The value of \(\int \frac{\left(x^2-1\right) d x}{x^3 \sqrt{2 x^4-2 x^2+1}}\) is

In a study about a pandemic, data of $900$ persons was collected. It was found that $190$ persons had symptom of fever, $220$ persons had symptom of cough, $220$ persons had symptom of breathing problem, $330$ persons had symptom of fever or cough or both, $350$ persons had symptom of cough or breathing problem or both, $340$ persons had symptom of fever or breathing problem or both, $30$ persons had all three symptoms (fever, cough and breathing problem). If a person is chosen randomly from these $900$ persons, then the probability that the person has at most one symptom is ______.

Let ${\text{S}}_{\text{n}}=\underset{\text{k}=1}{\overset{4\text{n}}{\Sigma }}{\left(-1\right)}^{\frac{\text{k}\left(\text{k}+1\right)}{2}}{\text{k}}^2$. Then S_n can take value(s)

Let \(a_{1}, a_{2}, a_{3}, \ldots . .\) be in harmonic progression with \(a_{1}=5\) and \(a_{20}=25\). The least positive integer \(n\) for which \(a_{n} < 0\) is

Statement I The curve \(y=-\frac{x^2}{2}+x+1\) is symmetric with respect to the line \(x=1\)
Statement II A parabola is symmetric about its axis.

Let ${\theta }_1,{\theta }_2,\dots ,{\theta }_{10}$ be positive valued angles (in radian) such that ${\theta }_1+{\theta }_2+\cdots +{\theta }_{10}=2\pi$. Define the complex numbers $z_1=e^{i{\theta }_1},z_k=z_{k-1}{e}^{i{\theta }_k}$ for $k=2,3,\dots ,10$, where $i=\sqrt{-1}$. Consider the statements $P$ and $Q$ given below:
\(P:\left|z_2-z_1\right|+\left|z_3-z_2\right|+\cdots+\left|z_{10}-z_9\right|\) \(+\left|z_1-z_{10}\right| \leq 2 \pi \)
\( Q:\left|z_2^2-z_1^2\right|+\left|z_3^2-z_2^2\right|+\cdots+\left|z_{10}^2-z_9^2\right|\) \(+\left|z_1^2-z_{10}^2\right| \leq 4 \pi\)

If the function $f:R\to R$ is defined by $f\left(x\right)=\left|x\right|\left(x-\sin x\right)$, then which of the following statements is TRUE?

The sides of a right-angled triangle are in arithmetic progression. If the triangle has area 24 square units, then what is the length of its smallest side?

Considering only the principal values of the inverse trigonometric functions, the value of $\frac{3}{2}{\cos }^{-1}\sqrt{\frac{2}{2+{\pi }^2}}+\frac{1}{4}{\sin }^{-1}\frac{2\sqrt{2}\pi }{2+{\pi }^2}+{\tan }^{-1}\frac{\sqrt{2}}{\pi }$ is _______. (If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places)  

The substituents \(R_{1}\) and \(R_{2}\) for nine peptides are listed in the table given below. How many of these peptides are positively charged at \(\mathrm{pH}=7.0\) ?

\(\begin{array}{|c|c|c|}\hline Peptide & \mathbf{R}_{\mathbf{1}} & \mathbf{R}_{\mathbf{2}} \\\hline I & \mathrm{H} & \mathrm{H} \\\hline II & \mathrm{H} & \mathrm{CH}_{3} \\\hline III & \mathrm{CH}_{2} \mathrm{COOH} & \mathrm{H} \\\hline IV & \mathrm{CH}_{2} \mathrm{CONH}_{2} & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} \\\hline V & \mathrm{CH}_{2} \mathrm{CONH}_{2} & \mathrm{CH}_{2} \mathrm{CONH}_{2} \\\hline VI & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} \\\hline VII & \mathrm{CH}_{2} \mathrm{COOH} & \mathrm{CH}_{2} \mathrm{CONH}_{2} \\\hline VIII & \mathrm{CH}_{2} \mathrm{OH} & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} \\\hline IX & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} & \mathrm{CH}_{3} \\\hline\end{array}\)

A solution when diluted with \(\mathrm{H}_2 \mathrm{O}\) and boiled, it gives a white precipitate. On addition of excess \(\mathrm{NH}_4 \mathrm{Cl} / \mathrm{NH}_4 \mathrm{OH}\), the volume of precipitate decreases leaving behind a white gelatinous precipitate. Identify the precipitate which dissolves in \(\mathrm{NH}_4 \mathrm{OH} / \mathrm{NH}_4 \mathrm{Cl}\).

Paragraph:
Read the following passage and answer the questions.
Let \(A B C D\) be a square of side length 2 units. \(C_2\) is the circle through vertices \(A, B, C, D\) and \(C_1\) is the circle touching all the sides of square \(A B C D\). \(L\) is the line through \(A\).Question:
A line \(M\) through \(A\) is drawn parallel to \(B D\). Points \(S\) moves such that its distances from the line \(B D\) are the vertex \(A\) are equal. If locus of \(S\) cuts \(M\) at \(T_2\) and \(T_3\) and \(A C\) at \(T_1\), then area of \(\Delta T_1 T_2 T_3\) is

For the cell $\mathrm{Zn} \left(\mathrm{s}\right)\left|{\mathrm{ZnSO}}_4 \left(\mathrm{aq}\right)\right|\left|{\mathrm{CuSO}}_4 \left(\mathrm{aq}\right)\right|\mathrm{Cu} \left(\mathrm{s}\right)$, when the concentration of ${\mathrm{Zn}}^{2+}$ is $10$ times the concentration of ${\mathrm{Cu}}^{2+}$, the expression for $\mathrm{\Delta G} \left(\mathrm{in}\mathrm{ }\mathrm{J}\mathrm{ }{\mathrm{mol}}^{-1}\right)$ is
$[\mathrm{F}=$Faraday's constant, $\mathrm{R}=$universal gas constant, $\mathrm{T}=$temperature, ${\mathrm{E}}^{\mathrm{o}}\mathrm{cell}=1.1\mathrm{V}]$