Assertion (A) & Reason (B) MCQ

MCQ 11 Mark

Assertion : The sum of the first hundred natural numbers, divisible by 5 , is 25250 .
Reason : The sum of first $n$ terms of an A.P. is given by $\frac{n}{2}(a+l)$, where, $l$ is the last term.

A
Both assertion and reason are correct and reason is the correct explanation of assertion.
✓
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

Correct option: B.

Both assertion and reason are correct but reason is not the correct explanation of assertion.

(b) Both Assertion and Reason are correct but Reason is not correct explanation of Assertion.
Explanation:
First 100 natural numbers divisible by 5 are $5,10,15,20, \ldots$
The above series is an A.P. with $a=5, d=10-5=5$ and $n=100$.
$\therefore S_n=\frac{\pi}{2}[2 \pi+(n-1) d]$
$\begin{array}{l}=\frac{100}{2}[2 \times 5+(100-1) \times 5] \\ =50 \times 505=25250 .\end{array}$

View full question & answer→

MCQ 21 Mark

Assertion : If fourth term of an A.P. is zero, then its $25^{\text {th }}$ term is three times its $11^{\text {th }}$ term.
Reason : The sum of first $n$ terms of an A.P. is given by $\frac{n}{2}(a+l)$, where, $l$ is the last term.

A
Both assertion and reason are correct and reason is the correct explanation of assertion.
✓
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

Correct option: B.

Both assertion and reason are correct but reason is not the correct explanation of assertion.

(b) Both Assertion and Reason are correct but Reason is not correct explanation of Assertion.
Explanation:
We have, $t_4=0$
$\begin{array}{l}\Rightarrow a+3 d=0 \\ \Rightarrow a=-3 d \ldots \text { (i) }\end{array}$
Now, $a_{11}=a+10 d=-3 d+10 d=7 d$
[Using (i)] ...(ii)
And $a_{25}=a+24 d=-3 d+24 d=21 d$
[Using (i)] ...(iii)
$\begin{array}{l}\Rightarrow a_{25}=21 d=3 \times 7 d \\ =3 \times a_{11}[ Using ( ii )]\end{array}$

View full question & answer→

MCQ 31 Mark

Assertion: The number of terms to be taken in the A.P. $9,17,25, \ldots$ So as to make a sum of 636 is 13.
Reason : The sum of first $n$ terms of an A.P. is given by $\frac{n}{2}[2 a+(n-1) d]$.

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:
For the given A.P., we have
$a=9, d=17-9=8 \text { and } S_n=636$
We know,
$\begin{array}{l}S_n=\frac{n}{2}[2 a+(n-1) A] \\ \Rightarrow 636=\frac{n}{2}[2 \times 9+(n-1) \times 8] \\ \Rightarrow 1272=n[8 n+10] \\ \Rightarrow 4 n^2+5 n-636=0\end{array}$
Using quadratic formula,
$\begin{array}{l}n=\frac{-5 \pm \sqrt{(5)^2-4 \times 4 \times(-636)}}{2 \times 4} \\ =\frac{-5 \pm \sqrt{10201}}{8} \\ =\frac{-5 \pm 101}{8}=\frac{96}{8}, \frac{-106}{8}=12, \frac{-53}{4}\end{array}$
Since, number of terms cannot be negative.
$\therefore n=12 \text {. }$

View full question & answer→

MCQ 41 Mark

Assertion : The sum of $2^{\text {nd }}$ and $7^{\text {th }}$ terms of an A.P. is 30 . If its $15^{\text {th }}$ term is 1 less than twice its $8^{\text {th }}$ term, then the A.P. is $1,5,9,13,17, \ldots$
Reason : The $n^{\text {th }}$ term of an A.P. is given by $a+(n-1) d$, where $a$ and $d$ are the first term and the common difference respectively.

✓
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

Correct option: A.

Both assertion and reason are correct and reason is the correct explanation of assertion.

(a) Both Assertion and Reason are correct and Reason is the correct explanation of Assertion.
Explanation:
We have,
$\begin{array}{l}t_2+t_7=30 \\ \Rightarrow(a+d)+(a+6 d)=30 \\ \Rightarrow 2 a+7 d=30 \ldots \text { (i) }\end{array}$
Also, $t_{15}=2 t_8-1$
$\Rightarrow a+14 d=2(a+7 d)-1$
$\Rightarrow a=1$
Putting $a=1$ in eq. (i), we get
d=4
$\therefore$ The A.P. is $1,1+4,1+2(4), \ldots$ i.e., $1,5,9, \ldots$

View full question & answer→

Mathematics Assertion (A) & Reason (B) MCQ

Test yourself on this topic