List of top Mathematics Questions

Let the tangents at two points A and B on the circle\( x^2 + y^2 – 4x + 3 = 0\) meet at origin O(0, 0). Then the area of the triangle OAB is

If \(\lim_{n\rightarrow \infty}\)(\(\sqrt{n^2-n-1}\) + nα+β)=0 then 8(α + β) is equal to

If \( \frac{d}{dx} \left( \frac{x^2+1}{(x^2+5)(x^2+9)} \right) = \frac{2x(x^2+1)}{(x^2+5)(x^2+9)} \left[ \frac{1}{f(x)} - \frac{1}{g(x)} - \frac{1}{h(x)} \right] \), then \( 2h(x)-f(x)-g(x) = \)

\( \int \frac{dx}{(1+\sqrt{x})^{2022}} = \)

If \( \int \frac{\sin(x-\pi/4)}{2+\sin 2x} dx = -\frac{1}{\sqrt{2}} \tan^{-1}(f(x)) + C \), then \(f(x) = \)

\( \int_{a-1}^{a} \frac{e^x(a-x)}{(x-a+1)^2} dx = \)

For \(n \in N\), If \( I_n = \int \frac{\sin nx}{\sin x} dx = \frac{2}{n-1}\sin((n-1)x) + I_{n-2} \) and \( \int_0^{\pi} \frac{\sin nx}{\sin x} dx = \frac{k\pi}{2} \) then k =

If a and b are respectively the order and degree of a differential equation \( y^2(y'')^2 + 3x(y')^{1/3} + x^2y^2 = \sin x \), then

An integrating factor of the differential equation \( (x^2+1)\frac{dy}{dx} + xy = x^3 \) is

Let f(x) = min {1, 1 + x sin x}, 0 ≤ x ≤ 2π. If m is the number of points, where f is not differentiable, and n is the number of points, where f is not continuous, then the ordered pair (m, n) is equal to

Let \(α, β\) be the roots of the equation \(x^2-\sqrt{2}x+\sqrt{6}=0\) and\( \frac{1}{α^2}+1\),\(\frac {1}{β^2}+1\) be the roots of the equation \(x^2 + ax + b = 0\) . Then the roots of the equation \(x^2 – (a + b – 2)x + (a + b + 2) = 0\) are

If the general solution of \( \frac{dy}{dx} = \frac{-y^2}{xy-y^2-x^2} \) is \( \tan^{-1}\left(\frac{x}{y}\right) = f(y) + C \), then \( f(e^3) = \)

\( \int_{-\pi/4}^{\pi/4} \cos^8 x \ dx = \)

If \( f(x) = \sin(\tan^{-1} x) \), then \( \int_0^1 xf''(x)dx = \)

If \( \int \frac{x^8+4}{x^4-2x^2+2} dx = Ax^5+Bx^3+Cx+K \), then \(5A+3B+C = \)

If an open cylinder of given surface area has maximum volume then its radius is

If \(x+y=k, x>0, y>0\) then \(x^2+y^2\) is minimum if

\( \int \sqrt{x+\sqrt{x^2+2}} \ dx = \)

The minimum distance of a point on the curve \( y = x^2 - 4 \) from the origin is

The normal to the curve y = f(x) at the points (3, 4) makes an angle \( \frac{3\pi}{4} \) with positive x-axis then \(f'(3) = \)

Equation of tangent to the curve \( y = x + \frac{4}{x^2} \) which is parallel to x-axis is

Assertion (A) : \( f(x)=|x| \) is differentiable at \( x=a \neq 0 \) and continuous but not differentiable at \( x=0 \) Reason (R) : If a function is differentiable at a point then it is continuous at that point. But converse is not true. .

If \( f(x) = \begin{cases 4 & -\infty<x<-\sqrt{5}
x^2-1 & -\sqrt{5} \le x \le \sqrt{5}
4 & \sqrt{5}<x<\infty \end{cases} \). If k is the number of points where f(x) is not differentiable then k -- 2 =}

\( \lim_{x \to a} \frac{x^3+a^3}{x+a} = 7 \implies a = \)