M.C.Q (1 Marks)

MCQ 11 Mark

Choose the correct answer from the given four options:
The famous mathematician associated with finding the sum of the first 100 natural numbers is:

✓
Pythagoras.
B
Newton.
C
Gauss.
D
Euclid.

Answer

Correct option: A.

Pythagoras.

Gauss is the famous mathematician associated with finding the sum of the first natural numbers i.e., 1, 2, 3 ............ 100.

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MCQ 21 Mark

Choose the correct answer from the given four options: The list of numbers $- 10, – 6, – 2, 2,...$ is:

A
An $AP$ with $d = -16$
✓
An $AP$ with $d = 4$
C
An $AP$ with $d = -4$
D
Not an $AP$

Answer

Correct option: B.

An $AP$ with $d = 4$

The given numbers are $-10,-6,-2,2 \ldots \ldots \ldots$
Here, $a_2=-10, a_2=-6, a_3=-2$ and $a_4=2$. $\qquad$
Since, $a_2-a_1=-6-(-10)$
$a_2-a_1=-6+10=4$
$a_3-a_2=-2-(-6)$
$a_4-a_3=2-(-2)$
$a_4-a_3=2+2=4$
Each successive term of given list has same difference i.e. $4.$
So, the given list forms an $AP$ with common difference, $d=4$.

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MCQ 31 Mark

Choose the correct answer from the given four options: In an $A P$ if $a=1, a_n=20$ and $S_n=399$, then $n$ is:

A
$19$
B
$21$
✓
$38$
D
$42$

Answer

Correct option: C.

$38$

$\because S_n=\frac{n}{2}[2 a+(n-1) d]$
$399=\frac{n}{2}[2 \times 1+(n-1) d]$
$798=2 n+n(n-1) d \ldots \ldots . .(i)$
and $an=20[\because an =a+(n-1) d]$
$\Rightarrow a+(n-1) d=20$
$\Rightarrow(n-1) d=19 \ldots \ldots$
Using $Eq. (ii)$ in $Eq. (i),$ we get
$\Rightarrow 798=2 n+19 n$
$\Rightarrow 798=21 n$
$\therefore n=\frac{498}{21}=38$

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MCQ 41 Mark

Choose the correct answer from the given four options: What is the common difference of an $AP$ in which $a_{18}-a_{14}=32 ?$

✓
$8$
B
$-8$
C
$-4$
D
$4$

Answer

Correct option: A.

$8$

$\text { Given, } a_{18}-a_{14}=32$
${\left[\therefore a_n=a+(n-1) d\right]}$
$\Rightarrow a+(18-1) d-[a+(14-1) d=32$
$\Rightarrow a+17 d-a-13 d=32$
$\Rightarrow 4 d=32$
$\Rightarrow d=8$
which is the required common difference of an $AP.$

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MCQ 51 Mark

Choose the correct answer from the given four options: If the common difference of an $AP$ is $5,$ then what is $a_{18}-a_{13} ?$

A
$5$
B
$20$
✓
$25$
D
$30$

Answer

Correct option: C.

$25$

Given, the common difference of $AP$ i.e., $d=5$
${\left[\therefore a_n=a+(n-1) d\right]}$
$\text {Now, } a_{18}-a_{13}=a+(18-1) d-[a+(13-1) d]$
$a_{18}-a_{13}=a+17 \times 5-a_{12} \times 5$
$a_{18}-a_{13}=85-60$
$a_{18}-a_{13}=25$

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MCQ 61 Mark

Choose the correct answer from the given four options:
If $7$ times the $7^{\text {th }}$ term of an $AP$ is equal to $11$ times its $11^{\text {th }}$ term, then its $18^{\text {th }}$ term will be:

A
$7$
B
$11$
C
$18$
✓
$0$

Answer

Correct option: D.

$0$

According to the question,
$7 a_7=11 a_{11}$
${\left[\therefore a_n=a+(n-1) d\right]}$
$\Rightarrow 7[a+(7-1) d]=11[a+(11-1) d]$
$\Rightarrow 7(a+6 d)=11(a+10 d)$
$\Rightarrow 7 a+42 d=11 a+110 d$
$\Rightarrow 4 a+68 d=0$
$\Rightarrow 2(2 a+34 d)=0$
$\Rightarrow 2 a 34 d=0[\therefore 2 \neq 0]$
$\Rightarrow a+17 d=0 \ldots \ldots .(i)$
$\Rightarrow 18^{\text {th }} \text { term of an AP, } a_{18}=a+(18-1) d$
$=a+17 d=0[\text { from } E q .(i)]$

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MCQ 71 Mark

Choose the correct answer from the given four options:
The first four terms of an AP, whose first term is -2 and common difference is -2, are:

A
-2, 0, 2, 4
B
-2, 4, – 8, 16
✓
-2, – 4, – 6, – 8
D
-2, – 4, – 8, –16

Answer

Correct option: C.

