Assertion (A) & Reason (B) MCQ

MCQ 11 Mark

Assertion : If $a: b=c: d$, then $b: a=d: c$ by invertendo
Reason (R): Invertendo property allows us to invert the terms of a proportion of form another true proportion.

A
Both assertion and reason are correct and reason is the correct explanation of assertion.
B
Both assertion and reason are correct but reason is not the correct explanation of assertion.
C
Assertion is correct but reason is incorrect.
D
Assertion is incorrect but reason is correct.

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MCQ 21 Mark

Assertion : Let $a: b$ be the duplicate ratio of $a+c$ and $b+c$. Then $b$ is the third proportion between $a$ and $c$.
Reason : If $y$ is the third proportion between $x$ and $z$, then $\frac{x}{y}-\frac{y}{z}$.

A
Both assertion and reason are correct and reason is the correct explanation of assertion.
B
Both assertion and reason are correct but reason is not the correct explanation of assertion.
C
Assertion is correct but reason is incorrect.
D
Assertion is incorrect but reason is correct.

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MCQ 31 Mark

Assertion : If $(4 a+5 b)(4 c-5 d)(4 a-5 b)(4 c+5 d)$, then $a, c, b, d$ are in proportion.
Reason: If $\frac{x}{y}=\frac{m}{n}$, then $x, y, m, n$ are in proportion.

A
Both assertion and reason are correct and reason is the correct explanation of assertion.
B
Both assertion and reason are correct but reason is not the correct explanation of assertion.
C
Assertion is correct but reason is incorrect.
✓
Assertion is incorrect but reason is correct.

Answer

Correct option: D.

Assertion is incorrect but reason is correct.

(d) Assertion is incorrect but reason is correct.
Explanation:
$\begin{array}{l}(4 a+5 b)(4 c-5 d)=(4 a-5 b)(4 c+5 d) \\ \Rightarrow \frac{4 a+5 b}{4 a-5 b}=\frac{4 c+5 d}{4 c-5 d}\end{array}$
Applying componendo and dividendo, we get
$\begin{array}{l}\frac{8 a}{10 b}=\frac{8 c}{10 d} \\ \Rightarrow \frac{a}{b}=\frac{c}{d}\end{array}$
$\therefore a . b . c$ and $d$ are in proportion

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MCQ 41 Mark

Assertion : If $y$ is the mean proportion between $x$ and $z$, then $x y z(x+y+z)^3=(x y+y z+z x)^3$.
Reason : If $y$ is the mean proportion between $x$ and $z$, then $y=\frac{x+z}{2}$.

A
Both assertion and reason are correct and reason is the correct explanation of assertion.
B
Both assertion and reason are correct but reason is not the correct explanation of assertion.
C
Assertion is correct but reason is incorrect.
D
Assertion is incorrect but reason is correct.

Answer

(c) Assertion is correct but reason is incorrect.
Explanation:
Since, $y$ is the mean proportion between $x$ and $z$.
$\therefore y^2=x z \ldots$ (i)
$\begin{array}{l}\text { Now, L.H.S. }=x y z(x+y+z)^3 \\ =(y)^2 y(x+y+z)^3[\text { Using (i)] } \\ =y^3(x+y+z)^3 \ldots \text { (ii) }\end{array}$
$\begin{array}{l}\text { R.H.S. }=(x y+y z+z x)^3 \\ =\left(x y+y z+y^2\right)^3 \\ =[y(x+z+y)]^3 \\ =y^3(x+y+z)^3 \ldots \text { (iii) }\end{array}$
From (ii) and (iii),

$x y z(x+y+z)^3=(x y+y z+z x)^3.$

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Mathematics Assertion (A) & Reason (B) MCQ

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