Mathematics Tes...

The sum of the roots of the equation \(x+1-2 \log _2\left(3+2^x\right)+2 \log _4\left(10-2^{-x}\right)=0\) is:

Let \(A=\left[\begin{array}{cc}1 & 2 \\ -1 & 4\end{array}\right]\). If \(A^{-1}=\alpha I+\beta A, \alpha, \beta \in R, I\) is a \(2 \times 2\) identity matrix, then \(4(\alpha-\beta)\) is equal to :

The differential equation satisfied by the system of parabolas \(y^2=4 a(x+a)\) is:

Find the value of \(\lim _{{x} \rightarrow \infty} \frac{{x}^{4}+3 {x}^{2}+5}{{x}^{4}+{x}^{2}-6}\)

The smallest positive integer \(n\) for which

\(\left(\frac{1-i}{1+i}\right)^{n^{2}}=1\)

where \(i=\sqrt{-1}\), is

For the vectors \(\overrightarrow{\mathrm{a}}=-4 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}\) and \(\overrightarrow{\mathrm{c}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}},\) if\(\overrightarrow{\mathrm{c}}=\mathrm{m} \overrightarrow{\mathrm{a}}+\mathrm{n} \overrightarrow{\mathrm{b}},\) then the value of \(\mathrm{m}+\mathrm{n}\) is:

If \(\tan ^{-1}\left(\frac{1-x}{1+x}\right)=\frac{1}{2} \tan ^{-1} x, x>0\), then \(x\) equals:

Let \(\mathrm{P}\) be a square matrix such that \(\mathrm{P}^2=\mathrm{I}-\mathrm{P}\). For \(\alpha, \beta, \gamma, \delta \in \mathrm{N}\), if \(\mathrm{P}^\alpha+\mathrm{P}^\beta=\gamma \mathrm{I}-29 \mathrm{P}\) and \(\mathrm{P}^\alpha-\mathrm{P}^\beta=\delta \mathrm{I}-13 \mathrm{P}\), then \(\alpha+\beta+\gamma-\delta\) is equal to :

Using the principal of mathematical induction, prove that \(1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots+\frac{1}{1+2+3+\ldots+n}=\frac{2 n}{n+1}\) for all:

If \(-2<2 x-1<2\) then the value of \(x\) lies in the interval: