Class 10th Maths notes
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📖 QUIZ EXPLANATION
❓ Question: If the discriminant of a quadratic equation is zero, the roots are:
✅ Correct Answer: Real and equal
💡 Explanation:
The correct answer is Real and equal.
Reason - When Δ = b² - 4ac = 0, the quadratic formula gives x = -b/2a twice, resulting in one repeated real root.
Concept - This means the parabola touches the x-axis at exactly one point (the vertex), making it a perfect square trinomial.
Fact - Examples include x² - 4x + 4 = 0, where Δ = 16 - 16 = 0, giving the double root x = 2.
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If the discriminant of a quadratic equation is zero, the roots are:
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📖 QUIZ EXPLANATION
❓ Question: The discriminant of the quadratic equation ax² + bx + c = 0 is:
✅ Correct Answer: b² - 4ac
💡 Explanation:
The correct answer is b² - 4ac.
Reason - The discriminant (Δ) is defined as b² - 4ac and determines the nature of roots in the quadratic formula.
Concept - The discriminant tells us about the roots: if Δ > 0 (two distinct real roots), Δ = 0 (one repeated real root), or Δ < 0 (no real roots).
Fact - This expression appears under the square root in the quadratic formula: x = (-b ± √(b² - 4ac))/2a.
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The discriminant of the quadratic equation ax² + bx + c = 0 is:
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📖 QUIZ EXPLANATION
❓ Question: The solution of x + y = 5 and x - y = 1 is:
✅ Correct Answer: (3, 2)
💡 Explanation:
The correct answer is (3, 2).
Reason - Adding the equations: (x + y) + (x - y) = 5 + 1 gives 2x = 6, so x = 3. Substituting: 3 + y = 5, so y = 2.
Concept - This uses the elimination method to solve simultaneous linear equations by eliminating one variable.
Fact - Verification: 3 + 2 = 5 ✓ and 3 - 2 = 1 ✓, confirming that (3, 2) satisfies both equations.
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📖 QUIZ EXPLANATION
❓ Question: A pair of linear equations has unique solution when:
✅ Correct Answer: a₁/a₂ ≠ b₁/b₂
💡 Explanation:
The correct answer is a₁/a₂ ≠ b₁/b₂.
Reason - When the ratios of coefficients are unequal, the lines intersect at exactly one point, giving a unique solution to the system.
Concept - This condition ensures that the two linear equations represent different, non-parallel lines that must intersect at one point.
Fact - If a₁/a₂ = b₁/b₂ but ≠ c₁/c₂, the lines are parallel (no solution), and if a₁/a₂ = b₁/b₂ = c₁/c₂, the lines are identical (infinitely many solutions).
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📖 QUIZ EXPLANATION
❓ Question: The zeroes of the polynomial x² - 5x + 6 are:
✅ Correct Answer: 2, 3
💡 Explanation:
The correct answer is 2, 3.
Reason - Setting x² - 5x + 6 = 0 and factoring gives (x - 2)(x - 3) = 0, so x = 2 or x = 3.
Concept - Zeroes are the values of x that make the polynomial equal to zero, and they correspond to the x-intercepts of the polynomial's graph.
Fact - This can be verified by substitution: (2)² - 5(2) + 6 = 4 - 10 + 6 = 0, and (3)² - 5(3) + 6 = 9 - 15 + 6 = 0.
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📖 QUIZ EXPLANATION
❓ Question: What is the degree of the polynomial 3x² + 2x + 1?
✅ Correct Answer: 2
💡 Explanation:
The correct answer is 2.
Reason - The degree of a polynomial is the highest power of the variable, and in 3x² + 2x + 1, the highest power of x is 2.
Concept - The degree determines many properties of the polynomial, including the maximum number of roots and the general shape of its graph.
Fact - This is a quadratic polynomial (degree 2), which means its graph is a parabola and it can have at most 2 real roots.
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📖 QUIZ EXPLANATION
❓ Question: The decimal expansion of a rational number is:
✅ Correct Answer: Either terminating or non-terminating repeating
💡 Explanation:
The correct answer is Either terminating or non-terminating repeating.
Reason - Rational numbers in decimal form either terminate (like 1/2 = 0.5) or have infinitely repeating patterns (like 1/3 = 0.333...).
Concept - The nature of decimal expansion depends on the prime factorization of the denominator when the fraction is in lowest terms - only factors of 2 and 5 give terminating decimals.
Fact - Examples include 3/4 = 0.75 (terminating) and 2/3 = 0.666... (non-terminating repeating), while irrational numbers like π have non-terminating, non-repeating expansions.
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📖 QUIZ EXPLANATION
❓ Question: Which of the following is an irrational number?
✅ Correct Answer: √2
💡 Explanation:
The correct answer is √2.
Reason - √2 is an irrational number because it cannot be expressed as a ratio of two integers and its decimal expansion is non-terminating and non-recurring.
Concept - Irrational numbers are real numbers that cannot be written as fractions (p/q where p and q are integers and q ≠ 0).
Fact - 2/3 and 0.5 are rational numbers, while π/π = 1, which is also rational. Only √2 among the given options is irrational.
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