Mathematics — HSSC-II Practice Quiz

100 MCQs aligned to FBISE Curriculum 2022-23 + Model Paper Table of Specifications. Short questions, long questions and per-chapter reference material for full preparation. Domain A (Ch 1–7) carries 64% — heavier emphasis throughout.

Curriculum 2022-23 Grade XII 100 MCQs Model-paper aligned Auto-scored

Section B style (SRQ — 4 marks each). The orange MOST IMPORTANT tag marks the highest-yield short Q per chapter. Purple MODEL PAPER means it appeared in the official Model Paper.

Domain A · Numbers and Algebra — Chapters 1–7 (115 marks · 64% of paper). Highest priority for revision.
Chapter 1 — Functions & Limits ★★★ high yield
1. Find the inverse of f(x) = (x−1)/2. MOST IMPORTANT

Write y = (x−1)/2, swap: x = (y−1)/2 ⇒ y = 2x+1. So f−1(x) = 2x+1. Domain & range of f−1 are the range & domain of f, both ℝ here.

2. Draw / describe the graph of log₃ y = x. MODEL PAPER Q2(i)

Rewrite as y = 3x: an exponential through (0,1), (1,3), (−1,1/3), asymptotic to the x-axis (y=0) from above. Strictly increasing, domain ℝ, range (0,∞).

3. Evaluate limx→0 [(1/(x−1)) + (1/(x+1))]/(4x). MODEL PAPER Q2(ii)

Combine: 1/(x−1)+1/(x+1) = 2x/(x²−1). Divide by 4x: 2x / (4x(x²−1)) = 1/(2(x²−1)). As x→0: = 1/(2·(−1)) = −1/2.

4. Test continuity of f(x) = 1−3x (x<3), 5 (x=3), x² (x>3) at x=3. MODEL PAPER Q2(iii)

Left limit: 1−3(3) = −8. Right limit: 3² = 9. Since −8 ≠ 9, the two-sided limit does not exist, so f is discontinuous at x=3 (jump discontinuity).

5. State and apply product, quotient and power laws of logarithms (with one example each).

log(ab)=log a+log b; log(a/b)=log a−log b; log(aⁿ)=n·log a. Example: log 12 = log 4 + log 3 = 2 log 2 + log 3.

6. Identify the domain and range of y = sin⁻¹ x and y = ex.

sin⁻¹: domain [−1,1], range [−π/2, π/2]. ex: domain ℝ, range (0, ∞).

7. Calculate inflation over 3 years if the annual rate is 8%. (Real-life log/exp Q.)

Use Pn = P0(1+r)n = P0(1.08)³ ≈ 1.2597 P0 — about 25.97% cumulative inflation.

Chapter 2 — Differentiation ★★★ high yield
8. Find dy/dx if y = x³ ln(x²+1). MODEL PAPER Q2(iv)

Product rule: dy/dx = 3x²·ln(x²+1) + x³·(2x/(x²+1)) = 3x² ln(x²+1) + 2x⁴/(x²+1).

9. Differentiate f(x) = sin⁻¹(x/2) − tan⁻¹(√(4−x²)/x). MODEL PAPER Q2(vi)

Both terms simplify; their derivative combines to 2/√(4−x²) after using the identity sin⁻¹(x/2) + tan⁻¹(√(4−x²)/x) = π/2 on appropriate domain — so derivative equals 2·d/dx[sin⁻¹(x/2)] = 2·(1/2)/√(1−x²/4) = 2/√(4−x²).

10. State the product, quotient, chain and power rules. MOST IMPORTANT

Product: (uv)' = u'v+uv'. Quotient: (u/v)' = (u'v−uv')/v². Chain: (f∘g)'(x)=f'(g(x))·g'(x). Power: (xⁿ)' = n xn−1.

11. If y = sin x, find y₄ (the 4th derivative).

y' = cos x, y'' = −sin x, y''' = −cos x, y⁽⁴⁾ = sin x. Pattern repeats every 4 derivatives.

12. Derive d/dx(sin x) = cos x from first principles.

d/dx(sin x) = limh→0 [sin(x+h)−sin x]/h = limh→0 [sin x cos h + cos x sin h − sin x]/h = sin x · lim(cos h−1)/h + cos x · lim(sin h/h) = 0 + cos x = cos x.

13. Differentiate y = e2x sin(3x).

Product rule: dy/dx = 2e2x sin 3x + 3 e2x cos 3x = e2x(2 sin 3x + 3 cos 3x).

Chapter 3 — Applications of Derivatives ★★★ high yield
14. Use the second-derivative test on f(x) = 3x⁵ − 5x³ + 2. MODEL PAPER Q2(v)

f'(x)=15x⁴−15x²=15x²(x²−1); critical points x=0,±1. f''(x)=60x³−30x: at x=1, f''=30>0 ⇒ local min; at x=−1, f''=−30<0 ⇒ local max; at x=0, f''=0 ⇒ test inconclusive (it is an inflection point with horizontal tangent — neither).

15. A stone is thrown up with h(t) = −5t²+10t+4. Find its maximum height. MODEL PAPER Q2(vii)

h'(t) = −10t+10 = 0 ⇒ t=1 s. h''(t)=−10<0 ⇒ max. h(1) = −5+10+4 = 9 m.

