List of top Mathematics Questions

If \( y (\cos x)^{\sin x} = (\sin x)^{\sin x} \), then the value of \( \frac{dy}{dx} \) at \( x = \frac{\pi}{4} \) is:
If \( x = \cos 2t + \log (\tan t) \) and \( y = 2t + \cot 2t \), then \( \frac{dy}{dx} \) is:
If \( y = 44x^{45} + 45x^{44} \), then \( y'' \) is:
The approximate value of \( \sqrt[3]{730} \) obtained by the application of derivatives is:
The general solution of the differential equation \[ (9x - 3y + 5) dy = (3x - y + 1) dx. \]
The general solution of the differential equation $$ \frac{dy}{dx} = \frac{2y^2 + 1}{2y^3 - 4xy + y}. $$ 
If \(P\left( \frac{\pi}{4} \right)\), \(Q\left( \frac{\pi}{3} \right)\) are two points on the circle \(x^2 + y^2 - 2x - 2y - 1 = 0\), then the slope of the tangent to this circle which is parallel to the chord \(PQ\) is:

The maximum value of the function \[ f(x) = 3\sin^{12}x + 4\cos^{16}x \] is ?

If \(A + B + C = 2S\), then \[ \sin(S - A)\,\cos(S - B)\;-\;\sin(S - C)\,\cos S \;=\;? \]
If \(\cos x + \cos y = \tfrac{2}{3}\) and \(\sin x - \sin y = \tfrac{3}{4}\), then \[ \sin(x - y) \;+\; \cos(x - y) \;=\;? \]
The solution set of the equation \(\cos^2(2x) + \sin^2(3x) = 1\) is \;?
If \(2\,\tan^{-1}x = 3\,\sin^{-1}x\) and \(x \neq 0\), then \(8x^2 +1 =\;?\)

Match the functions in List--I with their corresponding properties in List--II:

If \(z = 1 - \sqrt{3}\,i\), then \(z^3 - 3z^2 + 3z = \;?\)

The product of all the values of \(\bigl(\sqrt{3} - i\bigr)^{25}\) is ?

The number of common roots among the 12th and 30th roots of unity is ?

∫ √(2x2 - 5x + 2) dx = ∫ (41/60) dx,

and

-1/2 > α > 0, then α = ?

In a triangle \(ABC\), if \(\tan\tfrac{A}{2} : \tan\tfrac{B}{2} : \tan\tfrac{C}{2} = 15 : 10 : 6\), then \(\frac{a}{b - c} = \,?\)
Among the 4-digit numbers that can be formed using the digits \(\{1,2,3,4,5,6\}\) without repeating any digit, the number of such numbers which are divisible by 6 is \;?
If the number of circular permutations of 9 distinct things taken 5 at a time is \(n_1\), and the number of linear permutations of 8 distinct things taken 4 at a time is \(n_2\), then what is \(\frac{n_1}{n_2}\)?
The number of ways in which 4 different things can be distributed to 6 persons so that no person gets all the things is \;?

Among the 5 married couples, if the names of 5 men are matched with the names of their wives randomly, then the probability that no man is matched with the name of his own wife is ?

Three similar urns \(A,B,C\) contain \(2\) red and \(3\) white balls; \(3\) red and \(2\) white balls; \(1\) red and \(4\) white balls, respectively. If a ball is selected at random from one of the urns is found to be red, then the probability that it is drawn from urn \(C\) is ?

If \(4+3x-7x^2\) attains its maximum value \(M\) at \(x=\alpha\) and \(5x^2-2x+1\) attains its minimum value at \(x=\beta\), then \[ \frac{28\,(M - \alpha)}{5\,(m + \beta)} =\,? \] \textit{(Assume \(m\) is that minimum value of \(5x^2 -2x +1\) at \(x=\beta\))}
The number of normals that can be drawn through the point \((9,6)\) to the parabola \(y^2 = 4x\) is: