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Maths Assertion (A) & Reason (B) MCQ

Factorization Of Algebraic Expressions · CBSE STD 9 (CBSE - English Medium). 16 questions.

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Assertion (A) & Reason (B) MCQ

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16 questions · 10 auto-graded MCQ + 6 self-marked written.

MCQ 11 Mark
Statement-1 (A): The value of $\frac{(0.027)^3-(0.023)^3}{(0.027)^2-(0.027)(0.023)+(0.023)^2}$ is 0.05
Statement-2 $(R): \quad a^3-b^3=(a-b)\left(a^2-a b-b^2\right)$
  • A
    Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Stateme
  • B
    Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statemen
  • ✓
    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 21 Mark
Statement-1 (A): If $a, b, c$ are all non-zero such that $a+b+c=0$, then $\frac{a^2}{b c}+\frac{b^2}{c a}+\frac{c^2}{a b}=$
Statement-2 (R): If $a+b+c=9$ and $a^2+b^2+c^2=35$, then $a b+b c+c a=23$
  • A
    Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Stateme
  • ✓
    Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statemen
  • 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 Statemen
B
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MCQ 31 Mark
Statement-1 (A): If $a+b+c=5$ and $a b+b c+c a=10$, then $a^3+b^3+c^3-3 a b c=25$
Statement-2 (R): $a^3+b^3+c^3-3 a b c=(a+b+c)\left\{(a+b+c)^2-3(a b+b c+c a)\right\}$
  • A
    Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Stateme
  • B
    Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statemen
  • 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 41 Mark
Statement-1 (A): If $3 x=a+b+c$,
then$
(x-a)^3+(x-b)^3+(x-c)^3=3(x-a)(x-b)
$
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 Stateme
  • B
    Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statemen
  • 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 Stateme
A
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MCQ 51 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 is true, Statement-2 is true; Statement-2 is a correct explanation for Stateme
  • B
    Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statemen
  • 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 Stateme
A
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MCQ 61 Mark
Statement-1 (A): $\quad a^2+b^2+c^2-a b-b c-c a=0$ if and only if $a=b=c$.
Statement-2 $(R): \quad a^3+b^3+c^3-3 a b c=(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)$
  • A
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-6
  • 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
B. Statement-2 is true.$
\begin{array}{ll}
\text { Now, } & a^2+b^2+c^2-a b-b c-c a=0 \\
\Leftrightarrow & 2\left(a^2+b^2+c^2-a b-b c-c a\right)=0 \\
\Leftrightarrow & (a-b)^2+(b-c)^2+(c-a)^2=0 \Leftrightarrow a-b=0 \text { and } b-c=0 \text { and } c-a=0 \Leftrightarrow a=b=c
\end{array}
$
So, statement 1 is also true. Hence, options (b) is correct.
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MCQ 71 Mark
Statement-1 $(A)$ : The product of $\left(x^2+4 y^2+z^2+2 x y+x z-2 y z\right)$ and $(-z+x-2 y)$ is $x^3-8 y^3-z^3-6 x y z$
Statement-2 $(R): \quad a^3+b^3+c^3-3 a b c=(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)$
  • A
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-5
  • 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
A. Statement-2 is true, being a standard formula.
Now,
$(-z+x-2 y)\left(x^2+4 y^2+z^2+2 x y+x z-2 y z\right)$
$=\{x+(-2 y)+(-z)\}\left\{x^2+(-2 y)^2+(-z)^2-x(-2 y)-(-2 y)(-z)-x(-z)\right\}$
$=x^3+(-2 y)^3+(-z)^3-3 x(-2 y)(-z)$ [Using statement-2]
$=x^3-8 y^3-z^3-6 x y z$
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 81 Mark
Statement-1 $(A): \quad(a+b+c)\left\{(a-b)^2+(b-c)^2+(c-a)^2\right\}=2\left(a^3+b^3+c^3-3 a b c\right)$
Statement-2 (R): If $a+b+c=0$ then $(a+b)^3+(b+c)^3+(c+a)^3=-3 a b c$
  • A
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-4
  • 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
B.
$
\begin{aligned}
\text {} & (a+b+c)\left\{(a-b)^2+(b-c)^2+(c-a)^2\right\} \\
& =2(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)=2\left(a^3+b^3+c^3-3 a b c\right)
\end{aligned}
$
So, statement- 1 is true.
If $a+b+c=0$, then.
$
\begin{array}{ll}
& 2(a+b+c)=0 \\
\Rightarrow & (a+b)+(b+c)+(c+a)=0 \\
\Rightarrow \quad & (a+b)^3+(b+c)^3+(c+a)^3=3(a+b)(b+c)(c+a) \\
\Rightarrow \quad & (a+b)^3+(b+c)^3+(c+a)^3=3(-c)(-a)(-b)=-3 a b c
\end{array}
$
So, statement-2 is also true. Hence, option (b) is true.
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MCQ 91 Mark
Statement-1(A): $\quad a^3(b-c)^3+b^3(c-a)^3+c^3(a-b)^3=3(a-b)(b-c)(c-a)$
Statement-2 $(R)$ : If $a+b \div c=0$, then $a^3+b^3+c^3=3 a b c$
  • A
    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-1.
  • C
    Statement-1 is True, Statement-2 is False.
  • D
    Statement-1 is False, Statement-2 is True.
Answer
D. Statement-2 is true.
We observe that $a(b-c)+b(c-a)+c(a-b)=0$$
\therefore \quad a^3(b-c)^3+b^3(c-a)^3+c^3(a-b)^3=3 a b c(a-b)(b-c)(c-a)
$
So, statement-1 is not true. Hence, option (d) is correct.
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MCQ 101 Mark
Statement-1 $(A)$ : The value of $\frac{(0.093)^3+(0.007)^3}{(0.093)^2-(0.093)(0.007)+(0.007)^2}$ is 0.1.
Statement-2 (R): $\quad a^3+b^3=(a+b)\left(a^2-a b+b^2\right)$.
  • 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-1.
  • C
    Statement-1 is True, Statement-2 is False.
  • D
    Statement-1 is False, Statement-2 is True.
Answer
A. We observe that statement-2 is true.
Now, $\quad a^3+b^3=(a+b)\left(a^2-a b+b^2\right) \Rightarrow \frac{a^3+b^3}{a^2-a b+b^2}=a+b$
Replacing a by 0.093 and $b$ by 0.007 , we obtain
$
\frac{(0.093)^3+(0.07)^3}{(0.093)^2-(0.093)(0.07)+(0.07)^2}=0.093+0.07=0.1
$
So, statement-2 is also true and statement-2 is a correct explanation for statement-1. Hence, option (a) is correct.
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MCQ 111 Mark
Statement-1 (A): The value of $1000^3-900^3-100^3$ is $270,000,000$
Statement-2 (R): If $a+b+c=0$, then $a^3+b^3+c^3=3 a b c$
  • 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.
  • C
    Statement-1 is True, Statement-2 is False.
  • D
    Statement-1 is False, Statement-2 is True.
Answer
A. We know that$
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)
$
If $a+b+c=0$, then
$\begin{aligned} & a^3+b^3+c^3-3 a b c=0 \times\left(a^2+b^2+c^2-a b-b c-c a\right) \\ \Rightarrow \quad & a^3+b^3+c^3-3 a b c=0 \Rightarrow a^3+b^3+c^3=3 a b c\end{aligned}$
So, statement-2 is true. We find that: $1000+(-900)+(-100)=0$
Using statement-2, we obtain$
\begin{array}{ll}
& 1000^3+(-900)^3+(-100)^3=3 \times 1000 \times(-900) \times(-100) \\
\Rightarrow \quad & 1000^3-900^3-100^3=270,000,000
\end{array}
$
So, statement- 1 is also true.
We find that statement-2 is a correct explanation for statement- 1 . Hence, option (a) is correct.
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MCQ 121 Mark
Statement-1 (A): The value of $\frac{(0.027)^3+(0.023)^3}{(0.027)^2-(0.027)(0.023)+(0.023)^2}$ is 0.05
Statement-2 (R): $a^3-b^3=(a-b)\left(a^2-a b+b^2\right)$
  • 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 131 Mark
Statement-1 (A): If $a, b, c$ are all non-zero such that $a+b+c=0$, then $\frac{a^2}{b c}+\frac{b^2}{c a}+\frac{c^2}{a b}=3$
Statement-2 (R): If $a+b+c=9$ and $a^2+b^2+c^2=35$, then $a b+b c+c a=23$
  • 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 141 Mark
Statement-1 (A): If $a+b+c=5$ and $a b+b c+c a=10$, then $a^3+b^3+c^3-3 a b c=25$
Statement-2 (R): $a^3+b^3+c^3-3 a b c=(a+b+c)\left\{(a+b+c)^2-3(a b+b c+c a)\right\}$
  • 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 151 Mark
Statement-1 (A): Statement-1 (A): If $3 x=a+b+c$, then $(x-a)^3+(x-b)^3+(x-c)^3=3(x-a)(x-b)(x-c)$
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 161 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 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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