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SAAT Tahsili Math

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

Pie chart: Tahsili Curriculum Weighting showing 3rd Year (الصف الثالث) 50% (50), 2nd Year (الصف الثاني) 30% (30), 1st Year (الصف الأول) 20% (20)Tahsili Curriculum Weighting3rd Year (الصف الثالث) 50% (50%)2nd Year (الصف الثاني) 30% (30%)1st Year (الصف الأول) 20% (20%)
Figure 1: Tahsili Scientific Track Curriculum Weighting Across High School Years
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, ee ~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.

⚠️ Tahsili Exam Traps — Pacing & Calculator Assumptions

No calculator permitted: Every calculation in the Tahsili exam is designed for rapid mental computation, factoring, or algebraic estimation. If your scratch work requires multi-digit long division or difficult square roots, reconsider your approach — look for special factorable patterns or cancellation.

Equal weighting per item: Every question carries equal weight across all 4 subjects (Math, Physics, Chemistry, Biology). Spending over 2 minutes on a single math puzzle risks forfeiting easy questions in other sections.

A section of the Tahsili exam contains 22 questions to be completed in 25 minutes. If a student spends 4 minutes on the first 3 questions, what is the maximum average time in seconds they can spend on each remaining question?
Remaining questions: 22−3=1922 - 3 = 19. Remaining time: 25−4=2125 - 4 = 21 minutes =1260= 1260 seconds. Average time per question: 126019≈66.3 seconds. \frac{1260}{19} \approx 66.3 \text{ seconds}.
In an 88-question Tahsili exam where Grade 12 (3rd Year) accounts for 50%, Grade 11 (2nd Year) for 30%, and Grade 10 (1st Year) for 20%, how many total questions are drawn from Grade 11 and Grade 12 combined?
Combined percentage for Grades 11 and 12 = 30%+50%=80%30\% + 50\% = 80\%. Total questions = 0.80×88=70.4≈700.80 \times 88 = 70.4 \approx 70 questions.
📝 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=−b±b2−4ac2a x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a} . The discriminant b2−4ac b^{2}-4ac reveals how many real roots exist.

Quadratic Function Roots & Vertex — Coordinate plane with graph of f(x) = x² - 4x + 3123424xyf(x) = x² − 4x + 3
Figure 1: Parabola f(x) = x² - 4x + 3 showing roots at x = 1 and x = 3, and vertex at (2, -1)
Quadratic formula
−b±b2−4ac2a \dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}
Difference of squares
a2−b2=(a−b)(a+b) a^{2}-b^{2}=(a-b)(a+b)
Discriminant
b2−4ac b^{2}-4ac
Absolute value
∣x∣=k⇒x=±k |x|=k\Rightarrow x=\pm k
Worked example 1.A

Solve x2−5x+6=0 x^{2}-5x+6=0 . (x−2)(x−3)=0 (x-2)(x-3)=0 : x=2 or x=3. x=2\ \text{or}\ x=3.

Worked example 1.B

Solve x2+2x−8=0 x^{2}+2x-8=0 . (x+4)(x−2)=0 (x+4)(x-2)=0 : x=−4 or x=2. x=-4\ \text{or}\ x=2.

⚠️ Tahsili Exam Traps — Sign Changes in Inequalities & Extraneous Roots

Multiplying or dividing by a negative number: Remember to reverse the inequality sign when dividing or multiplying by a negative number (e.g., −2x<6⇒x>−3-2x < 6 \Rightarrow x > -3).

Extraneous roots in rational and radical equations: Always substitute candidate solutions back into the original expression. Values that make a denominator zero or produce an impossible negative square root (e.g., x=−2\sqrt{x} = -2) must be discarded.

For what value of kk is the matrix A=(k436)A = \begin{pmatrix} k & 4 \\ 3 & 6 \end{pmatrix} singular (has no inverse)?
A matrix is singular if its determinant is zero: det⁡(A)=(k)(6)−(4)(3)=0  ⟹  6k−12=0  ⟹  k=2. \det(A) = (k)(6) - (4)(3) = 0 \implies 6k - 12 = 0 \implies k = 2.
Find the value of mm such that the quadratic equation x2−6x+m=0x^2 - 6x + m = 0 has exactly one repeated real root.
A single repeated real root requires discriminant Δ=b2−4ac=0\Delta = b^2 - 4ac = 0: (−6)2−4(1)(m)=0  ⟹  36−4m=0  ⟹  m=9. (-6)^2 - 4(1)(m) = 0 \implies 36 - 4m = 0 \implies m = 9.
📝 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 y=f(x-h)+k shifts a graph h h right and k k up. A logarithm inverts an exponential: log⁡bx=y  ⟺  by=x \log_b x=y\iff b^{y}=x . The domain excludes values that break the rule (e.g. negative under a square root).

Reciprocal Function Asymptotes — Coordinate plane with graph of f(x) = 1/x-4-3-2-11234-4-224xyf(x) = 1/x
Figure 1: Parent reciprocal function f(x) = 1/x showing horizontal and vertical asymptotes

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

⊙ Explorer · Graph transformationsinteractive
vertex (h, k)—

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

Transformation
f(x−h)+k f(x-h)+k
Logarithm
log⁡bx=y  ⟺  by=x \log_b x=y\iff b^{y}=x
Exponent law
bmbn=bm+n b^{m}b^{n}=b^{m+n}
Inverse
swap x x and y y
Worked example 2.A

Solve 2x=8 2^{x}=8 . 23=8⇒x=3. 2^{3}=8\Rightarrow x=3.

Worked example 2.B

State the domain of x−2 \sqrt{x-2} . x≥2. x\ge 2.

⚠️ Tahsili Exam Traps — Composition Order & Restricted Domains

Composition is not commutative: In general, (f∘g)(x)≠(g∘f)(x)(f \circ g)(x) \neq (g \circ f)(x). Always substitute the inner function into the outer function, from right to left.

Domain of composite functions: An input xx must not only be in the domain of the inner function g(x)g(x), but g(x)g(x) must also yield a value acceptable to the domain of ff.

Find the inverse function f−1(x)f^{-1}(x) for f(x)=2x−53f(x) = \frac{2x - 5}{3}.
Set y=2x−53y = \frac{2x - 5}{3}, swap variables: x=2y−53  ⟹  3x=2y−5  ⟹  y=3x+52x = \frac{2y - 5}{3} \implies 3x = 2y - 5 \implies y = \frac{3x + 5}{2}. Thus f−1(x)=3x+52f^{-1}(x) = \frac{3x + 5}{2}.
Determine the horizontal and vertical asymptotes of the rational function g(x)=4x−12x+6g(x) = \frac{4x - 1}{2x + 6}.
Vertical asymptote: 2x+6=0  ⟹  x=−32x + 6 = 0 \implies x = -3. Horizontal asymptote from leading coefficient ratio: y=42=2y = \frac{4}{2} = 2.
📝 Chapter Quiz
3

Sequences & Series

SAAT Tahsili — Mathematics Ch.3 Sequences

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

▲ Reference — Sequences & sums

Arithmetic: an=a+(n−1)d a_n=a+(n-1)d , sum Sn=n2(2a+(n−1)d) S_n=\tfrac{n}{2}(2a+(n-1)d) . Geometric: an=ar n−1 a_n=ar^{\,n-1} ; an infinite geometric series sums to a1−r \dfrac{a}{1-r} when ∣r∣<1 |r|<1 .

Bar chart: Arithmetic vs Geometric Growth with categories: a₁ (2), a₂ (5), a₃ (8), g₁ (2), g₂ (4), g₃ (8)02468a₁2a₂5a₃8g₁2g₂4g₃8Arithmetic vs Geometric Growth
Figure 1: Comparison of terms in an arithmetic progression (common difference d = 3) vs geometric progression (common ratio r = 2)
Arithmetic term
a+(n−1)d a+(n-1)d
Arithmetic sum
n2(2a+(n−1)d) \tfrac{n}{2}(2a+(n-1)d)
Geometric term
ar n−1 ar^{\,n-1}
Infinite sum
a1−r, ∣r∣<1 \dfrac{a}{1-r},\ |r|<1
Worked example 3.A

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

Worked example 3.B

Find the sum of the infinite geometric series with a=4, r=12 a=4,\ r=\tfrac12 . 41−12=8. \frac{4}{1-\tfrac12}=8.

⚠️ Tahsili Exam Traps — Sum of Infinite Series Condition

Convergence condition: The formula for the infinite geometric series sum S=a11−rS = \frac{a_1}{1 - r} applies only if the common ratio satisfies ∣r∣<1|r| < 1. If ∣r∣≥1|r| \ge 1, the series diverges and has no finite sum.

Index count: When finding the number of terms between k=mk=m and k=nk=n, the total number of terms is n−m+1n - m + 1, not simply n−mn - m.

An arithmetic sequence has first term a1=7a_1 = 7 and common difference d=4d = 4. What is the 10th term a10a_{10}?
Using an=a1+(n−1)da_n = a_1 + (n - 1)d: a10=7+(10−1)(4)=7+36=43. a_{10} = 7 + (10 - 1)(4) = 7 + 36 = 43.
Find the sum of the infinite geometric series: 12+6+3+32+⋯ 12 + 6 + 3 + \frac{3}{2} + \cdots
First term a1=12a_1 = 12, ratio r=6/12=1/2r = 6/12 = 1/2. Since ∣r∣<1|r| < 1: S=a11−r=121−1/2=24. S = \frac{a_1}{1 - r} = \frac{12}{1 - 1/2} = 24.
📝 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

sin⁡30∘=12, cos⁡60∘=12, tan⁡45∘=1 \sin30^{\circ}=\tfrac12,\ \cos60^{\circ}=\tfrac12,\ \tan45^{\circ}=1 . The identity sin⁡2θ+cos⁡2θ=1 \sin^{2}\theta+\cos^{2}\theta=1 holds for all θ \theta . Convert with 180∘=π 180^{\circ}=\pi radians. The graphs of sin⁡ \sin and cos⁡ \cos have period 2π 2\pi .

Unit circle diagram showing terminal angle 45: Unit Circle CoordinatesUnit Circle Coordinatesxy45°(0.707, 0.707)
Figure 1: Standard Unit Circle showing coordinates (cos θ, sin θ) for reference angles

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

⊙ Explorer · The sine waveinteractive
amplitude—
period—

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

Special angles
sin⁡30∘=12 \sin30^{\circ}=\tfrac12
Identity
sin⁡2θ+cos⁡2θ=1 \sin^{2}\theta+\cos^{2}\theta=1
Radians
180∘=π 180^{\circ}=\pi
Period
sin⁡,cos⁡: 2π \sin,\cos:\ 2\pi
Worked example 4.A

Convert 180∘ 180^{\circ} to radians. π. \pi.

Worked example 4.B

A right triangle has opposite side 3 3 and hypotenuse 6 6 . Find θ \theta . sin⁡θ=36=12 \sin\theta=\tfrac36=\tfrac12 : θ=30∘. \theta=30^{\circ}.

⚠️ Tahsili Exam Traps — ASTC Quadrant Signs & Special Angle Values

Quadrant signs (ASTC): Remember the signs of trig functions across quadrants: All positive in Q1, Sine in Q2, Tangent in Q3, Cosine in Q4. For example, cos⁡(120∘)=−cos⁡(60∘)=−12\cos(120^\circ) = -\cos(60^\circ) = -\frac{1}{2}, not +12+\frac{1}{2}.

Pythagorean identity misuse: sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1, but note that sin⁡θ+cos⁡θ≠1\sin\theta + \cos\theta \neq 1!

Find the exact values of sin⁡(150∘)\sin(150^\circ) and cos⁡(150∘)\cos(150^\circ).
Reference angle: 180∘−150∘=30∘180^\circ - 150^\circ = 30^\circ. In Q2, sine is positive and cosine is negative: sin⁡(150∘)=12,cos⁡(150∘)=−32. \sin(150^\circ) = \frac{1}{2}, \qquad \cos(150^\circ) = -\frac{\sqrt{3}}{2}.
In triangle ABCABC, ∠A=30∘\angle A = 30^\circ, side a=6a = 6, and side b=63b = 6\sqrt{3}. Find sin⁡B\sin B.
By the Law of Sines: sin⁡Aa=sin⁡Bb  ⟹  1/26=sin⁡B63  ⟹  sin⁡B=6312=32. \frac{\sin A}{a} = \frac{\sin B}{b} \implies \frac{1/2}{6} = \frac{\sin B}{6\sqrt{3}} \implies \sin B = \frac{6\sqrt{3}}{12} = \frac{\sqrt{3}}{2}.
📝 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 =(x2−x1)2+(y2−y1)2 =\sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}} ; midpoint is the average of coordinates; gradient =ΔyΔx =\tfrac{\Delta y}{\Delta x} . A circle centred at the origin is x2+y2=r2 x^{2}+y^{2}=r^{2} . Interior angles of an n n -gon sum to (n−2)180∘ (n-2)180^{\circ} .

