If \(A=\{1,2,3,4, \cdots, n\}\) and \(B=\{x, y\}\), then the number of surjections from \(A\) to \(B\) is

The order and degree of differential equation \(\left[1+\left(\frac{d y}{d x}\right)^2\right]^2=\frac{d^2 y}{d x^2}\) respectively, are

If \(\hat{i}, \hat{j}, \hat{k}\) are unit vectors along X, Y, Z axes then,
(A) \(\hat{i} \times \hat{i}=1\)
(B) \(\hat{j} \cdot \hat{j}=0\)
(C) \(\hat{i} \times \hat{j}=\hat{k}\)
(D) \(\hat{k} \times \hat{i}=\hat{j}\)
(E) \(\hat{i} \cdot \hat{j}=0\)
Choose the correct answer from the options given below:

The corner points of the feasible region determined by the system of Linear Constraints are \((0,3),(1,2)\) and \((3,0)\). Let \(z=p x+q y\), where \(p, q>0\), be the objective function. Find the condition on \(p\) and \(q\) so that the minimum of \(z\) occurs at \((3,0)\) and \((1,2)\) :

If \(A\) is any event associated with sample space and If \(E_1, E_2, E_3\) are mutually exclusive and exhaustive events. Then which of the following are true?
(A) \(P(A)=P\left(E_1\right) P\left(E_1 \mid A\right)+P\left(E_2\right) P\left(E_2 \mid A\right)+P\left(E_3\right) P\left(E_3 \mid A\right)\)
(B) \(P(A)=P\left(A \mid E_1\right) P\left(E_1\right)+P\left(A \mid E_2\right) P\left(E_2\right)+P\left(A \mid E_3\right) P\left(E_3\right)\)
(C) \(P\left(E_i \mid A\right)=\frac{P\left(A \mid E_i\right) P\left(E_i\right)}{\sum_{i=1}^3 P\left(A \mid E_i\right) P\left(E_i\right)}, i=1,2,3\)
(D) \(P\left(A \mid E_i\right)=\frac{P\left(E_i \mid A\right) P\left(E_i\right)}{\sum_{i=1}^3 P\left(E_i \mid A\right) P\left(E_i\right)}, i=1,2,3\)
Choose the correct answer from the options given below:

If A and B are events such that \(P(A \mid B)=P(B \mid A)\), then :

Match List I with List II

LIST- I	LIST II
A. \(l x+m y+n z=d\) is	I. Equation of plane passing through a given point and normal to given vector
B. \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\) is	II. Equation of plane in normal form
C. \((\vec{r}-\vec{a}) \cdot \vec{n}=0\)	III. Plane passing through the intersection of two planes
\(D .\left(a_1 x+b_1 y+c_1 z+d_1\right)+\lambda\left(a_2 x+b_2 y+c_2 z+d_2\right)=0\)	IV. Intercept form of plane

Choose the correct answer from the options given below :

Let \(\left|\begin{array}{ll}3 & y \\ x & 1\end{array}\right|=\left|\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right|\) and \(x, y\) are natural numbers, then the number of solutions for the system is :

Let \(f:[-2,2] \rightarrow[-2,2]\) be a function defined by \(f(x)=x|x|\), then \(f\) is:

Read the passage carefully and answer the Questions
The decomposition of \(N_2O_5\ (g)\) is given as
\(N_2O_5(g) \rightarrow 2NO_2(g) + \frac{1}{2}O_2(g)\)
The above reaction is found to be of first order.
Choose the correct statement about the rate of the reaction:

Which of the following statements are correct about alkyl halides :
A. Alkyl halides are polar so they are soluble in water
B. RX on reaction with alc.KOH give alkenes
C. RX on reaction with \(AgNO_2\) give RONO
D. \(S_N1\) proceeds with inversion in configuration
E. R-X have higher boiling point than parent alkanes
Choose the correct answer from the options given below :

Arrange the given structures in correct sequence of spermatogenesis?
(A) Spermatozoa
(B) Secondary spermatocytes
(C) Spermatogonia
(D) Spermatids
Choose the correct answer from the options given below:

Which one of the following contributes the maximum percentage to the biodiversity of a vertebrate group?