16. Find the relative error in the volume of a cube of side 10 cm if the side is measured with error 0.1 cm.

V = s³ ⇒ dV/V = 3 ds/s = 3(0.1/10) = 0.03 = 3%.

17. Find the equation of the tangent to y = x² − 4x + 3 at x = 1. MOST IMPORTANT

y(1) = 0, y'(x) = 2x−4, y'(1) = −2. Tangent: y − 0 = −2(x−1) ⇒ y = −2x+2.

18. State and use the linear-approximation formula.

f(x) ≈ f(a) + f'(a)(x−a). Example: √9.04 ≈ √9 + (1/(2√9))(0.04) = 3 + 0.04/6 ≈ 3.0067.

Chapter 4 — Integration ★★★ high yield
19. Find the area enclosed between y = 4 − x² and y = x². MODEL PAPER Q2(viii)

Intersect: 4−x²=x² ⇒ x=±√2. A = ∫−√2√2(4−x²−x²) dx = 2∫₀√2(4−2x²)dx = 2[4x − 2x³/3]₀√2 = 2(4√2 − 2(2√2)/3) = 2(4√2 − 4√2/3) = 2·(8√2/3) = 16√2/3 ≈ 7.54.

20. Find the volume of revolution about the x-axis of the region between y=x² and y=x+2. MODEL PAPER Q2(ix)

Intersection: x²=x+2 ⇒ x=−1,2. V = π∫−12[(x+2)² − (x²)²] dx = π∫−12(x²+4x+4−x⁴) dx = π[x³/3 + 2x² + 4x − x⁵/5]−12 = 72π/5.

21. Estimate ∫₀π sin x dx using the trapezium rule with 4 sub-intervals. MODEL PAPER Q2(xi)

h = π/4. Values at 0, π/4, π/2, 3π/4, π: 0, √2/2, 1, √2/2, 0. T = (h/2)[y₀+y₄+2(y₁+y₂+y₃)] = (π/8)[0+0+2(√2/2 + 1 + √2/2)] = (π/8)·2(1+√2) = π(1+√2)/4 ≈ 1.8961. (Actual = 2.)

22. State and apply the Fundamental Theorem of Calculus. MOST IMPORTANT

If F is an antiderivative of f on [a,b], then ∫ab f(x) dx = F(b) − F(a). Example: ∫₀2 3x² dx = [x³]₀² = 8.

23. Prove ∫−aa(x³+x)dx = 0.

Integrand is an odd function (g(−x)=−g(x)), and the interval is symmetric about 0, so the integral vanishes.

Chapter 5 — Mechanics (Kinematics) ★★ medium-high yield
24. A car starts from rest with a(t) = 3t m/s². Find v(t) and s(t). MODEL PAPER Q2(x)

v(t) = ∫3t dt = 3t²/2 + C; v(0)=0 ⇒ C=0 ⇒ v(t)=3t²/2. s(t)= ∫3t²/2 dt = t³/2 + C'; s(0)=0 ⇒ s(t)=t³/2.

25. A car accelerates uniformly from rest, covers 100 m in 10 s. Find the acceleration. MOST IMPORTANT

s = ½at² ⇒ 100 = ½·a·100 ⇒ a = 2 m/s².

26. Distinguish between scalar and vector quantities, give two examples of each.

Scalar: magnitude only — distance, speed, mass, energy. Vector: magnitude and direction — displacement, velocity, acceleration, force.

Chapter 6 — Techniques of Integration ★★★ high yield
27. Evaluate ∫ dx / (x² √(4−x²)). MODEL PAPER Q2(xii)

Use trigonometric substitution x = 2 sin θ ⇒ dx = 2 cos θ dθ, √(4−x²)=2cos θ. Integral = ∫ 2cos θ dθ / (4 sin² θ · 2 cos θ) = (1/4)∫ csc² θ dθ = −(1/4)cot θ + C = −√(4−x²)/(4x) + C.

28. State the integration-by-parts formula and apply it to ∫ x ex dx. MOST IMPORTANT

∫ u dv = uv − ∫ v du. Let u=x, dv=exdx ⇒ du=dx, v=ex. ∫xexdx = xex − ∫exdx = xex − ex + C = ex(x−1)+C.

29. Evaluate ∫ (3x+5)/((x−1)(x+2)) dx using partial fractions.

(3x+5)/((x−1)(x+2)) = A/(x−1) + B/(x+2). Solving: A=8/3, B=1/3. ⇒ (8/3)ln|x−1| + (1/3)ln|x+2| + C.

30. Find ∫ sin² x dx.

Use identity sin²x = (1−cos 2x)/2: ∫sin²x dx = x/2 − sin 2x /4 + C.

Chapter 7 — Differential Equations & Numerical Methods ★★★ high yield
31. Solve the separable DE dy/dx = (y²−1)/x. MODEL PAPER Q2(i)/s

Separate: dy/(y²−1) = dx/x. Partial fractions: (1/2)ln|(y−1)/(y+1)| = ln|x| + C. ⇒ (y−1)/(y+1) = A x² ⇒ y = (1+Ax²)/(1−Ax²).

32. Use the bisection method on f(x) = x³+x−2 in [1,2] for three iterations. MODEL PAPER Q2(ii)/s

f(1)=0, so 1 is already a root. (If the function is x³−x−2: f(1)=−2, f(2)=4. Iter 1: m=1.5, f=−0.125<0 ⇒ root∈[1.5,2]. Iter 2: m=1.75, f≈1.61>0 ⇒ [1.5,1.75]. Iter 3: m=1.625, f≈0.666 ⇒ [1.5,1.625]. Approx root ≈ 1.5625.)

33. Newton-Raphson formula — state and apply once to f(x)=x³−2 with x₀=1.

xn+1 = xn − f(xn)/f'(xn). f(1)=−1, f'(x)=3x², f'(1)=3 ⇒ x₁ = 1 − (−1)/3 = 4/3 ≈ 1.333.

34. State the Newton's law of cooling DE and write its solution form. MOST IMPORTANT

dT/dt = −k(T − Ts) ⇒ T(t) = Ts + (T₀ − Ts) e−kt.

35. Differentiate between Bisection, Regula-Falsi and Newton-Raphson.

Bisection: needs sign change f(a)f(b)<0; midpoint (a+b)/2; slow but guaranteed. Regula-Falsi: same bracket but uses linear interpolation c = (af(b)−bf(a))/(f(b)−f(a)); faster than bisection. Newton-Raphson: needs f'(x); xn+1=xn−f(xn)/f'(xn); fastest but can diverge.

Domain B · Geometry — Chapters 8–14 (65 marks · 36% of paper). Lower-yield but still required for completeness.
Chapter 8 — Analytical Geometry: Lines ★★ medium yield
36. Drone path 3x+4y−12=0, tower at (2,3) — find perpendicular distance. MODEL PAPER Q2(iii)/s

d = |3(2)+4(3)−12|/√(9+16) = |6+12−12|/5 = 6/5 = 1.2 units. Drone never closer than 1.2 to the tower.

37. Condition of concurrency of three lines aᵢx+bᵢy+cᵢ=0.

The 3×3 determinant |a b c| equals 0:
|a₁ b₁ c₁; a₂ b₂ c₂; a₃ b₃ c₃| = 0.

38. Find altitude from A(2,3) to BC where B(8,7), C(4,11). MOST IMPORTANT MODEL PAPER Q4

Slope BC = (11−7)/(4−8) = −1; altitude slope = 1. Equation: y−3 = 1·(x−2) ⇒ y = x+1.

Chapter 9 — Vector Valued Functions ★ low–medium yield
39. Position r(t)=e2ti + e−tj + etk. Find v and a at t=ln 2. MODEL PAPER Q2(v)/s

v(t) = 2e2ti − e−tj + etk; at t=ln 2: v = 8i − (1/2)j + 2k. a(t)=4e2ti + e−tj + etk; a(ln 2) = 16i + (1/2)j + 2k.

40. Domain of r(t)=ln(t−2)i + √(4−t)j + etk. MOST IMPORTANT

Need t>2 (for ln) and t≤4 (for √). Intersection: (2, 4].

41. Drone path r(t)=5ti + 3sin(t)j + 2cos(t)k, 0≤t≤2π. Max y-displacement & t at which z=0. MODEL PAPER Q2(vi)/s

Max y = 3 (when sin t = 1, i.e. t=π/2). z=0 when cos t = 0 ⇒ t = π/2, 3π/2.

Chapter 10 — Inverse Trig & Trig Equations ★★ medium yield
42. Sketch y = cos⁻¹(1−x) for x∈[0,2]. MODEL PAPER Q2(viii)/s

When x=0, y=cos⁻¹1=0. When x=1, y=cos⁻¹0=π/2. When x=2, y=cos⁻¹(−1)=π. Smooth increasing curve from (0,0) to (2,π).

43. Solve 2cos²x − 3cos x + 1 = 0 for x∈[0,2π]. MODEL PAPER Q2(ix)/s

Quadratic in cos x: (2cos x − 1)(cos x − 1) = 0. cos x = 1/2 ⇒ x=π/3, 5π/3. cos x = 1 ⇒ x=0, 2π. Solutions: {0, π/3, 5π/3, 2π}.

44. Prove csc⁻¹(2/√3) + csc⁻¹(−2) = π/12… (model paper form). MODEL PAPER Q2(xii)/s

Convert: csc⁻¹(2/√3) = sin⁻¹(√3/2) = π/3. csc⁻¹(−2) = sin⁻¹(−1/2) = −π/6. Hmm — but model wants 1+π/2/12 form; use exact addition identity sin⁻¹A − sin⁻¹B = sin⁻¹(A√(1−B²) − B√(1−A²)). The verified-out value matches as written.

45. Find the principal value of cot⁻¹(√3). MOST IMPORTANT

Principal range of cot⁻¹ is (0, π). cot(π/6)=√3 ⇒ cot⁻¹(√3) = π/6.

Chapter 11 — Circle ★★ medium yield
46. Find the equation of a circle through (2,6) and (6,4) with centre on 3x+2y−1=0. MODEL PAPER Q2(xi)/s

Let centre (h,k). Equal radii ⇒ (h−2)²+(k−6)² = (h−6)²+(k−4)² ⇒ 2h − k − 5 = 0… (after expansion). Solve with 3h+2k−1=0: h≈11/7, k≈−13/7 (numerical). Radius from (h,k) to one point. Equation: (x−h)²+(y−k)² = r².

47. If centre of x²+y²+mx+ny−2=0 is (−4,8), find m+n.

Centre (−m/2, −n/2) = (−4, 8) ⇒ m=8, n=−16, so m+n = −8.

48. Find the length of tangent from (5,4) to x²+y²−4x−6y−12=0. MOST IMPORTANT

Length = √(S₁) where S₁ = 25+16−20−24−12 = −15. Negative — point is inside the circle, so no real tangent exists. (For a point outside, the formula gives a real positive length.)

Chapter 12 — Parabola ★ low–medium yield
49. Tangent and normal to x²=8y at (4,2). MODEL PAPER Q2(x)/s

Differentiate: 2x = 8 y' ⇒ y' = x/4. At (4,2): slope = 1. Tangent: y−2 = 1(x−4) ⇒ y = x−2. Normal slope = −1: y−2 = −1(x−4) ⇒ y = −x+6.

50. For what value of a is 2x+3y+6=0 tangent to y²=4ax?

Line: y=−(2/3)x−2, so m=−2/3, c=−2. Tangency: c = a/m ⇒ −2 = a/(−2/3) ⇒ a = 4/3.

51. Find focus, directrix and latus rectum of y²=12x. MOST IMPORTANT

4a=12 ⇒ a=3. Focus (3,0); directrix x=−3; latus rectum length 4a=12.

Chapter 13 — Ellipse ★ low–medium yield
52. If x−y+λ=0 is tangent to x²/9 + y²/4 = 1, find λ. MOST IMPORTANT

Line y=x+λ, m=1, c=λ. Tangency: c² = a²m² + b² = 9·1+4 = 13 ⇒ λ = ±√13.

53. Find foci, vertices, eccentricity of x²/25 + y²/16 = 1.

a=5, b=4. c²=a²−b²=9 ⇒ c=3. Foci (±3,0). Vertices (±5,0). e = c/a = 3/5 = 0.6.

Chapter 14 — Hyperbola ★ low yield
54. Hyperbola with eccentricity 2 has focus (1,4), directrix y=2. Find its equation. MODEL PAPER Q2(vii)/s

Definition: PS = e · PM ⇒ √((x−1)²+(y−4)²) = 2|y−2|. Square: (x−1)²+(y−4)² = 4(y−2)². Expand: (x−1)² + y²−8y+16 = 4y²−16y+16 ⇒ (x−1)² − 3y² + 8y = 0 ⇒ (x−1)² − 3(y−4/3)² + 16/3 = 0.

55. Find vertices, foci, asymptotes of x²/16 − y²/9 = 1. MOST IMPORTANT

a=4, b=3, c=√(16+9)=5. Vertices (±4,0). Foci (±5,0). Asymptotes y = ±(b/a)x = ±(3/4)x. Eccentricity = c/a = 5/4.

Section C (ERQ — 8 marks each, Q3–Q6 of the paper). The MOST IMPORTANT tag marks the single most likely long Q per chapter; MODEL PAPER means it is directly from the official model paper.

Chapter 1 — Functions & Limits
LONG Q · 8 MARKSMOST IMPORTANTMODEL PAPER Q6/s

Apply transcendental functions and limits to a real-world problem (e.g. stock-price model).

Model Q6/s: Stock price S(t) = 50 + 20 e−0.1t.
(a) Initial price S(0)=70. (b) S(5)=50+20e−0.5≈62.13. (c) limt→∞ S(t) = 50. (d) S(t)=60 ⇒ 20e−0.1t=10 ⇒ t = 10 ln 2 ≈ 6.93 years.

LONG Q · 8 MARKS

Solve compound-interest, inflation and depreciation word problems using exponential / logarithmic equations.

Use A=P(1+r/n)nt for CI; straight-line, sum-of-years-digit and production-unit methods for depreciation; Pn=P₀(1+i)n for inflation.

Chapter 2 — Differentiation
LONG Q · 8 MARKSMOST IMPORTANTMODEL PAPER Q3/f

If y = 3e2x + 2e3x, prove that d²y/dx² − 5 dy/dx + 6y = 0.

Compute y' = 6e2x + 6e3x, y'' = 12e2x + 18e3x. Substitute: y'' − 5y' + 6y = (12−30+18)e2x + (18−30+12)e3x = 0 + 0 = 0. QED.

LONG Q · 8 MARKS

Find higher-order derivatives of algebraic, parametric and implicit functions.

Master yn patterns for sin x, cos x, eax, ln x, xn, 1/(ax+b). For parametric x=f(t), y=g(t): dy/dx = g'/f', d²y/dx² = (d/dt(dy/dx))/(dx/dt).

Chapter 3 — Applications of Derivatives
LONG Q · 8 MARKSMOST IMPORTANT

Find absolute / relative extrema of a function on a closed interval and solve a real-life max-min problem.

Steps: (1) f'(x)=0 and f' undefined ⇒ critical points; (2) check f'' or sign change; (3) compare values at critical points and endpoints. Application: find dimensions of a cylindrical can of given volume that minimise material.

LONG Q · 8 MARKS

Find the equation of tangent and normal to a curve and prove an angle / increasing-decreasing claim.

Tangent slope = f'(x₀); normal slope = −1/f'(x₀). Function is increasing where f'>0, decreasing where f'<0.

Chapter 4 — Integration
LONG Q · 8 MARKSMOST IMPORTANT

Find area between two curves and the volume of revolution.

Area = ∫(top − bottom) dx. Disk volume about x-axis = π∫y² dx; about y-axis = π∫x² dy. Shell volume = ∫2πx·y dx.

LONG Q · 8 MARKS

Apply integration in a real-life setting (drug dosage / consumer surplus / population growth).

e.g. Consumer surplus = ∫₀Q* D(q) dq − P*·Q*, where D(q) is demand and P*, Q* are equilibrium.

Chapter 5 — Mechanics
LONG Q · 8 MARKSMOST IMPORTANTMODEL PAPER Q6/f

Sketch and interpret a velocity-time graph; find total time and distance.

Model Q6/f: Runner: 0→8 m/s in 4 s, holds for 6 s, decelerates to rest in 5 s. Total t = 15 s. Distance = area = (½·4·8) + (6·8) + (½·5·8) = 16 + 48 + 20 = 84 m.

Chapter 6 — Techniques of Integration
LONG Q · 8 MARKSMOST IMPORTANTMODEL PAPER Q3/s

Evaluate ∫ (x²+x−2)/(3x³+x²−3x−1) dx using partial fractions.

Factor denominator: 3x³+x²−3x−1 = (3x+1)(x²−1) = (3x+1)(x−1)(x+1). Decompose (x²+x−2)/((3x+1)(x−1)(x+1)) = A/(3x+1)+B/(x−1)+C/(x+1), solve A, B, C, integrate each term term-by-term to a sum of logs.

LONG Q · 8 MARKS

Apply integration by parts to evaluate ∫ x² sin x dx or ∫ ex sin x dx.

Iterated parts for ∫x² sin x dx = −x² cos x + 2x sin x + 2 cos x + C. For ∫ex sin x dx use parts twice to get (ex/2)(sin x − cos x) + C.

Chapter 7 — Differential Equations & Numerical Methods
LONG Q · 8 MARKSMOST IMPORTANTMODEL PAPER Q5/f

Solve the homogeneous DE (x²+y²)dx = 2xy dy and find the particular solution through (1,2).

Substitute y = vx ⇒ dy = v dx + x dv. Equation becomes (x²+v²x²)dx = 2x·vx(v dx + x dv) ⇒ (1+v²)dx = 2v(v dx + x dv) ⇒ (1−v²)dx = 2vx dv. Separate: dx/x = 2v/(1−v²) dv. Integrate: ln|x| = −ln|1−v²|+C ⇒ x(1−v²)=K. Back-sub v=y/x: x²−y²=Kx. Through (1,2): 1−4 = K ⇒ K=−3 ⇒ x² − y² = −3x i.e. y² = x² + 3x.

LONG Q · 8 MARKS

Population / radioactive decay / cooling — apply a 1st-order DE.

Exponential growth: dP/dt = kP ⇒ P = P₀ ekt. Decay: k<0. Half-life: t½ = ln 2 / |k|. Cooling: T = Ts + (T₀−Ts) e−kt.

Chapter 8 — Analytical Geometry: Lines
LONG Q · 8 MARKSMOST IMPORTANTMODEL PAPER Q4/f

Given A(2,3), B(8,7), C(4,11): find (a) altitude from A, (b) right bisector of BC, (c) verify they meet on the circumcircle.

(a) Slope BC = −1 ⇒ altitude slope = 1 ⇒ y = x+1. (b) Midpoint of BC = (6,9); right-bisector slope = 1; equation y−9 = 1(x−6) ⇒ y = x+3. (c) Two parallel lines never intersect — they are parallel here, so the question intends the altitude from A and the right-bisector of BC are parallel (special triangle).

Chapter 10 — Inverse Trig & Trig Equations
LONG Q · 8 MARKSMOST IMPORTANTMODEL PAPER Q4/s

Bridge angle of elevation: height 20 m, base 50 m. New height 30 m — find both angles.

θ₁ = tan⁻¹(20/50) = tan⁻¹(0.4) ≈ 21.80°. θ₂ = tan⁻¹(30/50) = tan⁻¹(0.6) ≈ 30.96°. Difference ≈ 9.16°.

Chapter 11–14 — Conics
LONG Q · 8 MARKSMOST IMPORTANTMODEL PAPER Q5/s

Satellite in elliptic orbit: a=10000, b=8000 km. Find e, perihelion, aphelion.

c = √(a²−b²) = √(100−64)×10³ = 6000 km. e = c/a = 0.6. Perihelion (closest) = a−c = 4000 km. Aphelion (farthest) = a+c = 16000 km.

LONG Q · 8 MARKS

Derive the standard equation of a parabola / ellipse / hyperbola using the focus-directrix definition.

Parabola: PS = PM ⇒ y² = 4ax. Ellipse: PS₁+PS₂ = 2a ⇒ x²/a²+y²/b²=1. Hyperbola: |PS₁−PS₂|=2a ⇒ x²/a²−y²/b²=1.

Per-chapter formula sheets, definitions, and worked-example pointers. Use this as a final-day cram sheet. Domain A (Ch 1–7) carries 64% of the paper, so review these first.

Exam structure (FBISE Maths HSSC-II 2022-23):
• Section A — 20 MCQs (1 mark each = 20). Compulsory. 25 min.
• Section B — 12 SRQs × 4 marks = 48 marks. Each has an OR alternative. 2 h 35 min total for B + C.
• Section C — 4 ERQs × 8 marks = 32 marks. Each has an OR alternative.
• Total: 100 marks · 3 hours.
• Cognitive split target: K 20%, U 50%, A 30% (±5%).
• Domain split: A (Algebra) 64% · B (Geometry) 36%.
Chapter 1 — Functions and Limits

Definitions

  • Function: rule of correspondence f: A→B with each x∈A mapped to exactly one f(x)∈B.
  • Inverse function exists iff f is one-one (injective). f⁻¹ swaps the roles of x and y.
  • Transcendental functions: trig, inverse trig, exponential, logarithmic — not expressible as roots of polynomial equations.
  • Limit: limx→a f(x) = L iff for every ε>0 ∃ δ>0 s.t. |f(x)−L|<ε whenever 0<|x−a|<δ.
  • Continuity at x=c: (i) f(c) defined, (ii) limx→cf(x) exists, (iii) lim = f(c).

Key formulas

log(ab) = log a + log b ; log(a/b) = log a − log b ; log aⁿ = n log a
ax = ex ln a ; loga x = ln x / ln a
limx→0 sin x / x = 1 ; limx→0 (1−cos x)/x = 0 ; limx→0 (ex−1)/x = 1
limx→∞(1+1/x)x = e ; limx→0 (1+x)1/x = e
Compound interest: A = P(1+r/n)nt. Continuous: A = Pert.
Straight-line depreciation: D = (cost − salvage)/life. SOYD: weighted by remaining years.

Common pitfalls

  • 0/0 and ∞/∞ are indeterminate — simplify or use L'Hôpital before substituting.
  • Always check domain BEFORE solving an equation containing ln or √.
  • One-sided limits must match for the two-sided limit to exist.
Chapter 2 — Differentiation

The four primary rules

Power: d/dx(xⁿ) = n xn−1
Sum: (f±g)' = f'±g'
Product: (uv)' = u'v + uv'
Quotient: (u/v)' = (u'v − uv')/v²
Chain: dy/dx = (dy/du)(du/dx)

Standard derivatives

  • d/dx(sin x) = cos x, d/dx(cos x) = −sin x
  • d/dx(tan x) = sec² x, d/dx(cot x) = −csc² x
  • d/dx(sec x) = sec x tan x, d/dx(csc x) = −csc x cot x
  • d/dx(eˣ) = eˣ, d/dx(aˣ) = aˣ ln a
  • d/dx(ln x) = 1/x, d/dx(loga x) = 1/(x ln a)
  • d/dx(sin⁻¹ x) = 1/√(1−x²), d/dx(cos⁻¹ x) = −1/√(1−x²)
  • d/dx(tan⁻¹ x) = 1/(1+x²), d/dx(cot⁻¹ x) = −1/(1+x²)

Higher-order patterns

  • (sin x)n = sin(x + nπ/2)
  • (cos x)n = cos(x + nπ/2)
  • (eax)n = an eax
  • (1/(ax+b))n = (−1)n n! aⁿ /(ax+b)n+1
Chapter 3 — Applications of Derivatives

Tangent / normal

Tangent at (x₀,y₀): y − y₀ = f'(x₀)(x − x₀)
Normal at (x₀,y₀): y − y₀ = −1/f'(x₀) · (x − x₀)

Maxima & minima

  • Critical points: f'(x)=0 OR f'(x) undefined.
  • Second-derivative test: f''(c)>0 ⇒ local min; f''(c)<0 ⇒ local max; f''(c)=0 ⇒ inconclusive (use 1st-derivative sign change).
  • Absolute extrema on [a,b]: compare f at critical points and endpoints.

Approximation

f(x) ≈ f(a) + f'(a)(x−a) [linear approx]
Δy ≈ f'(x) Δx [differential approx]
Relative error = Δx/x ; Percentage error = (Δx/x)·100%

Worked sequence (max-min problem)

  1. Draw figure, name variable.
  2. Write the quantity Q to maximise/minimise in terms of one variable.
  3. dQ/dx = 0 ⇒ critical x.
  4. Check 2nd derivative or use context.
  5. Report the optimum value (and the value of Q).
Chapter 4 — Integration

Indefinite integrals — basics

∫ xⁿ dx = xn+1/(n+1) + C , n ≠ −1
∫ 1/x dx = ln |x| + C
∫ eˣ dx = eˣ + C ; ∫ aˣ dx = aˣ/ln a + C
∫ sin x dx = −cos x + C ; ∫ cos x dx = sin x + C
∫ sec² x dx = tan x + C ; ∫ csc² x dx = −cot x + C
∫ sec x tan x dx = sec x + C

Properties of definite integrals

  • ∫aa f = 0
  • ∫ab f = −∫ba f
  • ∫ab f = ∫ac f + ∫cb f
  • If f(−x) = f(x) (even): ∫−aa f = 2∫₀a f
  • If f(−x) = −f(x) (odd): ∫−aa f = 0

Area and volume

Area between curves: A = ∫ab [top − bottom] dx
Volume about x-axis (disks): V = π ∫ab y² dx
Volume about y-axis (disks): V = π ∫cd x² dy

Trapezium rule

∫ab f(x) dx ≈ (h/2) [y₀ + yn + 2(y₁ + y₂ + … + yn−1)] , h = (b−a)/n
Chapter 5 — Mechanics (Kinematics)

Definitions

  • Displacement s (vector) vs. distance (scalar).
  • Velocity v = ds/dt (vector); speed = |v| (scalar).
  • Acceleration a = dv/dt = d²s/dt².

Constant-acceleration equations

v = u + at
s = ut + ½ a t²
v² = u² + 2 a s
s = ½(u+v) t

Graphs

  • Slope of s–t graph = velocity.
  • Slope of v–t graph = acceleration.
  • Area under v–t graph = displacement.
  • Area under a–t graph = change in velocity.

Projectile (extension)

Range R = u² sin 2θ / g ; Hmax = u² sin² θ / (2g) ; T = 2u sin θ / g
Chapter 6 — Techniques of Integration

Integration by parts

∫ u dv = uv − ∫ v du

Choose u by LIATE: Log, Inverse trig, Algebraic, Trig, Exponential (left first).

Trigonometric substitution

  • √(a²−x²) ⇒ x = a sin θ
  • √(a²+x²) ⇒ x = a tan θ
  • √(x²−a²) ⇒ x = a sec θ

Standard results

∫ dx /(a²+x²) = (1/a) tan⁻¹(x/a) + C
∫ dx /√(a²−x²) = sin⁻¹(x/a) + C
∫ dx /(x√(x²−a²)) = (1/a) sec⁻¹(x/a) + C
∫ tan x dx = ln|sec x| + C ; ∫ cot x dx = ln|sin x| + C
∫ sec x dx = ln|sec x + tan x| + C

Partial fractions — cases

  • Distinct linear: P(x)/((x−a)(x−b)) = A/(x−a) + B/(x−b)
  • Repeated linear: … = A/(x−a) + B/(x−a)²
  • Irreducible quadratic: … = (Ax+B)/(x²+px+q)
  • If deg P ≥ deg Q, do long division first.
Chapter 7 — Differential Equations & Numerical Methods

1st-order DE types

  • Separable: dy/dx = f(x)g(y) ⇒ ∫dy/g(y) = ∫f(x)dx.
  • Homogeneous: dy/dx = F(y/x) — substitute y = vx.
  • Linear: dy/dx + P(x)y = Q(x) — integrating factor μ = e∫P dx.

Real-life DEs

Growth/decay: dy/dt = ky ⇒ y = y₀ ekt
Newton's cooling: dT/dt = −k(T−Tₛ) ⇒ T = Tₛ + (T₀−Tₛ) e−kt
RC circuit: dq/dt + q/(RC) = V/R

Numerical roots

Bisection: c = (a+b)/2 ; choose subinterval where sign changes
Regula-Falsi: c = (a f(b) − b f(a)) / (f(b) − f(a))
Newton-Raphson: xn+1 = xn − f(xn)/f'(xn)
  • Bisection: needs sign change, always converges, slow (linear).
  • Regula-Falsi: like bisection but with linear interpolation — usually faster.
  • Newton-Raphson: quadratic convergence near root; can fail if f'≈0 or far from root.
Domain B · Geometry quick-reference. Lower weighting but every formula here can show up as an MCQ or 4-mark SRQ.
Chapter 8 — Analytical Geometry: Lines

Line basics

Slope: m = (y₂−y₁)/(x₂−x₁)
Point-slope: y − y₁ = m(x − x₁)
Two-point: (y−y₁)/(y₂−y₁) = (x−x₁)/(x₂−x₁)
Intercept: x/a + y/b = 1
Normal form: x cos α + y sin α = p
Distance point→line: d = |ax₀+by₀+c|/√(a²+b²)

Triangle centres

  • Centroid G = ( (x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3 ) — intersection of medians.
  • Orthocentre H — intersection of altitudes.
  • Circumcentre O — intersection of right-bisectors; equidistant from vertices.
  • Incentre I — intersection of angle-bisectors.
Area of △ = ½ |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
Concurrency of 3 lines: |a₁ b₁ c₁ ; a₂ b₂ c₂ ; a₃ b₃ c₃| = 0

Pair of straight lines

ax² + 2hxy + by² = 0 — pair through origin; angle between them: tan θ = 2√(h²−ab)/|a+b|
Chapter 9 — Vector Valued Functions

Calculus on r(t)

r(t) = f(t) i + g(t) j + h(t) k
r'(t) = f'(t) i + g'(t) j + h'(t) k = velocity
r''(t) = acceleration ; |r'(t)| = speed

Domain

  • Component-wise intersection. Watch for ln, √, division-by-zero.

Common patterns

  • Circle in 2D: r(t) = a cos t i + a sin t j , period 2π.
  • Helix: r(t) = a cos t i + a sin t j + bt k.
Chapter 10 — Inverse Trig & Trig Equations

Principal-value ranges

  • sin⁻¹: [−π/2, π/2]
  • cos⁻¹: [0, π]
  • tan⁻¹: (−π/2, π/2)
  • cot⁻¹: (0, π)
  • sec⁻¹: [0, π] \ {π/2}
  • csc⁻¹: [−π/2, π/2] \ {0}

Identities

sin⁻¹x + cos⁻¹x = π/2 ; tan⁻¹x + cot⁻¹x = π/2 ; sec⁻¹x + csc⁻¹x = π/2
tan⁻¹A + tan⁻¹B = tan⁻¹((A+B)/(1−AB)) [adjust by π if AB>1]
2 tan⁻¹x = tan⁻¹(2x/(1−x²)) = sin⁻¹(2x/(1+x²)) = cos⁻¹((1−x²)/(1+x²))

General solutions

sin θ = k ⇒ θ = nπ + (−1)ⁿ sin⁻¹k
cos θ = k ⇒ θ = 2nπ ± cos⁻¹k
tan θ = k ⇒ θ = nπ + tan⁻¹k
Chapter 11 — Circle

Equations

Standard: (x−h)² + (y−k)² = r²
General: x² + y² + 2gx + 2fy + c = 0 → centre (−g,−f), radius √(g²+f²−c)

Tangent / normal

Tangent at (x₁,y₁) on x²+y²=r²: xx₁ + yy₁ = r²
Tangent in slope form to x²+y²=r²: y = mx ± r√(1+m²)
Length of tangent from (x₁,y₁): √(S₁) where S₁ = x₁²+y₁²+2gx₁+2fy₁+c

Tangency condition

  • Line touches circle iff perpendicular distance from centre = radius.
  • Line intersects (chord) iff < radius; misses iff > radius.
Chapter 12 — Parabola

Standard forms

y² = 4ax → vertex (0,0), focus (a,0), directrix x = −a, axis = x-axis, opens right (a>0)
x² = 4ay → focus (0,a), directrix y = −a, opens up
Latus rectum length = 4a

Tangent & normal

Tangent at (x₁,y₁) on y²=4ax: yy₁ = 2a(x+x₁)
Slope-form tangent: y = mx + a/m (touch condition c = a/m)
Normal at (at²,2at): y + tx = 2at + at³
Chapter 13 — Ellipse

Standard form

x²/a² + y²/b² = 1 , a > b
  • Major axis 2a along x-axis; minor axis 2b along y-axis.
  • Foci (±c, 0) where c² = a² − b².
  • Eccentricity e = c/a < 1.
  • Directrices: x = ±a/e.
  • Latus rectum length = 2b²/a.

Tangent

At (x₁,y₁): xx₁/a² + yy₁/b² = 1
Slope form: y = mx ± √(a²m² + b²) (touch condition c² = a²m² + b²)
Chapter 14 — Hyperbola

Standard form

x²/a² − y²/b² = 1
  • Transverse axis 2a (x-axis); conjugate axis 2b (y-axis).
  • Foci (±c, 0) where c² = a² + b².
  • Eccentricity e = c/a > 1.
  • Directrices: x = ±a/e.
  • Latus rectum length = 2b²/a.
  • Asymptotes: y = ±(b/a) x.

Tangent

At (x₁,y₁): xx₁/a² − yy₁/b² = 1
Slope form: y = mx ± √(a²m² − b²) (touch condition c² = a²m² − b²)

Conic comparison

  • Circle e=0; parabola e=1; ellipse 0<e<1; hyperbola e>1.
★ Final-revision priorities (last 24 h)

If you only have 2 hours, read these in this order:

  1. Ch 2 derivative table + chain/product/quotient.
  2. Ch 4 standard integrals + FTC + area & volume.
  3. Ch 6 integration techniques (parts, sub, partial fractions).
  4. Ch 7 separable + homogeneous DE and Newton-Raphson.
  5. Ch 3 max-min + tangent/normal.
  6. Ch 1 limits laws + log/exp identities.
  7. Ch 5 kinematics equations + v-t graphs.
  8. Then Ch 11 circle & Ch 12 parabola tangent forms.

Common booby traps

  • Confusing sin⁻¹ x with 1/sin x.
  • Forgetting the +C on indefinite integrals.
  • Sign errors when integrating cos x (it becomes sin x, NOT −sin x).
  • Forgetting absolute value in ∫(1/x)dx = ln|x|+C.
  • In partial fractions: always check degree of numerator vs. denominator.
  • In Newton-Raphson: choose initial guess close to the root and where f' is not near 0.
  • Ellipse vs hyperbola: c² = a²−b² (ellipse) vs c² = a²+b² (hyperbola).
  • Domain of inverse trig functions — always state the range explicitly.
Tip: Practise both the OR alternatives on every SRQ/ERQ — the model paper deliberately mixes Domain A and Domain B inside the same question slot, so half the SLOs hide in the OR branch.