Embibe Experts Solutions for Chapter: Fundamentals of Mathematics, Exercise 1: Exercise 1

$P, Q$ and $R$ are three natural numbers such that $P$ and $Q$ are primes and $Q$ divides $PR$. Then out of the following the correct statement is:

The number of prime factors of ${\left(3\times 5\right)}^{12}{\left(2\times 7\right)}^{10}{\left(10\right)}^{25}$ is:

If $x, y$ are odd positive integers, then $x^2+y^2$ must be divisible by:

A $4$-digit number is formed by repeating a $2$-digit number such as $2525, 3232$ etc. Any number of this form is exactly divisible by:

Let $D$ be a recurring decimal of the form $D=0.a_1{a}_2{a}_1{a}_2{a}_1{a}_2\dots \dots ,$ where digits $a_1$ and $a_2$ lie between $0$ and $9.$ Further, at most one of them is zero. Which of the following numbers necessarily produces an integer, when multiplied by $D?$

Let $Q=\left(\frac{x}{y}\right)$ where $x$ and $y$ are positive real numbers. If both $x$ and $y$ are increased equally, then

The inequality $\left|2x-3\right|<1$ is valid, when $x$ lies in the interval

If the expression $x^2+2\lambda x+10-3\lambda$ is always positive for all real values of $x$, then a possible value of $\lambda$ is