Tahsili · ClipSAT

Home
Formulas

Open a chapter to see its formulas here.

⏱ Exam Timer: 00:00
Active Recall Deck 0
Active Recall Question
Click to show flashcard
Back side
FLASHCARDS

SAAT Tahsili Math

Provider · Qiyas — National Center for Assessment Type · Achievement (التحصيلي) Chapters · 8 Explorers · 3 interactive
i

Test format & strategy

SAAT Tahsili — Mathematics Format & Strategy

The Standardized Achievement Admission Test (SAAT — التحصيلي, "Tahsili") is administered by Qiyas and measures how well a student has mastered the secondary-school curriculum. For the science track it spans mathematics, physics, chemistry and biology; this subject focuses on the mathematics.

What to expect

Unlike the aptitude-based Qudrat, the Tahsili is curriculum-based: it rewards knowing the secondary syllabus — algebra, functions, sequences, trigonometry, coordinate geometry and introductory calculus. Questions are multiple choice and answered without a calculator.

SAAT Tahsili — Mathematics Syllabus Map
Chapter Topics tested Typical share
1 — Algebra Linear/quadratic equations, factoring, inequalities, systems ~20%
2 — Functions Domain/range, transformations, composites, inverses ~15%
3 — Sequences & Series Arithmetic/geometric nth term, finite/infinite sums ~10%
4 — Trigonometry Unit circle, special angles, Pythagorean identity, graphs ~15%
5 — Geometry Plane geometry, coordinate geometry, circle equations ~10%
6 — Limits & Derivatives Limit evaluation, power/chain/product rules, tangent slopes ~15%
7 — Integration Antiderivatives, definite integrals, area under curve ~10%
8 — Exponentials & Logs Log laws, solving exponential equations, \(e\) ~5%

Format: Multiple choice (MCQ), no calculator. Questions are in Arabic. Covers the Saudi secondary school curriculum (math track). Work efficiently by hand: simplify algebraically before computing, and estimate to check your answer.

Basis
secondary curriculum
Format
multiple choice
Calculator
not permitted
Span
algebra → calculus
▲ How to prepare

Work by topic, master the standard formulas, and practise recall under time pressure. The eight chapters below mirror the curriculum strands; each ends with worked examples, and the practice set blends all of them.

Practice Problem

A Tahsili math section has 50 questions. A student answers 42 correctly. If correct answers score +2 and incorrect score 0, what is the student's total score?

Show solution
Score\(=42\times2=84\) points out of a possible 100. (Unanswered or wrong: \(8\times0=0.\))

📝 Chapter Quiz
1

Algebra & Equations

SAAT Tahsili — Mathematics Ch.1 Algebra

Linear and quadratic equations, factoring, inequalities and systems — the algebraic backbone of the syllabus.

▲ Reference — Solving equations

Solve linear equations by isolating the variable. For quadratics, factor or use \( x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a} \). The discriminant \( b^{2}-4ac \) reveals how many real roots exist.

Quadratic formula
\( \dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a} \)
Difference of squares
\( a^{2}-b^{2}=(a-b)(a+b) \)
Discriminant
\( b^{2}-4ac \)
Absolute value
\( |x|=k\Rightarrow x=\pm k \)
Worked example 1.A

Solve \( x^{2}-5x+6=0 \). \( (x-2)(x-3)=0 \): \[ x=2\ \text{or}\ x=3. \]

Worked example 1.B

Solve \( x^{2}+2x-8=0 \). \( (x+4)(x-2)=0 \): \[ x=-4\ \text{or}\ x=2. \]

Practice Problem

Solve \(2x^2-5x-3=0\) by factoring, and verify using the quadratic formula.

Show solution
Factor: \((2x+1)(x-3)=0\Rightarrow x=-\frac{1}{2}\) or \(x=3.\) Verify: \(\Delta=25+24=49.\quad x=\frac{5\pm7}{4}\Rightarrow x=3\) or \(x=-\frac{1}{2}.\) ✓

📝 Chapter Quiz
2

Functions

SAAT Tahsili — Mathematics Ch.2 Functions

Domain and range, function evaluation, transformations, and the exponential and logarithmic families.

