Grade 13 — Term 1
17 study notes · 67 practice questions · Aligned with NIE syllabus
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This term in the Idasara Academy app has 17 lessons (NIE Units 21, 22, 23, 24). For each lesson, this page gives you the points you must know and one exam-style question with its answer. None of it needs an account.
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Lessons in This Term
The complex number system
What you must know
- Complex numbers CANNOT be ordered: Expressions like z1 > z2 or z1 < z2 have NO mathematical meaning in ℂ unless both numbers are strictly real.
- Square root of -a (for a > 0): √(-a) = i√a; be careful: √(-a) * √(-b) = (i√a)(i√b) = i^2 √(ab) = -√(ab), NOT +√(ab).
- Solving quadratic with negative discriminant: For ax^2 + bx + c = 0 with Δ = b^2 - 4ac < 0, roots are complex conjugates: x = [-b ± i√(4ac - b^2)] / (2a).
Try one question
What is the principal value of the imaginary unit expression i⁴⁷, where i = √(-1)?
- A. -i
- B. -1
- C. +1
- D. +i
Show the answer
Answer: A. -i. Since powers of i repeat in cycles of 4 (i¹ = i, i² = -1, i³ = -i, i⁴ = 1): 47 = 4 × 11 + 3. Therefore, i⁴⁷ = (i⁴)¹¹ · i³ = (1)¹¹ · (-i) = -i.
A model question written by Idasara on the exam pattern. It is not from a past paper.
Algebra of complex numbers
What you must know
- Denominator rationalization: Never leave an imaginary unit i in the denominator; multiply by complex conjugate z* to make denominator |z|^2.
- Sign rule for square roots: If b > 0 in √(a + ib), then x and y have the SAME sign; if b < 0, x and y have OPPOSITE signs.
- Multiplicative inverse: z^(-1) = 1/z = z* / |z|^2 = (x - iy) / (x^2 + y^2).
Try one question
What is the product of the two complex numbers z₁ = 2 + 3i and z₂ = 1 - 2i?
- A. -4 - i
- B. 8 + i
- C. 2 - 6i
- D. 8 - i
Show the answer
Answer: D. 8 - i. Multiplying directly: (2 + 3i)(1 - 2i) = 2(1) - 4i + 3i - 6i² = 2 - i - 6(-1) = 2 - i + 6 = 8 - i.
A model question written by Idasara on the exam pattern. It is not from a past paper.
Complex conjugate
What you must know
- Purely real condition via conjugate: z is purely real if and only if z = z*.
- Purely imaginary condition via conjugate: z is purely imaginary if and only if z = -z* (with z ≠ 0).
- Power of conjugate: (z^n)* = (z*)^n for any positive integer n.
Try one question
For any complex number z = x + iy with complex conjugate z̄ = x - iy, what is the value of the product z · z̄?
- A. 2iy
- B. 2x
- C. x² + y² (which equals |z|²)
- D. x² - y²
Show the answer
Answer: C. x² + y² (which equals |z|²). Computing: z · z̄ = (x + iy)(x - iy) = x² - (iy)² = x² - i² y² = x² - (-1)y² = x² + y². Since |z| = √(x² + y²), this is identical to |z|² (a purely real, non-negative quantity).
A model question written by Idasara on the exam pattern. It is not from a past paper.
Modulus of a complex number
What you must know
- Distance between two complex numbers: The Euclidean distance between points z1 and z2 in the Argand plane is |z1 - z2| = √[(x1 - x2)^2 + (y1 - y2)^2].
- Maximum and minimum values using Triangle Inequality: Given |z - z0| = r, the maximum value of |z| is |z0| + r and the minimum value of |z| is ||z0| - r|.
- Parallelogram identity in ℂ: |z1 + z2|^2 + |z1 - z2|^2 = 2(|z1|^2 + |z2|^2); sum of squares of diagonals of a parallelogram equals sum of squares of its four sides.
Try one question
What is the modulus of the complex number z = (3 + 4i) / (1 + i)?
- A. 25 / 2
- B. 5 / √2 (or 5√2 / 2)
- C. 7 / 2
- D. 5 / 2
Show the answer
Answer: B. 5 / √2 (or 5√2 / 2). Using the modulus quotient property |z₁ / z₂| = |z₁| / |z₂|: |3 + 4i| = √(3² + 4²) = √25 = 5. |1 + i| = √(1² + 1²) = √2. Thus, |z| = 5 / √2 = (5√2) / 2.
A model question written by Idasara on the exam pattern. It is not from a past paper.
The Argand diagram
What you must know
- Principal Argument Arg(z) Quadrant Rules: If x > 0: θ = arctan(y/x); If x < 0, y ≥ 0 (Quadrant II): θ = arctan(y/x) + π; If x < 0, y < 0 (Quadrant III): θ = arctan(y/x) - π; If x = 0, y > 0: θ = π/2; If x = 0, y < 0: θ = -π/2.
- Multiplication by i as a 90° rotation: Multiplying z by i rotates the vector OP counterclockwise by 90° (π/2 radians) without altering its modulus (|i*z| = |z|).
- Internal/external division in Argand plane: Point P dividing segment z1z2 in ratio m : n has complex number z = (m*z2 + n*z1) / (m + n).
Try one question
On an Argand diagram, multiplying a complex number z by the imaginary unit i represents which geometric transformation of the vector representing z?
- A. A clockwise rotation by 90° about the origin
- B. A doubling of the vector length along its original line
- C. An anticlockwise rotation by 90° (π/2 radians) about the origin without changing magnitude
- D. A reflection across the real horizontal axis
Show the answer
Answer: C. An anticlockwise rotation by 90° (π/2 radians) about the origin without changing magnitude. In polar/exponential form, i = cos(π/2) + i sin(π/2) = e^(iπ/2). Multiplying z = r e^(iθ) by i yields z · i = r e^(i(θ + π/2)), which rotates the position vector anticlockwise by π/2 (90°) about the origin with its modulus r unchanged.
A model question written by Idasara on the exam pattern. It is not from a past paper.
De Moivre's theorem
What you must know
- n-th roots of unity: Solutions to z^n = 1: z_k = cos(2kπ/n) + i sin(2kπ/n) = ω^k for k = 0, 1, ..., n - 1 where ω = e^(i 2π/n); sum of all n roots is 0: 1 + ω + ω^2 + ... + ω^(n-1) = 0.
- Cube roots of unity: 1, ω = (-1 + i√3)/2, and ω^2 = (-1 - i√3)/2; key identities: 1 + ω + ω^2 = 0 and ω^3 = 1.
- Factorization of x^n - 1: x^n - 1 = (x - 1) ∏_{k=1}^{n-1} (x - ω^k); for odd n, factors into real linear and quadratic factors.
Try one question
According to De Moivre's Theorem, what is the value of (cos θ + i sin θ)⁴?
- A. cos(4θ) + i sin(4θ)
- B. cos⁴ θ + i sin⁴ θ
- C. 4 cos θ + 4i sin θ
- D. cos(θ/4) + i sin(θ/4)
Show the answer
Answer: A. cos(4θ) + i sin(4θ). De Moivre's Theorem states that for any real θ and integer n: (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ). Setting n = 4 yields cos(4θ) + i sin(4θ).
A model question written by Idasara on the exam pattern. It is not from a past paper.
Loci and regions of a complex number
What you must know
- Circular arc locus: Arg[(z - z1)/(z - z2)] = α represents an arc of a circle passing through z1 and z2 subtending angle α; if α = ±π/2, the locus is a semicircle with diameter z1z2; if α = 0 or π, it represents the straight line through z1, z2 (excluding segment).
- Ellipse and Hyperbola loci: |z - z1| + |z - z2| = 2a (with 2a > |z1 - z2|) represents an Ellipse with foci z1, z2; ||z - z1| - |z - z2|| = 2a (with 2a < |z1 - z2|) represents a Hyperbola.
- Open vs closed boundary convention: When sketching inequalities such as |z - z0| < r or Arg(z) < α, use dashed lines for strict inequalities (<, >) and solid lines for non-strict inequalities (≤, ≥).
Try one question
In the complex plane, what geometric figure is described by the locus equation |z - (2 + 3i)| = 5?
- A. A straight line passing through (2, 3) with gradient 5
- B. A parabola with vertex at (2, 3)
- C. An ellipse with major axis 10
- D. A circle centered at (2, 3) with radius 5
Show the answer
Answer: D. A circle centered at (2, 3) with radius 5. The equation |z - z₀| = r represents the locus of points z whose distance from the fixed point z₀ = 2 + 3i is constant and equal to 5. In Cartesian coordinates (x - 2)² + (y - 3)² = 5² = 25, which is a circle with center (2, 3) and radius 5.
A model question written by Idasara on the exam pattern. It is not from a past paper.
Binomial expansion
What you must know
- Middle term determination: If n is even, there is ONE middle term at T_{n/2 + 1}; if n is odd, there are TWO middle terms at T_{(n+1)/2} and T_{(n+3)/2}.
- Greatest coefficient vs Greatest term: Greatest coefficient is always the middle coefficient nC_{n/2} (for even n); Greatest numerical term depends on values of a and b, found by solving T_{r+1} / T_r ≥ 1.
- Independent of x (constant) term: Set the overall power of x in T_{r+1} to 0 and solve for integer r (0 ≤ r ≤ n).
Try one question
In the binomial expansion of (a + b)ⁿ where n is a positive integer, what is the total number of terms in the expanded series?
- A. n
- B. 2ⁿ
- C. n - 1
- D. n + 1
Show the answer
Answer: D. n + 1. The Binomial Theorem gives (a + b)ⁿ = ∑_{r=0}^n ⁿC_r a^(n-r) b^r. Since the summation index r runs through all integer values from 0 up to n inclusive, there are exactly n + 1 terms in the expansion.
A model question written by Idasara on the exam pattern. It is not from a past paper.
Binomial theorem
What you must know
- Factoring for valid domain: To expand (a + bx)^n where |bx/a| < 1, factor out a^n: (a + bx)^n = a^n * [1 + (b/a)x]^n; expanding directly without factoring yields divergent nonsense.
- Calculating roots to decimal precision: To approximate √2, write √2 = √(49/25 * 50/49) = (7/5)√(1 + 1/49) = 1.4 * (1 + 1/49)^(1/2), where x = 1/49 is very small.
- Finite vs Infinite series distinction: When n is a positive integer, the series terminates after (n + 1) terms; when n is negative or fractional, the series has infinitely many terms and requires |x| < 1.
Try one question
What is the coefficient of x³ in the binomial expansion of (1 + 2x)⁵?
- A. 160
- B. 80
- C. 10
- D. 40
Show the answer
Answer: B. 80. The general term is T_{r+1} = ⁵C_r (1)^(5-r) (2x)^r = ⁵C_r · 2^r · x^r. For the term in x³, set r = 3: T₄ = ⁵C₃ · 2³ · x³ = [5! / (3! 2!)] · 8 · x³ = 10 · 8 · x³ = 80 x³. Thus, the coefficient is 80.
A model question written by Idasara on the exam pattern. It is not from a past paper.
Factorials
What you must know
- Why 0! = 1: From recurrence relation (n - 1)! = n! / n; substituting n = 1 gives 0! = 1! / 1 = 1.
- Factorial growth rate: Factorials grow faster than any exponential function a^n; n! >> 2^n for large n.
- Prime factorization of n! (Legendre's Formula): The exponent of prime p in n! is E_p(n!) = ⌊n/p⌋ + ⌊n/p^2⌋ + ⌊n/p^3⌋ + ...
Try one question
What is the exact numerical value of the factorial fraction 6! / (4! · 2!)?
- A. 20
- B. 15
- C. 30
- D. 12
Show the answer
Answer: B. 15. Expanding factorials: 6! / (4! · 2!) = (6 × 5 × 4!) / (4! × 2 × 1) = (6 × 5) / 2 = 30 / 2 = 15.
A model question written by Idasara on the exam pattern. It is not from a past paper.
Fundamental principle of counting
What you must know
- Restrictions must be filled FIRST: In constrained counting problems (e.g., forming even numbers, digits > 5000), always fill the constrained positions (unit digit, first digit) before filling unrestricted positions.
- With vs Without Repetition: Forming numbers/words with repetition allowed gives n^k possibilities; without repetition gives n*(n-1)*...*(n - k + 1) = nPk possibilities.
- Pigeonhole Principle: If n items are put into m containers and n > m, at least one container must contain more than one item.
Try one question
A student chooses an outfit consisting of one shirt and one pair of trousers. If there are 4 different shirts and 3 different pairs of trousers available, how many distinct outfits can be formed according to the Fundamental Principle of Counting?
- A. 12
- B. 10
- C. 7
- D. 24
Show the answer
Answer: A. 12. According to the Multiplication Principle of Counting, if a first task can be performed in m ways and a second independent task in n ways, the two tasks in succession can be performed in m × n ways: 4 × 3 = 12 distinct outfits.
A model question written by Idasara on the exam pattern. It is not from a past paper.
Permutations
What you must know
- Circular Permutations: Arranging n distinct objects in a circle has (n - 1)! ways (fixing one reference position); if clockwise and counterclockwise arrangements are indistinguishable (e.g., beads on a necklace), number of arrangements is (n - 1)! / 2.
- Complementary counting ('At least two together'): Total arrangements without restriction MINUS arrangements where no two are together (gap method).
- Rank of a word in dictionary: Form permutations alphabetically by determining how many words precede the given word starting with earlier letters.
Try one question
How many distinct 5-letter permutations can be formed using all the letters of the word 'MATHS' without repeating any letter?
- A. 24
- B. 120
- C. 60
- D. 720
Show the answer
Answer: B. 120. The word 'MATHS' consists of 5 distinct letters (M, A, T, H, S). The number of linear permutations of 5 distinct items is ⁵P₅ = 5! = 5 × 4 × 3 × 2 × 1 = 120.
A model question written by Idasara on the exam pattern. It is not from a past paper.
Combinations
What you must know
- Equality condition for combinations: If nCx = nCy, then either x = y or x + y = n.
- Division into groups: Number of ways to divide (m + n) distinct items into two groups of sizes m and n (m ≠ n) is (m+n)! / (m! n!); if group sizes are equal (m = n), divide by 2! to prevent overcounting.
- Geometric points configurations: From n points with no 3 collinear: Number of straight lines = nC2; Number of triangles = nC3; Number of diagonals of n-sided polygon = nC2 - n = n(n - 3)/2.
Try one question
From a group of 7 candidates, in how many distinct ways can a committee of 3 members be selected?
- A. 35
- B. 42
- C. 21
- D. 210
Show the answer
Answer: A. 35. Since the order of members within the committee does not matter, this is a combination problem: ⁷C₃ = 7! / (3! 4!) = (7 × 6 × 5) / (3 × 2 × 1) = 210 / 6 = 35 ways.
A model question written by Idasara on the exam pattern. It is not from a past paper.
Properties of matrices
What you must know
- Matrix multiplication compatibility condition: The product AB is defined if and only if number of columns in A equals number of rows in B; if A is m × p and B is p × n, then AB is m × n.
- Non-commutativity of matrix multiplication: In general, AB ≠ BA; matrix multiplication is associative (A(BC) = (AB)C) and distributive (A(B + C) = AB + AC), but strictly NON-COMMUTATIVE.
- Zero divisor property: AB = O does NOT imply A = O or B = O; non-zero matrices can multiply to produce the zero matrix.
Try one question
If matrix A has dimension 3 × 2 and matrix B has dimension 2 × 4, what is the dimension of the product matrix AB?
- A. Multiplication is not defined
- B. 3 × 4
- C. 4 × 3
- D. 2 × 2
Show the answer
Answer: B. 3 × 4. Matrix multiplication AB is defined because the number of columns of A (2) equals the number of rows of B (2). The resulting product matrix takes the number of rows of A and the number of columns of B, giving dimension 3 × 4.
A model question written by Idasara on the exam pattern. It is not from a past paper.
Special square matrices
What you must know
- Trace of a matrix: Sum of main diagonal elements: Tr(A) = a11 + a22 + ... + ann; Tr(AB) = Tr(BA).
- Triangular matrices: Upper Triangular matrix has all entries below main diagonal equal to zero (a_{ij} = 0 for i > j); Lower Triangular matrix has all entries above main diagonal zero.
- Symmetric and Skew-Symmetric definitions: A is Symmetric if A^T = A (a_{ij} = a_{ji}); A is Skew-Symmetric if A^T = -A (a_{ij} = -a_{ji}, implying all diagonal elements a_{ii} = 0).
Try one question
What is the determinant of the 2 × 2 matrix M = [[3, 2], [1, 4]]?
- A. 14
- B. 10
- C. 12
- D. 8
Show the answer
Answer: B. 10. For a 2 × 2 matrix [[a, b], [c, d]], the determinant is det(M) = ad - bc. Substituting: det(M) = (3)(4) - (2)(1) = 12 - 2 = 10.
A model question written by Idasara on the exam pattern. It is not from a past paper.
Transpose and inverse of a matrix
What you must know
- Singular vs Non-singular: If det(A) = 0, matrix A is Singular and has NO inverse; if det(A) ≠ 0, A is Non-singular and unique inverse exists.
- Order of product inverse: Crucial pitfall: (AB)^(-1) is B^(-1) * A^(-1), NEVER A^(-1) * B^(-1) due to non-commutativity.
- Orthogonal Matrix condition: A square matrix is Orthogonal if A^T * A = I, which implies A^(-1) = A^T and det(A) = ±1.
Try one question
What is the inverse matrix A⁻¹ of the 2 × 2 invertible matrix A = [[2, 1], [5, 3]]?
- A. [[2, -1], [-5, 3]]
- B. [[3, 1], [5, 2]]
- C. [[3, -1], [-5, 2]]
- D. [[-3, 1], [5, -2]]
Show the answer
Answer: C. [[3, -1], [-5, 2]]. Determinant det(A) = ad - bc = (2)(3) - (1)(5) = 6 - 5 = 1. The inverse of [[a, b], [c, d]] is (1/det(A)) [[d, -b], [-c, a]]. Substituting gives (1/1) [[3, -1], [-5, 2]] = [[3, -1], [-5, 2]].
A model question written by Idasara on the exam pattern. It is not from a past paper.
Simultaneous equations by matrices
What you must know
- Pre-multiplication requirement: In solving AX = B, you must PRE-MULTIPLY by A^(-1): A^(-1)(AX) = A^(-1)B; post-multiplying B*A^(-1) is undefined because dimensions do not match.
- Homogeneous linear systems: AX = O always has the trivial solution X = O (x = 0, y = 0); non-trivial (non-zero) solutions exist if and only if det(A) = 0.
- Consistency test: A linear system is Consistent if it possesses at least one solution; Inconsistent if no solution exists.
Try one question
Using matrix inversion to solve the simultaneous system: 2x + y = 7 and 5x + 3y = 19, what is the solution vector [x, y]ᵀ?
- A. x = 2, y = 3
- B. x = 1, y = 5
- C. x = 3, y = 1
- D. x = 4, y = -1
Show the answer
Answer: A. x = 2, y = 3. System is A X = B with A = [[2, 1], [5, 3]], X = [x, y]ᵀ, B = [7, 19]ᵀ. As det(A) = 6 - 5 = 1, A⁻¹ = [[3, -1], [-5, 2]]. Then X = A⁻¹ B = [[3, -1], [-5, 2]] [7, 19]ᵀ = [3(7) - 1(19), -5(7) + 2(19)]ᵀ = [21 - 19, -35 + 38]ᵀ = [2, 3]ᵀ. Thus, x = 2 and y = 3.
A model question written by Idasara on the exam pattern. It is not from a past paper.
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