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MATHS Assertion (A) & Reason (B) MCQ

Model Paper 9 · Rajasthan Board (RBSE) STD 11 Science (Rajasthan - English Medium). 2 questions.

Questions

Assertion (A) & Reason (B) MCQ

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2 questions · self-marked practice — reveal the answer and mark yourself.

MCQ 11 Mark
Assertion (A): If each of the observations $x _1, x _2, \ldots, x _{ n }$ is increased by a, where a is a negative or positive number, then the variance remains unchanged.
Reason (R): Adding or subtracting a positive or negative number to (or from) each observation of a group does not affect the variance.
  • A
    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
(a) Both A and R are true and R is the correct explanation of A.
Explanation: Assertion: Let $\bar{x}$ be the mean of $x _1, x _2 \ldots, x _{ n }$. Then, variance is given by
$\sigma_1^2=\frac{1}{n} \sum_{i=1}^n\left(x_i-\bar{x}\right)^2$
If a is added to each observation, the new observations will be
$y_i=x_i+a$
Let the mean of the new observations be $\overline{ y }$.
Then
$\begin{array}{l}\bar{y}=\frac{1}{n} \sum_{i=1}^n y_i=\frac{1}{n}
\sum_{i=1}^n\left(x_i+a\right)
\\ =\frac{1}{n}\left[\sum_{i=1}^n x_i+\sum_{i=1}^n a\right] \\ =\frac{1}{n} \sum_{i=1}^n
x_i+\frac{n a}{n}=\bar{x}+a\end{array}$
i.e. $\bar{y}=\bar{x}+a \ldots$...ii)
Thus, the variance of the new observations is $\sigma_2^2=\frac{1}{n} \sum_{i=1}^n\left(y_i-\bar{y}\right)^2=\frac{1}{n} \sum_{i=1}^n\left(x_i+a-\bar{x}-a\right)^2$ [using Eqs. (i) and (ii)]
Thus, the variance of the new observations is same as that of the original observations
Reason: We may note that adding (or subtracting) a positive number to (or from) each observation of a group does not affect the variance.
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MCQ 21 Mark
Assertion (A): The expansion of $(1+ x )^{ n }=n_{c_0}+n_{c_1} x+n_{c_2} x^2 \ldots+n_{c_n} x^n$.
Reason (R): If $x=-1$, then the above expansion is zero.
  • A
    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
(b) Both A and R are true but R is not the correct explanation of A.
Explanation: Assertion:
$(1+ x )^{ n }=n_{c_0}+n_{c_1} x+n_{c_2} x^2 \ldots+n_{c_n} x^n$
Reason:
$\begin{array}{l}(1+(-1))^{ n }=n_{c_0} 1^n+n_{c_1}(1)^{n-1}(-1)^1+n_{c_2}(1)^{n-2}(-1)^2+\ldots+{ }^n c_n(1)^{n-n}(-1)^n \\ =n_{c_8}-n_{c_1}+n_{c_2}-n_{c_3}+\ldots(-1)^{ n } n_{c_n}\end{array}$
Each term will cancel each other 
$\therefore(1+(-1))^{ n }=0$
Reason is also the but not the correct explanation of Assertion.
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