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Maths Assertion (A) & Reason (B) MCQ

Quadratic Equations · CBSE STD 10 (CBSE - English Medium). 11 questions.

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Assertion (A) & Reason (B) MCQ

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11 questions · timed · auto-graded

MCQ 11 Mark
Statement-1 (A) : If the difference of roots of the equation $x^2-2 p x+q=0$ is same as the difference of the roots of the equation $x^2-2 r x+s=0$, then $s-q=r^2-p^2$.
Statement-2 (R): The roots of the quadratic equation $a x^2+b x+c=0$ are given by $x=\frac{-b \pm \sqrt{D}}{2 a}$, where $D$ is the discriminant.
  • A
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
  • ✓
    Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1
  • C
    Statement-1 is True, Statement-2 is False.
  • D
    Statement-1 is False, Statement-2 is True.
Answer
Correct option: B.
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1
(B)Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1 
Clearly, statement- 2 is true. Let $\alpha, \beta$ be the roots of $x^2-2 p x+q=0$ and $\gamma, \delta$ be the roots of $x^2-2 r x+s=0$. Then, $\alpha+\beta=2 p, \alpha \beta=q, \gamma+\delta=2 r$ and $\gamma \delta=s$. It is given that
$
\begin{array}{ll}
& \alpha-\beta=\gamma-\delta \\
& (\alpha-\beta)^2=(\gamma-\delta)^2 \Rightarrow(\alpha+\beta)^2-4 \alpha \beta=(\gamma+\delta)^2-4 \gamma \delta \Rightarrow 4 p^2-4 q=4 r^2-4 s \Rightarrow s-q=r^2-p^2
\end{array}
$
So, statement-1 is true, but statement- 2 is not a correct explanation for statement-1.
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MCQ 21 Mark
Statement-1 (A): If $5+\sqrt{7}$ is a root of a quadratic equation with rational coefficients, then its other root is $5-\sqrt{7}$.
Statement-2 (R) : Surd roots of a quadratic equation with rational coefficients occur a conjugate pairs.
  • ✓
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1
  • C
    Statement-1 is True, Statement-2 is False.
  • D
    Statement-1 is False, Statement-2 is True.
Answer
Correct option: A.
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
(A)Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1. 
Statement-2, being a standard result, is true. Using statement-2, we find that $5-\sqrt{7}$ is، root of a quadratic equation with rational coefficients if its one root is $5+\sqrt{7}$. So, statement $1$ is also true and statement-2 is a correct explanation for statement-1. Hence, option (a) is correct.
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MCQ 31 Mark
Statement-1 (A): If $a c \neq 0$, then at least one of the two equations $a x^2+b x+c=0$ and $a x^2+b x-c=0$ has real and distinct roots.
Statement-2 (R): A quadratic equation has real and distinct roots if its discriminant is positive.
  • ✓
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1
  • C
    Statement-1 is True, Statement-2 is False.
  • D
    Statement-1 is False, Statement-2 is True.
Answer
Correct option: A.
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
(A)Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1. 
The roots of a quadratic equation $a x^2+b x+c=0$ are given by $x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}$ which are real and distinct, if $D=b^2-4 a c >0$. So, statement- 2 is true.
Let $D_1$ and $D_2$ be the discriminants of quadratic equations $a x^2+b x+c=0$ and $a x^2+b x-c=0$ respectively. Then, $D_1=b^2-4 a c$ and $D_2=b^2+4 a c$.
Now, $\quad a c \neq 0 \Rightarrow a c>0$ or $a c<0$
If $a c>0$, then $D_2>0$ but the sign of $D_1$ is undecided. Consequently, $a x^2+b x-c=0$ has real and distinct roots.
If $a c<0$, then $D_1>0$ but the sign of $D_2$ is undecided. Consequently, $a x^2+b x+c=0$ has real and distinct roots.
Hence, at least one of the two equations has real and distinct roots. So, statement-1 is true and statement-2 is a correct explanation for statement-1.
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MCQ 41 Mark
Statement-1 (A): If $a$ and $c$ are of opposite signs, then the quadratic equation $a x^2+b x+c=0$ has real and distinct roots.
Statement-2(R): If discrimmant Dof a quadratic equationt is not equal to zero, it has roal and distinct rooks.
  • A
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1
  • ✓
    Statement-1 is True, Statement-2 is False.
  • D
    Statement-1 is False, Statement-2 is True.
Answer
Correct option: C.
Statement-1 is True, Statement-2 is False.
(C)Statement-1 is True, Statement-2 is False. 
If $a$ and $c$ are of opposite signs, then $a c<0$. Therefore, $D=b^2-4 a c>0$ and hence $a x^2+b x+c$ has real and distinct roots. So, statement- 1 is true.
Clearly, statement- 2 is false, because roots of a quadratic equation are real and distinct only when its discriminant is positive.
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MCQ 51 Mark
Statement-1 (A) : If roots of the equation $(2 k-1) x^2+4 x-3=0$ are reciprocal of each other, then $k=-1$.
Statement- 2(R) : If $a=c$, then roots of $a x^2+b x+c=0$ are reciprocal of each other.
  • ✓
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1
  • C
    Statement-1 is True, Statement-2 is False.
  • D
    Statement-1 is False, Statement-2 is True.
Answer
Correct option: A.
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
(A)Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1. 
If roots of the equation $a x^2+b x+c=0$ are reciprocal of each other, then Product of roots $=1 \Rightarrow \frac{c}{a}=1 \Rightarrow c=a$.
So, statement-2 is true.
Using statement-2, the roots of $(2 k-1) x^2+4 x-3=0$ will be reciprocal of each other, if
$2 k-1--3 \Rightarrow k=-1$
So, statement-1 is also true and statement-2 is a correct explanation for statement-1.
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MCQ 61 Mark
Statement-1 (A) : If the roots of the equation $\frac{1}{x+p}+\frac{1}{x+q}=\frac{1}{r}$ are equal in magnitude and opposite in sign, then $p, r$, q are in $A . P$.
Statement-2 (R) ; The sum of the roots of the equation $a x^2+b x+c=0$ is $\frac{b}{a}$.
  • A
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1
  • ✓
    Statement-1 is True, Statement-2 is False.
  • D
    Statement-1 is False, Statement-2 is True.
Answer
Correct option: C.
Statement-1 is True, Statement-2 is False.
(C)Statement-1 is True, Statement-2 is False. 
The sum of the roots of the equation $a x^2+b x+c=0$ is $-\frac{b}{a}$. So, statement -2 is false.
If the roots of the equation $a x^2+b x+c=0$ are equal in magnitude but opposite in sign, then $b=0$ i.e. coefficient of $x=0$.
Now, $\quad \frac{1}{x+p}+\frac{1}{x+q}=\frac{1}{r} \Rightarrow x^2+(p+q-2 r) x+(p q-q r-r p)=0$
It's roots are equal in magnitude but opposite in sign.
$
\quad p+q-2 r=0 \Rightarrow p+q=2 r \Rightarrow p, r, q \text { are in A.P. }
$
So, statement -1 is true.
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MCQ 71 Mark
Statement-1 (A): If the equation $x^2-a x+b=0$ and $x^2+b x-a=0$ harve a common roots and $a+b+0$, then $a-b-1$
Statement-2 (R): A common rool of two equations salisfies both the equations.
  • ✓
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1
  • C
    Statement-1 is True, Statement-2 is False.
  • D
    Statement-1 is False, Statement-2 is True.
Answer
Correct option: A.
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
(A)Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1. 
Clearly, statement-2 is always true. Let $\alpha$ be a common root of the given equations. Then,
$\alpha^2-a \alpha+b=0$ and $\alpha^2+b \alpha-a=0 \Rightarrow\left(\alpha^2-a \alpha+b\right)-\left(\alpha^2+b \alpha-a\right)=0 \Rightarrow-\alpha(a+b)=-(a+b) \Rightarrow \alpha=1$
Putting $\alpha=1$ in $\alpha^2-a \alpha+b=0$, we obtain $a-b=1$.
So, statement-1 is true. Clearly, statement-2 is a correct explanation for statement-1.
Hence, option (a) is correct.
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MCQ 81 Mark
Statement-1 (A): If $a-b+c=0$, then $a x^2+b x+c=0$ has real roots.
Statement-2 (R): Roots of $x^2-x+1=0$ are not real.
  • A
    Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
  • ✓
    Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement-1 is true, Statement-2 is false.
  • D
    Statement-1 is false, Statement-2 is true.
Answer
Correct option: B.
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
B
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MCQ 91 Mark
Statement-1 (A): If $a+b+c=0$, then $a x^2+b x+c=0$ has real roots.
Statement-2 (R): If one root of a quadratic equation is real, then the other root is also real.
  • ✓
    Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement-1 is true, Statement-2 is false.
  • D
    Statement-1 is false, Statement-2 is true.
Answer
Correct option: A.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
A
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MCQ 101 Mark
Statement-1 (A): If $p, q, r$ and $s$ are real numbers and $p r=2(q+s)$, then at least one of the equations $x^2+p x+q=0$ and $x^2+r x+s=0$ has real roots.
Statement-2 (R): The sum of two real numbers is positive, then both the numbers are positive.
  • A
    Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
  • ✓
    Statement-1 is true, Statement-2 is false.
  • D
    Statement-1 is false, Statement-2 is true.
Answer
Correct option: C.
Statement-1 is true, Statement-2 is false.
C
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MCQ 111 Mark
Statement-1 (A): If $2+\sqrt{3}$ is a root of a quadratic equation with rational coefficients, then its other root is $2-\sqrt{3}$.
Statement-2 (R): Surd roots of a quadratic equation with rational coefficients occur in conjugate pairs.
  • ✓
    Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement-1 is true, Statement-2 is false.
  • D
    Statement-1 is false, Statement-2 is true.
Answer
Correct option: A.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
A
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