Show that (4+3 2 ) is irrational. - Vidyadip

Question

Show that $\big(4+3\sqrt2\big)$ is irrational.

✓

Answer

If possible, let $\big(4+3\sqrt2\big)$ be rational.
Then 4 and $3\sqrt2$ are rational.
$\Rightarrow4+3\sqrt2-4$ is rational $\dots(\because$ diference of two rationals is rational$)$
$\Rightarrow3\sqrt2$ is rational
$\Rightarrow\sqrt2$ is rational $\dots(\because$ 3 is rational$)$
This contradicts the fact that $\sqrt2$ is irrational.
The contradiction arises by assuming that $\big(4+3\sqrt2\big)$ is rational.
Hence, $\big(4+3\sqrt2\big)$ is irrational.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Question" from Real Numbers→Real Numbers — all question types→Maths STD 10 question papers→Maharashtra Board STD 10 question papers→Maharashtra Board question paper generator→

Similar questions

Find the mean, median and mode of the following data:

Class	0-10	10-20	20-30	30-40	40-50	50-60	60-70
Frequency	6	8	10	15	5	4	2

Solve for x and y:
$\frac{3}{\text{x}}-\frac{1}{\text{y}}+\text{9}=0,$
$\frac{2}{\text{x}}+\frac{3}{\text{y}}=\text{5}$ $(\text{x}\neq0,\ \text{y}\neq0).$

Find the roots of the following equations, if they exist, by applying the quadratic formula:
$x^2 + 6x - (a^2 + 2a -8) = 0$

If $\tan\theta=\frac{\text{a}}{\text{b}},$ show that $\frac{(\text{a}\sin\theta-\text{b}\cos\theta)}{(\text{a}\sin\theta+\text{b}\cos\theta)}=\frac{\big(\text{a}^2-\text{b}^2\big)}{\big(\text{a}^2 +\text{b}^2\big)}.$

The following table shows the marks scored by 80 students in an examination:

Marks	Less than 5	Less than 10	Less than 15	Less than 20	Less than 25	Less than 30	Less than 35	Less than 40
Number of students	3	10	25	49	65	73	78	80

Calculate the mean marks correct to 2 decimal places.

The following table gives the marks obtained by 50 students in a class test:

Marks	11-15	16-20	21-25	26-30	31-35	36-40	41-45	46-50
Number of students	2	3	6	7	14	12	4	2

Calculate the mean, median and mode for the above data.

If the points $P(-3, 9), Q(a, b)$ and $R(4, -5)$ are collinear and $a + b = 1$, find the values of $a$ and $b$.

Three cubes of a metal whose edges are in the ratio 3 : 4 : 5 are melted and converted into a single cube whose diagonal is $12\sqrt{3}\text{cm}$ Find the edges of the three cubes.

Draw a circle of radius 4.2cm. Draw a pair of tangents to this circle inclined to each other at an angle of 45°.

An aeroplane is flying at a height of 300m above the ground. Flying at this height the angles of depression on from the two aeoplane of two points on both banks of a river in opposite directions are 45° and 60° respectively. Find the width of the river. $\big[\text{Use}\sqrt{3}=1.732\big]$