Mathematics Tes...

In how many different ways can the letters of the word ALLAHABAD be arranged?

If \(z^{2}+z+1=0\), where \(z\) is a complex number, then the value of \(\left(z+\frac{1}{z}\right)^{2}+\left(z^{2}+\frac{1}{z^{2}}\right)^{2}+\left(z^{3}+\frac{1}{z^{3}}\right)^{2}+\ldots .+\left(z^{6}+\frac{1}{z^{6}}\right)^{2}\) is:

What is the value of \(k\) for which the sum of the squares of the roots of \(2 x^{2}-2(k-2) x-(k+1)=0\) is minimum?

Find the area of the parallelogram whose adjacent sides are \(\vec{a}=3 \hat{i}+\hat{j}+4 \hat{k}\) and \(\vec{b}=\hat{i}-\hat{j}+\hat{k}\).

What is (1 + tan α tan β)² + (tan α – tan β)² – sec² α sec² β equal to?

In how many different ways can the letters of the word 'RUMOUR' be arranged?

Let \(f(x)\) be a quadratic expression which is positive for all real \(x\). If \(g(x)=f(x)-f^{\prime}(x)+f^{\prime \prime}(x)\), then for any real \(x\):

Find the area of a triangle whose vertices are \(\mathrm{A}(1,1,1), \mathrm{B}(1,2,3)\) and \(\mathrm{C} (2,3,1)\).