Mathematics Tes...

Solve: 1 ≤ |x – 1| ≤ 3

If the determinant \(\left|\begin{array}{lll}{a}_{1} & {~b}_{1} & {c}_{1} \\ {a}_{2} & {~b}_{2} & {c}_{2} \\ {a}_{3} & {~b}_{3} & {c}_{3}\end{array}\right|=\Delta\), then \(\left|\begin{array}{lll}{x} {a}_{1} & {yb}_{1} & {zc}_{1} \\ {xa}_{2} & {yb}_{2} & {zc}_{2} \\ {xa}_{3} & {yb}_{3} & {zc}_{3}\end{array}\right|=?({x}, {y}, {z} \neq 0\) or \(1\) )

Find the sum of the series whose nth term is \((2 n-1)^{2}\).

Consider the following statements:

1. \(A=\{1,3,5\}\) and \(B=\{2,4,7\}\) are equivalent sets.

2. \(A=\{1,5,9\}\) and \(B=\{1,5,5,9,9\}\) are equal sets.

Which of the above statements is/are correct?

Let \(y=\log _e\left(\frac{1-x^2}{1+x^2}\right),-1

\(\text { The value of } \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{1+\sin ^2 x}{1+\pi^{\sin x}}\right) d x \text { is }\)

Using principle of mathematical induction, prove that for all \((2 \mathrm{n}+7)<(\mathrm{n}+3)^{2}\).

Evaluate the integral\(\int_{1}^{4} \frac{x}{x^{2}+1} d x\).

The expression \(\frac{\cot x+\operatorname{cosec} x-1}{\cot x-\operatorname{cosec} x+1}\) is equal to:

If \(\int_0^1 \frac{1}{\sqrt{3+x}+\sqrt{1+x}} d x=a+b \sqrt{2}+c \sqrt{3}\), where \(a, b, c\) are rational numbers, then \(2 a+3 b-4 c\) is equal to :