Assertion (A) & Reason (B) MCQ

MCQ 11 Mark

Assertion (A): $\left(\sum_{r=0}^{100}{ }^{500-r} C_{3}\right)+{ }^{400} C_{4}={ }^{501} C_{4}$
Reason (R): ${ }^{n} C_{r}+{ }^{n} C_{r-1}={ }^{n+1} C_{r}$

✓
Both A and R are true and R is the correct explanation of A .
B
Both $A$ and $R$ are true but $R$ is not the correct explanation of A.
C
$A$ is true but $R$ is false.
D
A is false but $R$ is true.

Answer

Correct option: A.

Both A and R are true and R is the correct explanation of A .

(a) Both $A$ and $R$ are true and $R$ is the correct explanation of $A$.
Explanation: $\sum_{r=0}^{100}{ }^{500-r} C_{3}+{ }^{400} \mathrm{C}_{4}$
$={ }^{500} \mathrm{C}_{3}+{ }^{499} \mathrm{C}_{3}+\ldots+{ }^{401} \mathrm{C}_{3}+{ }^{400} \mathrm{C}_{3}+{ }^{400} \mathrm{C}_{4}$
$={ }^{500} \mathrm{C}_{3}+{ }^{499} \mathrm{C}_{3}+\ldots{ }^{401} \mathrm{C}_{3}+{ }^{401} \mathrm{C}_{4}$
$={ }^{500} \mathrm{C}_{3}+{ }^{499} \mathrm{C}_{3}+\ldots+{ }^{402} \mathrm{C}_{4}$
Similarly,
${ }^{500} C_{3}+{ }^{500} C_{4}$
${ }^{501} \mathrm{C}_{4}=$ RHS

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MCQ 21 Mark

Assertion (A): If $P(A)=0.6$ and $P(B)=0.4$ and $P(A \mid B)=0.4$, then $P(A \cup B)=0.72$.
Reason (R): $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$.

✓
Both $A$ and $R$ are true and $R$ is the correct explanation of A .
B
Both $A$ and $R$ are true but $R$ is not the correct explanation of A.
C
$A$ is true but $R$ is false.
D
A is false but R is true.

Answer

Correct option: A.

Both $A$ and $R$ are true and $R$ is the correct explanation of A .

(a) Both A and R are true and R is the correct explanation of A .
Explanation: $P(A \mid B)=\frac{P(A \cap B)}{P(B)}$
$0.4=\frac{P(A \cap B)}{0.4}$
$\Rightarrow \mathrm{P}(\mathrm{A} \cap \mathrm{B})=0.16$
Now, $(A \cap B)=P(A)+P(B)-P(A \cap B)$
$=0.6+0.4-0.16$
$=1-0.28=0.72$

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Applied Maths Assertion (A) & Reason (B) MCQ

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