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Maths 3 Marks Question

P-1 Arithmetic Progression · Maharashtra Board STD 10 (MSBSHSE - Semi English Medium). 70 questions.

Questions · Page 2 of 2

3 Marks Question

Question 513 Marks
In an A.P. 17th term is 7 more than its 10th term. Find the common difference.
Answer
Given: t17 = 7 + t10 ……(1)
In t17, n = 17
In t10, n = 10
By using nth term of an A.P. formula,
tn = a + (n – 1)d
where n = no. of terms
a = first term
d = common difference
tn = nth term
Thus, on using formula in eq. (1) we get,
⇒ a + (17 – 1)d = 7 + (a + (10 – 1)d)
⇒ a + 16 d = 7 + (a + 9 d)
⇒ a + 16 d – a – 9 d = 7
⇒ 7 d = 7
$\Rightarrow d =\frac{7}{7}=1$
Thus, common difference “d” = 1
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Question 523 Marks
In year 2015, Mrs. Shaikh got a job with salary ₹ 1,80,000 per year. Her employer agreed to give ₹ 10,000 per year as increment. Then in how many years will her annual salary be ₹ 2,50,000?
Answer
YearFirst Year
(2015)		Second Year
(2016)		Third Year
(2017)		…
Salary (₹)[1,80,000][1,80,000+10,000] …

$\begin{array}{l}
a=1,80,000 \quad d=10,000 \quad n=? \quad t _{ n }=2,50,000 ₹ \\
t _{ n }=a+(n-1) d \\
2,50,000=1,80,000+(n-1) \times 10,000 \\
(n-1) \times 10000=70,000 \\
(n-1)=7 \\
n=8
\end{array}$
In the $8^{\text {th }}$ year her annual salary will be $₹ 2,50,000$.

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Question 533 Marks
Find the sum of all odd numbers from 1 to 150.
Answer
1 to 150 all odd numbers are $1,3,5,7, \ldots, 149$.
Which is an A.P.
Here $a=1$ and $d=2$. First let's find how many odd numbers are there from 1 to 150 , so find the value of $n$, if $t_{ n }=149$
$\begin{array}{ll}
t_{ n }=a+(n-1) d & \\
149=1+(n-1) 2 & \therefore 149=1+2 n-2 \\
& \therefore n=75
\end{array}$
Now let's find the sum of these 75 numbers $\quad 1+3+5+\ldots+149$.
$\begin{array}{c}
a=1 \text { and } d=2, n=75 \\
S _{ n }=\frac{n}{2}\left[ t _1+ t _{ n }\right] \\
S _{ n }=\frac{75}{2}[1+149] \\
S _{ n }=37.5 \times 150 \\
S _{ n }=5625
\end{array}$
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Question 543 Marks
How many two digit numbers are divisible by 4 ?
Answer
List of two digit numbers divisible by 4 is
$
12,16,20,24, \ldots, 96 \text {. }
$
Let's find how many such numbers are there.
$
t_{\mathrm{n}}=96, \quad a=12, \quad d=4
$
From this we will find the value of $n$.
$
\begin{aligned}
t_{\mathrm{n}} & =96, \therefore \text { By formula, } \\
96 & =12+(n-1) \times 4 \\
& =12+4 n-4 \\
\therefore 4 & =88 \\
\therefore n & =22
\end{aligned}
$
$\therefore$ There are 22 two digit numbers divisible by 4 .
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Question 553 Marks
Check whether 301 is in the sequence
$5,11,17,23, \ldots \text { ? }$
Answer
In the sequence $5,11,17,23, \ldots$
$
\begin{array}{l}
t_1=5, t_2=11, t_3=17, t_4=23, \ldots \\
t_2-t_1=11-5=6 \\
t_3-t_2=17-11=6
\end{array}
$
$\therefore$ This sequence is an A.P.
First term $a=5$ and $d=6$
If 301 is $n^{\text {th }}$ term, then.
$
\begin{array}{l}
t=a+(n-1) d=301 \\
\therefore 301=5+(n-1) \times 6 \\
=5+6 n-6 \\
\therefore 6 n=301+1=302 \\
\end{array}
$
$\therefore n=\frac{302}{6}$. But it is not an integer.
$\therefore 301$ is not in the given sequence.
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Question 563 Marks
The sixth term of an A.P. is 5 times the $1^{\text {st }}$ term and the eleventh term exceeds twice the fifth term by 3. Find the $8^{\text {th }}$ term.
Answer
$t_{8}$ = 33
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Question 573 Marks
The first and the last terms of an A.P. are 17 and 350 respectively. If the common difference is 9 , how many terms are there and what is their sum?
Answer
38,6973
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Question 583 Marks
Split 69 in three parts such that they are in A.P. and product of two smaller parts is 483.
Answer
$21,23,25$
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Question 593 Marks
Obtain the sum of 56 terms of an A.P. whose $19^{\text {th }}$ and $38^{\text {th }}$ terms are 52 and 148 respectively.
Answer
5600
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Question 603 Marks
In an A.P., if the $5^{\text {th }}$ and 12 th terms are 30 and 65 respectively, what is the sum of the first 20 terms.
Answer
1150
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Question 613 Marks
If $10^{ th }$ term and the $18^{ th }$ term of an A.P. are 25 and 41 respectively, then find the $38^{\text {th }}$ term.
Answer
$t_{38}=81$
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Question 653 Marks
For what value of $n$, the nth term of the following two A.P.s are equal?
$23,25,27,29$,....  and $-17,-10,-3,4$, ....
Answer
$n=9)
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Question 663 Marks
Find three consecutive terms in an A.P. whose sum is -3 and the product of their cubes is 512.
Answer
$-4,-1,2$ or $2,-1,-4$
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Question 703 Marks
Find four conseculive terms in an A.P. such that the sum of the middle two terms is 18 and product of the two end terms is 45.
Answer
$3,7,11,15$ or $15,11,7,3$
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