Practice the questions given in the worksheet on application problems on expansion of powers of binomials and trinomials.
1. Use (a ± b)\(^{2}\) = a\(^{2}\) ± 2ab + b\(^{2}\) to evaluate the following:
(i) (3.001)\(^{2}\)
(ii) (5.99)\(^{2}\)
(iii) 1001 × 999
(iv) 5.63 × 5.63 + 11.26 × 2.37 + 2.37 × 2.37
(v) 8.79 × 8.79 – 8.79 × 3.58 + 1.79 × 1.79
2. (i) If the sum of two numbers is 12 and the sum of their squares is 74, find the product of the numbers.
[Hint: a + b = 12, a\(^{2}\) + b\(^{2}\) = 74. To find ab.]
(ii) If the numbers x is 5 more than the number y and the sum of the squares of x and y is 37 then find the product of x and y.
(iii) The sum of two numbers is 14 and their difference is 2. Find the product of the two numbers.
[Hint: a + b = 14, a – b = 2. To find ab.]
3. (i) If the sum of three numbers is 10 and the sum of their squares is 38, find the sum of the products of the three numbers taking two at a time.
[Hint: a + b + c = 10, a\(^{2}\) + b\(^{2}\) + c\(^{2}\) = 38.
ab + bc + ca = \(\frac{1}{2}\){(a + b + c)\(^{2}\) – (a\(^{2}\) + b\(^{2}\) + c\(^{2}\))} = \(\frac{1}{2}\){10\(^{2}\) – 38}.]
(ii) If the sum of the squares of the squares of three numbers is equal to the square of their sum, prove that the sum of the products of the three numbers taking two at a time is equal to zero.
[Hint: x - y = 5, x\(^{2}\) + y\(^{2}\) = 37. To find xy.]
(iii)If the sum of the squares of three positive numbers is 14 and the sum of their products taking two at a time is 11, find the sum of the numbers.
[Hint: a\(^{2}\) + b\(^{2}\) + c\(^{2}\) = 14, ab + bc + ca = 11.
(a + b + c)\(^{2}\) = a\(^{2}\) + b\(^{2}\) + c\(^{2}\) + 2(ab + bc + ca) = 14 + 2 × 11 = 36.]
4. Find the value:
(i) (5.45)\(^{3}\) + (3.55)\(^{3}\)
(ii) (8.12)\(^{3}\) – (3.12)\(^{3}\)
(iii) 1.81 × 1.81 – 1.81 × 2.19 + 2.19 × 2.19
[Hint: Value = \(\frac{(1.81^{3} + (2.19)^{3}}{1.81 + 2.19}\)
= \(\frac{1}{4}\){(1.81 + 2.19)\(^{3}\) – 3 × 1.81 × 2.19(1.81 + 2.19)}
= \(\frac{1}{4}\){4\(^{3}\) – 12 × 1.81 × 2.19}]
(iv) 7.16 × 7.16 + 2.16 × 7.16 + 2.16 × 2.16
5. (i) If the sum of and product of two numbers are 7 and \(\frac{45}{4}\) respectively, find the sum of their cubes.
[Hint: Here, a + b = 7, ab = \(\frac{45}{4}\). To find a\(^{3}\) + b\(^{3}\).]
(ii) If the difference of two numbers is 10 and their product is – 24, find the difference of their cubes.
[Hint: Here, a - b = 10, ab = -24. To find a\(^{3}\) - b\(^{3}\).]
Answers for the worksheet on application problems on expansion of powers of binomials and trinomials are given below.
Answer:
1. (i) 9.006001
(ii) 35.8801
(iii) 999999
(iv) 64
(v) 49
2. (i) 35
(ii) 6
(iii) 48
3. (i) 31
(iii) 6
4. (i) 206.6175
(ii) 505.016
(iii) 4.1083
(iv) 71.3968
5. (i) \(\frac{427}{4}\)
(ii) 280
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