Assertion Reason

Mathematics

Chapter 4: Quadratic Equations

Class 10

Assertion Reason Mathematics Chapter 4: Quadratic Equations for Class 10th is very important as assertion reason and Case Study Based Passages have been introduced by CBSE in 2020. Assertion reason and Case Study Type of questions helps students to think Critically on every aspects of life.

Here is a collection of few questions for CBSE Class 10th Term 2 Exams. These Assertion Reason Questions are fully solved. 

Directions: Mark the option which is most suitable :
(a) If both assertion and reason are true and reason is the correct explanation of assertion.
(b) If both assertion and reason are true but reason is not the correct explanation of assertion.
(c) If assertion is true but reason is false.
(d) If both assertion and reason are false.

Question.1.
Assertion : `4x^{2}-12x+9=0` has repeated roots.
Reason : The quadratic equation `ax^{2}+bx+c=0` have repeated roots if discriminant `D>0`.

(c) Assertion (A) is true but reason (R) is false.
Assertion `4x^{2}-12x+9=0`
So `D=b^{2}-4ac`
⇒ `D=(-12)^{2}-4(4)(9)`
⇒ `D= 144-144 = 0`
Roots are repeated.

Question.2.
Assertion : The equation `x^{2}+3x+1=(x-2)^{2}` is a quadratic equation.
Reason : Any equation of the form `ax^{2}+bx+c=0` where `a\ne 0`, is called a quadratic equation.

(d) Assertion (A) is false but reason (R) is true.
We have, `x^{2}+3x+1=(x-2)^{2}`
⇒ `x^{2}+3x+1=x^{2}-4x+4`
⇒` 7x-3=0`
it is not of the form `ax^{2}+bx+c=0`
So, A is incorrect but R is correct.

Question.3.
Assertion : `(2x-1)^{2}-4x^{2}+5=0` is not a quadratic equation.
Reason : `x=0, 3` are the roots of the equation `2x^{2}-6x=0`

(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
Assertion and Reason both are true statements. But Reason is not the correct explanation.
Assertion `(2x-1)^{2}-4x^{2}+5=0`
⇒`-4x+6=0`
Reason `2x^{2}-6x=0`
⇒`2x(x-3)=0 \Rightarrow x=0`
and `x=3`

Question.4.
Assertion : The values of `x` are `-\frac{a}{2},a` for a quadratic equation `2x^{2}+ax-a^{2}=0`.
Reason : For quadratic equation `ax^{2}+bx+c=0`;
`x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}`

(d) Assertion (A) is false but reason (R) is true.
`2x^{2}+ax-a^{2}=0`
`x=\frac{-a\pm\sqrt{a^2+8a^{2}}}{4}`
⇒ `x=\frac{-a+3a}{4}`
⇒ `x=\frac{2a}{4}, \frac{-4a}{4}`
⇒ `x= \frac{a}{2},-a`
So, A is incorrect but R is correct.

Question.5.
Assertion : The equation `8x^{2}+3kx+2=0` has equal roots then the value of `k` is `\pm\frac{8}{3}`
Reason : The equation `ax^{2}+bx+c=0` has equal roots if `D=b^{2}-4ac=0`

(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
`8x^{2}+3kx+2=0`
Discriminant, D`=b^{2}-4ac`
⇒ `D=(3k)^{2}-4(8)(2)=(9k)^{2}-64`
For equal roots, D = 0
⇒`(9k)^{2}-64=`0
⇒`(9k)^{2}`=64
⇒`k^{2}=\frac{64}{9}`
⇒ `k=\pm\frac{8}{3}`
So, A and R both are correct and R explains A.

Question.6.
Assertion : The value of `k = 2`, if one root of the quadratic equation `6x^{2}-x-k=0` is `\frac{2}{3}`
Reason : The quadratic equation `ax^{2}+bx+c=0` has two roots.

(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
As one root is `\frac{2}{3}`, so `x=\frac{2}{3}`
Hence, substituring the value of `x`, we get
`6\times(\frac{2}{3})^{2}-\frac{2}{3}-k=0`
⇒ `6\times\frac{4}{9}-\frac{2}{3}=k`
⇒ `k=\frac{8}{3}-\frac{2}{3}`
⇒ `k = \frac{6}{3}=2`
k = 2
So, both A and R are correct but R does not explain A.

Question.7.
Assertion : The roots of the quadratic equation `x^{2}+2x+2=0` are imaginary.
Reason : If discriminant `D=b^{2}-4ac<0` then the roots of quadratic equation `ax^{2}+bx+c=0` are imaginary.

(b) (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
`x^{2}+2x+2=0`
Discriminant, `D=b^{2}-4ac`
⇒ `D=(2)^{2}-4(1)(c)`
⇒ `D=4-8=-4<0`
Roots are imaginary.
So, both A and R are correct and R explains A.

Question.8.
Assertion : If roots of the equation `x^{2}-bx+c=0` are two consecutive integers, then `b^{2}-4c=1`
Reason : If a, b, c are odd integer then the roots of the equation `4abcx^{2}+(b^{2}-4ac)x-b=0` are real and distinct.

(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
Assertion : Given equation
`x^{2}-bx+c=0`
Let `\alpha,\beta` be two roots such that
`|\alpha-\beta|=1`
⇒`(\alpha+\beta)^{2}-4\alpha\beta=1`
⇒ `b^{2}-4c=1`
Reason : Given equation
`4abcx^{2}+(b^{2}-4ac)x-b=0`
⇒ `D = (b^{2}-4ac)^{2}+16ab^{2}c`
⇒ `D = (b^{2}-4ac)^{2}>0`
Hence roots are real and unequal.

Question.9.
Assertion : The equation `9x^{2}+3kx+4=0` has equal roots for `k=\pm4`.
Reason : If discriminant ‘D’ of a quadratic equation is equal to zero then the roots of equation are real and equal. 

(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
Assertion `9x^{2}+3kx+4=0`
⇒ `D = b^{2}-4ac`
⇒ `D= (3k)^{2}-4(9)(4)`
⇒ `D = 9k^{2}-144`
For equal roots D = 0
⇒`9k^{2}=144`
⇒`k=\pm \frac{12}{3}`
⇒`k=\pm4`

Question.10.
Assertion : A quadratic equation `ax^{2}+bx+c=0`, has two distinct real roots, if `b^{2}-4ac>0`.
Reason : A quadratic equation can never be solved by using method of completing the squares.

(c) Assertion (A) is true but reason (R) is false.

Question.11.
Assertion : Sum and product of roots of `2x^{2}-3x+5=0` are `\frac{3}{2}` and `\frac{5}{2}` respectively.
Reason : If a and b are the roots of `ax^{2}+bx+c=0`, `a\neq 0`, then sum of roots `=\alpha+\beta=-\frac{b}{a}` and product of roots `=\alpha \beta=\frac{c}{a}`.

(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
Assertion and Reason both are correct and Reason is correct explanation.
Assertion `2x^{2}-3x+5=0`
So, `\alpha+\beta=-\frac{b}{a}=-\frac{-3}{2}=\frac{3}{2}`
and `=\alpha \beta=\frac{c}{a}=\frac{5}{2}`

Question.12.
Assertion : `2x^{2}-4x+3=0` is a quadratic equation.
Reason : All polynomials of degree n, when n is a whole number can be treated as quadratic equation.

(c) Assertion is correct statement but Reason is wrong statement.

Question.13.
Assertion : `3y^{2}+17y-30=0` have distinct roots.
Reason : The quadratic equation `ax^{2}+bx+c=0` have distinct roots (real roots) if D > 0.

(a) : `3y^{2}+17y-30=0`
Therefore, `D=b^{2}-4ac`
⇒ `D= (17)^{2}-4(3)(-30)`
⇒ `D= 289 + 360`
⇒ `D= 649 > 0`
So, roots are real and distinct.
Both Assertion and Reason are correct and Reason is the correct explanation of Assertion.

Question.14.
Assertion : `9x^{2}-3x-20=0 \Rightarrow (3x-5)(3x+4)=0` If the roots are calculated by splitting the middle term.
Reason : To factorise `ax^{2}+bx+c=0`, we write it in the form `ax^{2}+b_{1}x+b_{2}x+c=0` such that `b_{1}+b_{2}=b` and `b_{1}b_{2}=ac`.

(a) : We have, `9x^{2}-3x-20=0`
⇒ `9x^{2}–15x+12x–20=0`
[By splitting the middle term]
⇒ `3x(3x – 5) + 4(3x – 5) = 0`
⇒ `(3x + 4) (3x – 5) = 0`
Both Assertion and Reason are correct and Reason is the correct explanation of Assertion.

Question.15.
Assertion : The value of `k` for which the equation `kx^{2}-12x+4=0` has equal roots, is 9.
Reason : The equation `ax^{2}+bx+c=0, (a\neq 0)` has equal roots, if `b^{2}-4ac>0`.

(c) : Clearly, Reason is wrong.
Now, the given equation is `kx^{2}-12x+4=0`
If the roots are equal, then `(-12)^{2}-4(k)(4)=0`
⇒ `144 – 16k = 0`
⇒ `k = \frac{144}{16}=9`
Assertion is correct.

Question.16.
Assertion : Both the roots of the equation `x^{2}-x+1=0` are real.
Reason : The roots of the equation `ax^{2}+bx+c=0` are real if and only if `b^{2}-4ac \geq 0`.

(d) : Clearly, Reason is Correct.
Now, given quadratic equation is `x^{2}-x+1=0`
`D=b^{2}-4ac`
⇒ `D= (-1)^{2}-4(1)(1)`
⇒ `D= – 3 < 0`
Hence, the given quadratic equation has no real roots.
Assertion is wrong.

Question.17.
Assertion : `2\sqrt{2}` is a root of the quadratic equation `x^{2}-4\sqrt{2}x+8=0`.
Reason : The root of a quadratic equation satisfies it.

(a) : Clearly, Reason is correct
Now, We have, `x^{2}-4\sqrt{2}x+8=0`
Its root will be `2\sqrt{2}`, if it will satisfy the given equation.
Now, `(2\sqrt{2})^{2}-4\sqrt{2}(2\sqrt{2})+8`
⇒ `=8-16+8`
⇒ `=0`
Thus, `2\sqrt{2}` is a root of the given equation.
Both Assertion and Reason are correct and Reason is the correct explanation of Assertion.

Question.18.
Assertion : `\frac{1}{(x-1)(x-2)}+\frac{1}{(x-2)(x-3)}=\frac{2}{3}` (x≠1,2,3) is a quadratic equation.
Reason : An equation of the form `ax^{2}+bx+c=0`, where a, b, c ∈ R is a quadratic equation.

(c) : We have,
`\frac{1}{(x-1)(x-2)}+\frac{1}{(x-2)(x-3)}=\frac{2}{3}`
⇒ `\frac{x-3+x-1}{(x-1)(x-2)(x-3)}=\frac{2}{3}`
⇒ `\frac{2x-4}{(x-1)(x-2)(x-3)}=\frac{2}{3}`
⇒`\frac{2(x-2)}{(x-1)(x-2)(x-3)}=\frac{2}{3}`
⇒ `(x – 1)(x – 3) = 3`
⇒ `x^{2}-4x+3=3`
`x^{2}-4x=0`, which is of the form `ax^{2} + bx + c = 0`, where `a ≠ 0` `x^{2}-4x=0` is a quadratic equation.
Also, an equation of the form `ax^{2} + bx + c = 0, a, b, c ∈ R` and `a ≠ 0` is a quadratic equation.
Thus, Assertion is correct but Reason is wrong.

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