Number System
- The simplified value of
1 − 1 1 − 1 1 − 1 ........ 1 − 1 1 − 1 3 4 5 99 100
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2 × 3 × 4 ×......× 98 × 98 3 4 5 99 100 = 2 = 1 100 50 Correct Option: C
2 × 3 × 4 ×......× 98 × 98 3 4 5 99 100 = 2 = 1 100 50
- The sum of the squares of two positive numbers is 100 and difference of their squares is 28. Find the sum of the numbers :
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According to question
x2 + y2 = 100 ...(i)
x2 − y2 = 28 ...(ii)
Adding both the equations
x2 + y2
x2 − y2
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2x2 128
⇒ x2 = 64 ∴ x = 8
From the equation (i)
y2 = 100 − 64 ∴ y = 6
So, x + y = 8 + 6 = 14Correct Option: C
According to question
x2 + y2 = 100 ...(i)
x2 − y2 = 28 ...(ii)
Adding both the equations
x2 + y2
x2 − y2
-----------
2x2 128
⇒ x2 = 64 ∴ x = 8
From the equation (i)
y2 = 100 − 64 ∴ y = 6
So, x + y = 8 + 6 = 14
- The numbers 1, 3, 5,7 .., 99 and128 are multiplied together. The number of zeros at the end of the product must be :
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In 2m × 5n, the number of zeros
= n when m ≥ n
= m when m < n
Here, 128 = 27
In 1 × 3 × 5 × 7 × ... × 99 multiples of 5 are 5, 15, 25, 35, 45, 55, 65, 75, 85, 95 (= 510)
Clearly, 7 zeros will be found in the product.Correct Option: C
In 2m × 5n, the number of zeros
= n when m ≥ n
= m when m < n
Here, 128 = 27
In 1 × 3 × 5 × 7 × ... × 99 multiples of 5 are 5, 15, 25, 35, 45, 55, 65, 75, 85, 95 (= 510)
Clearly, 7 zeros will be found in the product.
- A number of friends decided to go on a picnic and planned to spend ₹108 on eatables. Three of them however did not turn up. As a consequence each one of the remaining had to contribute ₹3 extra. The number of them who attended the picnic was :
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Let the original number of friends be x.
∴ 108 + 108 = 3 x − 3 x ⇒ 108 x − x + 3 = 3 x(x − 3)
⇒ x (x – 3) = 108
⇒ x2 – 3x – 108 = 0
⇒ x2 –12 x + 9x –108 = 0
⇒ (x– 12) + 9 (x – 12) = 0
⇒ (x – 12) (x + 9) = 0
⇒ x = 12 as x ≠ – 9
∴ The number of friends who attended the picnic
= 12 – 3 = 9Correct Option: C
Let the original number of friends be x.
∴ 108 + 108 = 3 x − 3 x ⇒ 108 x − x + 3 = 3 x(x − 3)
⇒ x (x – 3) = 108
⇒ x2 – 3x – 108 = 0
⇒ x2 –12 x + 9x –108 = 0
⇒ (x– 12) + 9 (x – 12) = 0
⇒ (x – 12) (x + 9) = 0
⇒ x = 12 as x ≠ – 9
∴ The number of friends who attended the picnic
= 12 – 3 = 9
- If doubling a number and adding 20 to the result gives the same answer as multiplying the number by 8 and taking away 4 from the product, the number is :
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Let the number be x.
∴ According to question,
2x + 20 = 8x – 4
or 8x – 2x = 20 + 4
or 6x = 24
∴ x = 4Correct Option: C
Let the number be x.
∴ According to question,
2x + 20 = 8x – 4
or 8x – 2x = 20 + 4
or 6x = 24
∴ x = 4