**MATHMAA**

**SECTION - C**

15)

Proof:

Let us assume that √2 is an rational number.

√2 = \(\large \frac{p}{q}\) where HCF(p,q) = 1

Squaring on both sides,

2 = \(\large (\frac{p}{q})^{2}\)

2 = \(\large \frac{p^{2}}{q^{2}}\)

2\(q^{2}\) = \(p^{2}\)

So, 'p' is a even number.

Then, p = 2m where 'm' is any integer.

Squaring on both sides,

\(p^{2}\) = \(2m^{2}\)

\(p^{2}\) = 4\(m^{2}\)

\(2q^{2}\) = 4\(m^{2}\)

\(q^{2}\) = 2\(m^{2}\)

So 'q' is also an even number,

Therefore 2 is a common factor of 'p' and 'q', this contradicts our assumption that √2 is a rational number.

Hence √2 is an irrational number, so addition and multiplication with an irrational number is an irrational number.

\(\therefore\) 3 + 5√2 is an irrational number.

Hence proved.

16)

So, 3x³ + 10x² - 14x + 9 = 3x-2(x² + 4x - 2) + 5

Here the remainder is 5.

Therefore 5 should be subtracted from 3x³ + 10x² - 14x + 9 so that 3x - 2 divides it exactly.

SA1-Answers3

17)

5x + \(\large \frac{4}{y}\) = 9 ----------(1)

7x - \(\large \frac{2}{y} = 5 ----------(2)

(2)*2, we get

14x - \(\large \frac{4}{y}\) = 10 ----------(3)

(3)+(1)

14x - \(\large \frac{4}{y}\) + 5x - \(\large \frac{4}{y}\) = 10 + 9

19x = 19

x = 1

Plug x = 1 in (2),

7 - \(\large \frac{2}{y}\) = 5

\(\large \frac{2}{y}\) = 2

y = 1

Therefore x = 1 and y = 1.

18)

Let the fraction be \(\large \frac{x}{y}\).

Then,

x + y = 8 ----------(1)

\(\large \frac{x+3}{y+3}\) = \(\large \frac{3}{4}\)

4(x+3) = 3(y+3)

4x + 12 = 3y + 9

4x - 3y = -3 ----------(2)

(1)*3

3x + 3y = 24 ----------(3)

(2) + (3)

4x - 3y + 3x + 3y = -3 + 24

7x = 21

x = 3

Plug x=3 in (1), we get

y = 5

Therefore the fraction becomes \(\large \frac{3}{5}\).

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