Matric Mathematics Grade 12 Practice Questions 2026

Matric Mathematics is one of the most high-stakes NSC subjects — it unlocks university admission to engineering, actuarial science, computer science, and commerce programmes. Tour practice set covers all seven CAPS Grade 12 Mathematics topics across Paper 1 (algebra, functions, calculus, financial mathematics) and Paper 2 (analytical geometry, statistics, trigonometry, Euclidean geometry). Questions are written at NSC difficulty level with full working shown in every explanation.

Do not just check whether you got it right — read every explanation even for questions you answered correctly. Understanding the reasoning behind correct answers is what carries you through unfamiliar question variations in the actual exam.

Matric Mathematics consists of two 3-hour papers worth 150 marks each. Paper 1: algebra, functions, calculus, sequences. Paper 2: trigonometry, analytical geometry, statistics.

Priority: Functions and calculus together account for ~45% of Paper 1. Master these first.

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correct answers

Question 01 of 148

Derivative of f(x)=4x^3?

Explanation: Power rule: d/dx(4x^3)=3*4x^2=12x^2.

Question 02 of 148

Factorise: x^2-5x+6

Explanation: Find numbers multiplying to 6, adding to -5: -2 and -3. So (x-2)(x-3).

Question 03 of 148

sin(30 degrees)?

Explanation: sin(30)=1/2. Key: sin(30)=1/2, sin(45)=sqrt(2)/2, sin(60)=sqrt(3)/2.

Question 04 of 148

Midpoint of (2,4) and (6,10)?

Explanation: Midpoint=((2+6)/2,(4+10)/2)=(4,7).

Question 05 of 148

Solve: x^2-9=0

Explanation: x^2=9, x=+/-3. Both solutions required.

Question 06 of 148

log_10(1000)?

Explanation: 10^3=1000, so log_10(1000)=3.

Question 07 of 148

Common ratio in: 3,6,12,24?

Explanation: r=6/3=2. Each term multiplied by 2.

Question 08 of 148

Gradient of line y=3x-7?

Explanation: y=mx+c format. Gradient m=3, y-intercept c=-7.

Question 09 of 148

Discriminant b^2-4ac<0 means?

Explanation: Negative discriminant: roots are complex (non-real). Parabola does not cross x-axis.

Question 10 of 148

Integral of 3x^2?

Explanation: Power rule: integral of 3x^2=3*(x^3/3)+C=x^3+C.

Question 11 of 148

Turning point of f(x)=x^2-4x+1?

Explanation: x=-b/2a=4/2=2. f(2)=4-8+1=-3. Turning point: (2,-3).

Question 12 of 148

Period of y=cos(3x)?

Explanation: Period=360/b=360/3=120 degrees.

Question 13 of 148

Sum of first 10 terms: a=2, d=4?

Explanation: Sn=n/2*(2a+(n-1)d)=10/2*(4+36)=5*40=200.

Question 14 of 148

Simplify: x^5/x^2?

Explanation: Dividing same base: subtract exponents. x^5/x^2=x^3.

Question 15 of 148

Perpendicular gradient to slope -2?

Explanation: Perpendicular: m1*m2=-1. (-2)*m2=-1, m2=1/2.

Question 16 of 148

Amplitude of y=-4cos(x)?

Explanation: Amplitude=|A|=|-4|=4. Always positive.

Question 17 of 148

P(rolling prime on fair die)?

Explanation: Primes on die: 2,3,5. P=3/6=1/2.

Question 18 of 148

Circle equation: centre (2,3) radius 5?

Explanation: Standard form: (x-a)^2+(y-b)^2=r^2. So (x-2)^2+(y-3)^2=25.

Question 19 of 148

Solve simultaneously: y=x+2, y=2x-1?

Explanation: Set equal: x+2=2x-1. 3=x. y=3+2=5.

Question 20 of 148

tan(45 degrees)?

Explanation: tan(45)=sin(45)/cos(45)=1.

Question 21 of 148

Simplify: 2^3*2^4?

Explanation: Same base: add exponents. 2^3*2^4=2^7=128.

Question 22 of 148

Domain of f(x)=sqrt(x)?

Explanation: sqrt(x) requires x>=0. Domain: x>=0.

