Skill Check V: IITJEE Foundation Math

I. Consider the following relationship amongst various number systems: $\mathcal{N} \subset \mathcal{W} \subset \mathcal{Z} \subset \mathcal{Q} \subset \mathcal{R}$ .

Write the following numbers in the smallest set or subset in the above relationship:

(a) 8 (b) -8 (c) +478 (d) -2191 (e) -21.91 (f) +3.6 (g) 0 (h) +4.6 (i) $-6.\dot{7}$ (j) 8.292992999…(k) $\frac{3}{8}$ (l) $\frac{8}{2}$ (m) $0 \frac{0}{7}$ (n) $-3\frac{1}{5}$ (o) $\frac{22}{33}$ (p) $\sqrt{64}$ (r) $\sqrt{6.4}$ (s) $2+\sqrt{3}$ (t) $6\sqrt{4}$ (u) $4\sqrt{6}$

II. If $\frac{22}{7} = 3.1428571...$ is $\frac{22}{7}$ an irrational number?

III. Fill in the boxes with the correct real numbers in the following statements: (a) $2\sqrt{7}+\sqrt{7} = \Box+ 2\sqrt{7}$ (b) $3.\dot{8} + 4.65 = 4.65 + \Box$ (c) $\Box + 29 = 29 + 5\sqrt{10}$ (d) $3.\dot{9} + (4.69 +2.12) = (\Box + 4.69) + 2.12$ (e) $(\frac{7}{8} + \frac{3}{7})+\frac{6}{5} = (\frac{6}{5} + \frac{3}{7}) + \Box$ (f) $3\sqrt{2}(\sqrt{3}+2\sqrt{5}) = (3\sqrt{2}+2\sqrt{5}) + (\Box \times \Box)$ (g) $1\frac{3}{7} (2\frac{7}{11} + \Box) = (1\frac{3}{7} \times 1 \frac{8}{9}) + (1\frac{3}{7} + 2\frac{7}{11})$ (h) $2\frac{1}{3} + \Box =0$ (xi) $\frac{7}{-8} \times \Box = 1$ (i) $-7.35 + \Box = 0$

IV. Find the answers to the following expressions by using the properties of addition and multiplication of real numbers:

Before that, we can recapitulate the relevant properties here :

Properties of Real Numbers:
Closure Property: The sum, difference, product,or quotient of two real numbers is a real number.

Commutative Property of Addition and Multiplication:

A change in the order of addition or multiplication of two real numbers does not change their respective sum or product. (a) $x+y = y+x$ (b) $x \times y = y \times x$

Associative Property of Addition and Multiplication:

A change in the grouping of three real numbers while adding or multiplying does not change their respective sum or product :

$(a+b)+c = (a+b)+c$ and $a \times (b \times c) = (a \times b) \times c$

Distributive Property of Multiplication over Addition:

When a real number is multiplied by the sum of two or more real numbers, the product is the same as the sum of the individual products of the real number and each addend.

$m(a+b) = ma+mb$ . Clearly, multiplication has “distributed” over addition.

Identity Property of Real Numbers
The addition of zero or the multiplication with one does not change a real number. That is,

$a+0=0+a=a$ and $a \times 1 = a = 1 \times a$

Inverse Property of Real Numbers

Corresponding to every real number, there exists another real number of opposite sign such that the sum of the two real numbers is zero: $a+ a^{'}=0$ such that $a^{'}=-a$
Corresponding to every (non-zero) real number, there exists a real number, known as its reciprocal, such that the product of the two real numbers is 1. That is, $a \times \frac{1}{a} = 1$ , where $r \neq 0$ .

Now, in the questions below, identify the relevant properties:

(a) $283 +(717 + 386)$

(b) $(2154 - 1689) + 1689$

(d) $(6.784-3.297) + 3.297$

(e) $\frac{7}{13} + (\frac{6}{13}-1)$

(e) $0.25 \times (4.17 -0.17)$

(f) $(6.6 \times 6.6) + (6.6 \times 3.4)$

(g) $(\frac{2}{3} \times 5) - (\frac{2}{3} \times 2)$

(h) $(6.\dot{8} \times 5) - (6.\dot{8} \times 4)$

(i) $\frac{6}{7} \times \frac{7}{6} \times \frac{6}{7}$

V) Which of the following operations on irrational numbers are correct?

