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    • Class 10 Maths Chapter 4 - Quadratic Equations

    1. Assertion (A): The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0 .
      Reason (R): In a quadratic equation, aa, bb, and cc can be any real numbers.
      • A) Both A and R are true, and R is the correct explanation of A.
      • B) Both A and R are true, but R is not the correct explanation of A.
      • C) A is true, but R is false.
      • D) A is false, but R is true.
      • Answer: C) A is true, but R is false.
      • Explanation: While aa, bb, and cc are real numbers, aa cannot be zero in a quadratic equation.
    1. Assertion (A): The quadratic formula is used to find the roots of any quadratic equation.
      Reason (R): The quadratic formula is derived from the method of completing the square.
      • A) Both A and R are true, and R is the correct explanation of A.
      • B) Both A and R are true, but R is not the correct explanation of A.
      • C) A is true, but R is false.
      • D) A is false, but R is true.
      • Answer: A) Both A and R are true, and R is the correct explanation of A.
      • Explanation: The quadratic formula, x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} , is indeed derived from completing the square.
    1. Assertion (A): The discriminant of a quadratic equation is given by D=b2−4acD = b^2 - 4ac .
      Reason (R): The value of the discriminant determines the nature of the roots.
      • A) Both A and R are true, and R is the correct explanation of A.
      • B) Both A and R are true, but R is not the correct explanation of A.
      • C) A is true, but R is false.
      • D) A is false, but R is true.
      • Answer: A) Both A and R are true, and R is the correct explanation of A.
      • Explanation: The discriminant indicates whether the roots are real and distinct, real and equal, or complex.
    1. Assertion (A): If the discriminant of a quadratic equation is negative, the equation has two real roots.
      Reason (R): Negative discriminant indicates complex roots.
      • A) Both A and R are true, and R is the correct explanation of A.
      • B) Both A and R are true, but R is not the correct explanation of A.
      • C) A is true, but R is false.
      • D) A is false, but R is true.
      • Answer: D) A is false, but R is true.
      • Explanation: A negative discriminant means the roots are not real but complex.
    1. Assertion (A): A quadratic equation can have at most two distinct real roots.
      Reason (R): This is because the degree of the polynomial is 2.
      • A) Both A and R are true, and R is the correct explanation of A.
      • B) Both A and R are true, but R is not the correct explanation of A.
      • C) A is true, but R is false.
      • D) A is false, but R is true.
      • Answer: A) Both A and R are true, and R is the correct explanation of A.
      • Explanation: The degree of the polynomial being 2 implies it can have a maximum of two roots.
    1. Assertion (A): The roots of the equation x2+4x+4=0x^2 + 4x + 4 = 0 are real and distinct.
      Reason (R): The discriminant D=b2−4acD = b^2 - 4ac is positive.
      • A) Both A and R are true, and R is the correct explanation of A.
      • B) Both A and R are true, but R is not the correct explanation of A.
      • C) A is true, but R is false.
      • D) A is false, but R is true.
      • Answer: D) A is false, but R is true.
      • Explanation: The discriminant D=42−4⋅1⋅4=0D = 4^2 - 4 \cdot 1 \cdot 4 = 0 , indicating that the roots are real and equal, not distinct.
    1. Assertion (A): The quadratic equation x2+6x+9=0x^2 + 6x + 9 = 0 can be factored as (x+3)2=0(x + 3)^2 = 0 .
      Reason (R): This shows that the equation has one real root.
      • A) Both A and R are true, and R is the correct explanation of A.
      • B) Both A and R are true, but R is not the correct explanation of A.
      • C) A is true, but R is false.
      • D) A is false, but R is true.
      • Answer: A) Both A and R are true, and R is the correct explanation of A.
      • Explanation: Factoring shows that the equation has one repeated root, which is x=−3x = -3.
    1. Assertion (A): The sum of the roots of a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 is given by −ba-\frac{b}{a} .
      Reason (R): The sum of the roots can be derived from Vieta's formulas.
      • A) Both A and R are true, and R is the correct explanation of A.
      • B) Both A and R are true, but R is not the correct explanation of A.
      • C) A is true, but R is false.
      • D) A is false, but R is true.
      • Answer: A) Both A and R are true, and R is the correct explanation of A.
      • Explanation: Vieta's formulas state that for the roots r1r_1 and r2r_2 , r1+r2=−bar_1 + r_2 = -\frac{b}{a} .
    1. Assertion (A): The roots of the quadratic equation 2x2−3x+1=02x^2 - 3x + 1 = 0 are rational numbers.
      Reason (R): The discriminant DD of the equation is a perfect square.
      • A) Both A and R are true, and R is the correct explanation of A.
      • B) Both A and R are true, but R is not the correct explanation of A.
