Evaluate: \(106 \times 106 - 94 \times 94\)
Answer: A
This expression is in the form of \(a^2 - b^2\), which equals \((a+b)(a-b)\).
Here, a = 106 and b = 94.
\((106 + 94)(106 - 94) = (200)(12)\).
\(200 \times 12 = 2400\).
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What is the value of \(587 \times 999\)?
Answer: A
We can write 999 as (1000 - 1).
So, \(587 \times 999 = 587 \times (1000 - 1)\).
= \(587 \times 1000 - 587 \times 1\).
= 587000 - 587 = 586413.
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A boy was asked to multiply a number by 25. He instead multiplied the number by 52 and got the answer 324 more than the correct answer. What was the number?
Answer: A
Let the number be x.
Correct multiplication = 25x.
Incorrect multiplication = 52x.
Given, 52x - 25x = 324.
27x = 324.
x = \(324 \div 27 = 12\).
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What is the value of \((469 + 174)^2 - (469 - 174)^2\) / \((469 \times 174)\)?
Answer: B
This is based on algebraic identities. The numerator is in the form \((a+b)^2 - (a-b)^2\), which simplifies to \(4ab\).
Here, a=469 and b=174.
The expression becomes \(\frac{4ab}{ab} = 4\).
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The sum of three consecutive multiples of 3 is 72. What is the largest of the three numbers?
Answer: C
Let the three consecutive multiples of 3 be 3x, 3(x+1), and 3(x+2). A simpler way is to let them be n-3, n, n+3, where n is the middle number. Their sum is 3n = 72, so n=24.
The numbers are 21, 24, and 27.
The largest number is 27.
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How many natural numbers between 17 and 80 are divisible by 6?
Answer: B
The first multiple of 6 after 17 is 18 (6x3). The last multiple of 6 before 80 is 78 (6x13).
The numbers form an A.P.: 18, 24, ..., 78.
The number of terms can be found by the indices of the multiples: 3, 4, ..., 13.
Number of terms = (Last index - First index) + 1 = (13 - 3) + 1 = 11.
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Which of the following is a rational number?
Answer: C
A rational number is a number that can be expressed as a fraction p/q of two integers, where q is not zero. An irrational number cannot be expressed as such.
√2, √3, and √5 are irrational.
√4 = 2, which can be written as 2/1. Therefore, it is a rational number.
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A number exceeds its two-fifth by 75. The number is:
Answer: A
Let the number be x.
Two-fifth of the number is \(\frac{2}{5}x\).
The equation is \(x = \frac{2}{5}x + 75\).
\(x - \frac{2}{5}x = 75\).
\(\frac{3}{5}x = 75\).
\(x = 75 \times \frac{5}{3} = 25 \times 5 = 125\).
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What is the sum of the first 20 odd natural numbers?
Answer: C
The sum of the first 'n' odd natural numbers is given by the formula \(n^2\).
Here, n = 20.
The sum is \(20^2 = 400\).
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Find the value of \(0.6 + 0.66 + 0.066 + 6.606\)
Answer: A
Align the numbers vertically by their decimal points and add:
0.600
0.660
0.066
+ 6.606
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7.932
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