3-4-5 Right Triangle — Triangle CBA with sides 4, 5, 3 and angles 90°; right angle at CCBA45390°
Figure 1: Right-angled triangle with sides 3-4-5 illustrating the Pythagorean theorem and trigonometry ratios
Distance
Δx2+Δy2 \sqrt{\Delta x^{2}+\Delta y^{2}}
Midpoint
(x1+x22,y1+y22) \big(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2}\big)
Gradient
ΔyΔx \dfrac{\Delta y}{\Delta x}
Circle
x2+y2=r2 x^{2}+y^{2}=r^{2}
Worked example 5.A

Find the distance from (0,0) (0,0) to (3,4) (3,4) . 32+42=5. \sqrt{3^{2}+4^{2}}=5.

Worked example 5.B

Write the equation of a circle centred at the origin with radius 5 5 . x2+y2=25. x^{2}+y^{2}=25.

⚠️ Tahsili Exam Traps — Slope of Perpendicular Lines & Circle Equations

Perpendicular lines condition: The slopes satisfy m1⋅m2=−1m_1 \cdot m_2 = -1 (negative reciprocal, e.g., m=23  ⟹  m⊥=−32m = \frac{2}{3} \implies m_\perp = -\frac{3}{2}). A common mistake is flipping without changing the sign.

Circle equation radius: In (x−h)2+(y−k)2=R(x - h)^2 + (y - k)^2 = R, the radius is r=Rr = \sqrt{R}, not RR. If R=25R = 25, radius is 5.

Find the Euclidean distance between points A(1,−2)A(1, -2) and B(4,2)B(4, 2).
Distance formula: d=(4−1)2+(2−(−2))2=32+42=9+16=5. d = \sqrt{(4 - 1)^2 + (2 - (-2))^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5.
What is the center and radius of the circle defined by (x+3)2+(y−5)2=49(x + 3)^2 + (y - 5)^2 = 49?
Standard form (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2: Center is (−3,5)(-3, 5), and radius is r=49=7r = \sqrt{49} = 7.
📝 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

ddxxn=nxn−1 \dfrac{d}{dx}x^{n}=nx^{n-1} ; the derivative gives the slope of the tangent. Useful derivatives: ddxsin⁡x=cos⁡x \dfrac{d}{dx}\sin x=\cos x , ddxex=ex \dfrac{d}{dx}e^{x}=e^{x} . A limit of the form 00 \tfrac00 often factors and cancels.

Curve and Derivative Tangent — Coordinate plane with graph of f(x) = x³ - 3x-2-112-22xyf(x) = x³ − 3x
Figure 1: Curve f(x) = x³ - 3x showing tangent line at x = 1 and local extrema

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

⊙ Explorer · Secant → tangentinteractive
secant slope—

The secant slope is 2+h 2+h ; as h→0 h\to0 it tends to the tangent slope 2 2 (the dashed grey line).

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

Evaluate lim⁡x→3x2−9x−3 \displaystyle\lim_{x\to3}\frac{x^{2}-9}{x-3} . lim⁡x→3(x+3)=6. \lim_{x\to3}(x+3)=6.

Worked example 6.B

Differentiate 5x2−2x 5x^{2}-2x . 10x−2. 10x-2.

⚠️ Tahsili Exam Traps — Indeterminate Form 0/0 and Chain Rule

Direct substitution yielding 0/0: Getting 00\frac{0}{0} does not mean the limit does not exist; it is an indeterminate form. Factor numerator/denominator or apply L'Hôpital's rule immediately.

Chain rule omission: When differentiating f(g(x))f(g(x)), always remember to multiply by the inner derivative g′(x)g'(x). For example, ddx[(3x−1)4]=4(3x−1)3⋅3=12(3x−1)3\frac{d}{dx}[(3x - 1)^4] = 4(3x - 1)^3 \cdot 3 = 12(3x - 1)^3.

