Mathematics Tes...

The constraints –x + y ≤ 1, −x + 3y ≤ 9 and x, y ≥ 0 defines on-

What is the scalar projection of \(\vec{a}=\hat{i}-2 \hat{j}+\hat{k}\) on \(\vec{b}=4 \hat{i}-4 \hat{j}+7 \hat{k} ?\)

Find the shortest distance between the lines \(\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}\) and \(\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}\).

Find the value of$\sin \theta (1+\tan \theta )+\cos \theta (1+\cot \theta )$

Consider the following equations for two vectors \(\vec{a}\) and \(\vec{b}\):

1. \((\vec{a}+\vec{b}) \cdot(\vec{a}-\vec{b})=|\vec{a}|^{2}-|\vec{b}|^{2}\)

2. \((|\vec{a}+\vec{b}|)(|\vec{a}-\vec{b}|)=|\vec{a}|^{2}-|\vec{b}|^{2}\)

3. \(|\vec{a} \cdot \vec{b}|^{2}+|\vec{a} \times \vec{b}|^{2}=|\vec{a}|^{2}|\vec{b}|^{2}\)

Which of the above statement are correct?

Find the differentiation of the following function \(\sec ^{-1} \tan x .\)

For \(x \neq 0\), find the value of \(f(x)=\frac{x\left(3^{x}-1\right)}{1-\cos x}\):

The mean of 20 observations is \(15 .\) On checking, it was found that two observations were wrongly copied as 3 and 6 . If wrong observations are replaced by correct values 8 and 4 , then the correct mean is:

If \(m\left[\begin{array}{ll}-3 & 4\end{array}\right]+n\left[\begin{array}{ll}4 & -3\end{array}\right]=\left[\begin{array}{ll}10 & -11\end{array}\right]\), then find \(m\) and \(n\).