Class 10th Maths Chapter Assertion and Reason Questions
Assertion (A): The general form of a quadratic equation is
. Reason (R): In a quadratic equation, a,
b,
and c
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 a,
b,
and c
are real numbers, a
cannot be zero in a quadratic equation.
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, , is indeed
derived from completing the square.
Assertion (A): The discriminant of a quadratic equation is
given by . 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.
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.
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.
Assertion (A): The roots of the equation
are real and distinct. Reason (R): The discriminant 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 ,
indicating that the roots are real and equal, not distinct.
Assertion (A): The quadratic equation
can be factored as . 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=−3.
Assertion (A): The sum of the roots of a quadratic equation
is
given by . 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
and ,
.
Assertion (A): The roots of the quadratic equation
are rational numbers. Reason (R): The discriminant D
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 , which
is a perfect square.
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.
Assertion (A): The roots of the equation
are and
. 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 ,
giving the roots and
.
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.
Assertion (A): The product of the roots of the quadratic
equation
is . Reason (R): The product of the roots is given by
.
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
, which is
correct, but the explanation is accurate.
Assertion (A): The quadratic equation
has roots that are equal. Reason (R): The equation can be factored as .
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, .
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.
Assertion (A): The quadratic equation has no real solutions. Reason (R): The equation can be rewritten as
.
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
indicates no real solutions since the square of a real number cannot be
negative.
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.
Assertion (A): The equation
can be solved by factoring. Reason (R): The equation can be expressed as
.
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
.
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.
Assertion (A): The quadratic equation
can be simplified to . 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
.
Assertion (A): The quadratic equation
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 ,
which confirms complex roots.
Assertion (A): The roots of the quadratic equation
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
indicates that the roots are equal, not distinct.
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.
Assertion (A): The quadratic equation
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
indicates complex roots.
Assertion (A): The roots of the quadratic equation
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.