      • C) A is true, but R is false.
      • D) A is false, but R is true.
      • Answer: A) Both A and R are true, and R is the correct explanation of A.
      • Explanation: The discriminant D=(−3)2−4(2)(1)=9−8=1D = (-3)^2 - 4(2)(1) = 9 - 8 = 1 , which is a perfect square.
    1. Assertion (A): A quadratic equation cannot have more than two solutions.
      Reason (R): A quadratic function can be graphed as a parabola.
    • A) Both A and R are true, and R is the correct explanation of A.
    • B) Both A and R are true, but R is not the correct explanation of A.
    • C) A is true, but R is false.
    • D) A is false, but R is true.
    • Answer: A) Both A and R are true, and R is the correct explanation of A.
    • Explanation: The parabola can intersect the x-axis at most twice, resulting in a maximum of two solutions.
    1. Assertion (A): The roots of the equation x2−5x+6=0x^2 - 5x + 6 = 0 are 22 and 33 .
      Reason (R): The roots can be found using factoring or the quadratic formula.
    • A) Both A and R are true, and R is the correct explanation of A.
    • B) Both A and R are true, but R is not the correct explanation of A.
    • C) A is true, but R is false.
    • D) A is false, but R is true.
    • Answer: A) Both A and R are true, and R is the correct explanation of A.
    • Explanation: The equation factors to (x−2)(x−3)=0(x - 2)(x - 3) = 0 , giving the roots 22 and 33 .
    1. Assertion (A): If a quadratic equation has one root, it is called a double root.
      Reason (R): This occurs when the discriminant is zero.
    • A) Both A and R are true, and R is the correct explanation of A.
    • B) Both A and R are true, but R is not the correct explanation of A.
    • C) A is true, but R is false.
    • D) A is false, but R is true.
    • Answer: A) Both A and R are true, and R is the correct explanation of A.
    • Explanation: A double root occurs when the quadratic touches the x-axis at one point, corresponding to a discriminant of zero.
    1. Assertion (A): The product of the roots of the quadratic equation 3x2+6x+2=03x^2 + 6x + 2 = 0 is 23\frac{2}{3} .
      Reason (R): The product of the roots is given by ca\frac{c}{a} .
    • A) Both A and R are true, and R is the correct explanation of A.
    • B) Both A and R are true, but R is not the correct explanation of A.
    • C) A is true, but R is false.
    • D) A is false, but R is true.
    • Answer: D) A is false, but R is true.
    • Explanation: The product of the roots is actually 23\frac{2}{3} , which is correct, but the explanation is accurate.
    1. Assertion (A): The quadratic equation x2+2x+1=0x^2 + 2x + 1 = 0 has roots that are equal.
      Reason (R): The equation can be factored as (x+1)2=0(x + 1)^2 = 0 .
    • A) Both A and R are true, and R is the correct explanation of A.
    • B) Both A and R are true, but R is not the correct explanation of A.
    • C) A is true, but R is false.
    • D) A is false, but R is true.
    • Answer: A) Both A and R are true, and R is the correct explanation of A.
    • Explanation: The factoring confirms that there is one repeated root, x=−1x = -1 .
    1. Assertion (A): A quadratic equation may have complex roots.
      Reason (R): Complex roots occur when the discriminant is negative.
    • A) Both A and R are true, and R is the correct explanation of A.
    • B) Both A and R are true, but R is not the correct explanation of A.
    • C) A is true, but R is false.
    • D) A is false, but R is true.
    • Answer: A) Both A and R are true, and R is the correct explanation of A.
    • Explanation: A negative discriminant indicates that the roots are complex and non-real.
    1. Assertion (A): The quadratic equation x2+4=0x^2 + 4 = 0 has no real solutions.
      Reason (R): The equation can be rewritten as x2=−4x^2 = -4 .
    • A) Both A and R are true, and R is the correct explanation of A.
    • B) Both A and R are true, but R is not the correct explanation of A.
    • C) A is true, but R is false.
    • D) A is false, but R is true.
    • Answer: A) Both A and R are true, and R is the correct explanation of A.
    • Explanation: The equation x2=−4x^2 = -4 indicates no real solutions since the square of a real number cannot be negative.
    1. Assertion (A): A quadratic equation always has at least one root.
      Reason (R): This is due to the Fundamental Theorem of Algebra.
    • A) Both A and R are true, and R is the correct explanation of A.
    • B) Both A and R are true, but R is not the correct explanation of A.
    • C) A is true, but R is false.
    • D) A is false, but R is true.
    • Answer: D) A is false, but R is true.
    • Explanation: A quadratic equation can have two, one, or no real roots, but it will always have two roots in the complex number system.
    1. Assertion (A): The equation x2−6x+9=0x^2 - 6x + 9 = 0 can be solved by factoring.
      Reason (R): The equation can be expressed as (x−3)(x−3)=0(x - 3)(x - 3) = 0 .
    • A) Both A and R are true, and R is the correct explanation of A.
    • B) Both A and R are true, but R is not the correct explanation of A.
    • C) A is true, but R is false.
    • D) A is false, but R is true.
    • Answer: A) Both A and R are true, and R is the correct explanation of A.
    • Explanation: The equation factors to give the double root x=3x = 3 .
    1. Assertion (A): A quadratic equation can be solved using various methods such as factoring, completing the square, and using the quadratic formula.
      Reason (R): These methods yield the same roots for the quadratic equation.
    • A) Both A and R are true, and R is the correct explanation of A.
    • B) Both A and R are true, but R is not the correct explanation of A.
    • C) A is true, but R is false.
    • D) A is false, but R is true.
    • Answer: A) Both A and R are true, and R is the correct explanation of A.
    • Explanation: All methods are valid approaches to finding the same roots of a quadratic equation.
    1. Assertion (A): The quadratic equation 4x2−12x+9=04x^2 - 12x + 9 = 0 can be simplified to (2x−3)2=0(2x - 3)^2 = 0 .
      Reason (R): This indicates that there is one repeated root.
    • A) Both A and R are true, and R is the correct explanation of A.
    • B) Both A and R are true, but R is not the correct explanation of A.
    • C) A is true, but R is false.
    • D) A is false, but R is true.
    • Answer: A) Both A and R are true, and R is the correct explanation of A.
    • Explanation: The equation has a double root at x=32x = \frac{3}{2} .
    1. Assertion (A): The quadratic equation 2x2+3x+4=02x^2 + 3x + 4 = 0 has complex roots.
      Reason (R): The discriminant of this equation is negative.
    • A) Both A and R are true, and R is the correct explanation of A.
    • B) Both A and R are true, but R is not the correct explanation of A.
    • C) A is true, but R is false.
    • D) A is false, but R is true.
    • Answer: A) Both A and R are true, and R is the correct explanation of A.
    • Explanation: The discriminant D=32−4⋅2⋅4=9−32=−23D = 3^2 - 4 \cdot 2 \cdot 4 = 9 - 32 = -23 , which confirms complex roots.
    1. Assertion (A): The roots of the quadratic equation x2−2x+1=0x^2 - 2x + 1 = 0 are distinct.
      Reason (R): The discriminant of the equation is zero.
    • A) Both A and R are true, and R is the correct explanation of A.
    • B) Both A and R are true, but R is not the correct explanation of A.
    • C) A is true, but R is false.
    • D) A is false, but R is true.
    • Answer: D) A is false, but R is true.
    • Explanation: The discriminant D=0D = 0 indicates that the roots are equal, not distinct.
    1. Assertion (A): The quadratic equation can have two complex conjugate roots.
      Reason (R): This occurs when the coefficients of the equation are real, and the discriminant is negative.
    • A) Both A and R are true, and R is the correct explanation of A.
    • B) Both A and R are true, but R is not the correct explanation of A.
    • C) A is true, but R is false.
    • D) A is false, but R is true.
    • Answer: A) Both A and R are true, and R is the correct explanation of A.
    • Explanation: A negative discriminant results in two complex conjugate roots.
    1. Assertion (A): The quadratic equation 2x2+4x+4=02x^2 + 4x + 4 = 0 has two equal roots.
      Reason (R): The roots can be found by using the quadratic formula.
    • A) Both A and R are true, and R is the correct explanation of A.
    • B) Both A and R are true, but R is not the correct explanation of A.
    • C) A is true, but R is false.
    • D) A is false, but R is true.
    • Answer: D) A is false, but R is true.
    • Explanation: The discriminant D=42−4⋅2⋅4=16−32=−16D = 4^2 - 4 \cdot 2 \cdot 4 = 16 - 32 = -16 indicates complex roots.
    1. Assertion (A): The roots of the quadratic equation 3x2+5x+2=03x^2 + 5x + 2 = 0 can be found using the quadratic formula.
      Reason (R): The quadratic formula is applicable to any quadratic equation.
    • A) Both A and R are true, and R is the correct explanation of A.
    • B) Both A and R are true, but R is not the correct explanation of A.
    • C) A is true, but R is false.
    • D) A is false, but R is true.
    • Answer: A) Both A and R are true, and R is the correct explanation of A.
    • Explanation: The quadratic formula can be applied to find the roots of any quadratic equation.

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    I am Amit Thakur. I am currently pursuing my Post Graduation in Mathematics from Cotton University. I am dedicating this website to my students. Thank you.

    March 09, 2025 , ,

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