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State true or false.

Divide 32 into four parts which are the four terms of an AP such that the ratio of the product of the first and the fourth term to the product of the second and the third term is 7: 15. Then the parts are 2, 6, 10, 14 or 14, 10, 6 and 2.

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False.

True

answer is A.

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Detailed Solution

Given that 32 is divided into four parts of an AP.
The ratio of the product of the first and the fourth term to the product of the second and the third term is 7: 15.
Let the four parts in AP are given by

a − 3 d, a − d, a + d

and

a + 3 d

.
The sum of the terms is 32. So,

S 4 = 32

(a − 3 d) + (a − d) + (a + d) + (a + 3 d) = 32

4 a − 4 d + 4 d = 32

a = 8

From the given information,

(a − 3 d) (a + 3 d) : (a − d) (a + d) = 7 : 15

Substitute a = 8 in the above equation,

(8 − 3 d) (8 + 3 d) (8 − d) (8 + d) = 715

64 − 9 d 2 64 − d 2 = 715

960 − 135 d 2 = 448 − 7 d 2

135 d 2 − 7 d 2 = 960 − 448

128 d 2 = 512

d 2 = 4 d = ± 2

Taking a = 8, d = 2 and finding the four terms,

(8 − 3 × 2), (8 − 2), (8 + 2), and (8 + 3 × 2) 2, 6, 10, and 14

Taking a = 8, d = -2 and finding the four terms,

8 − 3 × − 2, 8 − − 2, {8 + (− 2)}, 8 + 3 − 2 = 8 + 6, 8 + 2, 8 − 2, a n d 8 − 6

= 14, 10, 6 and 2
The parts are 2, 6, 10, 14 or 14, 10, 6 and 2.
Hence the statement is true.
Hence option 1 is correct.

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