CBSE 50 MARKS PAPER ANSWER KEY

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📘 ANSWER KEY (NUMBER-WISE FORMAT)

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1.

HCF(a, b) = 1
HCF × LCM = a × b

⇒ 1 × LCM = ab
⇒ LCM = ab

✔ Correct Option: (D)

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2.

3 + √2 = Rational + Irrational = Irrational

✔ Correct Option: (A)

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3.

x + 1/x = 3
⇒ x² + 1 = 3x
⇒ x² − 3x + 1 = 0

a = 1, b = −3, c = 1

a − b + c = 1 − (−3) + 1 = 5

✔ Correct Option: (A)

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4.

P(Sure Event) = 1
All outcomes = {1, 2, 3, 4, 5, 6}

✔ Correct Option: (D)

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5.

Assertion: LCM is multiple of HCF ✔
Reason: HCF is highest common factor ✔

✔ Both true, R explains A

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6.

Assume √3 = p/q (co-prime)

3 = p²/q²
⇒ 3q² = p²

⇒ p = 3k

3q² = 9k²
⇒ q² = 3k²

⇒ q divisible by 3

❌ Contradiction

✔ √3 is irrational

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7.

Total = 200, Defective = 20

P(not buy) = 20/200 = 1/10

New total = 300
Good = 260

P(buy) = 260/300 = 13/15

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8.

HCF = 1, LCM = 15

HCF − LCM = 1 − 15 = −14

✔ Correct Option: (D)

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9.

2ⁿ ends in: 2, 4, 8, 6

❌ Never ends in 1

✔ Correct Option: (A)

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10.

(x + 2)³ = 2x(x² − 1)

x³ + 6x² + 12x + 8 = 2x³ − 2x
⇒ x³ − 6x² − 14x − 8 = 0 (Cubic)

x² + x − 2 = 2x + 1
⇒ x² − x − 3 = 0 (Quadratic)

✔ Correct Option: (B)

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11.

Total outcomes (3 dice) = 6³ = 216

✔ Correct Option: (D)

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12.

Prime factors of 36 = 2, 3 ✔
Both divide 6 ✔

✔ Both true, R explains A

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13.

36 = 2² × 3²
54 = 2 × 3³

LCM = 2² × 3³ = 108

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14.

Total outcomes = 10

(i) Prime > 2 → {3, 5, 7}
P = 3/10

(ii) Odd < 9 → {1, 3, 5, 7}
P = 4/10 = 2/5

(iii) Multiple of 4 → {4, 8}
P = 2/10 = 1/5

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15.

b = 35 ÷ 5 = 7
a = 210 ÷ 70 = 3

y = 420
x = 840

x = 2³ × 3 × 5 × 7

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16.

p = 2³ × 3² × 5
q = 2² × 3³

LCM = 2³ × 3³ × 5

✔ Correct Option: (D)

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17.

3ⁿ ends in: 3, 9, 7, 1

❌ Never ends in 5

✔ Correct Option: (B)

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18.

Prime number → exactly 2 factors

✔ Correct Option: (A)

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19.

x = √x

x² = x
⇒ x² − x = 0

a = 1, b = −1, c = 0

a + b + c = 0

✔ Correct Option: (A)

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20.

Sum ≥ 9 → 10 outcomes

P = 10/36

✔ Correct Option: (C)

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21.

a = Composite
b = Composite
c = Prime

✔ Answer: a and b

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22.

9s² − 6s + 1

(3s − 1)² = 0

Roots = 1/3, 1/3

Sum = 2/3 ✔

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23.

Total = 18

(i) White = 2/9
(ii) White or Red = 2/3
(iii) Not Red = 5/9

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24.

3x² + 6x + k

−2 + k/3 = −2/3

⇒ k = 4

✔ Correct Option: (D)

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25.

98 = 2 × 7²
28 = 2² × 7

HCF = 14
LCM = 196

n − 7m = 98

✔ Correct Option: (C)

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26.

Distinct zeroes = 2

✔ Correct Option: (C)

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27.

Red face cards = 6

P = 6/52 = 3/26

✔ Correct Option: (D)

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28.

√3 ≈ 1.732
√5 ≈ 2.236

13/7 = 1.857 ✔

✔ Correct Option: (D)

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29.

Assertion: False
Reason: True

✔ Answer: A false, R true

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30.

3x² + 4x − 4 = 0

(3x − 2)(x + 2) = 0

Roots = 2/3, −2

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31.

Favourable outcomes = 8

P = 8/36 = 2/9

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