(a - b)² =
Answer: A
\[ (a-b)^{2} = (a-b)(a-b) = a^{2} - ab - ab + b^{2} \]
\[ = a^{2} - 2ab + b^{2} \]
The middle term is essential: \( (a-b)^{2} \) is not \( a^{2} - b^{2} \), which factorises as \( (a+b)(a-b) \) instead.
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If \( a = 3 \), \( b = 2 \), \( c = 4 \) and \( x = 5 \), then the value of \( \dfrac{4ab - 2acx}{2ab + 3ac} \) is
Answer: A
Evaluate the numerator and denominator separately.
\[ 4ab = 4 \times 3 \times 2 = 24 \]
\[ 2acx = 2 \times 3 \times 4 \times 5 = 120 \]
\[ \text{numerator} = 24 - 120 = -96 \]
\[ 2ab = 12, \qquad 3ac = 36 \quad\Longrightarrow\quad \text{denominator} = 48 \]
\[ \frac{-96}{48} = -2 \]
Substituting into the whole numerator before dividing avoids the common error of cancelling term by term.
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On solving 2p - 3q - 4r + 6r - 2q + p, answer will be
Answer: C
Collect like terms.
\[ 2p + p = 3p \]
\[ -3q - 2q = -5q \]
\[ -4r + 6r = 2r \]
\[ \therefore\ 3p - 5q + 2r \]
Only terms in the same variable may be combined; \( p \), \( q \) and \( r \) stay separate.
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Algebraic expression of word expression "product of 8 and four times of a number c" is
Answer: C
Four times the number is \( 4c \), and the product of 8 with that is
\[ 8 \times 4c = 32c \]
The two multiplications compound; adding them instead would give \( 8 + 4c \).
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By solving algebraic expression 2(3m -4n +5) -3(2m -3n +4), answer should be
Answer: B
Expand both brackets and collect.
\[ 2(3m - 4n + 5) = 6m - 8n + 10 \]
\[ -3(2m - 3n + 4) = -6m + 9n - 12 \]
\[ (6m - 6m) + (-8n + 9n) + (10 - 12) = n - 2 \]
The \( m \) terms cancel entirely, which is why no \( m \) appears in the answer.
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In value 200x, variable is
Answer: C
In \( 200x \), the number 200 is the coefficient - a fixed value - while \( x \) is the variable, the symbol that can take different values.
Anything expressed purely in numbers, such as \( 10(20) \), is a constant.
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If Ana is x years old then 5 times Ana's age three years ago is
Answer: B
Three years ago Ana was \( x - 3 \) years old, and five times that is
\[ 5(x - 3) \]
The subtraction happens inside the bracket, before the multiplication: \( 5x - 3 \) would be five times her present age less three.
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Sum of 2a + 4b, 5a + 2c + 2d, -5a - 9b + 4c - d and 11b - 8c - 4d is
Answer: B
Collect each variable in turn.
\[ a: \ 2 + 5 - 5 = 2 \]
\[ b: \ 4 - 9 + 11 = 6 \]
\[ c: \ 2 + 4 - 8 = -2 \]
\[ d: \ 2 - 1 - 4 = -3 \]
\[ \therefore\ 2a + 6b - 2c - 3d \]
Taking one variable at a time across all four expressions is safer than adding them two at a time.
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If x = 10 and y = 4 then 5x + 4y =
Answer: A
\[ 5x + 4y = 5(10) + 4(4) = 50 + 16 = 66 \]
Both products must be formed before adding; stopping at \( 5x \) gives 50.
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Simplify a(c - b) - b(a - c)
Answer: B
Expand both brackets, watching the sign in front of the second.
\[ a(c-b) - b(a-c) = ac - ab - ab + bc \]
\[ = ac - 2ab + bc \]
The minus outside the second bracket flips both of its terms, which is what turns \( -bc \) into \( +bc \) and produces the \( -2ab \).
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