\(\int\left(\frac{1+x+x^2}{1+x^2}\right) e^{\tan ^{-1} x} d x=\)

Let \(A\) and \(B\) be two events in a random experiment with sample space such that \(P(B) \neq 0\) and \(P(A \mid B)=1\).
Which of the following is true?

The plane \(2 x-(1+\lambda) y+3 \lambda z=0, \lambda \in R\) passes through the intersection of planes :

Area (in sq. units) of the region bounded by the curves \(y = - 1, y = 2 x = y^3\) and \(x = 0\) is

In \(\triangle A B C\), which of the following is not true is:
(A) \(\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C A}=\overrightarrow{0}\)
(B) \(\overrightarrow{A B}+\overrightarrow{B C}-\overrightarrow{A C}=\overrightarrow{0}\)
(C) \(\overrightarrow{A B}+\overrightarrow{B C}-\overrightarrow{C A}=\overrightarrow{0}\)
(D) \(\overrightarrow{A B}-\overrightarrow{C B}+\overrightarrow{C A}=\overrightarrow{0}\)
(E) \(\overrightarrow{B A}-\overrightarrow{B C}-\overrightarrow{C A} \neq \overrightarrow{0}\)

The area (sq. units) bounded by the curve \(y=\sin x, \pi \leq x \leq 2 \pi\) and the \(x\)-axis is

If A and B are invertible matrices of order \(3,|A|=2\) and \(\left|(A B)^{-1}\right|=-\frac{1}{0}\), then the value of \(|B|\) is :

Which one of the following will show Tyndall effect ?

The remainder when (672 + 541 + 383 + 295 + 101 + 86) is divided by 3, is :

Read the following passage carefully and answer the given questions.
During anaerobic sludge digestion, bacteria produce a mixture of gases such as methane, hydrogen sulphide and carbon dioxide. These gases form biogas and can be used as source of energy as it is inflammable. The bioactive molecule, cyclosporin A, that is used as an immunosuppressive agent in organ-transplant patients, is produced by the fungus Trichoderma polysporum. You must have noticed that bottled fruit juices bought from the market are clearer as compared to those made at home. This is because the bottled juices are clarified by the use of pectinases and proteases. Statins produced by the yeast Monascus purpureus have been commercialised as blood-cholesterol lowering agents. It acts by competitively inhibiting the enzyme responsible for synthesis of cholesterol. The BOD test measures the rate of uptake of oxygen by micro-organisms in a sample of water and thus, indirectly, BOD is a measure of the organic matter present in the water.
During anaerobic sludge digestion, bacteria produce the biogas. The composition of this biogas is -

A cell of emf 10 V and the internal resistance 1 \( \Omega \) is being charged by an external supply of 100 V. An external resistance of 8 \( \Omega \) is connected to the charging circuit. The current in the circuit is

In an electrical circuit, the three resistors of 3 \( \Omega \), 6 \( \Omega \) and 9 \( \Omega \) are connected in series. The voltage drop across 3 \( \Omega \) is 5 V. The voltage drop across the combination is

Let \(L_1\) and \(L_2\) be two lines, represented as,
\(L_1: \vec{r}=\hat{i}+\hat{j}+\lambda(2 \hat{i}-\hat{j}+\hat{k})\) and \(L_2: \vec{r}=2 \hat{i}+\hat{j}-\hat{k}+\mu(4 \hat{i}-2 \hat{j}+2 \hat{k})\), where \(\lambda\) and \(\mu\) are scalars. Then which of the following are true?
(A)\(L_1\) is perpendicular to \(L_2\)
(B) \(L_1\) is parallel to \(L_2\)
(C) \(L_1\) passes through the point (1,1,0)
(D) \(L_2\) passes through the point (2,1,-1)
Choose the correct answer from the options given below: