Chapter 1–2 Real Numbers & Polynomials
Q1
HCF (36m², 18m) = 18m, where m is a prime number. Which statement about this is correct?
✅ Explanation
36m² = 2 × 18 × m × m and 18m = 18 × m. The common factors are 18 and m. So HCF = 18m. Note: HCF is always ≤ the smaller number (18m here).Q2
If the zeroes of a polynomial p(x) are −3 and 8, then the product of zeroes is:
✅ Explanation
Product of zeroes = α × β = (−3) × (8) = −24. Sum of zeroes = −3 + 8 = 5. For a polynomial ax² + bx + c, product of zeroes = c/a and sum = −b/a.Q3
The value of m for which 14 − 2√3 is proved irrational (given √3 is irrational) uses which method?
✅ Explanation
We assume 14 − 2√3 is rational (= a/b). This leads to √3 = (14b−a)/2b, which is rational — contradicting the given fact that √3 is irrational. This is proof by contradiction.Chapter 3–4 Linear & Quadratic Equations
Q4
The system 3x − 5y + 7 = 0 and −6x + 10y + 14 = 0 is:
✅ Explanation
a₁/a₂ = 3/−6 = −1/2, b₁/b₂ = −5/10 = −1/2, c₁/c₂ = 7/14 = 1/2. Since a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel → no solution → inconsistent.Q5
For the quadratic equation 3x² − 7x + m = 0 to have real and equal roots, the value of m is:
✅ Explanation
For equal roots, discriminant D = 0. D = b² − 4ac = (−7)² − 4(3)(m) = 49 − 12m = 0. So 12m = 49, giving m = 49/12.Q6
Two water taps together fill a tank in 8⁸⁄₉ hours. The larger tap takes 4 hours less than the smaller. The time taken by the smaller tap alone is:
✅ Explanation
Let smaller tap = x hrs, larger = (x−4) hrs. Together: 80/9 × [1/x + 1/(x−4)] = 1. This gives 9x² − 196x + 320 = 0. Solving: x = 20 (x = 16/9 rejected). Smaller tap = 20 hrs.Chapter 5 Arithmetic Progressions
Q7
For the A.P. −15/4, −10/4, −5/4, … the value of a₁₆ − a₁₂ is:
✅ Explanation
Common difference d = −10/4 − (−15/4) = 5/4. a₁₆ − a₁₂ = (a + 15d) − (a + 11d) = 4d = 4 × 5/4 = 5.Q8
The sum of first n terms of an AP is Sₙ = 2n² + 13n. The 10th term of this AP is:
✅ Explanation
aₙ = Sₙ − Sₙ₋₁ = (2n² + 13n) − [2(n−1)² + 13(n−1)] = 4n + 11. So a₁₀ = 4(10) + 11 = 51... wait: 40 + 11 = 51? Let me recheck: aₙ = 2n² + 13n − 2n² + 4n − 2 − 13n + 13 = 4n + 11. a₁₀ = 40 + 11 = 51. Hmm but the answer in the board paper was 31. Let me recheck: Sₙ = 2n² + 13n. aₙ = Sₙ − Sₙ₋₁ = 2n²+13n − [2(n-1)²+13(n-1)] = 2n²+13n − [2n²−4n+2+13n−13] = 2n²+13n−2n²−9n+11 = 4n+11. So a₁₀ = 51. The correct answer is 51.Chapter 6–7 Triangles & Coordinate Geometry
Q9
In △ABC, D divides BC in ratio 1:2. With A(1,5), B(−2,1), C(4,2), the coordinates of D are:
✅ Explanation
Section formula with m₁:m₂ = 1:2: x = (1×4 + 2×(−2))/(1+2) = (4−4)/3 = 0. y = (1×2 + 2×1)/(1+2) = 4/3. So D = (0, 4/3).Q10
The line joining P(−4, −2) and Q(10, 4) is divided by the y-axis in the ratio:
✅ Explanation
On y-axis, x = 0. Using section formula: (k×10 + 1×(−4))/(k+1) = 0 → 10k − 4 = 0 → k = 2/5. So ratio = 2:5.Chapter 9–12 Circles, Areas & Surface Areas
Q11
PQ and PR are tangents from P to a circle (radius 5 cm, centre O). If OP = 13 cm, then PQ =
✅ Explanation
Tangent ⊥ radius at point of contact. So OQ ⊥ PQ. By Pythagoras: PQ² = OP² − OQ² = 13² − 5² = 169 − 25 = 144. PQ = 12 cm.Q12
An arc of length 2.2 cm subtends an angle θ at the centre of a circle with radius 2.8 cm. The value of θ is:
✅ Explanation
Arc length = (θ/360°) × 2πr. So 2.2 = (θ/360°) × 2 × (22/7) × 2.8 = (θ/360°) × 17.6. θ = (2.2 × 360°)/17.6 = 45°.Q13
A conical cavity of maximum volume is carved from a solid hemisphere of radius 10 cm. Curved surface area of the cone is (π = 3.14):
✅ Explanation
For max volume cone in hemisphere: height = radius = 10 cm. Slant height ℓ = √(r²+h²) = √(100+100) = 10√2 cm. CSA = πrℓ = 3.14 × 10 × 10√2 = 314√2 cm².Q14
A camping tent is hemispherical with radius 1.4 m and has a door of area 0.50 m². The outer surface area of the tent is:
✅ Explanation
Outer surface = CSA of hemisphere − area of door = 2πr² − 0.50 = 2 × (22/7) × 1.4² − 0.5 = 12.32 − 0.5 = 11.82 m².Chapter 8 & 9 Trigonometry & Heights/Distances
Q15
Simplest form of sec A / √(sec²A − 1) is:
✅ Explanation
sec²A − 1 = tan²A (identity). So sec A / √tan²A = sec A / tan A = (1/cos A) / (sin A/cos A) = 1/sin A = cosec A.Q16
A wire from point A on ground reaches the top of pole BC at 60° elevation. If AB = 5√3 m, the length of the wire AC is:
✅ Explanation
cos 60° = AB/AC → 1/2 = 5√3/AC → AC = 10√3 m. Alternatively, tan 60° = BC/AB → BC = 5√3 × √3 = 15 m. Then AC = √(AB² + BC²) = √(75+225) = √300 = 10√3 m. ✓Chapter 13–14 Statistics & Probability
Q17
In step deviation method, x̄ = 64, h = 5, a = 62.5. The value of ū is:
✅ Explanation
x̄ = a + ū × h → 64 = 62.5 + ū × 5 → 1.5 = 5ū → ū = 0.3.Q18
Two dice are rolled together. The probability of getting outcome (x, y) where x > y is:
✅ Explanation
Favourable outcomes (x > y): (2,1),(3,1),(3,2),(4,1),(4,2),(4,3),(5,1),(5,2),(5,3),(5,4),(6,1),(6,2),(6,3),(6,4),(6,5) = 15 outcomes. P = 15/36 = 5/12.Q19
Meena's probability of winning a lottery is 0.08. If 800 tickets were sold, how many did she buy?
✅ Explanation
P(winning) = tickets bought / total tickets → 0.08 = n/800 → n = 0.08 × 800 = 64 tickets.Q20
Which of the following cannot be the probability of an event?
✅ Explanation
Probability must satisfy 0 ≤ P(E) ≤ 1. Option C: 10/0.2 = 50. Since 50 > 1, it cannot be a probability. All other options are between 0 and 1.