Question 23 of 148

Rearrange y=5x+10 for x?

Explanation: 5x=y-10, x=(y-10)/5.

Question 24 of 148

f'(x) represents?

Explanation: The derivative f'(x) gives the gradient (slope) of the tangent at any point x.

Question 25 of 148

Solve: 3^x=81?

Explanation: 81=3^4. So 3^x=3^4, x=4.

Question 26 of 148

y-intercept of y=2x^2-3x+5?

Explanation: y-intercept: set x=0. y=0-0+5=5.

Question 27 of 148

P(A and B) if independent: P(A)=0.4, P(B)=0.5?

Explanation: Independent: P(A and B)=P(A)*P(B)=0.4*0.5=0.2.

Question 28 of 148

Pythagorean identity?

Explanation: sin^2(x)+cos^2(x)=1 for ALL values of x. Fundamental identity.

Question 29 of 148

Area of triangle: base 8, height 5?

Explanation: Area=0.5*base*height=0.5*8*5=20.

Question 30 of 148

log_a(a^7)?

Explanation: log_a(a^n)=n. So log_a(a^7)=7. Inverse property.

Question 31 of 148

Domain of f(x)=1/x?

Explanation: f(x)=1/x undefined at x=0. Domain: all reals except x=0.

Question 32 of 148

Compound interest: P=1000, r=10%, n=3?

Explanation: A=P(1+r)^n=1000(1.1)^3=1000*1.331=1331.

Question 33 of 148

Distance formula between (x1,y1) and (x2,y2)?

Explanation: d=sqrt((x2-x1)^2+(y2-y1)^2). Square the differences, add, take root.

Question 34 of 148

Sum to infinity: a=4, r=0.5?

Explanation: S_inf=a/(1-r)=4/(1-0.5)=4/0.5=8.

Question 35 of 148

Gradient through (1,2) and (3,8)?

Explanation: m=(8-2)/(3-1)=6/2=3.

Question 36 of 148

Vertical asymptote of f(x)=1/(x-3)?

Explanation: f undefined when x-3=0, so x=3 is the vertical asymptote.

Question 37 of 148

Tn=2n-1 gives what sequence?

Explanation: T1=1, T2=3, T3=5, T4=7. These are odd numbers.

Question 38 of 148

Max value of f(x)=-x^2+4x-1?

Explanation: x=-b/2a=4/2=2. f(2)=-4+8-1=3. Maximum at turning point.

Question 39 of 148

Simplify: sqrt(50)?

Explanation: sqrt(50)=sqrt(25*2)=5sqrt(2).

Question 40 of 148

P(not rolling a 6) on fair die?

Explanation: P(not 6)=1-P(6)=1-1/6=5/6. Complementary probability.

Question 41 of 148

Angle in a semicircle is?

Explanation: Angle subtended by diameter at circumference is always 90 degrees (Thales' theorem).

Question 42 of 148

Rate of change interpretation of f'(x)?

Explanation: f'(x) = instantaneous rate of change = slope of tangent at point x.

Question 43 of 148

If f(x)=3x-2, find f(f(2))?

Explanation: f(2)=3(2)-2=4. f(f(2))=f(4)=3(4)-2=10.

Question 44 of 148

Simplify (2^4)^3?

Explanation: Power of power: multiply exponents. (2^4)^3=2^12.

Question 45 of 148

Mode of: 3,5,5,7,8,5,3?

Explanation: Mode = most frequent value. 5 appears 3 times.

Question 46 of 148

Percentage decrease from 200 to 150?

Explanation: % decrease=(50/200)*100=25%.

Question 47 of 148

Next term: 2,6,18,54,?

Explanation: Geometric: each term*3. 54*3=162.

Question 48 of 148

Sum of interior angles of pentagon?

Explanation: Sum=(n-2)*180=(5-2)*180=540 degrees.

Question 49 of 148

P^2+Q^2 if P=3 and Q=4?

Explanation: P^2+Q^2=9+16=25. Classic 3-4-5 triple.

Question 50 of 148

Compound interest formula?

Explanation: Compound interest: A=P(1+r)^n where P=principal, r=rate per period, n=periods.

Question 51 of 148

Solve the quadratic: 2x^2-8=0

Explanation: 2x^2=8, x^2=4, x=+/-2.

Question 52 of 148

Simplify: (3x^2)(4x^3)?

Explanation: Multiply coefficients: 3*4=12. Add exponents: x^2*x^3=x^5. Answer: 12x^5.

Question 53 of 148

What is the value of tan(60 degrees)?

Explanation: tan(60)=sin(60)/cos(60)=(sqrt(3)/2)/(1/2)=sqrt(3).

Question 54 of 148

Distance from (1,2) to (4,6)?

Explanation: d=sqrt((4-1)^2+(6-2)^2)=sqrt(9+16)=sqrt(25)=5.

Question 55 of 148

If f(x)=x^2-4, solve f(x)=0?

Explanation: x^2=4, x=+/-2.

Question 56 of 148

Derivative of f(x)=sin(x)?

Explanation: d/dx(sin x)=cos x. d/dx(cos x)=-sin x.

Question 57 of 148

What is the sum of the first n natural numbers?

Explanation: Sum=1+2+...+n=n(n+1)/2. Example: 1+2+3+4+5=15=5(6)/2.

Question 58 of 148

Which function has a maximum value?

Explanation: f(x)=-x^2+4 has negative leading coefficient: parabola opens downward, so has maximum.

Question 59 of 148

What is the effect of parameter b in y=sin(bx) if b>1?

Explanation: Period=360/b. If b>1, period decreases (graph compressed horizontally).

Question 60 of 148

Simplify: log_2(8)+log_2(4)?

Explanation: log_2(8)=3, log_2(4)=2. Sum=5. Or: log_2(32)=5.

Question 61 of 148

The straight line y=mx+c: what does c represent?

Explanation: In y=mx+c: m=gradient, c=y-intercept (where line crosses the y-axis).

Question 62 of 148

Factorise: 2x^2+5x+3?

Explanation: 2x^2+5x+3: factors are (2x+3)(x+1). Check: 2x^2+2x+3x+3=2x^2+5x+3.

Question 63 of 148

What is e (Euler's number) approximately?

Explanation: e≈2.718... Used in natural logarithms (ln) and exponential growth.

Question 64 of 148

The derivative of a constant is?

Explanation: d/dx(constant)=0. The gradient of a horizontal line is zero.

Question 65 of 148

Sum formula for geometric series to infinity requires |r|<1. Why?

Explanation: If |r|<1 each term is smaller than the previous, so the sum converges. If |r|>=1 it diverges to infinity.

Question 66 of 148

Solve: 2sin(x)=1 for 0<=x<=360?

Explanation: sin(x)=1/2. x=30 (first quad) and x=180-30=150 (second quad).

Question 67 of 148

What is standard form of a parabola opening upward?

Explanation: Parabola opening upward: a>0. Opening downward: a<0.

Question 68 of 148

A data set has mean 50 and standard deviation 10. A score of 70 is how many standard deviations above the mean?

Explanation: (70-50)/10=20/10=2 standard deviations above mean.

Question 69 of 148

Integration is the reverse of?

Explanation: Integration is the reverse (anti-derivative) of differentiation.

Question 70 of 148

What does the symbol sigma (Σ) represent in statistics?

Explanation: Sigma (Σ) represents summation — the sum of a series of values.

Question 71 of 148

What is a function?

Explanation: A function: each x-value (input) maps to exactly one y-value (output). Passes vertical line test.

Question 72 of 148

Vertical line test determines?

Explanation: Vertical line test: if any vertical line crosses the graph more than once, it is not a function.

Question 73 of 148

What is the x-coordinate of the vertex of y=2(x-3)^2+5?

Explanation: Vertex form y=a(x-h)^2+k: vertex at (h,k). Here h=3, so x-coordinate of vertex is 3.

Question 74 of 148

Simplify: (a^3)^4?

Explanation: Power of power: multiply exponents. (a^3)^4=a^12.

Question 75 of 148

What is the gradient of a line parallel to y=5x-3?

Explanation: Parallel lines have equal gradients. Gradient of y=5x-3 is 5.

Question 76 of 148

Radian measure: 180 degrees in radians?

Explanation: 180 degrees = pi radians. 360 degrees = 2pi radians.

Question 77 of 148

Probability: P(A and B)=0.12, P(A)=0.4. Find P(B|A)?

Explanation: P(B|A)=P(A and B)/P(A)=0.12/0.4=0.3.

Question 78 of 148

What is the range of the function y=-x^2+6x-5?

Explanation: Max at x=3: y=-9+18-5=4. Since parabola opens down, range is y<=4.

Question 79 of 148

Solve: |x-3|=5?

Explanation: |x-3|=5 means x-3=5 (x=8) or x-3=-5 (x=-2).

Question 80 of 148

What is an asymptote?

Explanation: Asymptote: line graph approaches but never reaches. Hyperbola y=1/x has x=0 and y=0 as asymptotes.

Question 81 of 148

Data: 3,5,7,9,11. What is the standard deviation?

Explanation: Mean=7. Deviations^2: 16,4,0,4,16. Variance=40/5=8. SD=sqrt(8)≈2.83.

Question 82 of 148

What is the effect of parameter k in y=f(x)+k?

Explanation: y=f(x)+k shifts the graph k units up (k>0) or down (k<0). Vertical translation.

Question 83 of 148

The equation x^2+y^2+4x-6y+4=0 represents a circle. Find its centre?

Explanation: Complete the square: (x+2)^2-4+(y-3)^2-9+4=0. Centre (-2,3), radius=3.

Question 84 of 148

What is the slope of a horizontal line?

Explanation: Horizontal line: y=constant, Δy=0. Slope=0/Δx=0.

Question 85 of 148

What is the slope of a vertical line?

Explanation: Vertical line: x=constant, Δx=0. Slope=Δy/0 = undefined (division by zero).

Question 86 of 148

Solve for x: log_3(x)=4?

Explanation: log_3(x)=4 means 3^4=x=81.

Question 87 of 148

What is a geometric mean of 4 and 16?

Explanation: Geometric mean=sqrt(4*16)=sqrt(64)=8.

Question 88 of 148

What is a linear function?

Explanation: Linear function: f(x)=mx+c, gives a straight line, constant gradient.

Question 89 of 148

Simplify: (x+3)^2?

Explanation: (x+3)^2=x^2+2(3)x+3^2=x^2+6x+9.

Question 90 of 148

What is the sum of exterior angles of any polygon?

Explanation: Sum of exterior angles of ANY convex polygon = 360 degrees.

Question 91 of 148

Area of a trapezium with parallel sides 5 and 9, height 4?

Explanation: Area=0.5*(a+b)*h=0.5*(5+9)*4=0.5*14*4=28.

Question 92 of 148

What is exponential decay?

Explanation: Exponential decay: y=a*b^x where 0

Question 93 of 148

Solve: 2^(x+1)=16?

Explanation: 16=2^4. So 2^(x+1)=2^4, x+1=4, x=3.

Question 94 of 148

What is the formula for the nth term of a quadratic sequence?

Explanation: Quadratic sequences have second differences constant: Tn=an^2+bn+c.

Question 95 of 148

If median>mean, the distribution is?

Explanation: If median>mean: distribution is negatively (left) skewed — tail extends left.

Question 96 of 148

What does the area under a velocity-time graph represent?

Explanation: Area under velocity-time graph = displacement. Gradient of velocity-time = acceleration.

Question 97 of 148

Simplify: (2x-1)(3x+4)?

Explanation: FOIL: 6x^2+8x-3x-4=6x^2+5x-4.

Question 98 of 148

What is the turning point of y=-(x-2)^2+7?

Explanation: Vertex form y=a(x-h)^2+k: vertex at (h,k)=(2,7). Maximum since a=-1<0.

Question 99 of 148

What is a rational number?

Explanation: Rational: can be written as a fraction p/q. Includes integers, fractions, terminating and recurring decimals.

Question 100 of 148

What is an irrational number?

Explanation: Irrational: cannot be expressed as p/q. Decimal is non-terminating and non-recurring.

Question 101 of 148

What is the cosine rule used for?

Explanation: Cosine rule: a^2=b^2+c^2-2bc*cos(A). Use when SAS (two sides, included angle) or SSS (three sides) known.

Question 102 of 148

What is the binomial theorem?

Explanation: Binomial theorem: (a+b)ⁿ = Σ C(n,r)aⁿ⁻ʳbʳ. C(n,r)=n!/(r!(n-r)!). E.g. (a+b)²=a²+2ab+b².

Question 103 of 148

What is the domain of f(x) = log(x-3)?

Explanation: log(x-3): argument x-3 must be >0. So x>3. Domain: (3,∞).

Question 104 of 148

If sin(x)=3/5 and x is in quadrant 2, what is cos(x)?

Explanation: sin²+cos²=1. cos²=1-9/25=16/25. cos=±4/5. In Q2, cos is negative: cos=-4/5.

Question 105 of 148

Derivative of f(x)=e^x?

Explanation: d/dx(eˣ)=eˣ. The exponential function is its own derivative.

Question 106 of 148

What is the integral of 1/x?

Explanation: ∫(1/x)dx = ln|x| + C. Tour is the integral rule for x⁻¹ (power rule doesn't apply when n=-1).

Question 107 of 148

What is the limit of (sin x)/x as x→0?

Explanation: lim(x→0) sin(x)/x = 1. Fundamental trigonometric limit. Proved geometrically or via L'Hôpital's rule.

Question 108 of 148

What is L'Hôpital's rule used for?

Explanation: L'Hôpital: if lim f(x)/g(x) is 0/0 or ∞/∞, then = lim f'(x)/g'(x). Apply until determinate form.

Question 109 of 148

The point (3,4) lies on a circle with centre (0,0). Equation of tangent at that point?

Explanation: Radius vector (0,0)→(3,4) has gradient 4/3. Tangent is perpendicular: gradient=-3/4. Through (3,4): y-4=-3/4(x-3) → 4y-16=-3x+9 → 3x+4y=25.

Question 110 of 148

What is the chain rule?

Explanation: Chain rule: derivative of composite function. d/dx[sin(x²)]=cos(x²)×2x. 'Derivative of outside × derivative of inside.'

Question 111 of 148

What is the product rule?

Explanation: Product rule: d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x). 'First × derivative of second + second × derivative of first.'

Question 112 of 148

What is the quotient rule?

Explanation: Quotient rule: d/dx[f/g] = (gf' - fg')/g². 'Low d-high minus high d-low over low-squared.'

Question 113 of 148

What is the second derivative test?

Explanation: Second derivative test at f'(c)=0: if f''(c)>0 → concave up → local minimum. If f''(c)<0 → concave down → local maximum.

Question 114 of 148

What is a point of inflection?

Explanation: Inflection point: f''(c)=0 and f'' changes sign. Not all f''=0 points are inflections (need sign change of f'').

Question 115 of 148

What is the fundamental theorem of calculus?

Explanation: FTC: area under f(x) from a to b = F(b)-F(a). Also: d/dx[∫ₐˣf(t)dt] = f(x). Integration is antidifferentiation.

Question 116 of 148

What is Euler's identity?

Explanation: Euler's identity: e^(iπ)+1=0. Called the most beautiful equation — connects e, i, π, 1, and 0.

Question 117 of 148

Solve: e^(2x) = 7?

Explanation: e^(2x)=7 → 2x=ln7 → x=ln7/2.

Question 118 of 148

What is a vector in 2D?

Explanation: 2D vector: (3,4) or 3i+4j. Magnitude=√(3²+4²)=5. Direction: tan⁻¹(4/3).

Question 119 of 148

What is the dot product of vectors a=(2,3) and b=(4,-1)?

Explanation: Dot product: a·b=2×4+3×(-1)=8-3=5. Scalar result. a·b=|a||b|cosθ.

Question 120 of 148

What is a geometric sequence with T1=2 and T4=54?

Explanation: T4=T1×r³. 54=2×r³. r³=27. r=3.

Question 121 of 148

What is the sum formula for a geometric series when r=1?

Explanation: When r=1: all terms equal a₁. Sn=n×a₁. The standard formula Sn=a(rⁿ-1)/(r-1) is undefined at r=1.

Question 122 of 148

Solve the system: 2x+y=7 and x-y=2?

Explanation: Add equations: 3x=9, x=3. y=7-2(3)=1. Check: 3-1=2 ✓

Question 123 of 148

What is the cosine of 180°?

Explanation: cos(180°)=-1. Key values: cos(0°)=1, cos(90°)=0, cos(180°)=-1, cos(270°)=0, cos(360°)=1.

Question 124 of 148

What is the sine rule for area of triangle?

Explanation: Area=½ab·sin(C). Use when you know 2 sides and included angle. Equivalent to ½×base×height.

Question 125 of 148

What is an arithmetic mean?

Explanation: Arithmetic mean: x̄=Σx/n. Sum all values, divide by number of values. Sensitive to outliers.

Question 126 of 148

What does a negative correlation coefficient indicate?

Explanation: Negative r (between -1 and 0): as x increases, y decreases. r=-1 perfect negative linear relationship.

Question 127 of 148

What is interpolation vs extrapolation?

Explanation: Interpolation: within known data range = more reliable. Extrapolation: beyond data range = increasingly unreliable.

Question 128 of 148

What is a percentile?

Explanation: Percentile: 50th percentile = median. 25th = Q1 (lower quartile). 75th = Q3 (upper quartile). IQR=Q3-Q1.

Question 129 of 148

What is an outlier's effect on mean vs median?

Explanation: Mean: pulled toward outlier. Median: resistant (just shifts one value slightly). Use median when outliers present.

Question 130 of 148

Simplify: sin²(x)/cos²(x)+sin²(x)/cos²(x)?

Explanation: sin²(x)/cos²(x)=tan²(x). Two of them: 2tan²(x).

Question 131 of 148

What is the general solution of tan(x)=1?

Explanation: tan has period 180°. tan(45°)=1. General: x=45°+n×180° (n∈ℤ).

Question 132 of 148

What is an irrational number example relevant to maths?

Explanation: √2≈1.41421... is irrational — non-terminating, non-repeating decimal. Cannot be expressed as p/q.

Question 133 of 148

What is the modulus/absolute value of -7?

Explanation: |-7|=7. Absolute value: distance from zero on number line. Always non-negative.

Question 134 of 148

What is the highest degree in f(x)=3x⁴-2x³+x-5?

Explanation: Degree of polynomial = highest power. 3x⁴ has power 4. Degree=4.

Question 135 of 148

What does the y-intercept tell you in real-world context?

Explanation: Y-intercept: value when x=0. In cost function C=50x+200, the y-intercept 200=fixed cost (even if 0 items produced).

Question 136 of 148

What is a rational function?

Explanation: Rational function: ratio of polynomials. f(x)=1/(x-2) has VA at x=2, HA at y=0. Domain excludes zeros of denominator.

Question 137 of 148

What is the difference between direct and inverse proportion?

Explanation: Direct: y/x=k (constant). Double x → double y. Inverse: xy=k. Double x → half y.

Question 138 of 148

What is the Fibonacci sequence formula?

Explanation: Fibonacci: 1,1,2,3,5,8,13... Each term is sum of two preceding. Appears in nature (shells, flowers, spirals).

Question 139 of 148

What is the number e (Euler's number)?

Explanation: e≈2.71828... Base of natural logarithm. Limit of (1+1/n)ⁿ as n→∞. Used in growth/decay models.

Question 140 of 148

What is ln(e)?

Explanation: ln(e)=1. Natural log is the inverse of eˣ. ln(eˣ)=x. Also ln(1)=0, ln(e²)=2.

Question 141 of 148

Factorise: a³+b³?

Explanation: Sum of cubes: a³+b³=(a+b)(a²-ab+b²). Difference of cubes: a³-b³=(a-b)(a²+ab+b²).

Question 142 of 148

What is the discriminant's role in determining nature of roots?

Explanation: Discriminant Δ=b²-4ac. For ax²+bx+c=0. Δ>0: 2 real distinct. Δ=0: 1 repeated. Δ<0: no real roots.

Question 143 of 148

What is the mean of the data set 2,4,4,4,5,5,7,9?

Explanation: Sum=2+4+4+4+5+5+7+9=40. Mean=40/8=5.

Question 144 of 148

What is the median of 2,4,4,4,5,5,7,9?

Explanation: 8 values (even): median=average of 4th and 5th = (4+5)/2=4.5.

Question 145 of 148

What does the standard deviation of 0 mean?

Explanation: SD=0: every data point equals the mean. No variability. E.g., data set {5,5,5,5} has SD=0.

