If \(\vec{a}, \vec{b}\) and \(\sqrt{3} \vec{a}-\vec{b}\) are three unit vectors, then the angle between \(\vec{a}\) and \(\vec{b}\) is :

Match List I with List II

LIST I	LIST II
A. \(\frac{d y}{d x}=e^{x-y}+x^2 e^{-y}\)	I. Linear Differential Equation of type \(\frac{d y}{d x}+P y=Q\)
B. \(\frac{d y}{d x}-2 y=\cos 3 x\)	II. Homogeneous Differential Equation
C. \(x^2 \frac{d y}{d x}=x^2-2 y^2+x y\)	III. Linear Differential Equation of type \(\frac{d x}{d y}+P x=Q\)
D. \(\left(x+2 y^3\right) \frac{d y}{d x}=y\)	IV. Variable separable

Choose the correct answer from the options given below:

The vector equation of line passing through (2, -1, 3) and perpendicular to the lines \(\frac{x-2}{3}=\frac{y-1}{1}=\frac{z+2}{2} \quad\) and \(\quad \frac{x+3}{-4}=\frac{y-5}{-3}=\frac{z+1}{2}\)
(Here \(\lambda\) is a parameter)

The least non-negative remainder when \(2^{101}\) is divided by \(5\) is:

A boat can row at the speed of 16 km/hr in still water. If the river is flowing at 8 km/hr, and it takes 8 hours for a round trip, then the distance between the two places is :

A die is thrown. If \(A\) is the event 'the number appearing is a multiple of 3 ' and \(B\) is the event 'the number appearing is even', then A and B are :

The following data shows the percentage of rural and urban Indian households who have a high speed internet connection.

Year(x)	Rural Household \((y \%)\)	Urban Household\((y \%)\)
2020	2	6
2021	3	7
2022	3	9
2023	4	15
2024	5	20

Based on the above data, which of the following statements are correct?
(A) A straight line trend by the method of least squares for rural Indians \(y_t=3.4+0.7(x-2022)\)
(B) A straight line trend by the method of least squares for urban Indians is \(y_t=11.4+4.3(x-2022)\)
(C) The forecast for the year 2025 for urban group using the trend equation is \(22.2 \%\)
(D) The forecast for the year 2025 for rural group using the trend equation is \(5.5 \%\)
Choose the correct answer from the options given below :

Match the complex given in List-I with type of hybridisation given in List-II.

List-I (Complex)	List-II (Hybridisation)
(A) \([Ni(CO)_4]\)	(I) \(sp^3d^2\)
(B) \([Ni(CN)_4]^{2-}\)	(II) \(dsp^2\)
(C) \([Co(NH_3)_6]^{3+}\)	(III) \(d^2sp^3\)
(D) \([CoF_6]^{3-}\)	(IV) \(sp^3\)

Choose the correct answer from the options given below:

The rate constant for a reaction is \(3.5 \mathrm{~mol}^{-1} \mathrm{~L} \mathrm{~s}-^{-1}\). The order of the reaction is:

Two long parallel wires separated by a distance of 2.50 cm, repel each other with a force of \( 4 \times 10^{-5} \text{ N/m} \). The current in one wire is 0.5 A. The current in the second wire will be

The particular solution of the differential equation \(x\left(1+y^2\right) d x-y\left(1+x^2\right) d y=0\), given that \(y(0)=1\) is :

A small amount of \(CaF_2\) is added in electrolytic reduction of \(Al_2O_3\) in fused cryolite :
A. To decrease the rate of oxidation of carbon at anode
B. To act as catalyst
C. To make fused mixture conducting
D. To lower the fusion temperature of melt
Choose the correct answer from the options given below :

Foetal ejection reflex triggers the release of: