List of top Mathematics Questions

\(\int\frac{2\cos 2x}{(1+\sin 2x)(1+\cos 2x)}dx=\)

\(\int\left(\frac{x}{x\cos x - \sin x}\right)^2 dx = \)

If \(\lim_{n\rightarrow\infty}[(1+\frac{1}{n^{2}})(1+\frac{4}{n^{2}})(1+\frac{9}{n^{2}})\cdots(1+\frac{n^{2}}{n^{2}})]^{\frac{1}{n}}=ae^{b}\), then a+b=

If \(I_{n}=\int_{0}^{\pi/4}\tan^{n}x~dx,\) then \(I_{13}+I_{11}=\)
The area (in square units) of the smaller region lying above the X-axis and bounded between the circle \[ x^2 + y^2 = 2ax \] and the parabola \[ y^2 = ax \]
The difference of the order and degree of the differential equation \[ \left(\frac{d^2y}{dx^2} \right)^{-7/2} - \left(\frac{d^3y}{dx^3} \right)^2 - \left(\frac{d^2y}{dx^2} \right)^{-5/2} - \left(\frac{d^4y}{dx^4} \right) = 0 \]
The solution of the differential equation \[ x dy - y dx = \sqrt{x^2 + y^2} dx \] when \( y(\sqrt{3}) = 1 \) is:
\(\int_{0}^{\pi}x \sin^4 x \cos^6 x dx = \)
The perpendicular distance from the point \( (-1,1,0) \) to the line joining the points \( (0,2,4) \) and \( (3,0,1) \) is:
The locus of the midpoints of the chords of the hyperbola \( x^2 - y^2 = a^2 \) which touch the parabola \( y^2 = 4ax \) is:
If the product of the eccentricities of the ellipse \( \frac{x^2}{16} + \frac{y^2}{b^2} = 1 \) and the hyperbola \( \frac{x^2}{9} - \frac{y^2}{16} = 1 \) is 1, then the value of \( b^2 \) is:
A bag contains 2 white, 3 green, and 5 red balls. If three balls are drawn one after the other without replacement, then the probability that the last ball drawn was red is:
If the roots of the equation \( x^3 - 13x^2 + Kx - 27 = 0 \) are in geometric progression, then \( K = \):
Find the general solution of the differential equation \( (x - y -1) dy = (x + y + 1) dx \).
Find the area enclosed by the curve \( y = x^3 - 19x + 30 \) and the X-axis.
Evaluate \( \int_{ \log 4}^{ \log 5} \frac{e^{2x} + e^x}{e^{2x} - 5e^x +6} dx \):
Evaluate the integral: \[ \int \frac{1}{x^5 \sqrt{x^5+1}} dx. \]
Evaluate the integral: \[ I = \int \frac{\csc x}{3\cos x + 4\sin x} dx. \]
Evaluate the integral: \[ I = \int (\tan^7 x + \tan x) dx. \]
Evaluate the integral: \[ I = \int \frac{x+1}{\sqrt{x^2 + x + 1}} dx. \]
By considering \( 1' = 0.0175 \), the approximate value of \( \cot 45^\circ 2' \) is:
A point moves on the curve \( y = x^3 - 3x^2 + 2x - 1 \) and its y-coordinate increases at a rate of 6 units per second. When the point is at (2,-1), the rate of change of its x-coordinate is:
Evaluate the limit: \[ \lim_{x \to \infty} \left( \frac{3x^2 - 2x + 3}{3x^2 + x - 2} \right)^{3x - 2} \]
Let \( P(a, 4, 7) \) and \( Q(3, \beta, 8) \) be two points. If the YZ-plane divides the join of the points P and Q in the ratio 2:3 and the ZX-plane divides the join of P and Q in the ratio 4:5, then the length of line segment PQ is:
If \( (a, b) \) and \( (c, d) \) are the internal and external centres of similitude of the circles \[ x^2 + y^2 + 4x - 5 = 0 \] and \[ x^2 + y^2 - 6y + 8 = 0 \] respectively, then \( (a + d)(b + c) \) is: