Athens Academy Islamabad · Unit Model Paper · Mathematics · Grade 9

Athens Academy Islamabad · Unit Model Paper · Mathematics · Grade 9 · Chapter 1

Athens Academy Islamabad · Unit Model Paper

Mathematics — SSC-I (Grade 9)

Chapter 1 · Real Numbers & Logarithms

Federal Board of Intermediate & Secondary Education · Session March 2027

Time allowed: 2 hours 40 minutes Total marks: 75 Sections B & C: 60 · Section A (portal): 15

SECTION AObjective — Multiple-Choice Questions Marks 15

How to prepare Section A

The objective section is not printed here — practise it on the Athens portal, where every attempt is timed and shuffled exactly like the board’s bubble-sheet section. Work through all three, in order:

Unit MCQs — full chapter bank Online Quiz — timed self-test Flashcards — laws & traps

Watch the chapter’s classic traps: a repeating decimal such as $0.\overline{45}$ is rational; $\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}$ only for non-negative $a,b$ ; and $\log_{b}(m+n)\neq \log_{b}m+\log_{b}n$ — the product law needs multiplication, not addition.

SECTION BShort-Response Questions Marks 36

Q.2 Attempt any NINE parts. Each part has an OR alternative — answer one side only. (9 × 4 = 36) ★ = high-yield.

☆(i)SLO M-09-01

Identify the property of real numbers used in each step:

(2+\sqrt{3})+\sqrt{5}=2+(\sqrt{3}+\sqrt{5})

and

7\times\tfrac{1}{7}=1

1×4

Solve using logarithms:

\dfrac{\sqrt[4]{34.7}}{2.981}

1×4

★(ii)SLO M-09-03

Simplify, giving the answer with a positive exponent:

\dfrac{4^{m}\times 15^{4m-2n+1}\times 9^{n-2m}}{10^{2m}\times 25^{m-n}}

1×4

Rationalise the denominator and simplify:

\dfrac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}

1×4

★(iii)SLO M-09-05

Find the value of

x

when

\log_{2}(x^{2}-1)=3

1×4

Without a calculator, evaluate

\log_{3}81+\log_{2}\tfrac{1}{8}

using the laws of logarithms.

1×4

☆(iv)SLO M-09-02

a=\sqrt{10}+3

, then find: (a)

a+\tfrac{1}{a}

(b)

a-\tfrac{1}{a}

(c)

a^{2}+\tfrac{1}{a^{2}}

2+1+1

Express

0.\overline{45}

(i.e.

0.454545\dots

) as a rational number

\tfrac{p}{q}

in lowest terms.

1×4

☆(v)SLO M-09-01

State whether each number is rational or irrational, giving a one-line reason:

\sqrt{49}

\pi

0.121121112\dots

\tfrac{22}{7}

1×4

Simplify by rationalising:

\dfrac{3\sqrt{2}}{\sqrt{6}-\sqrt{3}}

, giving your answer in simplest surd form.

1×4

★(vi)SLO M-09-04

Prove that

\sqrt{3}

is irrational.

Use the laws of exponents to prove that

\dfrac{x^{a}}{x^{b}}\cdot\dfrac{x^{b}}{x^{c}}\cdot\dfrac{x^{c}}{x^{a}}=1

☆(vii)SLO M-09-05

Simplify and express with radical sign removed from the denominator:

\dfrac{1}{\sqrt{7}-\sqrt{3}}+\dfrac{1}{\sqrt{7}+\sqrt{3}}

1×4

Given

\log 2 = 0.3010

and

\log 3 = 0.4771

, find

\log 48

using the laws of logarithms.

1×4

☆(viii)SLO M-09-02

Represent

\sqrt{5}

on the real number line by geometric construction, showing your right triangle clearly.

2+2

Change to logarithmic form and evaluate:

2^{5}=32

and

10^{-2}=0.01

. State the base and the logarithm in each case.

2+2

★(ix)SLO M-09-05

Simplify to a single power of

3

\dfrac{27^{n+1}\times 9^{-n}}{3^{n+2}}

1×4

\log_{10}2=0.3010

, find the number of digits in

2^{50}

(use

\log_{10}(2^{50})=50\log_{10}2

1×4

SECTION CExtended-Response Questions Marks 24

Attempt ALL questions Q.3–Q.5. Each has an OR choice — answer one side only. (3 × 8 = 24)

★Q.3SLO M-09-05

Prove the two laws of logarithms and apply them.

(a) For $m,n>0$ and base $b>0,\;b\neq 1$ , prove that $\log_{b}(mn)=\log_{b}m+\log_{b}n$ .

(b) Hence, using $\log 2=0.3010,\;\log 3=0.4771,\;\log 7=0.8451$ , evaluate $\log 42$ .

4+4

Prove the change-of-base rule and use it.

(a) Prove that $\log_{b}m=\dfrac{\log_{a}m}{\log_{a}b}$ for valid bases $a,b$ .

(b) Hence evaluate $\log_{8}32$ exactly, showing every step.

4+4

★Q.4SLO M-09-04

Radicals and rationalising — full working required.

(a) Rationalise and simplify $\dfrac{3+2\sqrt{2}}{3-2\sqrt{2}}$ , giving the answer in the form $a+b\sqrt{2}$ .

(b) If $x=\dfrac{1}{2-\sqrt{3}}$ , show that $x^{2}-4x+1=0$ .

4+4

Prove $\sqrt{2}$ is irrational, then classify.

(a) Prove, by contradiction, that $\sqrt{2}$ is not a rational number.

(b) Show that the sum of a rational and an irrational number is always irrational, and give one worked example.

5+3

☆Q.5SLO M-09-02

Laws of exponents — simplify completely.

Simplify to lowest terms, leaving no negative or zero exponents:

$\left(\frac{x^{a}}{x^{b}}\right)^{a+b}\cdot\left(\frac{x^{b}}{x^{c}}\right)^{b+c}\cdot\left(\frac{x^{c}}{x^{a}}\right)^{c+a}$

Scientific notation and logarithms in application.

(a) The distance light travels in one year is about $9.46\times 10^{15}$ m. Express $9.46\times 10^{15}$ in ordinary form and state how many significant figures it carries.

(b) A bacterial colony doubles every hour, starting from $500$ cells. Using $\log_{10}2=0.3010$ , find after how many whole hours the colony first exceeds $10^{6}$ cells.

4+4

MARKING SCHEMEFBISE SSC-I Mathematics Total 75

Section	Attempt	Marks
A — MCQs (prepare on portal)	All 15 · compulsory · bubble sheet	15 × 1 = 15
B — Short questions	Any 9 of the choices offered	9 × 4 = 36
C — Long questions	Any 3 of 3 (each with an OR)	3 × 8 = 24
Total	Mathematics — theory only, no practical	75

Sections B and C on this sheet are a Chapter-1 practice set; the real board paper draws its questions from the whole syllabus. Section A marks are earned on the portal quiz until exam day.

📃 Chapter 1 · Real Numbers — Chapter Model Paper

FBISE Grade 9 Mathematics — Chapter 1: Real Numbers Model Paper