Mathematics Tes...

Evaluate the integral \(\int_{-1}^{1} 5 x^{4} \sqrt{x^{5}+1} d x\).

Let \(f(x)\) be a polynomial of degree four having extreme values at \(x=1\) and \(x=2\). If \(\lim _{x \rightarrow 0}\left[1+\frac{f(x)}{x^2}\right]=3\), then \(f(2)\) is equal to :

Let \(\mathrm{L}\) denote the set of all straight lines in a plane. Let a relation \(\mathrm{R}\) be defined by \(\mathrm{lRm}\) if and only if \(\mathrm{l}\) is parallel to \(\mathrm{m}, \forall\) \(\mathrm{l,m \in L}\). Then \(\mathrm{R}\) is:

What is the value of \(\underset{{{{x} \rightarrow 0}}}{\lim} \frac{(1-\cos 2 {x})^{3}}{{x}^{6}}\)?

Let the matrix \(A=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right]\) and the matrix \(B_0=A^{49}+2 A^{98}\). If \(B_n=\operatorname{Adj}\left(B_{n-1}\right)\) for all \(n>1\), then \(\operatorname{det}\left(B_4\right)\) is equal to :

The number of ways in which 4 boys and 4 girls can be arranged in a row so, that no two girls and no two boys are together is:

Construct a \(3 \times 2\) matrix whose elements are given by \(a _{ ij }=\frac{1}{3}|2 i + j |\).

If a line is perpendicular to the line \(5 x-y=0\) and forms a triangle of area 5 square units with co-ordinate axes, then its equation is:

If \(f(x)=X^{5}-20 X^{3}+240 X\) then \(f'(x)\) is: