Find the value of \(0.3^2 + 0.1^2 - 2 \times 0.3 \times 0.1\).
Answer: A
The three terms are the expansion of a squared difference.
\[ a^{2} + b^{2} - 2ab = (a-b)^{2} \]
\[ (0.3 - 0.1)^{2} = 0.2^{2} = 0.04 \]
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A number is divided into two parts in such a way that 80% of the first part is 3 more than 60% of the second part, and 80% of the second part is 6 more than 90% of the first part. The number is:
Answer: C
Let the parts be x and y.
1) 0.8x = 0.6y + 3 => 8x = 6y + 30 => 4x - 3y = 15
2) 0.8y = 0.9x + 6 => 8y = 9x + 60 => 9x - 8y = -60
Solving these two linear equations. Multiply (1) by 8 and (2) by 3:
32x - 24y = 120
27x - 24y = -180
Subtracting the second from the first gives: 5x = 300 => x = 60.
Substitute x=60 into (1): 4(60) - 3y = 15 => 240 - 3y = 15 => 225 = 3y => y = 75.
The number is the sum of the parts: x + y = 60 + 75 = 135.
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What is the difference between the largest 4-digit number and the smallest 3-digit number?
Answer: A
The largest four-digit number is 9999 and the smallest three-digit number is 100.
\[ 9999 - 100 = 9899 \]
The smallest three-digit number is 100, not 99 - which would give 9900, the adjacent option.
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\(1/4\) of a number subtracted from \(1/3\) of the number gives 12. The number is:
Answer: B
Let the number be x.
\(\frac{1}{3}x - \frac{1}{4}x = 12\).
To subtract the fractions, find a common denominator, which is 12.
\(\frac{4x - 3x}{12} = 12\).
\(\frac{x}{12} = 12\).
\(x = 144\).
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Find the value of \(0.99^2\) using an algebraic identity.
Answer: A
Write 0.99 as \( 1 - 0.01 \) and use \( (a-b)^{2} \).
\[ (1 - 0.01)^{2} = 1 - 2(0.01) + (0.01)^{2} \]
\[ = 1 - 0.02 + 0.0001 = 0.9801 \]
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What is the value of the expression \(101 + 102 + 103 + ... + 200\)?
Answer: B
This is an arithmetic progression. Number of terms (n) = (Last Term - First Term) + 1 = (200 - 101) + 1 = 100.
Sum = \(\frac{n}{2}(\text{first term} + \text{last term})\)
Sum = \(\frac{100}{2} \times (101 + 200) = 50 \times 301 = 15050\).
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If the sum of two numbers is 22 and the sum of their squares is 404, then the product of the numbers is:
Answer: A
Expand the square of the sum.
\[ (a+b)^{2} = a^{2} + b^{2} + 2ab \]
\[ 22^{2} = 404 + 2ab \quad\Longrightarrow\quad 484 - 404 = 2ab \]
\[ ab = 40 \]
The two numbers turn out to be 2 and 20.
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The number 311311311311 is divisible by:
Answer: D
The number is formed by repeating '311' four times.
Divisibility by 3: Sum of digits = (3+1+1) × 4 = 5 × 4 = 20. Since 20 is not divisible by 3, the number is not divisible by 3.
Divisibility by 11: Sum of alternate digits. (1+3+1+3+1+3) - (1+1+1+1) = 12 - 4 = 8. No. Let's do it right to left. (1+1+1+1+1+1) - (3+3+3+3) = 6-12=-6. No. Let's do odd/even positions. (3+1+1+1+1+1)-(1+3+1+3+1+3) = 8-12=-4. It is not divisible by 11.
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What is the value of \(0.796 \times 0.796 - 0.204 \times 0.204\) / \(0.796 - 0.204\)?
Answer: C
The numerator is in the form \(a^2 - b^2\), which factors to \((a-b)(a+b)\).
So, the expression is \(\frac{(0.796 - 0.204)(0.796 + 0.204)}{0.796 - 0.204}\).
We can cancel out the \((0.796 - 0.204)\) term.
We are left with \(0.796 + 0.204 = 1.000 = 1\).
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The sum of the digits of a number is subtracted from the number. The resulting number is always divisible by:
Answer: D
This is a property related to the divisibility rule of 9. Let's take an example. Number = 42. Sum of digits = 6. Result = 42-6=36. 36 is divisible by 9.
Let the number be 10x+y. Sum of digits is x+y. The difference is (10x+y)-(x+y) = 9x. This is always divisible by 9.
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