If \(A=\left[a_{i j}\right]\) is a square matrix of order 2 such that, \(a_{i j}=\left\{\begin{array}{ll}2, & \text { when } i \neq j \\ 0, & \text { when } i=j\end{array}\right.\), then \(\operatorname{det}\left(A^2\right)\) is:

The equation of the normal to the curve \(y=\sin x\) at \((0,0)\) is :

If \(y=3 e^{2 x}+2 e^{3 x}\), then \(\frac{d^2 y}{d x^2}+6 y\) is equal to

If a matrix P is both symmetric and skew-symmetric, then

The value of the definite integral \(I=\int_0^2 x \sqrt{2-x} d x\) is :

Solution of the differential equation \(y \log _e y d x-x d y=0\) is (Where \(c\) is an arbitrary constant)

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 :

For the reaction \(x + y \rightarrow z\),
rate law expression is rate \(= k [x] [y]^3\).
1 If the volume of the vessel is reduced to of its original volume, then rate of reaction will 3:

A system consists of two protons and one electron. They are separated by a distance of 1 \(\text{\AA}\) from each other. If the potential energy at infinity is taken to be zero, the potential energy of the system is

Lactose on hydrolysis gives:

Match List - I with List - II.

LIST I	LIST II
A. \(\left[1+\left(\frac{d y}{d x}\right)^2\right]=\left(\frac{d^2 y}{d x^2}\right)^2\)	I. order \(=2\), degree \(=2\)
B. \(\frac{d^3 y}{d x^3}-3 \frac{d^2 y}{d x^2}+2\left(\frac{d y}{d x}\right)^4=y^4\)	II. order \(=3\), degree \(=2\)
C. \(\left(1+\frac{d y}{d x}\right)^3=\left(\frac{d^3 y}{d x^3}\right)^2\)	III. order \(=2\), degree \(=1\)
D. \(\left[1+\left(\frac{d y}{d x}\right)^2\right]^?=\frac{d^2 y}{d x^2}\)	IV. order \(=3\), degree \(=1\)

Choose the correct answer from the option given below:

Identify the incorrect statement about the human genome -

Answer the question on the basis of passage given below:
Aldehydes are generally more reactive than ketones in nucleophilic addition reactions due to steric and electronic reasons. Sterically, the presence of two relatively large substituents in ketones hinders the approach of nucleophile to carbonyl carbon more than aldehydes having only one such substituent Electronically aldehydes are more reactive than ketones because two alkyl groups reduce the elctrophilicity of the carbonyl more effectively than in former.
For preparation of phenylethanoic acid from benzyl alcohol, reagents required are
A. HBr
B. \(CO _2\)
C. \(K C N\)
D. \(\Delta / H_3 O^{+}\)
E. \(AgNO _3\)
Choose the correct answer from the options given below: