List of top Mathematics Questions

Let \(S\) be a symmetric matrix obtained from \[ A= \begin{bmatrix} 1 & 2 & -3\\ 2 & -2 & 1\\ 3 & 1 & -1 \end{bmatrix} \] and \(T\) be a skew-symmetric matrix obtained from \[ B= \begin{bmatrix} 4 & 2 & 0\\ 1 & -1 & 3\\ 0 & 2 & -3 \end{bmatrix} \] If trace of \(S=-4\) and the non-zero elements of \(T\) are \(-1,1\), then \(S+T=\)
If a real valued function \(f : A \to B\) defined by \(f(x)=|x|-[x]\) is a bijection, then \(A\) and \(B\) are respectively:
If \(A, B, C\) are vertices of a triangle with position vectors \(\vec{a}, \vec{b}, \vec{c}\) respectively, then find the position vector of the point \(D\) where the angle bisector from vertex \(A\) meets \(BC\).

If a function \( f : (-\infty, 2) \to \mathbb{R} \) is defined by \[ f(x)= \begin{cases} \dfrac{\alpha \left|x^2 - 3x + 2\right|}{x - 1}, & x < 1 \\[8pt] \dfrac{\sin([x] - x)}{x - [x]}, & x > 1 \\[8pt] \beta, & x = 1 \end{cases} \] and \( f \) is continuous at \( x = 1 \), then find \[ \frac{\alpha^2 + \beta^2}{|\alpha \beta|}. \] 

The general solution of the differential equation \[ (2xy+y^2)\,dy=(x^2-y^2)\,dx \] is:
The solution of the differential equation \[ \frac{dy}{dx} = \frac{3^{x+y}-2\cdot3^x} {3^{x+y}-2\cdot3^y} \] when \[ y(1)=2 \] is:
Evaluate: \[ \int \frac{3x\sec^2\!\sqrt{9x^2-12x+1} -\sec^2\!\sqrt{(3x-2)^2-3}} {\sqrt{9x^2-12x+1}} \,dx \]
Evaluate: \[ \int \frac{1}{\sin x \cos 2x}\,dx \]
Evaluate: \[ \int \frac{\cos 2x+\sin 4x} {\sqrt{3\sin 2x-2}} \,dx \]
If \[ \int_{2}^{3} \frac{3\log x} {3\log x+\log(125-75x+15x^2-x^3)} \,dx = k, \] then \[ 4k^2+2k+1= \]
Evaluate the definite integral: \[ \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \frac{x+\frac{\pi}{2}} {2-\sin^2 x} \,dx \]
Evaluate: \[ \int_{0}^{\pi} \sqrt{1+4\cos\frac{x}{2}} \left(\cos\frac{x}{2}-1\right)\,dx \]
If \(1^\circ \approx 0.01745\) radians, then the approximate value of \[ \sec 29^\circ \] is:
If \[ f(x)=x^3-19x+30 \] is a real valued function with domain \([-4,1]\), then the value of \(c\) according to Lagrange's Mean Value Theorem for \(f(x)\) is
If a tangent drawn at the point \(P(h,k)\), where \(h,k\in \mathbb{Z}\), on the curve \[ y=2x^3+3x^2-4x-1 \] passes through the point \(Q(2,8)\), then \(PQ=\)
Evaluate: \[ \int \frac{\sin4x}{\sin x}\,dx \]
If the base of an isosceles triangle is \(3\sqrt{2}\) feet and the two equal sides are increasing at the rate of \(1\) ft/s, then the rate of increase of its area (in sq.ft/sec) when the angle between the equal sides is a right angle is
The function \[ f(x)=x|x-1|+|x+2| \] is
If \(f:\mathbb{R}-\{0\}\rightarrow\mathbb{R}\) is a differentiable function such that \[ \frac{1}{3}f(x)+3f\!\left(\frac1x\right) = x-\frac{10}{3}, \] then find \[ f'(3)-f'\!\left(\frac13\right). \]
If a function \[ f(x)= \begin{cases} \dfrac{a}{|x|}, & x\le -1 \text{ or } x\ge 1,[6pt] \\ x^2+b, & -1<x<1, \end{cases} \] is differentiable on \(\mathbb{R}\), then \(a+b=\)
If \[ y=\tan^{-1}\left[\left(\frac{1-\cos 2\sqrt{x}}{1+\cos 2\sqrt{x}}\right)^{\frac12}\right], \qquad 0<x<\frac{\pi^2}{4}, \] then \(y(2y'+y)=\)
Evaluate \[ \lim_{x\to 0}\frac{\log(4+x)^x-\log 4^x}{\sin^2 x}. \]
If \[ (3y)^{2x}=5\left(2^{3x}\right), \] then \[ \left(\frac{dy}{dx}\right)_{x=1} = \]
If the foci of the ellipse \[ \frac{x^2}{25}+\frac{y^2}{k^2}=1 \qquad (k^2<25) \] and the hyperbola \[ \frac{x^2}{k}-\frac{y^2}{5}=1 \] are the same, then the product of the lengths of the latus rectum of the ellipse and that of the hyperbola is:
The area of the quadrilateral formed by the common tangents drawn to the circle \[ x^2+y^2=16 \] and the ellipse \[ 7x^2+25y^2=175 \] is: