List of top Mathematics Questions

In the expansion of \[ \left(9x-\frac{1}{3\sqrt{x}}\right)^{18}, \] if the coefficient of the term independent of \(x\) is \(221k\), then the value of \(k\) is:

The number of non-negative integer solutions of the equation \[ a + b + 2c = 22 \] is:

Evaluate \[ (0.2)^{\log_{\sqrt{5}}\alpha} + (0.04)^{\log_{5}\beta} \] if \[ \alpha = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots \] \[ \beta = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots \]

P is the point of intersection of the two half-lines \[ x-\sqrt{3}y=\alpha, \quad \alpha>0 \] Points \(A\) and \(B\) lie on these lines at a distance \(\alpha\) from \(P\). If the length of perpendicular from \(P\) on \(AB\) is \(\frac{\alpha}{2}\) and the radius of the circumcircle of \(\triangle PAB\) is \(R\), then find \[ \frac{\alpha^2}{R} \]

P is a point on \[ \frac{x^2}{9}+\frac{y^2}{4}=1 \] as \(P(3\cos\alpha,2\sin\alpha)\). Q is a point on \[ x^2+y^2-14x+14y+82=0 \] R is a point on line \[ x+y=5 \] If the centroid of triangle \(PQR\) is \[ \left(\cos\alpha+2,\;\frac{2\sin\alpha}{3}+3\right) \] find the sum of possible ordinates of \(R\).

If \(\left(2\alpha+1,\;\alpha^2-3\alpha,\;\frac{\alpha-1}{2}\right)\) is the image of \((\alpha,2\alpha,1)\) in the line \[ \frac{x-2}{3}=\frac{y-1}{2}=\frac{z}{1}, \] then the value of \(\alpha\) is:

If \(f(x)=(x-1)^4+1 \quad \forall x\in[1,\infty)\). Statement–1 : \(f(x)=f^{-1}(x)\) has only two solutions. Statement–2 : \(f^{-1}(x+1)=f(x)\) has no solution.

In the expansion of \[ \left(9x-\frac{1}{3\sqrt{x}}\right)^{18} \] if coefficient of the term independent of \(x\) is \(221k\), then the value of \(k\) is

If \(\hat u,\hat v\) are unit vectors and \[ |\hat u\times \hat v|=\frac{\sqrt3}{2} \] and \[ \vec A=\lambda\hat u+\hat v+\hat u\times \hat v \] then find \(\lambda\). (Angle between \(\hat u\) and \(\hat v\) is acute)

If \(z_1,z_2,z_3\) are roots of \[ x^3+ax^2+bx+c=0 \] Let \(z_1=1,\; z_2=1+i\sqrt2\) and \(a,b,c\in\mathbb{R}\). Then the value of \[ \int_{-1}^{1}(x^3+ax^2+bx+c)\,dx \] is

If \[ \alpha=\frac14+\frac18+\frac1{16}+\cdots \text{ up to infinity} \] \[ \beta=\frac13+\frac19+\frac1{27}+\cdots \text{ up to infinity} \] Then value of \[ (0.2)^{\log_5\alpha}+(0.04)^{\log_5\beta} \] is

If \[ f(x)=\int_{0}^{x}\tan(t-x)\,dt+\int_{0}^{x}f(t)\tan t\,dt \] then the value of \[ f''\left(\frac{\pi}{6}\right)-12f'\left(-\frac{\pi}{6}\right)+f\left(\frac{\pi}{6}\right) \] is

If \(f(x)\) satisfies the functional equation \[ f(x+y)=f(x)+2y^2+y+\alpha xy \] where \(x,y\) are whole numbers, such that \(f(0)=-1\) and \(f(1)=2\), then the value of \[ \sum_{i=1}^{5}\big(f(i)+\alpha\big) \] is

Let \[ S=\{z: z^2+4z+16=0,\; z\in\mathbb{C}\} \] then the value of \[ \sum_{z\in S}|z+\sqrt{3}i|^2 \] is

Evaluate \[ \int_{0}^{1}\cot^{-1}(1+x+x^2)\,dx \]

The maximum value of \[ E = 16\sin\frac{x}{2}\cos^3\frac{x}{2} \] where \(x\in[0,\pi]\), is

If function \(y(x)\) satisfies the differential equation \[ \frac{dy}{dx}+\left[\frac{6x^2+e^{-2x}(3x^2+2x^3+4)}{(x^3+2)(2+e^{-2x})}\right]y = e^{-2x}+2 \] such that \(y(0)=\frac{3}{2}\) and \[ y(1)=\alpha(e^{-2}+2) \] then \(\alpha\) is equal to

If the quadratic equation \[ (\lambda + 2)x^2 - 3\lambda x + 4\lambda = 0 \quad (\lambda \ne -2) \] has two positive roots then the number of possible integral values of \(\lambda\) is

From point \(B(4,8)\) on the parabola \(y^2 = 16x\), two perpendicular chords \(BA\) and \(BC\) are drawn. Given that the locus of the centroid of triangle \(BAC\) is another parabola with length of the latus rectum equal to \(\ell\), then \(3\ell\) is equal to

Area bounded between the curves \[ x = -2y^2 \] \[ x = 1 - 4y^2 \] is

If the system of equations \[ x + y + z = 5 \] \[ x + 2y + 3z = 9 \] \[ x + 3y + \lambda z = \mu \] has infinitely many solutions, then value of \( \lambda + \mu \) is

If \( \sum_{i=1}^{10} (x_i + 2)^2 = 180 \) and \( \sumᵢ=1¹0 (xᵢ - 1)² = 90 , then the Standard Deviation is equal to

If the matrix \[ \begin{bmatrix} 1 & 3 & 1 \\ 2 & 1 & \alpha \\ 0 & 1 & -1 \end{bmatrix} \] is singular. Given a function \( f(x) = \int_{0}^{x} (t^2 + 2t + 3)\, dt \), \( \forall x \in [1, \alpha] \). If \( m \) and \( n \) are the maximum and minimum values of the function \( f(x) \), then the value of \( 3(m - n) \) is:

If \(\sqrt{3}i, 1\) are the roots of \(x^{3} + ax^{2} + bx + c = 0\) where \(a, b, c \in \mathbb{R}\). Then \(\int_{-1}^{1} (x^{3} + ax^{2} + bx + c)dx\) is

The value of \( x \) which satisfies the equation \( \sin^{-1}\left(\frac{2}{3}\sqrt{1 - x^2}\right) = \cot^{-1}\left(2\sqrt{x}\right) \), is