▲ Reference — Functions & transformations

\( y=f(x-h)+k \) shifts a graph \( h \) right and \( k \) up. A logarithm inverts an exponential: \( \log_b x=y\iff b^{y}=x \). The domain excludes values that break the rule (e.g. negative under a square root).

Shift the V-shaped graph \( y=|x-h|+k \) below. Its corner — the vertex — sits exactly at \( (h,k) \).

⊙ Explorer · Graph transformationsinteractive
vertex (h, k)

The faint grey graph is the parent \( y=|x| \); the indigo one is the shifted version.

Transformation
\( f(x-h)+k \)
Logarithm
\( \log_b x=y\iff b^{y}=x \)
Exponent law
\( b^{m}b^{n}=b^{m+n} \)
Inverse
swap \( x \) and \( y \)
Worked example 2.A

Solve \( 2^{x}=8 \). \[ 2^{3}=8\Rightarrow x=3. \]

Worked example 2.B

State the domain of \( \sqrt{x-2} \). \[ x\ge 2. \]

Practice Problem

Let \(f(x)=x^2-4\) and \(g(x)=\sqrt{x+4}.\) Find \((g\circ f)(x)\) and its domain.

Show solution
\((g\circ f)(x)=g(f(x))=\sqrt{(x^2-4)+4}=\sqrt{x^2}=|x|.\) Domain: all real \(x\) (since \(x^2\ge0\) always). So domain \(=\mathbb{R}.\)

📝 Chapter Quiz
3

Sequences & Series

SAAT Tahsili — Mathematics Ch.3 Sequences

Arithmetic and geometric sequences, their \( n \)th terms, and the sums of finite and infinite series.

▲ Reference — Sequences & sums

Arithmetic: \( a_n=a+(n-1)d \), sum \( S_n=\tfrac{n}{2}(2a+(n-1)d) \). Geometric: \( a_n=ar^{\,n-1} \); an infinite geometric series sums to \( \dfrac{a}{1-r} \) when \( |r|<1 \).

Arithmetic term
\( a+(n-1)d \)
Arithmetic sum
\( \tfrac{n}{2}(2a+(n-1)d) \)
Geometric term
\( ar^{\,n-1} \)
Infinite sum
\( \dfrac{a}{1-r},\ |r|<1 \)
Worked example 3.A

Find the \( 10 \)th term of the arithmetic sequence with \( a=3,\ d=2 \). \[ 3+9\times2=21. \]

Worked example 3.B

Find the sum of the infinite geometric series with \( a=4,\ r=\tfrac12 \). \[ \frac{4}{1-\tfrac12}=8. \]

Practice Problem

The 3rd term of a geometric sequence is 12 and the 6th term is 96. Find the first term and the common ratio.

Show solution
\(a_3=ar^2=12\) and \(a_6=ar^5=96.\) Divide: \(r^3=8\Rightarrow r=2.\) \(a=12/r^2=12/4=3.\) First term \(a=3\), ratio \(r=2.\)

📝 Chapter Quiz
4

Trigonometry

SAAT Tahsili — Mathematics Ch.4 Trigonometry

Ratios of the special angles, the unit circle, the Pythagorean identity, and the behaviour of the sine and cosine graphs.

▲ Reference — Trig essentials

\( \sin30^{\circ}=\tfrac12,\ \cos60^{\circ}=\tfrac12,\ \tan45^{\circ}=1 \). The identity \( \sin^{2}\theta+\cos^{2}\theta=1 \) holds for all \( \theta \). Convert with \( 180^{\circ}=\pi \) radians. The graphs of \( \sin \) and \( \cos \) have period \( 2\pi \).

Change the amplitude and frequency of \( y=A\sin(Bx) \). The amber lines mark \( \pm A \); the period is \( \tfrac{2\pi}{B} \).

⊙ Explorer · The sine waveinteractive
amplitude
period

Larger \( A \) stretches the wave taller; larger \( B \) packs more cycles into the same width.

