List of top Mathematics Questions

All the solutions \((n,r)\) of the equation \( \frac{{}^nC_r}{{}^{n+1}C_r} = \frac{1}{3} \) can be obtained from one of the following equations given in the options for \( k=1,2,3, \dots \):

If the expansion of \[ \left(\frac{1+15x}{1-3x}\right) \] is valid and the coefficient of \(x^3\) in its expansion is \(k(3^3)\), then \(k=\)

Evaluate \[ \sum_{r=1}^{\infty} \frac{1\cdot3\cdot5\cdots(2r-1)} {2^{2r}r!} (\sqrt{3})^r : \]

If \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 - 3x + 1 = 0 \), then \( \frac{\alpha}{1-\alpha} + \frac{\beta}{1-\beta} + \frac{\gamma}{1-\gamma} = \)

The quadratic expression having zero's equal to the non-integral roots of \( ||3x-4|-6|=5 \) is:

The modulus of the product of all the values of \( (2+3i)^{3/5} \) is:

If \( x^5 + ax^4 + bx^3 + cx^2 + 5x + d = 0 \) is an odd order reciprocal equation of second type and \( \frac{1+\sqrt{3}i}{2} \) is a root, then \( b-c = \)

Consider the quadratic expression \( f(x) = x^2 + (10-a)x - 10a \). The sum of all values of 'a', such that the roots of \( f(x)=3 \) are integers is:

For a \( 3 \times 3 \) non singular matrix A, if \( \text{Adj}(\text{Adj}(\text{Adj}(\text{Adj}(A)))) = |A|^n A \), then \( n = \)

Among the solutions of the equation \( x^6 = 64 \), the sum of all those solutions whose real part is negative is:

On a matrix A when three elementary operations namely interchange of \( R_1 \) and \( R_2 \), \( R_2 \rightarrow R_2 - 2R_1 \), \( R_3 \rightarrow R_3 - 3R_1 \) are applied successively, A is transformed to \( \begin{bmatrix} 1 & 3 & 4 \\ 2 & 1 & 5 \\ 6 & 1 & 2 \end{bmatrix} \). Then \( \text{Tr}(A) = \)

If the two systems of equations \( x+y-2z=0, 4x+4y-8z=0, 3x+3y-6z=0 \) and \( x+y+z=3, 2x+2y-z=\lambda, x+y-\mu z=1 \) have the same set of solutions, then \( \lambda+\mu = \)

If \( \text{Arg}(\frac{z-1}{z+5}) = \pi/3 \) and \( |z-1|=|z+5| \), then \( |z|^2 = \)

If \( x_1+x_2+x_3+\dots+x_n=pn(n-1) \forall n \in \mathbb{N} \) and \( x_{j+1}-x_j = \text{constant} \, (j=1,2,\dots,n-1) \), then \( (\frac{x_n}{n-1})^2 = \)

The complete range of the function \( f(x) = [x]^2 + 5[x] + 6 \), where \([x]\) is the greatest integer function, is:

If a real valued function \( f:(1, 2] \rightarrow B \) defined by \( f(x) = \log_{10}(x-1) \) is a bijection, then \( B = \)

If \(\log_{10}3=0.4771,\ \log_{10}7=0.8450\), then the value of \(\log_3 7\) is

The L.C.M. of \(\frac{7}{3},\ \frac{5}{6},\ \frac{14}{9}\) is

\(\tan 1^{\circ} \cdot \tan 2^{\circ} \cdot \tan 3^{\circ} \cdots \tan 89^{\circ} =\)

If \(x=2+\sqrt3,\ y=2-\sqrt3\), then the value of \(x^{-3}+y^{-3}\) is

\((x-y)^3+(y-z)^3+(z-x)^3=\)

The order (characteristic) of 0.99 is-

The differential equation is $\frac{dy}{dx} + y\tan x = \sec x$ and $y(0) = 1$. Then the value of $y(\frac{\pi}{4}) =$

The general solution of the differential equation $\frac{dy}{dx} = e^{x-y} + x^{2}e^{-y}$ is

The differential equation is $\frac{dy}{dx} + \frac{y}{x} = 0$ and $y(1) = 2$. Then the value of $y(3) =$