9th Class Mathematics Unit 1 Real Numbers Numericals with Solutions

R1.1.Four options are given against each statement. Encircle the correct option. (i) \sqrt7 is: (a) integer (b) rational number (c) irrational number (d) natural number (ii) \pi and e are: (a) natural numbers (b) integers (c) rational numbers (d) irrational numbers (iii) If n is not a perfect square, then \sqrt n is: (a) rational number (b) natural number (c) integer (d) irrational number (iv) \sqrt3+\sqrt5 is: (a) whole number (b) integer (c) rational number (d) irrational number (v) For all x\in R, x=x is called: (a) reflexive property (b) transitive number (c) symmetric property (d) trichotomy property (vi) Let a,b,c\in R, then a>b and b>c \Rightarrow a>c is called ____ property: (a) trichotomy (b) transitive (c) additive (d) multiplicative (vii) 2^x\times8^x=64 then x= (a) 3/2 (b) 3/4 (c) 5/6 (d) 2/3 (viii) Let a,b\in R, then a=b and b=a is called ____ property: (a) reflexive (b) symmetric (c) transitive (d) additive (ix) \sqrt{75}+\sqrt{27}= (a) \sqrt{102} (b) 9\sqrt3 (c) 5\sqrt3 (d) 8\sqrt3 (x) The product of (3+\sqrt5)(3-\sqrt5) is: (a) prime number (b) odd number (c) irrational number (d) rational number

Result

(i) (c) irrational number

(ii) (d) irrational numbers

(iii) (d) irrational number

(iv) (d) irrational number

(v) (a) reflexive property

(vi) (b) transitive

(vii) 2^x × 2^3x=2⁶ ⇒ 4x=6 ⇒ x=tfrac32

(vii) (a) 3/2

(viii) (b) symmetric

(ix) sqrt{75}+sqrt{27}=5sqrt3+3sqrt3=8sqrt3

(ix) (d) 8sqrt3

(x) (3+sqrt5)(3-sqrt5)=9-5=4

(x) (d) rational number

R1.2.If a=\frac32, b=\frac53 and c=\frac75, then verify that (i) a(b+c)=ab+ac (ii) (a+b)c=ac+bc

Given

a=tfrac32, b=tfrac53, c=tfrac75

(i) LHS a(b+c)=frac32left(frac53+frac75right)=frac32left(4615right)=235

(i) RHS ab+ac=frac32 · frac53+frac32 · frac75=156+2110=235

Result

(i) a(b+c)=ab+ac=235

(ii) LHS (a+b)c=left(frac32+frac53right)frac75=196 · frac75=13330

(ii) RHS ac+bc=frac32 · frac75+frac53 · frac75=2110+3515=13330

(ii) (a+b)c=ac+bc=13330

R1.3.If a=\frac43, b=\frac52, c=\frac74, then verify the associative property of real numbers w.r.t addition and multiplication.

Given

a=tfrac43, b=tfrac52, c=tfrac74

Addition LHS (a+b)+c=left(frac43+frac52right)+frac74=236+frac74=6712

Addition RHS a+(b+c)=frac43+left(frac52+frac74right)=frac43+174=6712

Result

Addition (a+b)+c=a+(b+c)=6712

Multiplication LHS (a × b) × c=left(frac43 × frac52right) × frac74=103 × frac74=356

Multiplication RHS a × (b × c)=frac43 × left(frac52 × frac74right)=frac43 × 358=356

Multiplication (a × b) × c=a × (b × c)=356

R1.4.Is 0 a rational number? Explain.

Result

0=01

R1.5.State trichotomy property of real numbers.

Result

Exactly one of a<b, a=b, a>b holds for any a,bin R

R1.6.Find two rational numbers between 4 and 5.

Formula

q₁=frac12(4+5)=frac92,quad q₂=frac12left(frac92+5right)=194

frac92, 194

R1.7.Simplify the following: (i) \sqrt[5]{\frac{x^{15}y^{35}}{z^{20}}} (ii) \sqrt[3]{(27)^{2x}} (iii) \frac{6(3)^{n+2}}{3^{n+1}-3^n}

Result

(i) left(frac{x¹⁵y³⁵{z²⁰right)^1/5=x³y⁷z^-4=x³y⁷z⁴

(ii) (27)^2x/3=(3³)^2x/3=3^2x

(iii) 3ⁿ(6 × 9)3ⁿ(3-1)=542=27

R1.8.The sum of three consecutive odd integers is 51. Find the three integers.

Given

x, x+2, x+4

Working

x+(x+2)+(x+4)=51

3x+6=51 ⇒ 3x=45 ⇒ x=15

Result

15, 17, 19

R1.9.Abdullah picked up 96 balls and placed them into two buckets. One bucket has twenty-eight more balls than the other bucket. How many balls were in each bucket?

Given

x + (x+28) = 96

2x+28=96 ⇒ 2x=68 ⇒ x=34

Result

34 and 62 balls

R1.10.Salma invested Rs. 3,50,000 in a bank, which paid simple profit at the rate of 7\frac14\% per annum. After 2 years, the rate was increased to 8\% per annum. Find the amount she had at the end of 7 years.

Given

P=350000,quad R₁=7.25% (first 2 years),quad R₂=8% (next 5 years)

Working

Interest for first 2 years 350000 × 0.0725 × 2=50750

Interest for next 5 years 350000 × 0.08 × 5=140000

Total amount 350000+50750+140000

Result

Rs. 5,40,750

Unit 1: Real Numbers — Review Exercise