Special angles
\( \sin30^{\circ}=\tfrac12 \)
Identity
\( \sin^{2}\theta+\cos^{2}\theta=1 \)
Radians
\( 180^{\circ}=\pi \)
Period
\( \sin,\cos:\ 2\pi \)
Worked example 4.A

Convert \( 180^{\circ} \) to radians. \[ \pi. \]

Worked example 4.B

A right triangle has opposite side \( 3 \) and hypotenuse \( 6 \). Find \( \theta \). \( \sin\theta=\tfrac36=\tfrac12 \): \[ \theta=30^{\circ}. \]

Practice Problem

Given \(\sin\theta=\frac{3}{5}\) and \(\theta\) is in the first quadrant, find \(\cos\theta\), \(\tan\theta\), and \(\sin(2\theta).\)

Show solution
\(\cos\theta=\frac{4}{5}\) (Pythagorean identity: \(\cos^2\theta=1-\frac{9}{25}=\frac{16}{25}\)). \(\tan\theta=\frac{3/5}{4/5}=\frac{3}{4}.\) \(\sin(2\theta)=2\sin\theta\cos\theta=2\cdot\frac{3}{5}\cdot\frac{4}{5}=\frac{24}{25}.\)

📝 Chapter Quiz
5

Geometry & Coordinate Geometry

SAAT Tahsili — Mathematics Ch.5 Geometry

Plane-geometry facts together with the coordinate toolkit: distance, midpoint, gradient, and the equation of a circle.

▲ Reference — Points & shapes

Distance \( =\sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}} \); midpoint is the average of coordinates; gradient \( =\tfrac{\Delta y}{\Delta x} \). A circle centred at the origin is \( x^{2}+y^{2}=r^{2} \). Interior angles of an \( n \)-gon sum to \( (n-2)180^{\circ} \).

Distance
\( \sqrt{\Delta x^{2}+\Delta y^{2}} \)
Midpoint
\( \big(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2}\big) \)
Gradient
\( \dfrac{\Delta y}{\Delta x} \)
Circle
\( x^{2}+y^{2}=r^{2} \)
Worked example 5.A

Find the distance from \( (0,0) \) to \( (3,4) \). \[ \sqrt{3^{2}+4^{2}}=5. \]

Worked example 5.B

Write the equation of a circle centred at the origin with radius \( 5 \). \[ x^{2}+y^{2}=25. \]

Practice Problem

Find the equation of the circle with centre \((3,-2)\) that passes through \((7,1).\)

Show solution
Radius \(r=\sqrt{(7-3)^2+(1+2)^2}=\sqrt{16+9}=5.\) Equation: \((x-3)^2+(y+2)^2=25.\)

📝 Chapter Quiz
6

Limits & Derivatives

SAAT Tahsili — Mathematics Ch.6 Derivatives

The introductory calculus on the syllabus: evaluating limits and differentiating with the basic rules.

▲ Reference — Differentiation

\( \dfrac{d}{dx}x^{n}=nx^{n-1} \); the derivative gives the slope of the tangent. Useful derivatives: \( \dfrac{d}{dx}\sin x=\cos x \), \( \dfrac{d}{dx}e^{x}=e^{x} \). A limit of the form \( \tfrac00 \) often factors and cancels.

The derivative is a limit of slopes. Shrink \( h \) below: the secant through \( (1,1) \) and \( (1+h,(1+h)^2) \) rotates toward the tangent, and its slope approaches \( f'(1)=2 \).

⊙ Explorer · Secant → tangentinteractive
secant slope

The secant slope is \( 2+h \); as \( h\to0 \) it tends to the tangent slope \( 2 \) (the dashed grey line).

