Match List - I with List - II.

LIST - I	LIST - II
(A) \((\hat{i} \times \hat{j})+(\hat{j} \times \hat{i})\)	(I) 1
(B) \(\hat{i} \cdot \hat{i}+\hat{j} \cdot \hat{j}+\hat{k} \cdot \hat{k}\)	(II) \(\hat{k}-\hat{j}\)
(C) \(\hat{i} \times(\hat{i}+\hat{j}+\hat{k})\)	(III) \(\overrightarrow{0}\)
(D) \(\hat{i} \cdot(\hat{i}+\hat{j}+\hat{k})\)	(IV) 3

Choose the correct answer from the options given below :

The integral \(\int e^x\left(\frac{x-1}{2 x^2}\right)\)dx is equal to

Match List - I with List - II

List - I (Differential Equations)	List - II (Order and degree)
(A) \(ydx + xlog(y/x)dy - 2xdy = 0\)	(I) Order: 2, degree:1
(B) \(\left(\frac{d^3 y}{d x^3}\right)^2+3 \frac{d^2 y}{d x^2}+2\left(\frac{d y}{d x}\right)^4=y^2\)	(II) Order :1, degree:1
(C) \(\frac{d y}{d x}+\log \left(\frac{d y}{d x}\right)+x=y\)	(III) Order: 3, degree:2
(D) \(\left(\frac{d s}{d t}\right)^4+2 s \frac{d^2 s}{d t^2}=0\)	(IV) Order: 1, degree: Not defined

Choose the correct answer from the options given below:

If \(x=4 t\) and \(y=\frac{4}{t}\), then \(\frac{d^2 y}{d x^2}\) is

The value of the integral \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(x+x^3+x^5\right) d x\) is:

Let \(y=138.86+7.64(x-2021)\) be a straight line of best fit by using least square method.

\(YEAR(X)\)	2018	2019	2020	2021	2022	2023	2024
\(PROFIT(y)\)	144	130	126	144	138	156	164

Then the trend value for the year 2024 is :

The area (in sq. units) of the region bounded by the curve \(y=x^5\), the \(x\)-axis and the ordinates \(x=-1\) and \(x=1\) is equal to:

Answer the question on the basis of passage given below :
A colloid is a heterogeneous system in which a substance is dispersed (dispersed phase) as fine particles in another substance called dispersion medium.
Their particle size is intermediate between that of pure solutions (less than 1 nm) and suspensions (greater than 1000 nm). They are known for interesting properties such as enormous surface area, Tyndall effect, Brownian motion etc.
They are widely used in industries and day to day life.
Some prominent uses of colloids are those in smoke precipitators, purification of water, medicines, cleansers etc.
Which of the following electrolyte exhibits maximum flocculating power in the coagulation of a positive metal hydroxide sol?

A pendulum of length 'l' having a bob of mass 'm' carrying charge 'q' is suspended from a stationary rigid support in a region of uniform electric field 'E' as shown in the figure. If the system is in equilibrium, the nature and magnitude of 'q' will be :

Wood spirit is known as

The integral \(I=\int e^x\left(\frac{x-1}{3 x^2}\right) d x\) is equal to:

Which of the following is correct about the compound annual growth rate?

Two point charges, +2 \( \mu \text{C} \) and -3 \( \mu \text{C} \), are separated by a distance of 20 cm in the air. At what point between the two charges and on the line joining the two charges is the electric potential zero?
(Take the potential at infinity to be zero)