1. The degree of a quadratic polynomial is:
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Answer: C. 2
Explanation: A quadratic polynomial is of the general form ax² + bx + c (a ≠ 0), so its degree is 2.
2. If α and β are the zeroes of the polynomial p(x) = ax² + bx + c, then α + β is equal to:
A. -b/a
B. c/a
C. b/a
D. -c/a
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Answer: A. -b/a
Explanation: The sum of zeroes of a quadratic polynomial ax² + bx + c is given by -b/a.
3. If α and β are the zeroes of the polynomial p(x) = ax² + bx + c, then the product αβ is equal to:
A. -b/a
B. c/a
C. -c/a
D. a/c
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Answer: B. c/a
Explanation: The product of zeroes of a quadratic polynomial ax² + bx + c is given by c/a.
4. The graph of a quadratic polynomial y = ax² + bx + c is always a:
A. Straight line
B. Circle
C. Parabola
D. Ellipse
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Answer: C. Parabola
Explanation: The graph of any quadratic function forms a U-shaped or inverted U-shaped curve called a parabola.
5. How many zeroes can a linear polynomial have at most?
A. 0
B. 1
C. 2
D. Infinite
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Answer: B. 1
Explanation: A polynomial of degree n can have at most n real zeroes. For a linear polynomial (degree 1), the maximum number of zeroes is 1.
6. The zeroes of the polynomial p(x) = x² - 3 are:
A. 3, -3
B. √3, -√3
C. 3, 0
D. √3, 3
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Answer: B. √3, -√3
Explanation: Setting x² - 3 = 0 ⇒ x² = 3 ⇒ x = ±√3.
7. The zeroes of the quadratic polynomial x² + 7x + 10 are:
A. 2, 5
B. -2, -5
C. -2, 5
D. 2, -5
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Answer: B. -2, -5
Explanation: x² + 7x + 10 = (x + 2)(x + 5) = 0 ⇒ x = -2, -5.
8. A quadratic polynomial whose sum and product of zeroes are -3 and 2 respectively is:
A. x² + 3x + 2
B. x² - 3x + 2
C. x² + 3x - 2
D. x² - 3x - 2
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Answer: A. x² + 3x + 2
Explanation: Formula is x² - (Sum)x + (Product) = x² - (-3)x + 2 = x² + 3x + 2.
9. If 1 is a zero of the polynomial p(x) = ax² - 3(a - 1)x - 1, then the value of a is:
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Answer: A. 1
Explanation: p(1) = a(1)² - 3(a - 1)(1) - 1 = 0 ⇒ a - 3a + 3 - 1 = 0 ⇒ -2a + 2 = 0 ⇒ a = 1.
10. If the graph of p(x) intersects the X-axis at 3 distinct points, then the number of zeroes of p(x) is:
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Answer: D. 3
Explanation: The number of zeroes of p(x) is equal to the number of points where the graph intersects the X-axis.
11. If one zero of the quadratic polynomial x² + 3x + k is 2, then the value of k is:
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Answer: B. -10
Explanation: Substitute x = 2 into 2² + 3(2) + k = 0 ⇒ 4 + 6 + k = 0 ⇒ k = -10.
12. The degree of the constant polynomial 5 is:
A. 0
B. 1
C. 5
D. Not defined
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Answer: A. 0
Explanation: A non-zero constant polynomial can be written as 5x⁰, so its degree is 0.
13. The degree of the zero polynomial is:
A. 0
B. 1
C. Any natural number
D. Not defined
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Answer: D. Not defined
Explanation: The zero polynomial p(x) = 0 has an undefined degree because 0 can be multiplied by x to any power.
14. If α and β are zeroes of p(x) = x² - 5x + 6, then the value of α + β - αβ is:
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Answer: B. -1
Explanation: α + β = -(-5)/1 = 5, and αβ = 6/1 = 6. Thus, α + β - αβ = 5 - 6 = -1.
15. If the zeroes of the quadratic polynomial ax² + bx + c are reciprocal to each other, then:
A. a = c
B. b = c
C. a = b
D. a = -c
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Answer: A. a = c
Explanation: If zeroes are α and 1/α, product of zeroes = α × (1/α) = 1. Since αβ = c/a, we get c/a = 1 ⇒ a = c.
16. If one zero of the quadratic polynomial 2x² - 8x - m is 5, then the other zero is:
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Answer: B. -1
Explanation: Sum of zeroes α + β = -(-8)/2 = 4. Given α = 5 ⇒ 5 + β = 4 ⇒ β = -1.
17. Which of the following expressions is NOT a polynomial?
A. x² + 2x + 1
B. x + 1/x
C. 3x³ - 5x + 2
D. x/2 + 5
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Answer: B. x + 1/x
Explanation: 1/x = x⁻¹, which has a negative exponent. Polynomial exponents must be non-negative integers.
18. The number of zeroes of a cubic polynomial is at most:
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Answer: C. 3
Explanation: A cubic polynomial has degree 3, so it can have at most 3 real zeroes.
19. If α, β are zeroes of x² - p(x + 1) - c such that (α + 1)(β + 1) = 0, then c is equal to:
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Answer: A. 1
Explanation: Expanding: x² - px - (p + c). Here α + β = p and αβ = -(p + c). (α + 1)(β + 1) = αβ + (α + β) + 1 = -(p + c) + p + 1 = 1 - c = 0 ⇒ c = 1.
20. The quadratic polynomial whose zeroes are 2 and -2 is:
A. x² + 4
B. x² - 4
C. x² - 2
D. x² + 2
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Answer: B. x² - 4
Explanation: Polynomial = (x - 2)(x + 2) = x² - 4.
21. If the sum of zeroes of the polynomial kx² + 2x + 3k is equal to their product, then k is equal to:
A. 1/3
B. -2/3
C. 2/3
D. -1/3
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Answer: B. -2/3
Explanation: Sum = -2/k, Product = 3k/k = 3. Given -2/k = 3 ⇒ k = -2/3.
22. If α, β are zeroes of 2x² + 5x + k such that α² + β² + αβ = 21/4, then k is:
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Answer: B. 2
Explanation: α + β = -5/2, αβ = k/2. α² + β² + αβ = (α + β)² - αβ = 25/4 - k/2 = 21/4 ⇒ k/2 = 1 ⇒ k = 2.
23. If the graph of a polynomial does not intersect the X-axis at all, then the number of real zeroes is:
A. 0
B. 1
C. 2
D. Infinite
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Answer: A. 0
Explanation: Real zeroes correspond to the points of intersection with the X-axis. If there are no intersection points, there are 0 real zeroes.
24. The zeroes of x² - 2x - 8 are:
A. 4, -2
B. -4, 2
C. 4, 2
D. -4, -2
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Answer: A. 4, -2
Explanation: x² - 2x - 8 = (x - 4)(x + 2) = 0 ⇒ x = 4, -2.
25. If α and β are the zeroes of x² + x + 1, then 1/α + 1/β is equal to:
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Answer: B. -1
Explanation: 1/α + 1/β = (α + β) / αβ = (-1/1) / (1/1) = -1.
26. A polynomial of degree 4 is called a:
A. Cubic polynomial
B. Quadratic polynomial
C. Biquadratic (or Quartic) polynomial
D. Linear polynomial
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Answer: C. Biquadratic (or Quartic) polynomial
Explanation: A polynomial of degree 4 is known as a biquadratic or quartic polynomial.
27. If one zero of 3x² + 8x + k is reciprocal of the other, then value of k is:
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Answer: A. 3
Explanation: Product of zeroes = c/a ⇒ α × (1/α) = k/3 ⇒ 1 = k/3 ⇒ k = 3.
28. What is the value of p(2) for the polynomial p(x) = x³ - 2x² + 3x - 5?
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Answer: A. 1
Explanation: p(2) = (2)³ - 2(2)² + 3(2) - 5 = 8 - 8 + 6 - 5 = 1.
29. If α and β are zeroes of 4x² - 4x + 1, then α + β is:
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Answer: A. 1
Explanation: α + β = -(-4)/4 = 4/4 = 1.
30. If the zeroes of x² + (a + 1)x + b are 2 and -3, then:
A. a = -7, b = -1
B. a = 5, b = -1
C. a = 0, b = -6
D. a = 2, b = -6
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Answer: C. a = 0, b = -6
Explanation: Sum of zeroes = 2 + (-3) = -1 = -(a + 1) ⇒ a + 1 = 1 ⇒ a = 0. Product = 2 × (-3) = -6 = b ⇒ b = -6.
