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Maths Assertion (A) & Reason (B) MCQ

Exponents Of Real Numbers · CBSE STD 9 (CBSE - English Medium). 9 questions.

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Assertion (A) & Reason (B) MCQ

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9 questions · timed · auto-graded

MCQ 11 Mark
Statement-1 (A): If $(16)^{2 x+3}=(64)^{x+3}$, then $4^{2 x-2}=256$. Statement-2 (R): If $a \neq 0, \pm 1$, then $a^m=a^n \Rightarrow m=n$ and $\left(a^m\right)^n=a^{m n}$.
  • ✓
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-3
  • B
    Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-3
  • 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-3
(a)
Clearly, statement- 2 is true as it is one of the laws of exponents. $ \begin{array}{ll} \text { Now, } & (16)^{2 x+3}=(64)^{x+3} \Rightarrow\left(2^4\right)^{2 x+3}=\left(2^6\right)^{x+3} \Rightarrow 2^{4(2 x+3)}=2^{6(x+3)} \Rightarrow 4(2 x+3)=6(x+3) \\ \Rightarrow & 8 x+12=6 x+18 \Rightarrow 2 x=6 \Rightarrow x=3 \\ \therefore & 4^{2 x - 2}=4^{6 - 2}=4^4=256 \end{array} $
So, statement-1 is also true. Also, statement-2 is a correct explanation for statement-1. Hence option (a) is correct.
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MCQ 21 Mark
Statement-1 $(A): \left\{\left(a^{-1}+b^{-1}\right)\left(a^{-1}-b^{-1}\right)\right\} \div\left\{\left(\frac{1}{a^{-1}}-\frac{1}{b^{-1}}\right)\left(\frac{1}{a^{-1}}+\frac{1}{b^{-1}}\right)\right\}=1$.
Statement-2 ( $R$ ): For any $a \neq 0, a^{-m}=\frac{1}{a^m}$ and $a^m=\frac{1}{a^{-m}}$.
  • A
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-2
  • B
    Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-2
  • C
    Statement-1 is True, Statement-2 is False
  • ✓
    Statement-1 is False, Statement-2 is True
Answer
Correct option: D.
Statement-1 is False, Statement-2 is True
(d)
We observe that statement-2 is true.
Now,
\begin{array}{l} \left\{\left(a^{-1}+b^{-1}\right)\left(a^{-1}-b^{-1}\right)\right\} \div\left\{\left(\frac{1}{a^{-1}}-\frac{1}{b^{-1}}\right)\left(\frac{1}{a^{-1}}+\frac{1}{b^{-1}}\right)\right\} \\ =\left\{\left(\frac{1}{a}+\frac{1}{b}\right)\left(\frac{1}{a}-\frac{1}{b}\right)\right\} \div\{(a-b)(a+b)\} \\ =\left\{\left(\frac{a+b}{a b}\right)\left(\frac{b-a}{a b}\right)\right\} \div\{(a-b)(a+b)\}=-\frac{(a-b)(a+b)}{a^2 b^2} \div(a-b)(a+b)=-\frac{1}{a^2 b^2} \end{array}
So, statement-1 is not true. hence, option (d) is correct.
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MCQ 31 Mark
Statement-1 (A): $\sqrt{\frac{81}{64} \sqrt{\frac{81}{64} \sqrt{\frac{81}{64} \sqrt{\frac{81}{64}}}}} \cdots \cdot x=\frac{9}{8}$,
Statement-2 (R): For any positive real number $x: \sqrt{x \sqrt{x \sqrt{x \sqrt{x \sqrt{x}}}}} \ldots x=x$.
  • 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)
$\begin{array}{l}\text { Let } y=\sqrt{x \sqrt{x \sqrt{x \sqrt{x \sqrt{x}}}}} \ldots x . \text { Then } \\y^2=x \sqrt{x \sqrt{x \sqrt{x \sqrt{x}}}} \ldots \infty \\\Rightarrow \quad y^2=x y \Rightarrow y^2-x y=0 \Rightarrow y(y-x)=0 \Rightarrow y-x=0 \Rightarrow y=x . \quad[\because y \neq 0]\end{array}$
So, statement- 2 is true. Using statement-2, we find that $\sqrt{\frac{81}{64} \sqrt{\frac{81}{64} \sqrt{\frac{81}{64} \sqrt{\frac{81}{64}}}}}, \ldots \infty=\frac{81}{64}$
So, statement-1 is not true. Hence, option (c) is correct.
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MCQ 41 Mark
Statement-1 (A): If m, n are positive integers, then for any positive real number a, $\{\sqrt[m]{\sqrt[n]{a}}\}^{m n}=a$
Statement-2 (R): If m, n, p are rational numbers and a is any positive real number, then $\left(\left(a^m\right)^n\right)^p=a^{m n p}$.
  • ✓
    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 51 Mark
Statement-1 (A): $\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+}}}} \ldots \ldots \ldots \ldots \infty=3$.
Statement-2 (R): $\sqrt{x+\sqrt{x+\sqrt{x+}}} \ldots \ldots \ldots \ldots \infty=x, x>0$.
  • 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 61 Mark
Statement-1 (A): $\sqrt{5 \sqrt{5 \sqrt{5 \sqrt{5}}}} \cdots \cdots \ldots \ldots=5 \sqrt{5}$.
Statement-2 (R): $\sqrt{x \sqrt{x \sqrt{x \sqrt{x}}}} \ldots \ldots \ldots \ldots \infty=x, x>0$.
  • 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.
  • C
    Statement-1 is true, Statement-2 is false.
  • ✓
    Statement-1 is false, Statement-2 is true.
Answer
Correct option: D.
Statement-1 is false, Statement-2 is true.
D
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MCQ 71 Mark
Statement-1 (A): $\sqrt{7 \sqrt{7 \sqrt{7 \sqrt{7}}}}=\sqrt[16]{7^{15}}$.
Statement-2 (R): $\sqrt{a \sqrt{a \sqrt{a \ldots \ldots \ldots .}}} n$ terms $=a^{\frac{2^n-1}{2^n}}$.
  • ✓
    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 81 Mark
Statement-1 (A): If $a^x=b^y=c^z=a b c$, then $x y+y z+z x=x y z$.
Statement-2 (R): If $a^n=k$, then $a=k^{1 / n}$.
  • ✓
    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 91 Mark
Statement-1 (A): $\left[\left\{\left(\frac{1}{7^2}\right)^{-2}\right\}^{-1 / 3}\right]^{1 / 4}=7^{-1 / 3}$
Statement-2 (R): $\left(\left(a^m\right)^n\right)^s=a^{m n s}, a>0$
  • ✓
    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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