Algebraic expressions

On substituting \(q=4\), \(q^3=4^3=64\) and \(2q=2\times4=8\). Thus, \(\frac{q^3-2q}{4}=\frac{64-8}{4}=\frac{56}{4}=14\). Therefore, 14 is the correct option. A result such as 16 can arise from an error in finding or subtracting \(2q\). Exam tip: evaluate the exponent first, simplify the numerator, and then divide.

If \(n\) is odd, the four consecutive odd numbers are \(n\), \(n+2\), \(n+4\), and \(n+6\). Their sum is \(n+(n+2)+(n+4)+(n+6)=4n+12\). \(4n+6\) is not the sum of all four terms. Exam tip: consecutive odd numbers differ by 2.

Simplify the brackets first: \(3(4x-5)=12x-15\). In the square bracket, \(2x-3(1-x)=2x-3+3x=5x-3\). Therefore, the complete expression is \(12x-15-(5x-3)=12x-15-5x+3=7x-12\). The option \(7x-18\) can result from incorrectly treating \(-(-3)\) as \(-3\) in the final subtraction. Exam tip: when removing a bracket preceded by a minus sign, change the signs of every term inside it.

Given x=1 and y=3, we get x+y=4. Therefore, 3(x+y)^2=3×4^2=3×16=48, and 4xy=4×1×3=12. Hence, the value is 48−12=36. A value such as 38 can result from an error while squaring or multiplying. Exam tip: after substitution, evaluate brackets first, then powers, multiplication, and subtraction.

Combine like terms: the a-terms give 4a+9a-6a=7a, and the b-terms give -5b-2b=-7b. Hence, the simplified form is 7a-7b. In 19a-7b, the term -6a has not been subtracted while combining the a-terms. Exam tip: add or subtract only terms having the same variable and exponent.

The perimeter of a rectangle is \(2(\text{length}+\text{breadth})\). Thus, \(2[(4x-7)+(3x+2)] = 2(7x-5) = 14x-10\). Therefore, the correct expression is \(14x-10\). The expression \(7x-5\) is only the sum of the length and breadth, not the perimeter. Exam tip: A rectangle has two equal lengths and two equal breadths, so multiply \(l+b\) by 2.

Substituting x=1 gives 6(1)^3-9(1)^2+4(1)-8. Since every positive power of 1 is 1, this becomes 6-9+4-8=-7. Therefore, the correct answer is -7. The value -5 can result from an error while adding or subtracting the terms. Exam tip: after substitution, retain the sign of every term and simplify step by step.

First simplify the innermost bracket: \(2x-(x-6)=2x-x+6=x+6\). Then \(7x-3(x+6)=7x-3x-18=4x-18\). Hence, the correct answer is \(4x-18\). The option \(4x+18\) results from incorrectly multiplying \(-3\) by \(+6\). Exam tip: When removing brackets preceded by a minus sign, check the sign of every term carefully.

To subtract, change the sign of every term in the second expression: \((5x^2-3x+8)-(2x^2-7x-6)=5x^2-3x+8-2x^2+7x+6\). Combining like terms gives \(3x^2+4x+14\). Therefore, option A is correct. In option C, the sign of \(-7x\) has not been changed to \(+7x\). Exam tip: When a minus sign appears before brackets, reverse the signs of all terms inside the brackets.

Substituting x=2 gives 2(2+5)-3(2-1). Thus, 2×7-3×1=14-3=11, so the correct answer is 11. A value such as 13 may result from not subtracting the term 3(2-1) correctly. Exam tip: after substitution, evaluate brackets first, then multiplication and subtraction.

First simplify the innermost brackets: \(2x-(x-2)=2x-x+2=x+2\). The expression becomes \(2(5x-3)-4(x+2)\). Expanding gives \(10x-6-4x-8=6x-14\). Hence, \(6x-14\) is correct. A choice such as \(14x-14\) can result from not distributing \(-4\) across the complete term \((x+2)\). Exam tip: simplify nested brackets from the innermost bracket outward.

A minus sign before the second bracket changes the sign of every term inside it: \(2x^2-3x+5-x^2-4x+7\). Combining like terms gives \((2x^2-x^2)+(-3x-4x)+(5+7)=x^2-7x+12\). Therefore, option A is correct. In option D, the constant term has been given the wrong sign. Exam tip: when a bracket is preceded by a minus sign, change the sign of every term inside it.

Substitute \(x=-2\): \(3x^3-2x^2+5x-1=3(-2)^3-2(-2)^2+5(-2)-1\). Since \((-2)^3=-8\) and \((-2)^2=4\), the value is \(-24-8-10-1=-43\). A value such as \(-35\) can result from mishandling powers or negative signs. Exam tip: check the signs of odd and even powers of a negative number separately.

Only like terms can be added or subtracted. The \(a^2b\) terms give \(4a^2b+5a^2b=9a^2b\), and the \(ab^2\) terms give \(-3ab^2+7ab^2=4ab^2\). Therefore, the simplified expression is \(9a^2b+4ab^2\). \(16a^3b^3\) is incorrect because it would result from multiplication, whereas this question requires addition of terms. Exam tip: Check both the variables and their exponents before combining terms.

In option A, opening the brackets gives \(3(2p-5)-(p+3)=6p-15-p-3=5p-18\). Hence, it is correct. Option B simplifies to \(7p-18\), since \(2(3p-9)+p=6p-18+p\). Exam tip: When removing brackets preceded by a minus sign, change the sign of every term inside the bracket.

Substituting the given values, \(m^2n=3^2\times(-2)=9\times(-2)=-18\), and \(2mn^2=2\times3\times(-2)^2=2\times3\times4=24\). Therefore, \(m^2n-2mn^2=-18-24=-42\). The option \(-18\) is only the value of the first term; it does not subtract the second term. Exam tip: the square of a negative number is positive, so \((-2)^2=4\).

On expanding the brackets, the expression becomes 2x+6y-6x+3y+4x. Combining like terms, the coefficient of x is 2-6+4=0, while the coefficient of y is 6+3=9. Hence, the simplified expression is 9y. The option 6x+9y results from combining the x-terms incorrectly. Exam tip: When a bracket is multiplied by a negative number, apply the sign change to every term inside it.

The terms \(kx\) and \(6x\) are like terms, so \(kx+6x=(k+6)x\). For this to equal \(11x\), we need \(k+6=11\). Therefore, \(k=5\). If \(k=6\), the expression becomes \(12x\), not \(11x\). Exam tip: add the coefficients when combining like terms.

Combining like terms gives \(7u^2v+3u^2v=10u^2v\) and \(-4uv^2-9uv^2=-13uv^2\). Thus, the simplified expression is \(10u^2v-13uv^2\), so the coefficient of \(uv^2\) is \(-13\). The value \(-4\) is the coefficient of only one term, not the combined like terms. Exam tip: first collect all terms with exactly the same variable part before identifying a coefficient.

The governing concept is translation of verbal mathematical language into an algebraic expression. “The sum of the squares of x and y” means x^2 + y^2, because each variable is squared before the addition. “Subtracting 2xy from” that sum means the entire 2xy term is taken away: x^2 + y^2 − 2xy. Therefore option C is correct. Option A incorrectly uses x^2 − y^2, which is a difference rather than a sum. Option B adds 2xy instead of subtracting it, while option D reverses the order and represents 2xy − x^2 − y^2.