2-3x+1=0$, then

Let $\alpha$ and $\beta$ be the roots of equation $p{x}^2+qx+r=0,\ne 0.$ If p,q,r are in A.P. and $\frac{1}{\alpha }+\frac{1}{\beta }=4,$ then the value of $|\alpha -\beta |$is :

If roots of the equation $a{x}^2+bx+c=0$ are $\alpha ,\beta$, then the equation whose roots are $\frac{1+\alpha }{1-\alpha },\frac{1+\beta }{1-\beta },$ where $\alpha \ne 1,\beta \ne 1$ is

The value of $a$ for which the equation $(a^2-a-2)x^2+(a^2-4)x+(a^2-3a+2)=0$ have more than two roots

If every pair of equations $x^2+ax+bc=0,$ $x^2+bx+ac=0,$ $x^2+cx+ab=0$ has a common root, then product of these common roots is

${\sqrt{2x}}^2+x+\sqrt{2}=0$

For every possible $x\in R$, If $\frac{x^2+2x+a}{x^2+4x+3a}$ can take all real values, then

A quadratic equation with rational coefficients if one of its roots is ${\cot }^2{18}^{\circ }$ is