1.

Let \(S=\left\{\begin{pmatrix} 0 & 1 & c \\ 1 & a & d\\ 1 & b & e \end{pmatrix}:a,b,c,d,e\in\left\{0,1\right\}\ \text{and} |A|\in \left\{-1,1\right\}\right\}\), where |A| denotes the determinant of A. Then the number of elements in S is _______.

2.

Let f(x) be a continuously differentiable function on the interval (0, ∞) such that f(1) = 2 and
\(\lim\limits_{t→x}\frac{t^{10}f(x)-x^{10}f(t)}{t^9-x^9}=1\)
for each x > 0. Then, for all x > 0, f(x) is equal to

• \(\frac{31}{11x}-\frac{9}{11}x^{10}\)
• \(\frac{9}{11x}+\frac{13}{11}x^{10}\)
• \(\frac{-9}{11x}+\frac{31}{11}x^{10}\)
• \(\frac{13}{11x}+\frac{9}{11}x^{10}\)

3.

A closed vessel contains 10 g of an ideal gas X at 300 K, which exerts 2 atm pressure. At the same temperature, 80 g of another ideal gas Y is added to it and the pressure becomes 6 atm. The ratio of root mean square velocities of X and Y at 300 K is

• 2√2:√3
• 2√2:1
• 1:2
• 2:1

4.

A group of 9 students, s₁, s₂,…., s₉, is to be divided to form three teams X, Y and Z of sizes 2, 3, and 4, respectively. Suppose that s₁ cannot be selected for the team X and s₂ cannot be selected for the team Y. Then the number of ways to form such teams, is _______.

5.

Let \(S=\left\{a+b\sqrt2:a,b\in Z\right\},T_1=\left\{(-1+\sqrt2)^n:n\in \N\right\},\ \text{and}\ T_2=\left\{(1+\sqrt2)^n:n\in\N\right\}\). Then which of the following statements is (are) TRUE ?

6.

The option(s) with correct sequence of reagents for the conversion of P to Q is(are)