\(\int_{-1}^1 \frac{x^3+|x|+1}{x^2+2|x|+1} d x\) is equal to

If \(\left|\begin{array}{ccc}x-4 & 2 x & 2 x \\ 2 x & x-4 & 2 x \\ 2 x & 2 x & x-4\end{array}\right|=(A+B x)(x-A)^2\), then the ordered pair (A, B) is (where A and B are real numbers):

\(\int \frac{1}{(x+1)(x+2)} d x\) is equal to

The equation of the normal to the curve \(y=\frac{x-7}{(x-2)(x-3)}\) at (7,0) is:

The number of vectors of unit length perpendicular to both the vectors \(\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}\) and \(\vec{b} = \hat{i} - \hat{j} + 3\hat{k}\) is:

The probability that X can take the values \(x_i\) has the following form:
\(P\left(X=x_i\right)=\left\{\begin{array}{ll}0.2 & \text { if } x_i=0 \\ k x_i & \text { if } x_i=1 \text { or } 2 \\ k\left(5-x_i\right) & \text { if } x_i=3 \\ 0 & \text { otherwise }\end{array}\right.\)
The value of P(X = 2) is:

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

Applying the Kirchhoff's rule to the loop (a) ABCA and (b) BCDEB, two correct equations are given

(A) \(8-4 I_2-2\left(I_2+I_3\right)-I_1=0\)
(B) \(4-2\left(I_2+I_3\right)-4\left(I_2+I_3-I_1\right)=0\)
(C) \(4+2\left(I_2-I_3\right)-4\left(I_2+I_3-I_1\right)=0\)
(D) \(8+4 I_2+2\left(I_2-I_3\right)-I_1=0\)
Choose the correct answer from the options given below:

If an electron and a photon have energy, then,
(A) Electrons and photons both have wavelengths
(B) Electron and photon both have momentum
(C) Both electron and photon do not have momenta
(D) The electron has energy and momentum and the photon has energy only
Choose the correct answer from the options given below:

For an objective function \(z=a x+b y\) where \(a, b>0\).
The corner points of feasible region determined by a set of constraints are \((0,20),(10,10),(30,30),(0,40)\).
The condition on \(a\) and \(b\) such that max \(z\) occurs at both the points \((30,30)\) and \((0,40)\) is:

Number of unpaired electrons in the square planar [Pt(CN)₄]^2- are :

The properties which depend on the number of solute particles irrespective of their nature relative to the total number of particles in the solution are called colligative properties. These are relative lowering of vapour pressure of the solvent, depression of freesing point of the solvent, elevation of boiling point of the solvent and osmotic pressure of the solution. All of these properties help in calculating the observed molar mass of the solute.
Determination of osmotic pressure is considered best for the determination of molecular mass of the solute.
Based on solute-solvent interactions, arrange the following in order of increasing solubility in n-octane.
(A) Cycloheptane
(B) NaCl
(C) \(CH _3 OH\)
(D) \(CH _3 CN\)
Choose the correct answer from the options given below :

Which statement is not true for a detergent molecule ?