Simplification
- The sum of three positive numbers is 18 and their product is 162. If the sum of two numbers is equal to the third number, then the sum of squares of the numbers is
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Let three positive integers be x, y and z.
According to the question,
x + y + z = 18 ..... (i)
xyz = 162 ..... (ii)
and x + y = z ..... (iii)
From equation (i),
z + z = 18 ⇒ 2z = 18 ⇒ z = 9
∴ xyz = 162
⇒ xy × 9 = 162⇒ xy = 162 = 18 ..... (iv) 9
∴ (x – y)2 = (x + y)2 – 4xy
= (9)2 – 4 × 18
= 81 – 72 = 9
∴ x – y = 3
∴ x + y + x – y = 9 + 3
⇒ 2x = 12 ⇒ x = 6
∴ x + y + z = 18
⇒ 6 + y + 9 = 18
⇒ y = 18 – 15 = 3
∴ x2 + y2 + z2
= (6)2 + (3)2 + (9)2
= 36 + 9 + 81 = 126Correct Option: B
Let three positive integers be x, y and z.
According to the question,
x + y + z = 18 ..... (i)
xyz = 162 ..... (ii)
and x + y = z ..... (iii)
From equation (i),
z + z = 18 ⇒ 2z = 18 ⇒ z = 9
∴ xyz = 162
⇒ xy × 9 = 162⇒ xy = 162 = 18 ..... (iv) 9
∴ (x – y)2 = (x + y)2 – 4xy
= (9)2 – 4 × 18
= 81 – 72 = 9
∴ x – y = 3
∴ x + y + x – y = 9 + 3
⇒ 2x = 12 ⇒ x = 6
∴ x + y + z = 18
⇒ 6 + y + 9 = 18
⇒ y = 18 – 15 = 3
∴ x2 + y2 + z2
= (6)2 + (3)2 + (9)2
= 36 + 9 + 81 = 126
- A number of boys raised Rs. 12,544 for a famine fund, each boy has given as many rupees as there were boys. The number of boys was :
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Number of boys = √12544 = 112
Illustration :
Correct Option: B
Number of boys = √12544 = 112
Illustration :
- If 5416 * 6 is a perfect square, then the digit at ‘*’ is :
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∵ 1466 × 6 = 8796
∴ * = 9Correct Option: A
∵ 1466 × 6 = 8796
∴ * = 9
- The least number that should be subtracted from the number 32146 to make it a perfect square is :
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√32146 > 179
179 × 179 = 32041
∴ Required answer
= 32146 – 32041 = 105Correct Option: B
√32146 > 179
179 × 179 = 32041
∴ Required answer
= 32146 – 32041 = 105
- The sum of the perfect squares between 120 and 300 is
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Required sum = 121 + 144 + 169 + 196 + 225 + 256 + 289 = 1400
Correct Option: A
Required sum = 121 + 144 + 169 + 196 + 225 + 256 + 289 = 1400