Question 146 of 148

What is conditional probability P(A|B)?

Explanation: P(A|B)=P(A∩B)/P(B). Restricts sample space to outcomes where B occurred. Bayes' theorem: P(A|B)=P(B|A)P(A)/P(B).

Question 147 of 148

What is Bayes' theorem?

Explanation: Bayes': P(A|B)=P(B|A)×P(A)/P(B). Updates prior probability P(A) with likelihood P(B|A) to get posterior P(A|B).

Question 148 of 148

What is the remainder theorem?

Explanation: Remainder theorem: p(x)÷(x-a) → remainder = p(a). If p(a)=0, then (x-a) is a factor (factor theorem).

Question 149 of 150

What is the value of sin(30°)?

Explanation: sin(30°) = 0.5 (or 1/2). Tour is one of the special angles to memorise. Triangle: 30°-60°-90°, sides in ratio 1:√3:2. sin(30°)=1/2, cos(30°)=√3/2, tan(30°)=1/√3.

Question 150 of 150

Simplify: (x²-9)/(x-3)

Explanation: x²-9 = (x-3)(x+3). Dividing by (x-3) cancels that factor (x≠3): result is x+3. Tour is the difference of squares factorisation.

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DBE Past Papers - Mathematics

Official NSC Mathematics past papers and memos from 2014 onwards. Free download.

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Khan Academy - Mathematics

Free video lessons on every Grade 12 Mathematics topic with exercises.

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Siyavula Mathematics Grade 12

Free CAPS-aligned textbook with worked examples and exercises.

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Mind the Gap Mathematics

Official SA government study guide for Grade 12 Mathematics. Free PDF.

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e-classroom

SA educational resource with exam papers, summaries, and revision videos.

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Desmos Graphing Calculator

Free online graphing calculator to visualise functions and explore calculus.

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The quadratic formula solves ax² + bx + c = 0: x = [−b ± √(b² − 4ac)] ÷ 2a. The discriminant (b² − 4ac) determines the nature of roots: positive = two real roots, zero = one repeated root, negative = no real roots (complex roots). Always check the discriminant before solving.

An arithmetic sequence has a constant difference (common difference d) between consecutive terms. Example: 3, 7, 11, 15 (d=4). nth term: a + (n−1)d. A geometric sequence has a constant ratio (r) between consecutive terms. Example: 2, 6, 18, 54 (r=3). nth term: a·r^(n−1). Sum to infinity of geometric series: a÷(1−r) when |r|<1.

Differential calculus studies the rate of change of functions. The derivative f'(x) gives the instantaneous rate of change (gradient of the tangent) at any point. Rules: power rule (d/dx[xⁿ] = nxⁿ⁻¹), product rule, chain rule, quotient rule. Setting f'(x) = 0 finds turning points; f''(x) determines if they are maxima or minima.

The Remainder Theorem states that when a polynomial f(x) is divided by (x − a), the remainder equals f(a). If f(a) = 0, then (x − a) is a factor of f(x) — tour is the Factor Theorem. Use these together to factorise cubic polynomials by finding a root by inspection first.

Method 1 (Substitution): Make one variable the subject in one equation, substitute into the other. Method 2 (Elimination): Multiply equations to make coefficients equal, then add or subtract to eliminate one variable. For non-linear simultaneous equations (one linear, one quadratic): use substitution — substitute the linear equation into the quadratic.

For a right-angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. SOHCAHTOA is the mnemonic. The Pythagorean identity: sin²θ + cos²θ = 1. Key special angles: sin 30° = 0.5, cos 60° = 0.5, tan 45° = 1, sin 90° = 1, cos 0° = 1.

sin(A+B) = sinA·cosB + cosA·sinB. cos(A+B) = cosA·cosB − sinA·sinB. Double angle: sin 2A = 2 sinA cosA. cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A. These are used to simplify expressions and prove identities.

Area = ½ ab sin C, where a and b are two sides and C is the included angle between them. Tour applies to any triangle, not just right-angled ones. The Sine Rule: a/sinA = b/sinB = c/sinC (for finding unknown sides/angles when you have a side-angle pair). The Cosine Rule: c² = a² + b² − 2ab cosC (for when you have two sides and the included angle, or all three sides).

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