Evaluate the limit: lim⁡x→3x2−9x−3 \lim_{x \to 3} \frac{x^2 - 9}{x - 3}
Factoring difference of squares: lim⁡x→3(x−3)(x+3)x−3=lim⁡x→3(x+3)=3+3=6. \lim_{x \to 3} \frac{(x - 3)(x + 3)}{x - 3} = \lim_{x \to 3} (x + 3) = 3 + 3 = 6.
Find the slope of the tangent line to the curve y=2x3−5x+1y = 2x^3 - 5x + 1 at x=2x = 2.
Derivative: y′=6x2−5y' = 6x^2 - 5. Evaluate at x=2x = 2: m=6(2)2−5=24−5=19. m = 6(2)^2 - 5 = 24 - 5 = 19.
📝 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

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

Area Under Curve — Coordinate plane with graph of f(x) = 4 - x²-2-1121234xyf(x) = 4 − x²
Figure 1: Definite integral representing the geometric area bounded by f(x) = 4 - x² and the x-axis
Power rule
xn+1n+1+C \dfrac{x^{n+1}}{n+1}+C
Definite
F(b)−F(a) F(b)-F(a)
Constant
∫k dx=kx+C \displaystyle\int k\,dx=kx+C
Area
∫abf dx \displaystyle\int_a^b f\,dx
Worked example 7.A

Evaluate ∫023x2 dx \displaystyle\int_0^2 3x^{2}\,dx . [x3]02=8. \big[x^{3}\big]_0^2=8.

Worked example 7.B

Find the area under y=x y=x from 0 0 to 4 4 . ∫04x dx=[x22]04=8. \int_0^4 x\,dx=\Big[\tfrac{x^{2}}{2}\Big]_0^4=8.

⚠️ Tahsili Exam Traps — Constant of Integration & Limits Evaluation Order

Constant of integration +C: Indefinite integrals always require +C+ C. On Tahsili multiple choice questions, options often differ only by the presence or absence of +C+ C.

Definite integral order: Always evaluate the upper limit first minus the lower limit: ∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a). Reversing limits changes the sign: ∫abf(x) dx=−∫baf(x) dx\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx.

Evaluate the definite integral: ∫03(2x+1) dx \int_0^3 (2x + 1)\,dx
Antiderivative is x2+xx^2 + x. Evaluate limits: [x2+x]03=(32+3)−0=12. [x^2 + x]_0^3 = (3^2 + 3) - 0 = 12.
Find the area bounded by the curve y=3x2y = 3x^2, the x-axis, and the vertical lines x=1x = 1 and x=2x = 2.
Area integral: A=∫123x2 dx=[x3]12=23−13=8−1=7. A = \int_1^2 3x^2\,dx = [x^3]_1^2 = 2^3 - 1^3 = 8 - 1 = 7.
📝 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⁡bx=y \log_b x=y means by=x b^{y}=x ; log⁡(xy)=log⁡x+log⁡y \log(xy)=\log x+\log y ; log⁡(xn)=nlog⁡x \log(x^{n})=n\log x ; ln⁡e=1 \ln e=1 .

Exponential and Logarithmic Inverses — Coordinate plane with graph of y = 2^x-2-1123246xyy = 2ˣ
Figure 1: Graphs of inverse pair y = 2^x and y = log₂(x) reflected across line y = x
Definition
by=x b^{y}=x
Product
log⁡x+log⁡y \log x+\log y
Power
nlog⁡x n\log x
Natural
ln⁡e=1 \ln e=1
Worked example 8.A

Evaluate log⁡216 \log_2 16 . 24=16⇒4. 2^{4}=16\Rightarrow 4.

⚠️ Tahsili Exam Traps — Logarithm Properties & Negative Arguments

Logarithm argument must be strictly positive: log⁡b(x)\log_b(x) is only defined for x>0x > 0 and b>0,b≠1b > 0, b \neq 1. Expressions like log⁡2(−4)\log_2(-4) or log⁡2(0)\log_2(0) are undefined.

Common logarithmic laws misconception: log⁡(A+B)≠log⁡A+log⁡B\log(A + B) \neq \log A + \log B! The product rule is log⁡(A⋅B)=log⁡A+log⁡B\log(A \cdot B) = \log A + \log B.

