Simplification
- 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 :
- 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
- three numbers are such that their sum is 50, product is 3750 and the sum of their reciprocals is 31/50 .Find the sum of the squares of the three numbers.
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x + y + z = 50 ; xyz = 3750
∴ 1 + 1 + 1 = yz + zx + xy x y z xyz = 31 150 ⇒ xy + yz + zx = 31 xyz 150 = 31 × 3750 = 775 150
∴ (x + y + z )2 = x2 + y2 + z2 + 2(xy + yz + zx)
⇒ (50)2 + x2 + y2 + z2 + 2 × 775
⇒ 2500 = x2 + y2 + z2 + 1550
⇒ x2 + y2 + z2 = 2500 – 1550
= 950Correct Option: C
x + y + z = 50 ; xyz = 3750
∴ 1 + 1 + 1 = yz + zx + xy x y z xyz = 31 150 ⇒ xy + yz + zx = 31 xyz 150 = 31 × 3750 = 775 150
∴ (x + y + z )2 = x2 + y2 + z2 + 2(xy + yz + zx)
⇒ (50)2 + x2 + y2 + z2 + 2 × 775
⇒ 2500 = x2 + y2 + z2 + 1550
⇒ x2 + y2 + z2 = 2500 – 1550
= 950
- The greatest perfect square number of 6 digits is
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Largest 6-digit number = 999999
∴ Required perfect square number = 999999 – 1998 = 998001Correct Option: B
Largest 6-digit number = 999999
∴ Required perfect square number = 999999 – 1998 = 998001
- If a perfect square, not divisible by 6, be divided by 6, the remainder will be
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Remainder on dividing 32 = 9 by 6 = 3
Remainder on dividing 42 = 16 by 6 = 4
Remainder on dividing 52 = 25 by 6 = 1Correct Option: C
Remainder on dividing 32 = 9 by 6 = 3
Remainder on dividing 42 = 16 by 6 = 4
Remainder on dividing 52 = 25 by 6 = 1
