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Chapter 4: Quadratic Equations

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This page provides comprehensive Chapter 4: Quadratic Equations - Assertion Reason Worksheet - SJMaths. Assertion and Reason type questions for Class 10 Quadratic Equations. Practice for CBSE Board Exams.

Assertion-Reason Worksheet

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Directions:

In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:

  • (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
  • (B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
  • (C) Assertion (A) is true but Reason (R) is false.
  • (D) Assertion (A) is false but Reason (R) is true.
  1. Question 1:
    Assertion (A): The equation $x(x+1) + 8 = (x+2)(x-2)$ is a quadratic equation.
    Reason (R): An equation of the form $ax^2 + bx + c = 0, a \neq 0$ is a quadratic equation.
    (A)
    (B)
    (C)
    (D)
    Solution: (D)
    Step 1: Simplify A: $x^2 + x + 8 = x^2 - 4 \Rightarrow x + 12 = 0$.
    Step 2: This is a linear equation, not quadratic. So A is false.
    Step 3: R is the correct definition of a quadratic equation.
  2. Question 2:
    Assertion (A): The roots of the equation $x^2 + x + 1 = 0$ are imaginary.
    Reason (R): If discriminant $D = b^2 - 4ac < 0$, then the roots of quadratic equation are imaginary.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Step 1: Calculate $D$ for $x^2 + x + 1 = 0$. $D = 1^2 - 4(1)(1) = 1 - 4 = -3$.
    Step 2: Since $D < 0$, roots are imaginary. A is true.
    Step 3: R correctly states the condition. R explains A.
  3. Question 3:
    Assertion (A): If the roots of $2x^2 - 4x + k = 0$ are equal, then $k = 2$.
    Reason (R): For equal roots, discriminant $D = 0$.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Step 1: For equal roots, $D = b^2 - 4ac = 0$.
    Step 2: $(-4)^2 - 4(2)(k) = 0 \Rightarrow 16 - 8k = 0 \Rightarrow 8k = 16 \Rightarrow k = 2$.
    Step 3: Both A and R are true, and R explains A.
  4. Question 4:
    Assertion (A): The roots of the quadratic equation $x^2 - 4x + 5 = 0$ are real and distinct.
    Reason (R): If $D > 0$, then roots are real and distinct.
    (A)
    (B)
    (C)
    (D)
    Solution: (D)
    Step 1: Calculate $D = (-4)^2 - 4(1)(5) = 16 - 20 = -4$.
    Step 2: Since $D < 0$, roots are not real. A is false.
    Step 3: R is a true statement about discriminants.
  5. Question 5:
    Assertion (A): The product of roots of the quadratic equation $2x^2 - 3x + 5 = 0$ is $5/2$.
    Reason (R): The product of roots of $ax^2 + bx + c = 0$ is given by $c/a$.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Step 1: Using R, product $= c/a = 5/2$.
    Step 2: A matches the calculation. R is the correct formula.
  6. Question 6:
    Assertion (A): $x = 2$ is a root of the equation $2x^2 - 3x - 2 = 0$.
    Reason (R): If a value $x = \alpha$ satisfies the equation, then it is a root.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Step 1: Substitute $x=2$: $2(2)^2 - 3(2) - 2 = 8 - 6 - 2 = 0$.
    Step 2: Since LHS = RHS, $x=2$ is a root. A is true.
    Step 3: R is the definition of a root.
  7. Question 7:
    Assertion (A): The equation $3x^2 + 4\sqrt{3}x + 4 = 0$ has equal roots.
    Reason (R): $D = b^2 - 4ac$ for this equation is zero.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Step 1: Calculate $D = (4\sqrt{3})^2 - 4(3)(4) = 48 - 48 = 0$.
    Step 2: Since $D=0$, roots are equal. A is true.
    Step 3: R states $D=0$, which is the reason for equal roots.
  8. Question 8:
    Assertion (A): Every quadratic equation has exactly two real roots.
    Reason (R): A quadratic equation can have at most two roots.
    (A)
    (B)
    (C)
    (D)
    Solution: (D)
    Step 1: Quadratic equations can have real or imaginary roots. They don't always have *real* roots. A is false.
    Step 2: R is true; degree 2 implies at most 2 roots.
  9. Question 9:
    Assertion (A): If the sum of roots is 3 and product is 2, the equation is $x^2 - 3x + 2 = 0$.
    Reason (R): A quadratic equation is given by $x^2 - (\text{Sum})x + \text{Product} = 0$.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Step 1: Using R, $x^2 - 3x + 2 = 0$.
    Step 2: A matches the formula. R is correct.
  10. Question 10:
    Assertion (A): The values of $x$ satisfying $x^2 = 4$ are $2$ and $-2$.
    Reason (R): The degree of a quadratic equation is 2.
    (A)
    (B)
    (C)
    (D)
    Solution: (B)
    Step 1: $x^2 = 4 \Rightarrow x = \pm 2$. A is true.
    Step 2: R is true, but it doesn't directly explain why the specific values are 2 and -2 (that comes from square root property). So B is correct.
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