Solve for xx: 32x−1=27 3^{2x - 1} = 27
Base 3 equality: 27=33  ⟹  2x−1=3  ⟹  2x=4  ⟹  x=227 = 3^3 \implies 2x - 1 = 3 \implies 2x = 4 \implies x = 2.
Find the value of: log⁡2(32)−log⁡3(81) \log_2(32) - \log_3(81)
Evaluate logarithms: log⁡2(32)=5\log_2(32) = 5 and log⁡3(81)=4\log_3(81) = 4. Subtract: 5−4=15 - 4 = 1.
📝 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 =favourabletotal =\tfrac{\text{favourable}}{\text{total}} ; independent events multiply.

Normal distribution bell curve (mean 100, sd 15): Normal Distribution showing shaded probability regionNormal Distribution557085100115130145
Figure 1: Standard Normal Distribution curve showing the Empirical Rule (68-95-99.7%)
Mean
∑xn \tfrac{\sum x}{n}
Median
middle value
Probability
favtotal \tfrac{\text{fav}}{\text{total}}
Independent
P(A)P(B) P(A)P(B)
Worked example 9.A

Find the probability of an even number on a fair die. 36=12. \frac{3}{6}=\frac12.

⚠️ Tahsili Exam Traps — Independent vs Mutually Exclusive Events

Mutually exclusive vs Independent: Mutually exclusive means P(A∩B)=0P(A \cap B) = 0, so P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B). Independent means the occurrence of one does not affect the other, so P(A∩B)=P(A)⋅P(B)P(A \cap B) = P(A) \cdot P(B).

Permutations vs Combinations: Use permutations nPrnPr when order matters (e.g., arrangements, positions); use combinations nCrnCr when order does not matter (e.g., committee selection).

A set of test scores follows a normal distribution with mean μ=80\mu = 80 and standard deviation σ=5\sigma = 5. What percentage of students scored between 75 and 85?
The range [75,85][75, 85] is [μ−σ,μ+σ][\mu - \sigma, \mu + \sigma]. By the Empirical Rule, approximately 68% of scores fall within one standard deviation.
In how many ways can a committee of 3 students be chosen from a group of 7 students?
Order does not matter, so use combinations: (73)=7×6×53×2×1=35 ways. \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \text{ ways}.
📝 Chapter Quiz
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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 3x-7=11 .

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

Solve x2=81 x^{2}=81 .

Basic
x=±9. x=\pm 9.
3

Solve x2−5x+6=0 x^{2}-5x+6=0 .

Intermediate

(x−2)(x−3)=0 (x-2)(x-3)=0 :

x=2, 3. x=2,\ 3.
4

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

Intermediate

6+3y=12 6+3y=12 :

y=2. y=2.
5

Solve ∣x∣=5 |x|=5 .

Intermediate
x=±5. x=\pm 5.
6

Factor x2−9 x^{2}-9 .

Intermediate
(x−3)(x+3). (x-3)(x+3).
7

Solve x2+2x−8=0 x^{2}+2x-8=0 .

Advanced

(x+4)(x−2)=0 (x+4)(x-2)=0 :

x=−4, 2. x=-4,\ 2.
8

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

Basic
2(3)+1=7. 2(3)+1=7.
9

State the domain of x−2 \sqrt{x-2} .

Intermediate
x≥2. x\ge 2.
10

If f(x)=x2 f(x)=x^{2} , find f(−3) f(-3) .

Intermediate
(−3)2=9. (-3)^{2}=9.
11

Solve 2x=8 2^{x}=8 .

Intermediate
x=3. x=3.
12

Evaluate log⁡216 \log_2 16 .

Intermediate

24=16 2^{4}=16 :

4. 4.
13

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

Intermediate
(2, 3). (2,\,3).
14

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

Advanced

Swap and solve: x=2y x=2y :

f−1(x)=x2. f^{-1}(x)=\tfrac{x}{2}.
15

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

Basic
14. 14.
16

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

Intermediate
3+9×2=21. 3+9\times2=21.
17

Find the common ratio of 2, 6, 18,… 2,\,6,\,18,\ldots

Intermediate
3. 3.
18

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

Intermediate
52(2⋅2+4⋅3)=52(16)=40. \tfrac{5}{2}\big(2\cdot2+4\cdot3\big)=\tfrac52(16)=40.
19

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

Intermediate
3⋅24=48. 3\cdot2^{4}=48.
20

Find the sum of the infinite geometric series a=4, r=12 a=4,\ r=\tfrac12 .