Power rule
\( \dfrac{d}{dx}x^{n}=nx^{n-1} \)
Sine
\( \dfrac{d}{dx}\sin x=\cos x \)
Exponential
\( \dfrac{d}{dx}e^{x}=e^{x} \)
Tangent slope
\( f'(a) \)
Worked example 6.A

Evaluate \( \displaystyle\lim_{x\to3}\frac{x^{2}-9}{x-3} \). \[ \lim_{x\to3}(x+3)=6. \]

Worked example 6.B

Differentiate \( 5x^{2}-2x \). \[ 10x-2. \]

Practice Problem

Find \(\dfrac{dy}{dx}\) for \(y=(3x^2+1)^4\) using the chain rule, then find the slope of the tangent at \(x=1.\)

Show solution
Chain rule: \(\frac{dy}{dx}=4(3x^2+1)^3\cdot6x=24x(3x^2+1)^3.\) At \(x=1\): \(24(1)(3+1)^3=24\cdot64=1536.\)

📝 Chapter Quiz
7

Integration & Applications

SAAT Tahsili — Mathematics Ch.7 Integration

Antiderivatives, definite integrals, and the area under a curve — the closing strand of school calculus.

▲ Reference — Integration

\( \displaystyle\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+C \) (for \( n\neq-1 \)). A definite integral \( \displaystyle\int_a^b f=F(b)-F(a) \) measures the signed area under \( f \).

Power rule
\( \dfrac{x^{n+1}}{n+1}+C \)
Definite
\( F(b)-F(a) \)
Constant
\( \displaystyle\int k\,dx=kx+C \)
Area
\( \displaystyle\int_a^b f\,dx \)
Worked example 7.A

Evaluate \( \displaystyle\int_0^2 3x^{2}\,dx \). \[ \big[x^{3}\big]_0^2=8. \]

Worked example 7.B

Find the area under \( y=x \) from \( 0 \) to \( 4 \). \[ \int_0^4 x\,dx=\Big[\tfrac{x^{2}}{2}\Big]_0^4=8. \]

Practice Problem

Evaluate \(\displaystyle\int_1^3(2x^2-3x+1)\,dx.\)

Show solution
Antiderivative: \(F(x)=\dfrac{2x^3}{3}-\dfrac{3x^2}{2}+x.\) \(F(3)=18-\frac{27}{2}+3=\frac{42-27}{2}+3=\frac{15}{2}+3=\frac{21}{2}.\) \(F(1)=\frac{2}{3}-\frac{3}{2}+1=\frac{4-9+6}{6}=\frac{1}{6}.\) Integral\(=\frac{21}{2}-\frac{1}{6}=\frac{63-1}{6}=\frac{62}{6}=\frac{31}{3}\approx10.33.\)

📝 Chapter Quiz
8

Exponentials & Logarithms

SAAT Tahsili — Mathematics Ch.8 Exp & Logs

Exponential growth and its inverse, the logarithm, appear throughout the mathematics section.

▲ Reference — Exp & logs

\( \log_b x=y \) means \( b^{y}=x \); \( \log(xy)=\log x+\log y \); \( \log(x^{n})=n\log x \); \( \ln e=1 \).

Definition
\( b^{y}=x \)
Product
\( \log x+\log y \)
Power
\( n\log x \)
Natural
\( \ln e=1 \)
Worked example 8.A

Evaluate \( \log_2 16 \). \[ 2^{4}=16\Rightarrow 4. \]

Practice Problem

Solve for \(x\): \(\log_2(x+3)+\log_2(x-1)=5.\)

Show solution
Combine: \(\log_2[(x+3)(x-1)]=5\Rightarrow(x+3)(x-1)=32.\) \(x^2+2x-3=32\Rightarrow x^2+2x-35=0\Rightarrow(x+7)(x-5)=0.\) Reject \(x=-7\) (makes \(x-1=-8<0\)). \(x=5.\)

📝 Chapter Quiz
9

Statistics & Probability

SAAT Tahsili — Mathematics Ch.9 Statistics

Averages, spread and the probability of simple events round out the syllabus.

▲ Reference — Data & chance

The mean is total over count; the median is the middle value; probability \( =\tfrac{\text{favourable}}{\text{total}} \); independent events multiply.

Mean
\( \tfrac{\sum x}{n} \)
Median
middle value
Probability
\( \tfrac{\text{fav}}{\text{total}} \)
Independent
\( P(A)P(B) \)
Worked example 9.A

Find the probability of an even number on a fair die. \[ \frac{3}{6}=\frac12. \]

Practice Problem

A data set has values: 8, 12, 10, 15, 10, 14, 11. Find the mean, median, mode, and range.

