For an electric dipole in a non-uniform electric field with dipole moment parallel to the field, force F and torque t are :

The value of the determinant \(\left|\begin{array}{ccc}a \cos \theta & b \sin \theta & 0 \\ -b \sin \theta & a \cos \theta & 0 \\ 0 & 0 & c\end{array}\right|\) is :

Maximize \(z=2 x-y\)
Subject to constraints
\(x+y \leq 4, \quad x+3 y \leq 6, \quad x \geq 0, \quad y \geq 0 .\)
The corner points of the feasible region are:

The function \(f(x)=x^2 e^{-2 x}\) increases on

The integral \(\int e^x(\tan x+\log \sec x) d x\) is equal to :

Let \(A=\{a, b, c\}\). Then number of relations containing \((a, b)\) and \((b, c)\) which are reflexive and transitive but not symmetric is :

The maximum value of the objective function \(Z=2 x+y\) of an LPP, subject to the constraints \(x \leq 6, y \leq 2, x-y \leq 0\), \(x \geq 0, y \geq 0\), is:

The angle between the pair of lines given by \(\vec{r}=\hat{i}+2 \hat{j}-3 \hat{k}+\lambda(\hat{i}-2 \hat{j}+2 \hat{k})\) and \(\vec{r}=5 \hat{i}+\hat{j}+\hat{k}+\mu(3 \hat{i}-2 \hat{j}+6 \hat{k})\)

A coil when given supply of 20 V D.C. draws a current of 4 A. When the same coil is given supply of 20 V A.C. with 80 Hz, draw a current of 2 A. The self Inductance of the coil is :

In a circuit, resistance \( R_1 \) is connected in series with the parallel combination of resistances \( R_2 \) and \( R_3 \). The currents passing through the resistances \( R_1, R_2 \) and \( R_3 \) are \( I_1, I_2 \) and \( I_3 \), respectively. What will be the ratio \( \frac{I_3}{I_1} \) of currents in terms of the resistances connected in the circuit?

Read the passage carefully and answer the questions based on the passage:
Alcohols are versatile compounds as they can act as both nucleophile and electrophile. They undergo reactions based on cleavage of O-H and C-O bonds. The cleavage of O-H bond reflects the acidic character of alcohols. However they are weaker acids than water, phenol and carboxylic acids. Alkoxide ion formed by loss of proton from the OH group of alcohols reacts with water to form the corresponding conjugate acid - base pair. They act as Bronsted base due to presence of unshared electron pairs on oxygen.Both alcohols and phenol react with carboxylic acids, acid chlorides and anhydrides to form esters. The introduction of acetyl group in alcohols or phenol is known as acetylation. Alcohols react with hydrogen halides to form alkyl halides which form the basis of Lucas test. They also undergo acid catalyzed dehydration to form alkenes. Alcohols can be easily oxidized with variety of oxidizing agents. The nature of the product depends on the type of oxidizing agent and the nature of alcohol.
Pyridinium chlorochromate is used for:

Enzymes that cut the DNA at specific locations are known as :

Antibiotic resistance gene in a vector helps in the selection of: