Sylhet Board · SSC Mathematics · 2022
Chapter Coverage
| Q | Chapter | Topic | Marks |
|---|---|---|---|
| 1 | Ch 1 – Real Numbers | Surd operations, hill measurements | 10 |
| 2 | Ch 3 – Algebraic Expressions | if x+y=7, xy=12, find algebraic values | 10 |
| 3 | Ch 5 – Simultaneous Equations | Two hill villages, distance and population | 10 |
| 4 | Ch 9 – Geometric Series | Population of hill town, exponential growth | 10 |
| 5 | Ch 11 – Circles | Tangent-secant from hill observation | 10 |
| 6 | Ch 13 – Mensuration | Hemispherical hill cave: surface and volume | 10 |
| 7 | Ch 14 – Trigonometry | Angle of depression from hill top to road | 10 |
| 8 | Ch 15 – Statistics | Rainfall data, cumulative frequency | 10 |
Two hill trail lengths are √180 m and √45 m.
(a) [2m] Simplify each surd fully.
(b) [4m] Find 2√180 − 3√45 and √180 × √45.
(c) [4m] Show that √(180/45) = 2 and explain in terms of surd laws.
√45 = √(9×5) = 3√5
√180×√45 = 6√5×3√5 = 18×5 = 90
Using surd law: √(a/b) = √a/√b = 6√5/(3√5) = 2 ✓
This is the surd quotient law: √(m/n) = √m/√n for positive m, n.
The length and width of a hill plot satisfy: x + y = 7 and xy = 12.
(a) [2m] Find x² + y².
(b) [4m] Find x³ + y³.
(c) [4m] Find the values of x and y individually. Then evaluate x⁴ + y⁴.
x=4, y=3 (or x=3, y=4)
x⁴+y⁴ = (x²+y²)²−2(xy)² = 25²−2×144 = 625−288 = 337
Village A and Village B are in the Sylhet hills. The sum of their populations is 4200. Village A has 3 times the population of Village B minus 600.
(a) [2m] Set up a system of equations.
(b) [4m] Solve for the individual populations.
(c) [4m] If Village A grows by 10% and Village B shrinks by 5%, find the new total population.
A = 3B − 600
A = 3(1200)−600 = 3000
New total = 3300+1140 = 4440
A Sylhet hill town has population 10,000 and grows at 5% per year (geometric growth).
(a) [2m] Write the population as a GP with first term and ratio.
(b) [4m] Find the population after 4 years. (1.05⁴ = 1.2155)
(c) [4m] How many complete years until the population doubles? (log 2 = 0.3010, log 1.05 = 0.0212)
Pₙ = 10000 × 1.05^(n−1)
1.05ⁿ = 2
n·log 1.05 = log 2
n = 0.3010/0.0212 = 14.2 → 15 years (complete years)
From an external point T on a hilltop, a tangent TA touches a circle at A, and a secant TBC passes through the circle (B and C on circle). TB = 4 cm, TC = 9 cm.
(a) [2m] State the tangent-secant theorem.
(b) [4m] Find the length of tangent TA using the theorem.
(c) [4m] Find BC (the chord). If the circle radius is r and OA⊥TA, find r given OT=10 cm.
TA = 6 cm
OA ⊥ TA (radius to tangent point), OT = 10, TA = 6
OA² = OT²−TA² = 100−36 = 64 → r = OA = 8 cm
A hemispherical cave entrance in a Sylhet hill has radius 3.5 m. (π = 22/7)
(a) [2m] Find the curved surface area of the hemisphere.
(b) [4m] Find the total surface area (including flat circular base).
(c) [4m] Find the volume of the hemisphere.
Total SA = 77+38.5 = 115.5 m²
= (2/3)×134.75 = 89.83 m³
From the top of a 100 m hill, a person sees a road below. The angle of depression to the near end of the road is 60° and to the far end is 30°.
(a) [2m] Draw and label the diagram.
(b) [4m] Find the horizontal distances to both ends of the road from the base of the hill. (tan 30°=1/√3, tan 60°=√3)
(c) [4m] Find the length of the road section visible.
Far end: tan 30° = 100/d₂ → d₂ = 100√3 ≈ 173.2 m
Annual rainfall (mm) in Sylhet over 6 years: 3200, 3650, 2900, 4100, 3500, 3750.
(a) [2m] Find the mean annual rainfall.
(b) [4m] Find the variance.
(c) [4m] Find the standard deviation. Compare with Barisal mean of 248 mm/month (multiply Sylhet mean by appropriate factor for comparison).
Mean = 21100/6 = 3516.67 mm/year
Squared: 100,279, 17,769, 380,282, 340,274, 278, 54,443
Sum ≈ 893,325
Variance = 893325/6 ≈ 148,888 mm²
Sylhet monthly mean = 3516.67/12 ≈ 293 mm/month vs Barisal 248 mm/month
Sylhet receives about 18% more monthly rainfall than Barisal.