Show solution
Ordered: 8, 10, 10, 11, 12, 14, 15. Mean\(=(8+10+10+11+12+14+15)/7=80/7\approx11.4.\) Median\(=11\) (4th value). Mode\(=10.\) Range\(=15-8=7.\)

📝 Chapter Quiz
§

Practice set

Fifty curriculum-style problems across all chapters — algebra through calculus — tagged by difficulty. Click any problem for a full worked solution.

1

Solve \( 3x-7=11 \).

Basic
\[ 3x=18,\qquad x=6. \]
2

Solve \( x^{2}=81 \).

Basic
\[ x=\pm 9. \]
3

Solve \( x^{2}-5x+6=0 \).

Intermediate

\( (x-2)(x-3)=0 \):

\[ x=2,\ 3. \]
4

If \( 2x+3y=12 \) and \( x=3 \), find \( y \).

Intermediate

\( 6+3y=12 \):

\[ y=2. \]
5

Solve \( |x|=5 \).

Intermediate
\[ x=\pm 5. \]
6

Factor \( x^{2}-9 \).

Intermediate
\[ (x-3)(x+3). \]
7

Solve \( x^{2}+2x-8=0 \).

Advanced

\( (x+4)(x-2)=0 \):

\[ x=-4,\ 2. \]
8

If \( f(x)=2x+1 \), find \( f(3) \).

Basic
\[ 2(3)+1=7. \]
9

State the domain of \( \sqrt{x-2} \).

Intermediate
\[ x\ge 2. \]
10

If \( f(x)=x^{2} \), find \( f(-3) \).

Intermediate
\[ (-3)^{2}=9. \]
11

Solve \( 2^{x}=8 \).

Intermediate
\[ x=3. \]
12

Evaluate \( \log_2 16 \).

Intermediate

\( 2^{4}=16 \):

\[ 4. \]
13

Find the vertex of \( y=(x-2)^{2}+3 \).

Intermediate
\[ (2,\,3). \]
14

Find the inverse of \( f(x)=2x \).

Advanced

Swap and solve: \( x=2y \):

\[ f^{-1}(x)=\tfrac{x}{2}. \]
15

Find the next term: \( 2,\,5,\,8,\,11,\ldots \)

Basic
\[ 14. \]
16

Find the \( 10 \)th term of the arithmetic sequence \( a=3,\ d=2 \).

Intermediate
\[ 3+9\times2=21. \]
17

Find the common ratio of \( 2,\,6,\,18,\ldots \)

Intermediate
\[ 3. \]
18

Find the sum of the first 5 terms of the arithmetic sequence \( a=2,\ d=3 \).

Intermediate
\[ \tfrac{5}{2}\big(2\cdot2+4\cdot3\big)=\tfrac52(16)=40. \]
19

Find the \( 5 \)th term of the geometric sequence \( a=3,\ r=2 \).

Intermediate
\[ 3\cdot2^{4}=48. \]
20

Find the sum of the infinite geometric series \( a=4,\ r=\tfrac12 \).

Advanced
\[ \frac{4}{1-\tfrac12}=8. \]
21

Find \( 1+2+3+\cdots+20 \).

Intermediate
\[ \frac{20\times21}{2}=210. \]
22

Evaluate \( \sin 30^{\circ} \).

Basic
\[ \tfrac12. \]
23

Evaluate \( \cos 60^{\circ} \).

Basic
\[ \tfrac12. \]
24

Evaluate \( \tan 45^{\circ} \).

Intermediate
\[ 1. \]
25

Convert \( 180^{\circ} \) to radians.

Intermediate
\[ \pi. \]
26

Simplify \( \sin^{2}\theta+\cos^{2}\theta \).

Intermediate
\[ 1. \]
27

A right triangle has opposite side \( 3 \) and hypotenuse \( 6 \). Find \( \theta \).

Intermediate

\( \sin\theta=\tfrac36=\tfrac12 \):

\[ \theta=30^{\circ}. \]
28

Evaluate \( \cos 90^{\circ} \).

Advanced
\[ 0. \]
29

State the period of \( \sin x \).

