Mathematics — MCQ Practice Questions & Answers

Question 1

If the sum of the roots of the quadratic equation 2x² + (k-3)x + (k-5) = 0 is equal to half of their product, then the value of k is:

Accepted Answer

Correct answer: A. 11 — For a quadratic equation ax² + bx + c = 0, sum of roots = -b/a and product of roots = c/a. Here, sum = -(k-3)/2 and product = (k-5)/2. Given that sum = (1/2) × product, we have -(k-3)/2 = (1/2) × (k-5)/2. Solving: -(k-3)/2 = (k-5)/4, which gives -2(k-3) = k-5, leading to -2k+6 = k-5, thus k = 11.

Answer

B. 9

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C. 7

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D. 5

Question 2

If x, y, z are in arithmetic progression and x + y + z = 15, then the middle term is:

Accepted Answer

Correct answer: B. 5 — When three numbers are in arithmetic progression, they can be written as (a-d), a, (a+d) where a is the middle term. Their sum is (a-d) + a + (a+d) = 3a = 15, giving a = 5. Therefore, the middle term y = 5.

Answer

A. 3

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C. 7

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D. 9

Question 3

If p and q are roots of x² - 5x + 6 = 0, then the value of (1/p + 1/q) is:

Accepted Answer

Correct answer: A. 5/6 — Using Vieta's formulas: p + q = 5 and pq = 6. We need 1/p + 1/q = (p+q)/pq = 5/6. This approach avoids finding individual roots and uses the relationship between roots directly.

Answer

B. 6/5

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C. 1/2

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D. 2

Question 4

If α and β are roots of 3x² - 4x + k = 0 and α² + β² = 10/9, then k equals:

Accepted Answer

Correct answer: C. 1/3 — From Vieta's formulas: α + β = 4/3 and αβ = k/3. Using α² + β² = (α+β)² - 2αβ, we get 10/9 = (4/3)² - 2(k/3) = 16/9 - 2k/3. Solving: 10/9 = 16/9 - 2k/3 gives 2k/3 = 6/9 = 2/3, so k = 1/3.

Answer

A. 1

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B. 2/3

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D. 4/3

Question 5

If a, b, c are in geometric progression with common ratio r, and a + b + c = 39, while a² + b² + c² = 651, then the value of r is:

Accepted Answer

Correct answer: B. 2 — In GP: b = ar and c = ar². Given a + ar + ar² = 39 and a² + a²r² + a²r⁴ = 651. From the first equation: a(1 + r + r²) = 39. From the second: a²(1 + r² + r⁴) = 651. Dividing these equations and solving yields r = 2, which satisfies both conditions when a = 3.

Answer

A. 1/2

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C. 3

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D. 1/3

Question 6

A circle is inscribed in a square of side 8 cm. If a smaller square is inscribed in the circle, what is the area of the smaller square?

Accepted Answer

Correct answer: A. 32 cm² — The inscribed circle in the square has diameter 8 cm, so radius = 4 cm. When a square is inscribed in this circle, its diagonal equals the diameter (8 cm). If the diagonal of the smaller square is 8 cm, then using diagonal = side√2, we get side = 8/√2 = 4√2 cm. Therefore, area = (4√2)² = 32 cm².

Answer

B. 64 cm²

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C. 16 cm²

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D. 48 cm²

Question 7

In a right-angled triangle, the altitude drawn to the hypotenuse divides it into segments of lengths 9 cm and 16 cm. What is the length of the altitude?

Accepted Answer

Correct answer: A. 12 cm — By the geometric mean altitude theorem, when an altitude is drawn to the hypotenuse of a right triangle, the altitude is the geometric mean of the two segments of the hypotenuse. Therefore, altitude = √(9 × 16) = √144 = 12 cm. This is a fundamental property used frequently in competitive exams.

Answer

B. 15 cm

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C. 18 cm

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D. 20 cm

Question 8

Two concentric circles have radii r and R (where r < R). The area of the ring formed between them is equal to the area of the inner circle. What is the ratio R:r?

Accepted Answer

Correct answer: A. √2:1 — The area of the ring = π(R² - r²). Given that this equals the area of the inner circle = πr², we have R² - r² = r², which gives R² = 2r², so R = r√2. Therefore, the ratio R:r = √2:1.

Answer

B. 2:1

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C. √3:1

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D. 3:1

Mathematics - MCQ Practice Questions