Mathematics Tes...

The total number of subsets of a finite set A has 56 more elements than the total number of subsets of another finite set B. What is the number of elements in set A?

If \(f(x)=x^{3}+3 x^{2}+3 x-7\), then find the value of \(\frac{d f(x)}{d x}\) at \(x=2\).

Consider the following L.P.P. Maximize \(Z=3 x+2 y\) Subject to the constraints:

\(\mathrm{x}+2 \mathrm{y} \leq 10 \)

\(3 \mathrm{x}+\mathrm{y} \leq 15 \)

\(\mathrm{x}, \mathrm{y} \geq 0\)

Find the maximum value of \(Z\).

What is \(\left|\begin{array}{ccc}-\mathrm{a}^{2} & \mathrm{ab} & a c \\ \mathrm{ab} & -\mathrm{b}^{2} & \mathrm{bc} \\ \mathrm{ac} & \mathrm{bc} & -\mathrm{c}^{2}\end{array}\right|\) equal to?

\(f(x)=2 x-\tan ^{-1}x-\log \left\{x+\sqrt{x^{2}+1}\right\}\) is monotonically increasing when:

Given \(\overrightarrow{{A}}=2 \hat{{i}}-\hat{{j}}+2 \hat{{k}}\) and \(\overrightarrow{{B}}=-\hat{{i}}-2 \hat{{j}}+2 \hat{{k}}(\hat{i}, \hat{j}\) and \(\hat{{k}}\) are unit vectors along \({x}, {y}\) and \({z}\) axes respectively). The unit vector in the direction of \(\overrightarrow{{A}}-\overrightarrow{{B}}\) is:

Consider the following linear programming problem:

Maximize \(z=6 x+10 y\)

Subject to \(x \leq 4\)

\(y \leq 6\)

\(3 x+2 y \leq 18\)

\(x \geq 0, y \geq 0\)

The maximum value of the objective function is:

What is the value of the determinant\(\left|\begin{array}{ccc}x+2 & x+3 & x-1 \\ x+6 & x+8 & x+4 \\ x+9 & x+11 & x+7\end{array}\right|\).

What is the value of \(m\) if the vectors \(2 \hat{ i }-\hat{ j }+\hat{ k }, \hat{ i }+2 \hat{ j }-3 \hat{ k }\) and \(3 \hat{ i }+ m \hat{ j }+5 \hat{ k }\) are coplanar?