1. H.C.F of 6 and 20 is (i) 2 (ii) 5 (iii) 120 (iv) None of these. 2. Zeroes of the quadratic polynomial x^2 - 2x - 8 are (i) 2, -4 (ii) -2, 4 (iii) 2, 4 (iv) None of these. 3. If... 1. H.C.F of 6 and 20 is (i) 2 (ii) 5 (iii) 120 (iv) None of these. 2. Zeroes of the quadratic polynomial x^2 - 2x - 8 are (i) 2, -4 (ii) -2, 4 (iii) 2, 4 (iv) None of these. 3. If a1/a2 = b1/b2 = c1/c2, then the pair of linear equations has a (i) Unique solution (ii) no solution (iii) Infinite solution (iv) None of these.
Understand the Problem
The question is a mathematics examination paper containing multiple-choice questions about the highest common factor (HCF), the zeros of a quadratic polynomial, and the conditions for solutions of linear equations.
Answer
HCF: \(2\); Zeros: \(4, -2\); Solutions: Infinite solutions.
Answer for screen readers
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The HCF of 6 and 20 is (2).
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The zeros of the polynomial (x^2 - 2x - 8) are (4) and (-2).
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The condition leads to infinite solutions.
Steps to Solve
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Finding the HCF of 6 and 20
To find the highest common factor (HCF), we can list the factors of both numbers and find the greatest one that they have in common.
- Factors of 6: 1, 2, 3, 6
- Factors of 20: 1, 2, 4, 5, 10, 20
The common factors are 1 and 2. Therefore, the HCF is 2.
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Finding the zeros of the quadratic polynomial (x^2 - 2x - 8)
We use the quadratic formula (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).
For (x^2 - 2x - 8):
- (a = 1)
- (b = -2)
- (c = -8)
Calculate the discriminant: $$D = b^2 - 4ac = (-2)^2 - 4(1)(-8) = 4 + 32 = 36$$
Now, plug in the values: $$x = \frac{-(-2) \pm \sqrt{36}}{2(1)} = \frac{2 \pm 6}{2}$$
This gives:
- (x = \frac{8}{2} = 4)
- (x = \frac{-4}{2} = -2)
Thus, the zeros are (4) and (-2).
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Understanding the conditions for solutions of linear equations
Given the equation ( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} ):
- This condition indicates that the two linear equations are dependent and represent the same line.
- The pair of linear equations has infinite solutions.
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The HCF of 6 and 20 is (2).
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The zeros of the polynomial (x^2 - 2x - 8) are (4) and (-2).
-
The condition leads to infinite solutions.
More Information
- The HCF (highest common factor) finds the largest number that divides two or more numbers.
- The zeros of a polynomial are the points at which it intersects the x-axis.
- Linear equations can have unique solutions, no solution, or infinite solutions based on their relations.
Tips
- A common mistake is miscalculating the HCF by overlooking factors.
- Another mistake is incorrectly applying the quadratic formula, especially in computing the discriminant.
- Misunderstanding the relationship between the coefficients can lead to confusion about the nature of solutions in linear equations.
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