-2, – 4, – 6, – 8

Let the first four terms of an AP are a, a + d, a + 2d and a + 3d.
Given, that first term, a = -2 and common difference, d = -2, then have an AP as follows-
= -2, -2 - 2, - 2 + 2(-2), -2 + 3(-2)
= -2, -4, -6, -8

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MCQ 81 Mark

Choose the correct answer from the given four options: The $4^{\text {th }}$ term from the end of the $AP: -11,-8,-5, \ldots, 49$ is:

A
$37$
✓
$40$
C
$43$
D
$58$

Answer

Correct option: B.

$40$

We know that, the $n ^{\text {th }}$ term of an $AP$ from the end is
$a_n=1-(n-1) d$
Here, $I=$ Last term and $I=49 [$given$]$
Common difference, $d =-8-(-11)$
$d=-8+11$
$d=3$
From $Eq. (i), a_4=49-(4-1) 3$
$a_4=49-9$
$a_4=40$

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MCQ 91 Mark

Choose the correct answer from the given four options:
The sum of first five multiples of 3 is:

✓
45
B
55
C
65
D
75

Answer

Correct option: A.

The first five multipies of 3 are 3, 6, 9, 12 and 15.
Here, first term, a = 3, common difference, d = 6 - 3 = 3
and number of terem, n = 5
$\bigg[\therefore\text{S}_{\text{n}}= \frac{\text{n}}{2}\left\{2\text{a}+\big(\text{n}-1\big)\text{d}\right\}\bigg] $
$\therefore\text{S}_{\text{5}}= \frac{\text{5}}{2}\big[2\text{a}+\big(\text{5}-1\big)\text{d}\big]$
$\text{S}_{5}=\frac{5}{2}\big[2\times3+4\times3\big]$
$\text{S}_{5}=\frac{5}{2}\big(6+12\big)=5\times9=45$

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MCQ 101 Mark

Choose the correct answer from the given four options: In the first term of an $AP$ is $-5$ and the common difference is $2, $ then the sum of the first $6$ terms is:

✓
$0$
B
$5$
C
$6$
D
$15$

Answer

Correct option: A.

$0$

Given, $a = -5$ and $d = 2$
$\bigg[\because\text{S}_{\text{n}}=\frac{\text{n}}{2}\left\{2\text{a}+\big(\text{n}-1\big)\text{d}\right\}\bigg]$
$\text{S}_{\text{6}}= \frac{6}{2}\big[2\text{a}+\big(6-1\big)\text{d}\big] $
$S_6=3[2(-5)+5(2)]$
$S_6=3(-10+10)$
$S_6=0$

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MCQ 111 Mark

Choose the correct answer from the given four options: Which term of the $AP: 21, 42, 63, 84 , ...........$ is $210$ ?

A
$9^{\text {th }}$
✓
$10^{\text {th }}$
C
$11^{\text {th }}$
D
$12^{\text {th }}$

Answer

Correct option: B.

$10^{\text {th }}$

Let nth term of the given $AP$ be $210$
Here, first term, $a=21$
and common difference, $d=42-21=21$ and $a_n=210$
$\therefore a_n=a+(n-1) d$
$\Rightarrow 210=21+(n-1) 21$
$\Rightarrow 210=21+21 n-21$
$\Rightarrow 210=21 n$
$\Rightarrow n=10$

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MCQ 121 Mark

Choose the correct answer from the given four options: The sum of first $16$ terms of the $AP \ 10, 6, 2, ...... $ is:

✓
$-320$
B
$320$
C
$-325$
D
$-400$

Answer

Correct option: A.

$-320$

Given: $A P$ is $0,6,2, \ldots \ldots \ldots$
Here, first term $a=10$, common difference, $d=-4$
${\left[\therefore S_n=\frac{n}{2}\{2 a+(n-1) d\}\right]}$
$S_{16}=\frac{16}{2}[2 a+(16-1) d]$
$S_{16}=8[2 \times 10+15(-4)]$
$S_{16}=8(20-60)=8(-40)$
$S_{16}=-320$

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MCQ 131 Mark

Choose the correct answer from the given four options: In an $A P$, if $d=-4, n=7, a_n=4$, then $a$ is:

A
$6$
B
$7$
C
$20$
✓
$28$

Answer

Correct option: D.

