List of top Mathematics Questions

Consider the following statements
Statement-I: A function \( f: A \rightarrow B \) is said to be one-one if and only if \[ f(x) = f(y) \Rightarrow x = y \]
Statement-II: A relation \( f: A \rightarrow B \) is said to be a function if \[ x = y \Rightarrow f(x) \neq f(y) \]
Then which one of the following is true?
The general solution of the differential equation \( \frac{dy}{dx} + \frac{\sec x}{\cos x + \sin x}y = \frac{\cos x}{1+\tan x} \) is
The differential equation of the family of circles passing through the origin and having centre on X-axis is
The general solution of the differential equation \( \frac{dy}{dx} = \frac{x+y}{x-y} \) is
If \( k \in N \) then \( \lim_{n\to\infty} \left[ \frac{1}{n+1} + \frac{1}{n+2} + \frac{1}{n+3} + \dots + \frac{1}{kn} \right] = \) (Note: The last term should be \( \frac{1}{n+ (k-1)n} = \frac{1}{kn} \) or sum up to \(n+(k-1)n\). The given form \(1/kn\) as the endpoint of the sum means sum from \(r=1\) to \((k-1)n\). The sum is usually \( \sum_{r=1}^{(k-1)n} \frac{1}{n+r} \). If the last term is \( \frac{1}{kn} \), it means \( n+r = kn \implies r = (k-1)n \). So it's \( \sum_{r=1}^{(k-1)n} \frac{1}{n+r} \).) Let's assume the sum goes up to \( \frac{1}{n+(k-1)n} = \frac{1}{kn} \). So the sum is \( \sum_{r=1}^{(k-1)n} \frac{1}{n+r} \). No, this seems to be \( \frac{1}{n+1} + \dots + \frac{1}{n+(kn-n)} \). The sum should be written as \( \sum_{i=1}^{(k-1)n} \frac{1}{n+i} \). The dots imply the denominator goes up. The last term is \( \frac{1}{kn} \). This means the sum is actually \( \frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{n+(k-1)n} \). The number of terms is \( (k-1)n \).
\( \int_{-1}^{4} \sqrt{\frac{4-x}{x+1}} \ dx = \)
\( \int \sqrt{x^2+x+1} \ dx \)
\( \int_{5\pi}^{25\pi} |\sin 2x + \cos 2x| \ dx = \)
\( \int_{0}^{\pi/4} \frac{\cos^2 x}{\cos^2 x + 4\sin^2 x} dx = \)
If \( \int \frac{\cos^3 x}{\sin^2 x + \sin^4 x} dx = c - \operatorname{cosec} x - f(x) \), then \( f\left(\frac{\pi}{2}\right) = \)
\( \int \left( \sum_{r=0}^{\infty} \frac{x^r 2^r}{r!} \right) dx = \)
Which one of the following functions is monotonically increasing in its domain?
If the area of a right angled triangle with hypotenuse 5 is maximum, then its perimeter is
\( \int \frac{13\cos 2x - 9\sin 2x}{3\cos 2x - 4\sin 2x} dx = \)
If \( \beta \) is an angle between the normals drawn to the curve \( x^2+3y^2=9 \) at the points \( (3\cos\theta, \sqrt{3}\sin\theta) \) and \( (-3\sin\theta, \sqrt{3}\cos\theta) \), \( \theta \in \left(0, \frac{\pi}{2}\right) \), then
\( \int \frac{dx}{12\cos x + 5\sin x} = \)
If \( \sqrt{x-xy} + \sqrt{y-xy} = 1 \), then \( \frac{dy}{dx} = \)
If the tangent drawn at the point \( (x_1,y_1) \), \(x_1,y_1 \in N \) on the curve \( y = x^4 - 2x^3 + x^2 + 5x \) passes through origin, then \( x_1+y_1 = \)
If \( y = \tan^{-1}\left(\frac{x}{1+2x^2}\right) + \tan^{-1}\left(\frac{x}{1+6x^2}\right) \), then \( \frac{dy}{dx} = \)
If g is the inverse of the function f(x) and \( g(x) = x + \tan x \) then, \( f'(x) = \)
\([x]\) denotes the greatest integer less than or equal to x. If \(\{x\}=x-[x]\) and \( \lim_{x\to 0} \frac{\sin^{-1}(x+[x])}{2-\{x\}} = \theta \), then \( \sin\theta + \cos\theta = \)
If Q \( (\alpha, \beta, \gamma) \) is the harmonic conjugate of the point P(0,-7,1) with respect to the line segment joining the points (2,-5,3) and (-1,-8,0), then \( \alpha - \beta + \gamma = \)
If the distance between the foci of a hyperbola H is 26 and distance between its directrices is \( \frac{50}{13} \), then the eccentricity of the conjugate hyperbola of the hyperbola H is
On a line with direction cosines l, m, n, \( A(x_1, y_1, z_1) \) is a fixed point. If \( B=(x_1+4kl, y_1+4km, z_1+4kn) \) and \( C=(x_1+kl, y_1+km, z_1+kn) \) (\(k>0\)) then the ratio in which the point B divides the line segment joining A and C is
If the line of intersection of the planes \(2x+3y+z=1\) and \(x+3y+2z=2\) makes an angle \( \alpha \) with the positive x-axis, then \( \cos \alpha = \)