Question
Prove that $\big(2\sqrt3-1\big)$ is an irrational number.
✓
Answer
Let $\text{x}=2\sqrt3-1$ be a rational number.
$\text{x}=2\sqrt3-1$
$\Rightarrow\text{x}^2=\big(2\sqrt3-1\big)^2$
$\Rightarrow\text{x}^2=\big(2\sqrt3\big)^2+1^2-2\big(2\sqrt3\big)(1)$
$\Rightarrow\text{x}^2=12+1-4\sqrt3$
$\Rightarrow\text{x}^2-13=-4\sqrt3$
$\Rightarrow\frac{13-\text{x}^2}{4}=\sqrt3$
Since $x$ is a rational number, $x^2$ is also a rational number.
$\Rightarrow 13 − x^2$ is a rational number
$\Rightarrow\frac{13-\text{x}^2}{4}$ is a rational number
$\Rightarrow\sqrt3$ is a rational number
But $\sqrt3$ is an irrational number, which is a contradiction.
Hence, our assumption is wrong.
Thus, $\big(2\sqrt3-1\big)$ is an irrational number.
$\text{x}=2\sqrt3-1$
$\Rightarrow\text{x}^2=\big(2\sqrt3-1\big)^2$
$\Rightarrow\text{x}^2=\big(2\sqrt3\big)^2+1^2-2\big(2\sqrt3\big)(1)$
$\Rightarrow\text{x}^2=12+1-4\sqrt3$
$\Rightarrow\text{x}^2-13=-4\sqrt3$
$\Rightarrow\frac{13-\text{x}^2}{4}=\sqrt3$
Since $x$ is a rational number, $x^2$ is also a rational number.
$\Rightarrow 13 − x^2$ is a rational number
$\Rightarrow\frac{13-\text{x}^2}{4}$ is a rational number
$\Rightarrow\sqrt3$ is a rational number
But $\sqrt3$ is an irrational number, which is a contradiction.
Hence, our assumption is wrong.
Thus, $\big(2\sqrt3-1\big)$ 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
A survey regarding the heights (in cm) of 50 girls of a class was conducted and the following data was obtained:
Find the mean, median and mode of the above data.
| Height in cm | 120-130 | 130-140 | 140-150 | 150-160 | 160-170 | Total |
| Number of girls | 2 | 8 | 12 | 20 | 8 | 50 |
Solve the following systems of equations by using the method of cross multiplication:
$2ax + 3by = (a + 2b),$
$3ax + 2by = (2a + b).$
→$2ax + 3by = (a + 2b),$
$3ax + 2by = (2a + b).$
Write the step of construct the tangents to a circle from an external point.
A hollow sphere of external and internal diameters 8cm and 4cm, respectively is melted into a solid cone of base diameter 8cm. Find the height of the cone.
The height of 50 girls of class X of a school are recorded as follows:
Draw a 'more than type' ogive for the above data.
|
Height (in cm)
|
135-140
|
140-145
|
145-150
|
150-155
|
155-160
|
160-165
|
|
Number of girls
|
5
|
8
|
9
|
12
|
14
|
2
|
If the mean of the following frequency distribution is 54, find the value of p.
|
Class
|
0-20
|
20-40
|
40-60
|
60-80
|
80-100
|
|
Frequency
|
7
|
p
|
10
|
9
|
13
|
$A(6, 1), B(8, 2)$ and $C(9, 4)$ are the vertices of a parallelogram ABCD. If E is the midpoint of DC, find the area of $\triangle\text{ADE}.$
→On a circular table cover of radius 42cm, a design is formed by a girl leaving an equilateral triangle ABC in the middle, as shown in the figure. Find the covered area of the design. $\Big[\text{Use }\sqrt{3}=1.73\text{ and }\pi=\frac{22}{7}\Big]$
→Prove that $\big(4-5\sqrt2\big)$ is an irrational number.
→Solve the following quadratic equation:
$\frac{1}{\text{2a}+\text{b}+\text{2x}}=\frac{1}{\text{2a}}+\frac{1}{\text{b}}+\frac{1}{\text{2x}}$
→$\frac{1}{\text{2a}+\text{b}+\text{2x}}=\frac{1}{\text{2a}}+\frac{1}{\text{b}}+\frac{1}{\text{2x}}$