$28$

In an $AP, \ a_n=a+(n-1) d$
$\Rightarrow 4= a +(7-1)(-4) ($by given conditions$)$
$\Rightarrow 4= a +6(-4)$
$\Rightarrow 4+24= a$
$\Rightarrow a=28$

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MCQ 141 Mark

Choose the correct answer from the given four options: In an $AP,$ if $a = 3.5, d = 0, n = 101,$ then $a_n$ will be:

A
$0$
✓
$3.5$
C
$103.5$
D
$104.5$

Answer

Correct option: B.

$3.5$

For an $AP\ a_n=a+(n-1) d=3.5+(101-1) \times 0$
$[$by given conditions$]$
$\therefore$ = 3.5

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MCQ 151 Mark

Choose the correct answer from the given four options:
Two APs have the same common difference. The first term of one of these is $–1$ and that of the other is$ – 8$. Then the difference between their $4^{th}$ terms is:

A
$-1.$
B
$-8$
✓
$7$
D
$-9$

Answer

Correct option: C.

$7$

Let the common diference of two $AP$ s are $d _1$ and $d _2$, respectively.
Bycondition, $d _1= d _2= d$
Let the first term of first AP $\left(a_1\right)=-1$
and the first term of second $A P\left(a_2\right)-8$
we know that, the $n^{\text {th }}$ term of an $A P, T_1=a+(n-1) d$
$\therefore 4^{\text {th }}$ term of first $A P, T_4=a+(4-1) d=-1+3 d$
and $4^{\text {th }}$ term of second $A P, T_4=a_2+(4-1) d=-8+3 d$
Now, the difference between there $4^{\text {th }}$ term is i.e.,
${\left[T_4-T^{\prime} 4\right]=(-1+3 d)-(-8+3 d)}$
$=-1+3 d+8-3 d=7$

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MCQ 161 Mark

Choose the correct answer from the given four options:
The $11^{\text {th }}$ term of the AP: $-5, \frac{-5}{2}, 0, \frac{5}{2}, \ldots \ldots \ldots$ is:

A
$-20$
✓
$20$
C
$-30$
D
$30$

Answer

Correct option: B.

$20$

Given $AP ,-5, \frac{-5}{2}, 0, \frac{5}{2}$
Here, $a=-5$
$d=\frac{-5}{2}+5=\frac{5}{2}$
$a 11=a+(11-1) d$
${\left[\therefore a_n=a+(n-1) d\right]}$
$=-5+(10) \times \frac{5}{2}$
$=-5+25=20$

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MCQ 171 Mark

Choose the correct answer from the given four options:
The $21^{\text {st }}$ term of the $AP$ whose first two terms are $-3$ and $4$ is:

A
$17$
✓
$137$
C
$143$
D
$-143$

Answer

Correct option: B.

$137$

Main concept used: $a_n=a+(n-1) d$
Here, $a=a_1=-3, a_2=4$
$\therefore d=a_2-a_1=4-(-3)=4+3=7$
Hence, $d=7$
Now, $a_n=a+(n-1) d$
$\Rightarrow a_{21}=-3+(21-1) \times 7$
$\Rightarrow a_{21}=-3+20 \times 7$
$\Rightarrow a_{21}=-3+140$
$\Rightarrow a_{21}=137$

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MCQ 181 Mark

Choose the correct answer from the given four options:
If the $2^{\text {nd }}$ term of an $AP$ is $13$ and the $5^{\text {th }}$ term is $25$ , what is its $7^{\text {th }}$ term?

A
$30$
✓
$33$
C
$37$
D
$38$

Answer

Correct option: B.

$33$

Given, $a _2=13$ and $a _5=25$
${\left[\because a_n=-a+(n-1) d\right]}$
$\Rightarrow a+(2-1) d=25$
and $a+(5+1) d=25$
$\Rightarrow a+d=13$
and $a+4 d=25$
on subtracting Eq. $(i)$ from Eq. $(ii)$, we get
$\Rightarrow 3 d=25-13=12 \Rightarrow d=4$
From Eq. $(i), a=13-4=9$
$\therefore a_7=a+(7-1) d$
$\Rightarrow a_7=9+6 \times 4$
$\Rightarrow a_7=33$

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Maths M.C.Q (1 Marks)

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