The 10th term of an A.P. is 31 and 20th term is 71. Find the 30th term.
20. Find the value of 'p' for which one root of the quadratic equation px? - 14x + 8 =0 is 6 times
the other
Answers
1. The 10th term of an A.P. is 31 and 20th term is 71. Find the 30th term.
Solution: Let the first term of the A.P. be a and the common difference be d.
Then the 10th term is (a + 9d)
and the 20th term is (a + 19d).
Given,
a + 9d = 31 ..... (1)
a + 19d = 71 ..... (2)
Using (2) - (1), we get
10d = 40
i.e., d = 4
Then a = 31 - 9 (4)
i.e., a = 31 - 36
i.e., a = - 5
Hence the 30th term is (a + 29d)
= - 5 + 29 (4)
= - 5 + 116
= 111
2. Find the value of 'p' for which one root of the quadratic equation px² - 14x + 8 =0 is 6 times
the other.
Solution: The given equation is
px² - 14x + 8 = 0 ..... (1)
Let the two roots be k and 6k
Since k, 6k are roots of (1),
k + 6k = - (- 14)/p
or, 7k = 14/p ..... (2)
& k . 6k = 8/p
or, 6k² = 8/p ..... (3)
Dividing (3) by (2), we get
6k² / 7k = 8/p / 14/p
or, 6k/7 = 4/7
or, k = 2/3
From (2), we write
7 (2/3) = 14/p
or, 21/3 = 14/p
or, p = 14 * 3/21
or, p = 2
Therefore the value of p is 2.
Answer:
1) The 30th term of the AP is 111
2) The value of p in the quadratic equation is 3
Step-by-step explanation:
1) The nth term of an AP is given by:
aₙ = a₁ + (n - 1)d
d = common difference
a₁ = the first term
n = Number of terms
10th term:
31 = a₁ + (10 - 1)d
a₁ + 9d = 31
20th term
71 = a₁ + (20 - 1)d
a₁ + 19d = 71
We solve the equations simultaneously to get the value of d and a₁
a₁ + 9d = 31.................1)
a₁ + 19d = 71...............ii)
Subtracting ii from i we have:
10d = 40
d = 4
a₁ = 31 - 9 × 4
a₁ = -5
The 30th term is given by:
a₃₀ = -5 + (30 - 1)4
= -5 + 116 = 111
The 30th term is 111
2) px² - 14x + 8 = 0
We have that:
We have that:
a = p, b = -14, c = 8
Let one root of the quadratic equation be k, the other root will be 6k
The sum of the roots of the quadratic equation is given by:
= -b/a = -(-14/p) = 14/p
k + 6k = 14/p
7k = 14/p
p = 14/7k
p= 2/k
The products of the roots is given by:
= c/a
k × 6k = 8/p
6k² = 8/p
p = 8/6k²
p = 4/3k²
Equating the two values of p we have:
4/3k² = 2/k
4k = 6k²
k = 4/6 = 2/3
p = 2/k
p = 2 × 3/2 = 3