The integral \(\int \log x d x\) is equal to to:

Let A and B be two events. Then which of the following statements are TRUE?
(A) \(P(B \mid A)=\frac{P(A \cap B)}{P(A)}\), provided \(P(A) \neq 0\)
(B) \(P\left(B^{\prime}\right)=1+P(B)\)
(C) \(P(A \cup B)=P(A)+P(B)+P(A \cap B)\)
(D) \(P(A \cap B)=P(A) \cdot P(B)\) if A and B are independent events
Choose the correct answer from the options given below:

Let \(\Delta=\left|\begin{array}{ccc}a+b x & c+d x & p+q x \\ a x+b & c x+d & p x+q \\ u & v & w\end{array}\right|=\left(k-x^2\right)\left|\begin{array}{ccc}a & c & p \\ b & d & q \\ u & v & w\end{array}\right|\) then the value of k is:

Two pipes P and Q can fill a tank in 26 minutes and 52 minutes respectively.
Both the pipes are opened together for some time and then pipe P is closed.
If the tank is filled in 26 minutes, then after how many minutes pipe P is closed?

Match \(List - I\) with \(List - II\)

\(List - I\)	\(List - II\)
General solution	Differential Equation
(A) \(y=c x\)	(I) \(y^{\prime}=\frac{1}{2 \sqrt{x}}-1\)
(B) \(y=e^{-3 x}+c\)	(II) \(y^{\prime}+3 y=0\)
(C) \(x+y=\sqrt{x}+c\)	(III) \(y^{\prime}=\frac{y^2}{1-x y}(x y \neq 1)\)
(D) \(x y=\log y+c\)	(IV) \(x y^{\prime}=y\)

(c is an arbitrary constant and \(y^{\prime}=\frac{d y}{d x}\) )
Choose the correct answer from the options given below:

If \(\left|\begin{array}{ccc}x-2 & 1 & 2 \\ 1 & 2 x & 1 \\ -1 & 1 & 1\end{array}\right|=3\) then the values of x are:

The angle between the lines \(\vec{r}=3 \hat{i}+2 \hat{j}-4 \hat{k}+\lambda(\hat{i}+2 \hat{j}+2 \hat{k})\) and \(\vec{r}=5 \hat{j}-2 \hat{k}+\mu(3 \hat{i}+2 \hat{j}+6 \hat{k})\) is:

Silencing of specific mRNA by binding of complementary dsRNA, that prevents translation of mRNA is known as:

Increase in population is due to:

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 microorganisms in a sample of water and thus, indirectly, BOD is a measure of the organic matter present in the water.
The bottled juices are clarified by the combination of -

Answer the question on the basis of passage given below :
Living systems are made up of various complex bio-molecules like carbohydrates, proteins, nucleic acids, lipids etc. Proteins and carbohydrates are essential constituents of our food. Carbohydrates are primarily produced by plants and form a very large group of naturally occurring organic compounds. Some common examples are cane sugar, glucose, starch etc.
Carbohydrates are stored in animals as:

Which one of the followings can helps in destroying the tumor by activating the immune system of the patient?

Antibiotic, which is bacteriostatic among the following is: