CDS Elementary Mathematics 2021 Question Paper

If the number 413283P759387 is divisible by 13, then what is the value of $P$?

What is the remainder when $2^{1000000}$ is divided by 7?

How many pairs of $(x,y)$ can be chosen from the set $\{2,3,6,8,9\}$ such that $\frac{x}{y}+\frac{y}{x}=2$, where $x\ne y$?

Consider the pairs of prime numbers $(m,n)$ between 50 and 100 such that $m-n=6$. How many such pairs are there?

How many terms are there in the following product?
$(a_1+a_2+a_3)(b_1+b_2+b_3+b_4)(c_1+c_2+c_3+c_4+c_5)$

What is the remainder when $27^{27}-15^{27}$ is divided by 6?

If $a+b+c=0$, then which of the following are correct?

1. $a^3+b^3+c^3=3abc$
2. $a^2+b^2+c^2=-2(ab+bc+ca)$
3. $a^3+b^3+c^3=-3ab(a+b)$
   Select the correct answer using the code given below.

If $p=\frac{\sqrt{3q+2}+\sqrt{3q-2}}{\sqrt{3q+2}-\sqrt{3q-2}}$, then what is the value of $p^2-3pq+2$?

What is the unit digit in the expansion of $67^{32}$?

What is the value of $x$, if $\frac{b+\sqrt{b^2-2bx}}{b-\sqrt{b^2-2bx}}=a$?