Real Numbers — Class 10 Mathematics (Standard) PYQs

Assertion (A) : $\text{H.C.F. }(36{\text{ m}}^2,18\text{ m})=18\text{ m}$, where m is a prime number.
Reason (R) : H.C.F. of two numbers is always less than or equal to the smaller number.

Prove that $4-2\sqrt{5}$ is an irrational number given that $\sqrt{5}$ is irrational.

The dimensions of a window are $156\text{cm}\times 216\text{cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

If $\text{HCF}(98,28)=m$ and $\text{LCM}(98,28)=n$, then the value of $n - 7m$ is :

If $(-1{)}^n+(-1{)}^8=0$, then $n$ is :

Which of the following is a rational number between $\sqrt{3}$ and $\sqrt{5}$ ?

Prove that $\frac{1}{\sqrt{5}}$ is an irrational number.

The greatest number which divides $70$ and $125$, leaving remainders $5$ and $8$ respectively, is :

Three sets of Physics, Chemistry and Mathematics books have to be stacked in such a way that all the books are stored subject-wise and the height of each stack is the same. The number of Physics books is $144$, the number of Chemistry books is $180$ and the number of Mathematics books is $192$. Assuming that the books are of same thickness, determine the number of stacks of Physics, Chemistry and Mathematics books.

Prove that $\sqrt{5}$ is an irrational number.