List of top Mathematics Questions

If $x>\sqrt{3}$ and $\frac{x^2+1}{(x^2+2)(x^2+3)}$ is expanded in terms of powers of $x$, then the coefficient of $x^{-8}$ is

If $-\frac{2}{3}<x<\frac{2}{3}$, then the value of the $5^{th}$ term in the expansion of $\frac{1}{\sqrt{2-3x}}$ when $x = \frac{1}{2}$ is

If the coefficients of $x^{10}$ and $x^{11}$ in the expansion of $(1 + \alpha x + \beta x^2)(1+x)^{11}$ are 396 and 144 respectively, then $\alpha^2 + \beta^2 =$

If \(1, \omega, \omega^2\) are the cube roots of unity, then \( \left(1\left(2+\frac{1}{\omega}\right)\left(2+\frac{1}{\omega^2}\right) + 2\left(3+\frac{1}{\omega}\right)\left(3+\frac{1}{\omega^2}\right) + 3\left(4+\frac{1}{\omega}\right)\left(4+\frac{1}{\omega^2}\right) + \dots + 10 \text{ terms}\right) = \)

Evaluate the infinite series: \[ \left|\begin{array}{ccc} 2 & 1 & \frac{1}{3} \\ 3 & 1 & 1 \\ \end{array}\right| + \left|\begin{array}{ccc} 1 & \frac{1}{3} & \frac{1}{2} \\ 3 & 1 & 1 \\ \end{array}\right| + \left|\begin{array}{ccc} 1 & \frac{1}{4} & \frac{1}{9} \\ 3 & 1 & 1 \\ \end{array}\right| + \left|\begin{array}{ccc} 1 & \frac{1}{4} & \frac{1}{27} \\ 3 & 1 & 1 \\ \end{array}\right| + \cdots = ? \]

The range of the real valued function \( f(x) = \cos^{-1} \left( \dfrac{3}{\sqrt{9x^2 - 12x + 22}} \right) \) is

If a function \( f(x) \) is given as: \[ f(x) = \begin{cases} \frac{\sqrt{1+ax^2+bx^3}-\sqrt{1-ax^2-bx^3}}{x^2}, & x<0 \\ 5, & x = 0 \\ \frac{\tan3x-\sin3x}{bx^3}, & x>0 \end{cases} \] and is continuous at \( x = 0 \), then the geometric mean of \( a \) and \( b \) is:

The general solution of the differential equation \( \sec(x - y + 1) dy = dx \) is:

The general solution of the differential equation \[ \frac{dy}{dx} = \frac{2xy-4x+y-2}{2xy+x-4y-2} \] is:

The differential equation for which \( y^2 = 4a(x + a) \) (where \( a \) is a parameter) is the general solution is:

Evaluate the integral: \[ I = \int_0^{3\pi/2} \frac{\cos^5 x}{\cos^3 x+\sin^3 x}dx \]

Evaluate the integral: \[ \int \left[\frac{1}{\cos x} - \frac{1}{\sin x} - \frac{1}{\sin x + 3\cos x}\right] dx \]

Evaluate the integral: \[ I = \int_{\pi/6}^{\pi/3} \cos^{-4} x \, dx \]

Evaluate the integral: \[ \int \operatorname{Cos}^{-1} \left( \frac{1-x^2}{1+x^2} \right) dx \]

Find the area (in square units) of the region bounded by the lines \( x=0 \), \( x=\frac{\pi}{2} \), and the curves \( f(x) = \sin x \), \( g(x) = \cos x \):

Evaluate the integral: \[ I = \int_0^x \frac{t^2}{\sqrt{a^2 + t^2}} dt \]

Evaluate the integral: \[ \int \frac{\sec^2 x}{(\sec x+\tan x)^{5/2}}dx \]

Evaluate the integral: \[ \int \frac{x+1}{x^3 - 1}dx \]

If the lengths of the tangent, subtangent, normal, and subnormal for the curve \( y = x^2 + x - 1 \) at the point \( (1,1) \) are \( a, b, c, \) and \( d \) respectively, then their increasing order is:

Evaluate the integral: \[ \int \frac{x^4-16x^2+2x+8}{x^3-4x^2+2}dx \]

P and Q are the ends of a diameter of the circle \( x^2+y^2=a^2(a>\frac{1}{\sqrt{2}}) \). \( s \) and \( t \) are the lengths of the perpendiculars drawn from P and Q onto the line \( x+y=1 \) respectively. When the product \( st \) is maximum, the greater value among \( s, t \) is:

If the volume of a sphere is increasing at the rate of 12 \( \text{cm}^3/\text{sec} \), then the rate (in \( \text{cm}^2/\text{sec} \)) at which its surface area is increasing when the diameter of the sphere is 12 cm is:

Let \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) be such that \( x = 0 \) is the only real root of \( P'(x) = 0 \). If \( P(-1)<P(1) \), then in the interval \( [-1,1] \):

If \( y = \log(\sec(\tan^{-1}x)) \) for \( x>0 \), then \( \frac{dy}{dx} \) at \( x = 1 \) is:

The quadratic equation whose roots are \[ l = \lim_{\theta \to 0} \left( \frac{3\sin\theta - 4\sin^3\theta}{\theta} \right) \] \[ m = \lim_{\theta \to 0} \left( \frac{2\tan\theta}{\theta(1-\tan^2\theta)} \right) \] is: