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Maths Assertion (A) & Reason (B) MCQ

Algebraic Identities [NEW] · Gujarat Board (GSEB) STD 9 (GSEB - 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): The square root of $\frac{1}{a b c}\left(a^2+b^2+c^2\right)+2\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$ is $\sqrt{\frac{a}{b c}}+\sqrt{\frac{b}{c a}}+\sqrt{\frac{c}{a b}}$.
Statement-2 (R): $a^3+b^3+c^3-3 a b c=(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)$
  • 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 21 Mark
Statement-1 (A): $\sqrt{(a+b+c)^2+(a-b+c)^2+2\left(b^2-a^2-c^2-2 a c\right)}=2 b$
Statement-2 (R): $(x+y+z)^2=x^2+y^2+z^2+2(x y+y z+z x)$
  • ✓
    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-2 is true, being a standard formula, using statement-2, we obtain
$(a+b+c)^2+(a-b+c)^2+2\left(b^2 \quad a^2-c^2-2 a c\right)=2\left(a^2+b^2+c^2+2 a c\right)+2\left(b^2-a^2-c^2-2 a c\right)=4 b^2$
$\therefore \sqrt{(a+b+c)^2+(a-b+c)^2+2\left(b^2-a^2-c^2-2 a c\right)}=2 b$
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MCQ 31 Mark
Statement-1 (A): If $a+b+c=6, a b+b c+c a=11$, then $a^2+b^2+c^2=14$
Statement-2 (R): $(a+b+c)^2=a^2+b^2+c^2+2(a b+b c+c a)$
  • ✓
    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.
  • 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 41 Mark
Statement-1 (A): If $a+b+c=0$, then $a^3+b^3+c^3=3 a b c$
Statement-2 (R): $a^3+b^3+c^3-3 a b c=(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)$
  • ✓
    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): $\frac{\left(x^2-y^2\right)^3+\left(y^2-z^2\right)^3+\left(z^2-x^2\right)^3}{(x-y)^3+(y-z)^3+(z-x)^3}=(x+y)(y+z)(z+x)$
Statement-2 (R): If $a+b+c=0$, then $a^3+b^3+c^3=3 a b c$.
  • ✓
    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 61 Mark
Statement-1 (A): $a+b+c=6$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{3}{2}$, then $\frac{a}{b}+\frac{a}{c}+\frac{b}{a}+\frac{b}{c}+\frac{c}{a}+\frac{c}{b}=6$
Statement-2 $(R): (a+b+c)^2=a^2+b^2+c^2+2(a b+b c+c a)$
  • 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-2 is true.
We have,$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{3}{2} \Rightarrow 2(a b+b c+c a)=3 a b c \Rightarrow a b+b c+c a=\frac{3}{2} a b c$
Thus, we have,
$a+b+c=6$ and $a b+b c+c a=\frac{3}{2} a b c$
$\Rightarrow(a+b+c)(a b+b c+c a)=6 \times \frac{3}{2} a b c$
$\Rightarrow a\left(b^2+c^2\right)+b\left(c^2+a^2\right)+c\left(a^2+b^2\right)+3 a b c=9 a b c \Rightarrow a\left(b^2+c^2\right)+b\left(c^2+a^2\right)+c\left(a^2+b^2\right)=6 a b c$
$\therefore \frac{a}{b}+\frac{a}{c}+\frac{b}{a}+\frac{b}{c}+\frac{c}{a}+\frac{c}{b}=\left(\frac{a}{b}+\frac{b}{a}\right)+\left(\frac{a}{c}+\frac{c}{a}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)=\frac{a^2+b^2}{a b}+\frac{a^2+c^2}{c a}+\frac{b^2+c^2}{b c}$
$=\frac{a\left(b^2+c^2\right)+b\left(c^2+a^2\right)+c\left(a^2+b^2\right)}{a b c}=\frac{6 a b c}{a b c}=6$
So, statement-1 is true. But, statement-2 is not a correct explanation for statement-1
Hence, option (b) is correct
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MCQ 71 Mark
Statement-1 (A): $(a+b+c)^2=a^2+b^2+c^2-2(a b+b c+c a)$
Statement-2 (R): $a^3+b^3+c^3-3 a b c=(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)$
  • 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 81 Mark
Statement-1 $(A): (a-b)^3+(b-c)^3+(c-a)^3=3(a-b)(b-c)(c-a)$
Statement-2 $(R)$; If $a+b+c=0$, then $a^3+b^3+c^3=3 a b c$
  • ✓
    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-2, being a standard result, is true. We find that
$(a-b)+(b-c)+(c-a)=0$
Therefore, using statement-2, we obtain
$(a-b)^3+(b-c)^3+(c-a)^3=3(a-b)(b-c)(c-a)$
So, statement-1 is true. Also, statement-2 is a correct explanation for statement-1. Hence, option (a) is correct.
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MCQ 91 Mark
Statement-1 (A): $a^3+\frac{3}{8} a x+\frac{1}{64} x^3-\frac{1}{8}=\left(a+\frac{x}{4}-\frac{1}{2}\right)\left(a^2+\frac{x^2}{16}+\frac{1}{4}-\frac{a x}{4}+\frac{x}{8}+\frac{a}{2}\right)$
Statement-2 (R): $a^3+b^3+c^3+3 a b c=(a+b+c)\left(a^2+b^2+c^2+a b+b c+c a\right)$
  • 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 101 Mark
Statement-1 (A): $a^3+b^3+3 a b-1=(a+b-1)\left(a^2+b^2+a+b-a b+1\right)$
Statement-2 (R): $ a^3+b^3+c^3-3 a b c=(a+b+c)\left(a^2+b^2+c^2+a b+b c+c a\right)$
  • 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-2 is not true, because
$a^3+b^3+c^3-3 a b c=(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)$
Using this formula, we obtain
$a^3+b^3+(-1)^3-3 a b(-1)=(a+b+(-1))\left(a^2+b^2+(-1)^2-a b-a(-1)-b(-1)\right)$
$\Rightarrow \quad a^3+b^3+3 a b-1=(a+b-1)\left(a^2+b^2+a+b-a b+1\right)$
So, statement-1 is true. Hence, option (c) is correct.
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MCQ 111 Mark
Statement-1 $(A): a^2+b^2+c^2-a b-b c-c a=0$ if and only if $a=b=c$.
Statement-2 (R):$(a+b+c)^2=a^2+b^2+c^2+2 a b+2 b c+2 c a$
  • 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-2, being a standard result, is true.
Now, $\quad a^2+b^2+c^2-a b-b c-c a=0$
$ \Rightarrow \quad 2 a^2+2 b^2+2 c^2-2 a b-2 b c-2 c a=0$[Multiplying both sides by 2]
$\Rightarrow \quad\left(a^2+b^2-2 a b\right)+\left(b^2+c^2-2 b c\right)+\left(c^2+a^2-2 c a\right)=0$
$\Rightarrow \quad(a-b)^2+(b-c)^2+(c-a)^2=0$
$\Rightarrow\quad a-b=0$ and $b-c=0$ and $c-a=0 \Rightarrow a=b=c$.
So, statement-1 is also true. Hence, option (b) is correct.
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