The differential equation for \(y=A \cos (\alpha x)+B \sin (\alpha x)\), where A and B are arbitrary constants is:

Let \(L_1\) and \(L_2\) be two lines, represented as,
\(L_1: \vec{r}=\hat{i}+\hat{j}+\lambda(2 \hat{i}-\hat{j}+\hat{k})\) and \(L_2: \vec{r}=2 \hat{i}+\hat{j}-\hat{k}+\mu(4 \hat{i}-2 \hat{j}+2 \hat{k})\), where \(\lambda\) and \(\mu\) are scalars. Then which of the following are true?
(A)\(L_1\) is perpendicular to \(L_2\)
(B) \(L_1\) is parallel to \(L_2\)
(C) \(L_1\) passes through the point (1,1,0)
(D) \(L_2\) passes through the point (2,1,-1)
Choose the correct answer from the options given below:

Match List I with List II

List-I (Inequality)	List-II (Solution Set)
(A) \(2 x-3<x+2 \leq 3 x+5, x \in R\)	(I) \(x \in(-1, \infty)\)
(B) \(|2 x+3|<7, x \in R\)	(II) \(x \in(-\infty, 120]\)
(C) \(\frac{1}{2}\left(\frac{3}{5} x+4\right) \geq \frac{1}{3}(x-6), x \in R\)	(III) \(x \in(-5,2)\)
(D) \(\frac{|x+1|}{x+1}>0, x \in R -\{1\}\)	(IV) \(x \in\left[-\frac{3}{2}, 5\right)\)

Choose the correct option

In a 100 metre race, \(A\) beats \(B\) by 10 metres and \(C\) by 13 metres. By how many meters will \(B\) beat \(C\) in a 180 metre race?

The particular solution of the differential equation \(\frac{d y}{d x}=8 y x\) when \(y=1\) at \(x=0\) is:

If \(A=\left|\begin{array}{ccc}x & -3 & 4 \\ 3 & y & -5 \\ -4 & z & 0\end{array}\right|\) is a Skew-Symmetric matrix and \(a d j A=\left[a_{i j}\right]_{3 \times 3}\), then \(a_{11}+a_{22}+a_{33}\) is equal to:

During lockdown, Amit was suffering from heart problem. After proper check-up it was deducted that he has \(45\%\) chance of having heart attack, but yoga and meditation assumed to reduce the chance of heart attack by \(25\%\) and \(20\%\) respectively. He has option to choose any of the two options with equal probabilities. If Amit choose yoga to reduce the chance of heart attack, then the chance of heart attack is:During lockdown, Amit was suffering from heart problem. After proper check-up it was deducted that he has \(45\%\) chance of having heart attack, but yoga and meditation assumed to reduce the chance of heart attack by \(25\%\) and \(20 \%\) respectively. He has option to choose any of the two options with equal probabilities.
If Amit choose yoga to reduce the chance of heart attack, then the chance of heart attack is:

Flower colour in Antirrhinum sp. is an example of:

A unit vector perpendicular to the vectors \(\hat{i}-\hat{j}\) and \(\hat{i}+\hat{j}\) is

If the total energy is transferred by the electromagnetic waves on a surface be \(U\) in time \(t\), then the total momentum delivered to the surface is

A pair of adjacent coils has a mutual inductance of 2.0 H. If the current in one coil changes from 0 to 25 A in 0.8 s, the change of flux linkage with the other coil is

A square matrix \(B=\left[b_{i j}\right]_{n \times n}\) where
\(b_{i j}=0\) when \(i \neq j\)
\(b_{i j}=k\) when \(i=j\) for some constant \(k\) is called:

Acquired immunity is characterised by memory. Identify the sequence in which it happens :
(A) Subsequent encounter of the body with same pathogen
(B) Primary response is produced when the body encounter pathogens for the first time
(C) T-lymphocytes help B-lymphocytes to produce an army of proteins
(D) Body exhibits highly intensified anamnestic response
Choose the correct answer from the options given below: