CQ1. A surveyor marks two posts at A(2, −1) and B(6, 2) on a map grid. Using the distance formula, how far apart are the posts (in map units)?
Answer: (B) 5
CQ2. A bridge is planned midway between towns P(4, −1) and Q(7, 2). What are the mid-point coordinates of the bridge site?
Answer: (A) (11/2, 1/2)
CQ3. A ramp runs from floor point A(2, 3) to platform B(6, −7) in a coordinate diagram (y vertical). What is the slope of the ramp?
Answer: (A) −5/2
CQ4. A straight road meets the axes at intercepts given by 3x + 2y − 12 = 0. What are the x- and y-intercepts?
Answer: (D) (4 and 6)
CQ5. A circular park has equation x² + y² − 6x − 8y = 0. What is the radius of the park boundary?
Answer: (C) 5
CQ6. A parabolic arch is modelled by y² = 12x (units metres). Where is the focus of this parabola?
Answer: (A) (3, 0)
CQ7. In a stock system, A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]]. What is the product AB?
Answer: (A) [[4, 6], [10, 12]]
CQ8. A parallelogram is spanned by column vectors of [[2, 1], [0, 3]]. What is the absolute area ( |det| )?
Answer: (C) 6
CQ9. For the linear system AX = B with A = [[1, 2], [2, 4]], why is there no unique solution via A⁻¹?
Answer: (A) det A = 0 (singular)
CQ10. A coding machine uses f(x) = 2x + 1 on real inputs. Is f one-one (injective)?
Answer: (A) Yes
CQ11. A robot arm angle sensor reports sin θ = 1/2 and returns the principal arcsine. What angle (radians) does sin⁻¹(1/2) give?
Answer: (A) π/6
CQ12. Near a camera pivot, a small angle x (radians) satisfies lim_{x→0} (sin 5x)/x. What is this limit?
Answer: (B) 5
CQ13. The side of a square grows as s = t³ (metres). Instantaneous rate of change of side at general t is ds/dt. What is d/dt (t³)?
Answer: (A) 3t²
CQ14. A pendulum angle (radians) is θ = sin 3t. What is dθ/dt?
Answer: (A) 3 cos 3t
CQ15. A bacterial count is modelled by N = e^{2t}. What is dN/dt?
Answer: (A) 2e^{2t}
CQ16. A path y = x² is lit at the point (1, 1). What is the slope of the tangent line there?
Answer: (B) 2
CQ17. Accumulated water volume increases at rate x³ (arbitrary units). An antiderivative of x³ is:
Answer: (A) x⁴/4 + C
CQ18. Net change of a quantity with rate 2x from x = 0 to x = 1 is ∫₀¹ 2x dx. Its value is:
Answer: (B) 1
CQ19. Two drones are at (3, 5, −1) and (9, 2, −4). What is the straight-line distance between them?
Answer: (A) √54 = 3√6
CQ20. Forces represented by vectors a = i and b = j act at a joint. What is a · b?
Answer: (A) 0
3 marks · Analyse procedures and apply lesson formulas to new scenarios · Try first, then show model answer.
AQ1. Three sensor posts sit at A(−1, −1), B(2, 3) and C(−2, 6). (a) Compute AB², BC², AC². (b) Explain why the triangle is isosceles right-angled at B. (c) How would area = 0 change the conclusion?
Model Answer
(a) AB² = (3)²+(4)² = 25; BC² = (−4)²+(3)² = 25; AC² = (−1)²+(7)² = 50.
(b) AB = BC and AB²+BC² = AC² ⇒ isosceles right-angled at B (L13 distance / Pythagoras).
(c) Area 0 ⇔ collinear — no triangle (L13 area formula).
AQ2. Point R divides the join of P(2, −5) and Q(−3, 5) internally in ratio 1 : 4. (a) Find R. (b) Explain why internal division uses plus signs. (c) How would external division change the formula?
Model Answer
(a) R = ((1·(−3)+4·2)/(1+4), (1·5+4·(−5))/5) = (1, −3).
(b) Internal division is a weighted average — weights m₂ on P and m₁ on Q with positive sum (L13 section formula).
(c) External uses minus: (m₁x₂ − m₂x₁)/(m₁ − m₂) (L13).
AQ3. A safety fence must pass through (1, 2) with slope 3. (a) Write its equation. (b) Find perpendicular distance from (0, 0) to the line 3x − y − 1 = 0. (c) Why absolute value appears in the distance formula?
Model Answer
(a) y − 2 = 3(x − 1) ⇒ y = 3x − 1 ⇒ 3x − y − 1 = 0.
(b) d = |0−0−1|/√(9+1) = 1/√10.
(c) Distance is unsigned length; |ax+by+c| removes side sign (L14).
AQ4. A circular fountain has equation x² + y² + 2x − 8 = 0. (a) Find centre and radius by completing the square. (b) Why must g² + f² − c > 0 for a real circle? (c) Does (3, 0) lie on the fountain rim?
Model Answer
(a) (x+1)² + y² = 9 ⇒ centre (−1, 0), r = 3.
(b) r² = g²+f²−c must be positive for positive radius (L15).
(c) (3+1)²+0 = 16 ≠ 9 — not on the circle.
AQ5. Compare parabola y² = 4ax, ellipse x²/25 + y²/16 = 1 and hyperbola x²/9 − y²/16 = 1 via eccentricity e. (a) State e for each type. (b) Compute e for the given ellipse. (c) Why does e classify conics?
Model Answer
(a) Parabola e = 1; ellipse e < 1; hyperbola e > 1 (L16).
(b) a = 5, b = 4, e = √(1 − 16/25) = 3/5.
(c) e = distance-to-focus / distance-to-directrix; the constant ratio shapes the curve (L16 definition).
AQ6. Let A = [[0, 1], [0, 0]] and B = [[0, 0], [1, 0]]. (a) Compute AB and BA. (b) What does AB ≠ BA show about matrix multiplication? (c) State the order condition for AB to exist.
Model Answer
(a) AB = [[1, 0], [0, 0]]; BA = [[0, 0], [0, 1]].
(b) Matrix multiplication is not commutative in general (L20).
(c) Columns of A must equal rows of B (inner dimensions match).
AQ7. For A = [[1, 2], [3, 4]]: (a) Find det A. (b) Write A⁻¹ using (1/det) adj A. (c) Explain why det A ≠ 0 is required for a unique solution of AX = B.
Model Answer
(a) det A = 4 − 6 = −2.
(b) A⁻¹ = (−1/2) [[4, −2], [−3, 1]] = [[−2, 1], [3/2, −1/2]].
(c) If det = 0, A is singular — no inverse; system has 0 or infinitely many solutions (L21–L22).
AQ8. Let f(x) = x + 1 and g(x) = 2x. (a) Find (f ∘ g)(x) and (g ∘ f)(x). (b) Why is order important? (c) Find f⁻¹ if f is considered bijective on R.
Model Answer
(a) fog = 2x + 1; gof = 2(x + 1) = 2x + 2.
(b) Composition applies rightmost first; fog ≠ gof generally (L23).
(c) y = x + 1 ⇒ x = y − 1 ⇒ f⁻¹(x) = x − 1.
AQ9. (a) State the range of sin⁻¹ x. (b) Evaluate sin⁻¹(sin 2π/3) and explain why it is not 2π/3. (c) Why does sin⁻¹ x + cos⁻¹ x = π/2 on [−1, 1]?
Model Answer
(a) [−π/2, π/2] (L24).
(b) sin(2π/3) = √3/2; principal arcsine is π/3 ≠ 2π/3 because 2π/3 is outside the range.
(c) Complementary angles on principal branches: cos⁻¹ x = π/2 − sin⁻¹ x (L24).
AQ10. For f(x) = (x² − 1)/(x − 1) when x ≠ 1 and f(1) = 3: (a) Find lim_{x→1} f(x). (b) Is f continuous at 1 with this definition? (c) What three conditions define continuity at a point?
Model Answer
(a) Factor: (x−1)(x+1)/(x−1) → lim = 2.
(b) lim = 2 ≠ f(1) = 3 ⇒ not continuous at 1.
(c) f
(a) defined, lim exists, and lim = f
(a) (L25).
AQ11. Differentiate y = x² sin x. (a) Identify product rule parts. (b) Write y′. (c) How would chain rule appear if y = sin(x²) instead?
Model Answer
(a) u = x², v = sin x.
(b) y′ = 2x sin x + x² cos x (L26 product, L27 trig).
(c) Outer sin, inner x²: y′ = cos(x²)·2x (chain rule).
AQ12. For y = xˣ (x > 0): (a) Take ln both sides. (b) Differentiate to find y′. (c) Why is log differentiation preferred here?
Model Answer
(a) ln y = x ln x.
(b) y′/y = ln x + 1 ⇒ y′ = xˣ (ln x + 1).
(c) Variable base and exponent; log converts product/power structure (L28).
AQ13. f(x) = x³ − 3x. (a) Find critical points. (b) Use f′′ to classify local max/min. (c) In a related-rates setup, why differentiate the geometric relation with respect to time t?
Model Answer
(a) f′ = 3x² − 3 = 0 ⇒ x = ±1.
(b) f′′ = 6x; f′′(−1) < 0 local max; f′′(1) > 0 local min (L29).
(c) Chain rule links rates: dy/dt = (dy/dx)(dx/dt) (L29 related rates).
AQ14. (a) Evaluate ∫ x eˣ dx by parts. (b) Compute ∫₀¹ eˣ dx using FTC. (c) Why is +C omitted in definite integrals?
Model Answer
(a) u = x, dv = eˣ dx ⇒ ∫ = x eˣ − ∫ eˣ = eˣ(x − 1) + C (L30).
(b) eˣ from 0 to 1 = e − 1 (L31).
(c) F
(b) − F
(a) cancels the constant (L31 FTC).
AQ15. (a) Solve dy/dx = y (general solution). (b) Solve dy/dx + y = eˣ using integrating factor. (c) State order and degree of (y″)³ + y′ = 0.
Model Answer
(a) dy/y = dx ⇒ ln|y| = x + C₁ ⇒ y = A eˣ (L32 separable).
(b) IF = eˣ; d/dx(y eˣ) = e^{2x}; y = ½ eˣ + C e^{−x}.
(c) Order 2, degree 3 (L32).
AQ16. A line has direction ratios 2, −1, 2. (a) Find direction cosines. (b) Write the corresponding unit vector. (c) Why must l² + m² + n² = 1?
Model Answer
(a) r = 3; DC = (2/3, −1/3, 2/3) (L33).
(b) â = (2/3)i − (1/3)j + (2/3)k (L34).
(c) Cosines of angles with axes are components of a unit direction vector (L33–L34).
AQ17. Plane π: x + y + z − 3 = 0. (a) Give a normal vector. (b) Distance from origin to π. (c) When are two planes perpendicular?
Model Answer
(a) n = <1, 1, 1> (L35).
(b) d = |−3|/√3 = √3.
(c) Normals n₁ · n₂ = 0 (L34–L35).
AQ18. (a) Write the vector equation of the line through (1, 2, 3) parallel to <1, 0, 0>. (b) What does “skew lines” mean? (c) State the shortest-distance formula structure for skew lines r = a₁ + λb₁ and r = a₂ + μb₂.
Model Answer
(a) r = <1, 2, 3> + λ <1, 0, 0> (L36).
(b) Non-parallel, non-intersecting, non-coplanar.
(c) SD = |(a₂−a₁)·(b₁×b₂)| / |b₁×b₂| (L36).
AQ19. Maximise z = 2x + 5y over feasible vertices (0, 0), (4, 0), (3, 3), (0, 2). (a) Evaluate z at each vertex. (b) State the optimum. (c) Why is the optimum at a corner for linear z?
Model Answer
(a) z = 0, 8, 21, 10.
(b) Maximum 21 at (3, 3) (L37).
(c) Level lines of linear objective last touch a polygonal feasible region at a vertex (or whole edge if parallel) — corner point theorem (L37).
AQ20. Statement: “If a number is divisible by 6, then it is divisible by 3.” (a) Write the contrapositive. (b) Write the converse. (c) Which is logically equivalent to the original, and why?
Model Answer
(a) If not divisible by 3, then not divisible by 6.
(b) If divisible by 3, then divisible by 6.
(c) Contrapositive ~q ⇒ ~p is equivalent to p ⇒ q; converse is not (L38).
5 marks · Link 2–3 lessons · Deep application and analysis of existing maths concepts only.
ZQ1. A circular garden has equation x² + y² = 25. A straight path has slope −1 and passes through the centre. (a) Write the path equation. (b) Find intersection points of path and circle. (c) Show the chord is a diameter and find its length. Link distance and circle definitions.
Model Answer
(a) Centre (0,0), slope −1 ⇒ y = −x.
(b) x² + x² = 25 ⇒ x = ±5/√2; points (5/√2, −5/√2), (−5/√2, 5/√2).
(c) Mid-point of chord is origin = centre ⇒ diameter; length 10 via distance formula. Uses L13 distance, L14 slope form, L15 circle (notes/PYQ themes).
ZQ2. Ellipse x²/25 + y²/9 = 1. (a) Find a, b, e and foci. (b) Using distance, verify for vertex (5, 0) that sum of distances to foci is 2a. (c) Contrast with parabola definition (e = 1).
Model Answer
(a) a = 5, b = 3, e = 4/5, foci (±4, 0).
(b) Distances to (±4,0): |5−4| + |5−(−4)| = 1+9 = 10 = 2a.
(c) Ellipse: sum to two foci constant; parabola: equal distance to one focus and directrix, e = 1 (L16, L13 distance).
ZQ3. System: x + y = 3, x − y = 1. (a) Write as AX = B. (b) Compute det A and A⁻¹. (c) Solve X = A⁻¹B and verify by substitution. Analyse when this method fails.