(a) $6\sqrt{5} - 4\sqrt{3}=2\sqrt{2}$

(b) $\sqrt{7} \times \sqrt{7} = 7$

(d) $\sqrt{7} \times \sqrt{7} = 49$

(e) $\sqrt{7} + \sqrt{2} = \sqrt{9}$

(f) $2 \sqrt{8} \times 3\sqrt{2} =24$

(g) $8\sqrt{2} + 8 \sqrt{2} =32$

(h) $2\sqrt{3}= 3\sqrt{6} = \frac{2}{3\sqrt{2}}$

(i) $5+\sqrt{3} = 5\sqrt{3}$

(j) $3\sqrt{20} \div 3\sqrt{5}=2$

VI) Find the rationalizing factors of the following irrational numbers:

(a) $\sqrt{10}$

(b) $\sqrt{7}$

(d) $3\sqrt{7}$

(e) $-2\sqrt{8}$

(f) $-6\sqrt{7}$

(g) $\frac{1}{\sqrt{2}}$

(h) $\frac{2}{\sqrt{3}}$

(i) $2\sqrt{3}=4\sqrt{3}$

(j) $7\sqrt{5} - 2\sqrt{5}$

(k) $1+\sqrt{2}$

(l) $3-\sqrt{5}$

(m) $3\sqrt{2}+6$

(n) $4\sqrt{7} + 6\sqrt{2}$

(o) $3\sqrt{6}-2\sqrt{3}$

VII) Rationalize the denominators of the following numbers:

(a) $\frac{1}{\sqrt{3}}$

(b) $\frac{3}{\sqrt{3}}$

(d) $\frac{8}{\sqrt{6}}$

(e) $\frac{3}{2\sqrt{5}}$

(f) $\frac{\sqrt{5}}{\sqrt{7}}$

(g) $\frac{3\sqrt{3}}{3\sqrt{5}}$

(h) $\frac{3}{\sqrt{5}-sqrt{3}}$

(i) $\frac{5}{\sqrt{3}+\sqrt{2}}$

(j) $\frac{17}{4\sqrt{6}+3\sqrt{5}}$

(k) $\frac{3}{3+\sqrt{3}}$

(l) $\frac{11}{3\sqrt{5}-2\sqrt{3}}$

(m) $\frac{\sqrt{5}}{3\sqrt{5}-3\sqrt{2}}$

(n) $\frac{\sqrt{3}+1}{\sqrt{3}-1}$

(o) $\frac{\sqrt{5}-sqrt{2}}{\sqrt{5}+\sqrt{2}}$

VIII. Find the additive inverse of each of the following irrational numbers:

(i) $\sqrt{7}$ (ii) $3\sqrt{5}$ (iii) $-6\sqrt{7}$ (iv) $5+\sqrt{7}$ (v) $3\sqrt{7} - 2\sqrt{8}$

IX. Find the multiplicative inverse of each of the following irrational numbers:

(i) $\sqrt{6}$ (ii) $\frac{1}{2\sqrt{7}}$ (iii) $\frac{3\sqrt{8}}{2\sqrt{7}}$ (iv) $\frac{4}{3+\sqrt{2}}$ (v) $\frac{2\sqrt{5}+3\sqrt{6}}{5\sqrt{8}-4\sqrt{7}}$

X. Illustrate the closure property of addition of real numbers using the irrational numbers $\sqrt{5}$ and $2\sqrt{5}$ .

XI. Illustrate that the closure property does not apply on subtraction of real numbers using two rational numbers: $2\frac{1}{7}$ and $-3\frac{2}{5}$ .

XII. Illustrate the distributive property of multiplication over addition of real numbers using three irrational numbers: $3\sqrt{7}$ , $-2\sqrt{7}$ and $\sqrt{7}$ .

Regards,

Nalin Pithwa

Mathematics Hothouse