Chapter 1β2 Real Numbers & Polynomials
Q1
HCF (36mΒ², 18m) = 18m, where m is a prime number. Which statement about this is correct?
β
Explanation
36mΒ² = 2 Γ 18 Γ m Γ m and 18m = 18 Γ m. The common factors are 18 and m. So HCF = 18m. Note: HCF is always β€ the smaller number (18m here).Q2
If the zeroes of a polynomial p(x) are β3 and 8, then the product of zeroes is:
β
Explanation
Product of zeroes = Ξ± Γ Ξ² = (β3) Γ (8) = β24. Sum of zeroes = β3 + 8 = 5. For a polynomial axΒ² + bx + c, product of zeroes = c/a and sum = βb/a.Q3
The value of m for which 14 β 2β3 is proved irrational (given β3 is irrational) uses which method?
β
Explanation
We assume 14 β 2β3 is rational (= a/b). This leads to β3 = (14bβa)/2b, which is rational β contradicting the given fact that β3 is irrational. This is proof by contradiction.Chapter 3β4 Linear & Quadratic Equations
Q4
The system 3x β 5y + 7 = 0 and β6x + 10y + 14 = 0 is:
β
Explanation
aβ/aβ = 3/β6 = β1/2, bβ/bβ = β5/10 = β1/2, cβ/cβ = 7/14 = 1/2. Since aβ/aβ = bβ/bβ β cβ/cβ, the lines are parallel β no solution β inconsistent.Q5
For the quadratic equation 3xΒ² β 7x + m = 0 to have real and equal roots, the value of m is:
β
Explanation
For equal roots, discriminant D = 0. D = bΒ² β 4ac = (β7)Β² β 4(3)(m) = 49 β 12m = 0. So 12m = 49, giving m = 49/12.Q6
Two water taps together fill a tank in 8βΈββ hours. The larger tap takes 4 hours less than the smaller. The time taken by the smaller tap alone is:
β
Explanation
Let smaller tap = x hrs, larger = (xβ4) hrs. Together: 80/9 Γ [1/x + 1/(xβ4)] = 1. This gives 9xΒ² β 196x + 320 = 0. Solving: x = 20 (x = 16/9 rejected). Smaller tap = 20 hrs.Chapter 5 Arithmetic Progressions
Q7
For the A.P. β15/4, β10/4, β5/4, β¦ the value of aββ β aββ is:
β
Explanation
Common difference d = β10/4 β (β15/4) = 5/4. aββ β aββ = (a + 15d) β (a + 11d) = 4d = 4 Γ 5/4 = 5.Q8
The sum of first n terms of an AP is Sβ = 2nΒ² + 13n. The 10th term of this AP is:
β
Explanation
aβ = Sβ β Sβββ = (2nΒ² + 13n) β [2(nβ1)Β² + 13(nβ1)] = 4n + 11. So aββ = 4(10) + 11 = 51... wait: 40 + 11 = 51? Let me recheck: aβ = 2nΒ² + 13n β 2nΒ² + 4n β 2 β 13n + 13 = 4n + 11. aββ = 40 + 11 = 51. Hmm but the answer in the board paper was 31. Let me recheck: Sβ = 2nΒ² + 13n. aβ = Sβ β Sβββ = 2nΒ²+13n β [2(n-1)Β²+13(n-1)] = 2nΒ²+13n β [2nΒ²β4n+2+13nβ13] = 2nΒ²+13nβ2nΒ²β9n+11 = 4n+11. So aββ = 51. The correct answer is 51.Chapter 6β7 Triangles & Coordinate Geometry
Q9
In β³ABC, D divides BC in ratio 1:2. With A(1,5), B(β2,1), C(4,2), the coordinates of D are:
β
Explanation
Section formula with mβ:mβ = 1:2: x = (1Γ4 + 2Γ(β2))/(1+2) = (4β4)/3 = 0. y = (1Γ2 + 2Γ1)/(1+2) = 4/3. So D = (0, 4/3).Q10
The line joining P(β4, β2) and Q(10, 4) is divided by the y-axis in the ratio:
β
Explanation
On y-axis, x = 0. Using section formula: (kΓ10 + 1Γ(β4))/(k+1) = 0 β 10k β 4 = 0 β k = 2/5. So ratio = 2:5.Advertisement
Chapter 9β12 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 =
β
Explanation
Tangent β₯ radius at point of contact. So OQ β₯ PQ. By Pythagoras: PQΒ² = OPΒ² β OQΒ² = 13Β² β 5Β² = 169 β 25 = 144. PQ = 12 cm.Q12
An arc of length 2.2 cm subtends an angle ΞΈ at the centre of a circle with radius 2.8 cm. The value of ΞΈ is:
β
Explanation
Arc length = (ΞΈ/360Β°) Γ 2Οr. So 2.2 = (ΞΈ/360Β°) Γ 2 Γ (22/7) Γ 2.8 = (ΞΈ/360Β°) Γ 17.6. ΞΈ = (2.2 Γ 360Β°)/17.6 = 45Β°.Q13
A conical cavity of maximum volume is carved from a solid hemisphere of radius 10 cm. Curved surface area of the cone is (Ο = 3.14):
β
Explanation
For max volume cone in hemisphere: height = radius = 10 cm. Slant height β = β(rΒ²+hΒ²) = β(100+100) = 10β2 cm. CSA = Οrβ = 3.14 Γ 10 Γ 10β2 = 314β2 cmΒ².Q14
A camping tent is hemispherical with radius 1.4 m and has a door of area 0.50 mΒ². The outer surface area of the tent is:
β
Explanation
Outer surface = CSA of hemisphere β area of door = 2ΟrΒ² β 0.50 = 2 Γ (22/7) Γ 1.4Β² β 0.5 = 12.32 β 0.5 = 11.82 mΒ².Chapter 8 & 9 Trigonometry & Heights/Distances
Q15
Simplest form of sec A / β(secΒ²A β 1) is:
β
Explanation
secΒ²A β 1 = tanΒ²A (identity). So sec A / βtanΒ²A = 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 60Β° elevation. If AB = 5β3 m, the length of the wire AC is:
β
Explanation
cos 60Β° = AB/AC β 1/2 = 5β3/AC β AC = 10β3 m. Alternatively, tan 60Β° = BC/AB β BC = 5β3 Γ β3 = 15 m. Then AC = β(ABΒ² + BCΒ²) = β(75+225) = β300 = 10β3 m. βChapter 13β14 Statistics & Probability
Q17
In step deviation method, xΜ = 64, h = 5, a = 62.5. The value of Ε« is:
β
Explanation
xΜ = a + Ε« Γ h β 64 = 62.5 + Ε« Γ 5 β 1.5 = 5Ε« β Ε« = 0.3.Q18
Two dice are rolled together. The probability of getting outcome (x, y) where x > y is:
β
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?
β
Explanation
P(winning) = tickets bought / total tickets β 0.08 = n/800 β n = 0.08 Γ 800 = 64 tickets.Q20
Which of the following cannot be the probability of an event?
β
Explanation
Probability must satisfy 0 β€ P(E) β€ 1. Option C: 10/0.2 = 50. Since 50 > 1, it cannot be a probability. All other options are between 0 and 1.Advertisement