Model Answer
(a) A = [[1,1],[1,−1]], X = [x,y]ᵀ, B = [3,1]ᵀ.
(b) det = −2; A⁻¹ = (−1/2)[[−1,−1],[−1,1]] = [[1/2, 1/2],[1/2, −1/2]].
(c) X = [2, 1]ᵀ. Method fails if det A = 0 (L20–L22).
ZQ4. Let f(x) = sin x on [−π/2, π/2]. (a) Why is f bijective onto [−1, 1]? (b) Identify f⁻¹. (c) Using lim_{x→0} sin x / x = 1, find lim_{x→0} sin⁻¹ x / x. Explain continuity of sin⁻¹ at 0.
Model Answer
(a) Strictly increasing continuous, onto [−1,1] (L23–L24 ranges).
(b) f⁻¹ = sin⁻¹.
(c) Let u = sin⁻¹ x → 0 as x → 0; sin u / u → 1 ⇒ u / sin u → 1 ⇒ sin⁻¹ x / x → 1. sin⁻¹ continuous on [−1,1] so at 0 (L25).
ZQ5. A quantity y = e^{sin x}. (a) Find dy/dx using chain rule. (b) Find d/dx (x eˣ) with product rule. (c) At a critical point of a differentiable f, explain the roles of f′ = 0 and the second-derivative test.
Model Answer
(a) y′ = e^{sin x} cos x (L28 chain).
(b) eˣ + x eˣ = eˣ(x+1) (L26 product, L28).
(c) f′ = 0 necessary for interior extrema (Fermat); f′′ < 0 local max, f′′ > 0 local min when f′′ ≠ 0 (L29).
ZQ6. (a) Find general antiderivative of 2x. (b) Evaluate net change ∫₀² 2x dx. (c) Solve the IVP dy/dx = 2x, y(0) = 3 and relate to (a)–(b).
Model Answer
(a) x² + C (L30).
(b) [x²]₀² = 4 (L31).
(c) y = x² + 3; from 0 to 2, y increases by 4, matching the definite integral of the rate (FTC / DE, L32).
ZQ7. Points A(1, 0, 0), B(0, 1, 0), C(0, 0, 1). (a) Find vectors AB and AC. (b) Compute AB × AC and interpret |AB × AC|. (c) Find a Cartesian equation of plane ABC and its distance from origin.
Model Answer
(a) AB = <−1,1,0>, AC = <−1,0,1>.
(b) AB×AC = <1,1,1>; area of parallelogram = √3; triangle area √3/2 (L34).
(c) Plane: x + y + z = 1; distance from O: 1/√3 (L35).
ZQ8. Line L₁: r = <0,0,0> + λ <1,1,0>; L₂: r = <0,0,1> + μ <1,−1,0>. (a) Are directions parallel? (b) Compute b₁ × b₂. (c) Find shortest distance and interpret coplanarity if SD = 0.
Model Answer
(a) <1,1,0> not scalar multiple of <1,−1,0> — not parallel.
(b) b₁×b₂ = <0,0,−2>.
(c) a₂−a₁ = <0,0,1>; (a₂−a₁)·(b₁×b₂) = −2; SD = 2/2 = 1. If SD = 0 with non-parallel directions, lines intersect (or coplanar intersecting); skew need SD > 0 (L36).
ZQ9. Maximise profit z = 3x + 2y subject to x ≤ 100, y ≤ 200, x + y ≤ 250, x ≥ 0, y ≥ 0 (pens/pencils model from lesson themes). (a) List corner points of the feasible region. (b) Evaluate z at corners and state optimum. (c) How do linear inequalities relate to half-planes in L14?
Model Answer
(a) Corners: (0,0), (100,0), (100,150), (50,200), (0,200) [from intersections of active constraints].
(b) z values: 0; 300; 3·100+2·150=600; 3·50+2·200=550; 400 → max 600 at (100,150).
(c) Each linear inequality is a closed half-plane bounded by a straight line (L14); intersection is the feasible polygon (L37).
ZQ10. Claim: “Every invertible 2×2 matrix has non-zero determinant.” (a) Is this a mathematical statement? (b) Write it as an implication p ⇒ q. (c) Prove via contrapositive using det and invertibility from L21–L22. (d) Give a counter-example structure if the claim were “every matrix is invertible.”
Model Answer
(a) Yes — definite truth value (L38).
(b) p: A is invertible 2×2; q: det A ≠ 0.
(c) Contrapositive: if det A = 0 then A is not invertible — matches L21–L22 singular matrices have no inverse.
(d) Counter-example: zero matrix or [[1,2],[2,4]] has det 0 — disproves universal “every matrix invertible” (L38 quantifier negation).