Question
Prove that $\sqrt{5}$ is an irrational number.
✓
Answer
Let $\sqrt{5}$ be a rational number.
$\therefore \sqrt{5}=\frac{p}{q}$, where $q \neq 0$ and let $p \& q$ be co-prime.
$5 q ^2= p ^2 \Rightarrow p ^2$ is divisible by $5 \Rightarrow p$ is divisible by $5.....$ (i)
$\Rightarrow p =5 a$, where ' a ' is some integer
$25 a^2=5 q^2 \Rightarrow q^2=5 a^2 \Rightarrow q^2$ is divisible by $5 \Rightarrow q$ is divisible by 5 ........ (ii)
(i) and (ii) leads to contradiction as ' $p$ ' and ' $q$ ' are co-prime.
$\therefore \sqrt{5}$ is an irrational number.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Prove that:
$\frac{\cos(90^\circ-\theta)\sec(90^\circ-\theta)\tan\theta}{\text{cosec}(90^\circ-\theta)\sin(90^\circ-\theta)\cot(90^\circ-\theta)}+\frac{\cos\theta\sin(90^\circ-\theta)}{\cot\theta}=2$
→$\frac{\cos(90^\circ-\theta)\sec(90^\circ-\theta)\tan\theta}{\text{cosec}(90^\circ-\theta)\sin(90^\circ-\theta)\cot(90^\circ-\theta)}+\frac{\cos\theta\sin(90^\circ-\theta)}{\cot\theta}=2$
If the $9^{\text {th }}$ term of an AP is zero, prove that its $29^{\text {th }}$ term is twice its $19^{\text {th }}$ term.
→Show that every positive integer is either even or odd.
In the given figure, the radii of two concentric circles are $13 \ cm$ and $8 \ cm. AB$ is a diameter of the bigger circle and $BD$ is a tangent to the smaller circle touching it at $D$. Find the length of $AD$.
→??The following table gives the frequency distribution of the percentage of marks obtained by 2300 students in a competitive examination.
|
Marks obtained (in per cent)
|
11-20
|
21-30
|
31-40
|
41-50
|
51-60
|
61-70
|
71-80
|
|
Number of students
|
141
|
221
|
439
|
529
|
495
|
322
|
153
|
- Convert the given frequency distribution into the continuous form.
- Find the median class and write its class mark.
- Find the modal class and write its cumulative frequency.
In figure, two tangents $R Q$ and $R P$ are drawn from an external point $R$ to the circle with centre $O$. If $\angle P R Q=120^{\circ}$, then prove that $O R=P R+R Q$.
→In a corner of a rectangular field with dimensions 35m × 22m, a well with 14m inside diameter is dug 8m deep. The earth dug out is spread evenly over the remaining part of the field. Find the rise in the level of the field.
If $\sin \theta=\frac{3}{5}$, evaluate $\frac{\operatorname{cosec} \theta-\cot \theta}{2 \cot \theta}$
→A sum of ₹ 2800 is to be used to award four prizes. If each prize after the first is ₹ 200 less than the preceding prize, find the value of each of the prizes.
If A, B and C are the angle of a $\triangle\text{ABC},$ prove that $\tan\Big(\frac{\text{C+A}}{2}\Big)=\cot\frac{\text{B}}{2}.$
→