Question 11
In an A. P. the first term is 8, nth term is 33 and the sum to first n terms is 123.Find n and d, the common differences
Solution
33=8+(6−1)d
d=5
Question 12
A sum of Rs 700 is to be used to give seven cash prizes to students of a school for their overall academic performances. If each price is Rs 20 less than its preceding prize, find the value of each prize
Solution
Let first prize is a,then seven prizes will be
a, a-20,a- 40,a-60,a-80,a-100,a-120
Now
a+a−20+a−40+a−60+a−80+a−100+a−120=700
7a=1120
a=160
So prizes will be
160,140,120,100,80,60,40
Question 13
Ronit is moving towards a stationary source of sound at a constant speed. At his initial position, he hears a pitch of 5 hertz. If the pitch from the source increases by 4 hertz for every 5 m Ramit travels, find the distance he would have travelled towards the source when the pitch is exactly 149 hertz
Solution
Writes an arithmetic progressions (AP) for the increase in pitch every 5 m as:
5 hertz, 9 hertz, 13 hertz, 17 hertz, …
Identifies the first term of the AP, a as 5 and the common difference, 4 as 4.
Finds the n th term of the AP when the pitch is exactly 149 hertz as:
5 + 4( n – 1) = 149
n=37
Finds the distance travelled as 36 x 5 = 180 m
Question 14
if the 7th term of an ap is 27 and the 25th term is 20 more than the 20th term, find the A.P
Solution
a7=27
a25=20+a20
a+24d=20+a+19d
24d-19d=20
5d=20 d=4
=a+6d=27
=a+6×4=27
a+24=27
a=3
The A.P. is a, a+d, a+2d, ……
I.e. 3,7,11,15,19 ……
Question 15
which term of the ap : 3,8,13,18 …. is 68?
Solution
a25 – a11=0
Question 16
The sum of roots of the equation 2x2-px+7=0 is 8. The quadratic equation kx2px+2p=0 has two equal roots. Find value of k
Solution
k =0; k=2
Question 17
Solve for x by splitting the middle term 6x2 + 17x + 5 = 0
Solution
(3x+1) (2x+5) = 0 ; x = -1/3, -5/2
Question 18
In a class test, the sum of kavita’s marks in Science and Maths is 30. Had she got 2 marks more in science and 3 marks less in Maths then the product of the marks would have been 210. Find her marks in two subjects separately.
Solution
Two possible answers for this given question is:
If Kavita scored 13 mark in mathematics, then marks scored in Science =
(30 – 13) = 17.
If Kavita scored 12 mark in mathematics, then marks scored in Science =
(30 – 12) = 18.
Question 19
Rahul and Manish are workers in a factory. Both are very hardworking and one day they are assigned a piece of work in which Rahul takes 6 days less than the time taken by Manish to finish a piece of work. If both Rahul and Manish together can finish it in 4 days, find the time taken by Manish to finish the work
Solution
x2 − 14x + 24 = 0
x2 − 12x − 2x + 24 = 0
(x − 12) (x – 2) = 0
x = 12 or x = 2, but x cannot be less than 6.
So, x = 12
Therefore, Manish can finish the work in 12 days.
Question 20
The speed of a boat in still water is 8 km/hr. It can go 15 km upstream and 22 km downstream in 5 hours. Find the speed of the stream.
Solution
5x2 – 7x + 296 – 320 = 0
5x2-7x – 24 = 0
(5x + 8)(x- 3) = 0
x – 3 =0 [Speed can not be negative 5x + 8 0]
x = 3
Hence, the speed of the stream is 3 km/hr
SELF ASSESSMENT
Q1. If the sum of 7 terms of an A. P. is 49 and that of 17 terms is 289, find the sum
of n terms.
Q2. If there are (2n+1) terms in AP then prove that the ratio of the sum of odd terms and the sum of even terms is (n+1) : n .
Q3. which of the ARITHMATIC PROGRESSION 7,14,21 …. will be 98 more than its
35th term?
Q4. There are 20 rows of seats in a concert hall with 20 seats in the first row, 21
seats in the second row, 22 seats in the third row, and so on.In total, how
many seats are there in the concert hall?
Q5. In an AP
Sm=n , Sn = m then S(m+n) = ?
Q6. Solve the equation : 1+4+7+10+ …… + x = 287