Advanced
41−12=8. \frac{4}{1-\tfrac12}=8.
21

Find 1+2+3+⋯+20 1+2+3+\cdots+20 .

Intermediate
20×212=210. \frac{20\times21}{2}=210.
22

Evaluate sin⁡30∘ \sin 30^{\circ} .

Basic
12. \tfrac12.
23

Evaluate cos⁡60∘ \cos 60^{\circ} .

Basic
12. \tfrac12.
24

Evaluate tan⁡45∘ \tan 45^{\circ} .

Intermediate
1. 1.
25

Convert 180∘ 180^{\circ} to radians.

Intermediate
π. \pi.
26

Simplify sin⁡2θ+cos⁡2θ \sin^{2}\theta+\cos^{2}\theta .

Intermediate
1. 1.
27

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

Intermediate

sin⁡θ=36=12 \sin\theta=\tfrac36=\tfrac12 :

θ=30∘. \theta=30^{\circ}.
28

Evaluate cos⁡90∘ \cos 90^{\circ} .

Advanced
0. 0.
29

State the period of sin⁡x \sin x .

Intermediate
2π. 2\pi.
30

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

Basic
180∘−110∘=70∘. 180^{\circ}-110^{\circ}=70^{\circ}.
31

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

Intermediate
3.14×25=78.5. 3.14\times25=78.5.
32

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

Intermediate
32+42=5. \sqrt{3^{2}+4^{2}}=5.
33

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

Intermediate
(4, 6). (4,\,6).
34

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

Intermediate
8−23−1=3. \frac{8-2}{3-1}=3.
35

Find the sum of the interior angles of a pentagon.

Intermediate
(5−2)×180∘=540∘. (5-2)\times180^{\circ}=540^{\circ}.
36

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

Advanced
x2+y2=25. x^{2}+y^{2}=25.
37

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

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

Evaluate lim⁡x→2(3x+1) \displaystyle\lim_{x\to2}(3x+1) .

Basic
3(2)+1=7. 3(2)+1=7.
39

Evaluate lim⁡x→3x2−9x−3 \displaystyle\lim_{x\to3}\frac{x^{2}-9}{x-3} .

Intermediate

Factor and cancel:

lim⁡x→3(x+3)=6. \lim_{x\to3}(x+3)=6.
40

Differentiate x3 x^{3} .

Basic
3x2. 3x^{2}.
41

Differentiate 5x2−2x 5x^{2}-2x .

Intermediate
10x−2. 10x-2.
42

Find the slope of the tangent to y=x2 y=x^{2} at x=2 x=2 .

Intermediate

y′=2x y'=2x , at x=2 x=2 :

4. 4.
43

Differentiate sin⁡x \sin x .

Intermediate
cos⁡x. \cos x.
44

Find the critical points of y=x3−3x y=x^{3}-3x .

Advanced

y′=3x2−3=0⇒x2=1 y'=3x^{2}-3=0 \Rightarrow x^{2}=1 :

x=±1. x=\pm 1.
45

Differentiate ex e^{x} .

Intermediate
ex. e^{x}.
46

Evaluate ∫2x dx \displaystyle\int 2x\,dx .

Basic
x2+C. x^{2}+C.
47

Evaluate ∫023x2 dx \displaystyle\int_0^2 3x^{2}\,dx .

Intermediate
[x3]02=8. \big[x^{3}\big]_0^2=8.
48

Evaluate ∫x2 dx \displaystyle\int x^{2}\,dx .

Intermediate
x33+C. \tfrac{x^{3}}{3}+C.
49

Evaluate ∫01(2x+1) dx \displaystyle\int_0^1 (2x+1)\,dx .

Intermediate
[x2+x]01=2. \big[x^{2}+x\big]_0^1=2.
50

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

Advanced
∫04x dx=[x22]04=8. \int_0^4 x\,dx=\Big[\tfrac{x^{2}}{2}\Big]_0^4=8.
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