Intermediate
\[ 2\pi. \]
30

Two angles of a triangle are \( 40^{\circ} \) and \( 70^{\circ} \). Find the third.

Basic
\[ 180^{\circ}-110^{\circ}=70^{\circ}. \]
31

Find the area of a circle with \( r=5 \). Take \( \pi=3.14 \).

Intermediate
\[ 3.14\times25=78.5. \]
32

Find the distance from \( (0,0) \) to \( (3,4) \).

Intermediate
\[ \sqrt{3^{2}+4^{2}}=5. \]
33

Find the midpoint of \( (2,4) \) and \( (6,8) \).

Intermediate
\[ (4,\,6). \]
34

Find the gradient of the line through \( (1,2) \) and \( (3,8) \).

Intermediate
\[ \frac{8-2}{3-1}=3. \]
35

Find the sum of the interior angles of a pentagon.

Intermediate
\[ (5-2)\times180^{\circ}=540^{\circ}. \]
36

Write the equation of a circle centred at the origin with radius \( 5 \).

Advanced
\[ x^{2}+y^{2}=25. \]
37

A right triangle has legs \( 5 \) and \( 12 \). Find the hypotenuse.

Intermediate
\[ \sqrt{25+144}=13. \]
38

Evaluate \( \displaystyle\lim_{x\to2}(3x+1) \).

Basic
\[ 3(2)+1=7. \]
39

Evaluate \( \displaystyle\lim_{x\to3}\frac{x^{2}-9}{x-3} \).

Intermediate

Factor and cancel:

\[ \lim_{x\to3}(x+3)=6. \]
40

Differentiate \( x^{3} \).

Basic
\[ 3x^{2}. \]
41

Differentiate \( 5x^{2}-2x \).

Intermediate
\[ 10x-2. \]
42

Find the slope of the tangent to \( y=x^{2} \) at \( x=2 \).

Intermediate

\( y'=2x \), at \( x=2 \):

\[ 4. \]
43

Differentiate \( \sin x \).

Intermediate
\[ \cos x. \]
44

Find the critical points of \( y=x^{3}-3x \).

Advanced

\( y'=3x^{2}-3=0 \Rightarrow x^{2}=1 \):

\[ x=\pm 1. \]
45

Differentiate \( e^{x} \).

Intermediate
\[ e^{x}. \]
46

Evaluate \( \displaystyle\int 2x\,dx \).

Basic
\[ x^{2}+C. \]
47

Evaluate \( \displaystyle\int_0^2 3x^{2}\,dx \).

Intermediate
\[ \big[x^{3}\big]_0^2=8. \]
48

Evaluate \( \displaystyle\int x^{2}\,dx \).

Intermediate
\[ \tfrac{x^{3}}{3}+C. \]
49

Evaluate \( \displaystyle\int_0^1 (2x+1)\,dx \).

Intermediate
\[ \big[x^{2}+x\big]_0^1=2. \]
50

Find the area under \( y=x \) from \( 0 \) to \( 4 \).

Advanced
\[ \int_0^4 x\,dx=\Big[\tfrac{x^{2}}{2}\Big]_0^4=8. \]

Test generator

Build a randomised test from this subject’s question bank. Pick the number of questions and a difficulty, generate, then reveal the worked answers when you’re ready — or print a clean copy to hand out.

Full official exam paper

Based on the Saudi SAAT Tahsili Mathematics section: 50 MCQ questions in 50 minutes. Covers Saudi secondary curriculum: algebra, geometry, trigonometry, statistics, and calculus basics.

Downloads

Take the Tahsili material offline. The PDF prints straight from your browser; a native Word packet with editable equation-editor math is available on request.

PDF · ready now

Tahsili notes & practice

All eight chapters plus every worked solution, formatted as a clean print document.

DOCX · on request

Native Word packet

Editable .docx with real Word equations (OMML) — trigonometry and calculus included as editable objects.

Official · Past Papers

Past Papers & Official Tests

Officially released exams and mark schemes from the examining body.

Worksheet Library

9 original ClipSAT worksheets with full answer keys, each available in English and Arabic — free to download and print.

Loading worksheet library…

Mock Exam

--:--