Algebraic expressions

Correct answer: A. \(4x-23\)

Explanation: Expand the brackets first: \(3(2x-5)=6x-15\) and \(-2(x+4)=-2x-8\). Combining like terms gives \(6x-15-2x-8=4x-23\). Therefore, the correct answer is \(4x-23\). The result \(4x-7\) can arise from an error while multiplying \(-2\) by \(4\). Exam tip: when a bracket is preceded by a negative coefficient, apply it carefully to every term inside the bracket.

Correct answer: B. 17

Explanation: Given \(a=-2\), we have \(a^2=(-2)^2=4\). Substituting in the expression, \(4a^2-3a-5=4(4)-3(-2)-5=16+6-5=17\). Therefore, 17 is correct. The value 9 can result from handling the sign of \(-3a\) incorrectly. Exam tip: Always use brackets around a negative substituted value before squaring or multiplying.

Correct answer: C. (3p^2+5p+3)

Explanation: Direct answer: Option C, \(3p^2+5p+3\). To simplify, combine only like terms: terms with \(p^2\) go together, terms with \(p\) go together, and constants go together. First, \(5p^2-2p^2=3p^2\). Next, \(-3p+8p=5p\). Finally, \(7-4=3\). Putting these results together gives \(3p^2+5p+3\). Option A has the wrong coefficient of \(p^2\) and also the wrong constant. Option B changes the signs and does not correctly combine the terms. Option C is correct because every like-term calculation is correct. Option D incorrectly adds the two quadratic coefficients and gives the wrong coefficient of \(p\). A common mistake is combining unlike terms; \(p^2\), \(p\), and constants must remain separate.

Correct answer: C. (-3x+12)

Explanation: Changing signs of the subtracted expression gives (2x+3-5x+9=-3x+12). Change signs carefully while removing brackets.

Correct answer: A. 17

Explanation: Substituting the given values, \(2x^2-xy+4y=2(3)^2-(3)(-1)+4(-1)\). Therefore, \(18+3-4=17\), so 17 is correct. Since \(y=-1\), the term \(-xy\) becomes \(-(3\times-1)=+3\); mishandling this sign can lead to the nearby option 21. Exam tip: always substitute a negative value using brackets.

Correct answer: B. \(3n+6\)

Explanation: If \(n\) is even, the three consecutive even numbers are \(n\), \(n+2\), and \(n+4\). Their sum is \(n+(n+2)+(n+4)=3n+6\). Therefore, option B is correct. \(n+6\) adds 6 only to the first number; it does not represent the sum of all three numbers. Exam tip: consecutive even numbers always differ by 2.

Correct answer: A. \(6m+3\)

Explanation: On opening the brackets, \(7m-2(3m-4)+5(m-1)=7m-6m+8+5m-5\). Combining like terms gives \((7m-6m+5m)+(8-5)=6m+3\). Therefore, the correct answer is \(6m+3\). The option \(6m-13\) results from handling the signs of the constant terms incorrectly. Exam tip: When a minus sign or coefficient occurs before brackets, multiply it by every term inside the brackets.

Correct answer: C. (20)

Explanation: The direct answer is C: 20. Substitute the given value carefully. First, if r=4, then r squared is about multiplying 4 by 4, so r²=16. Next, calculate the two terms in the numerator: 3r²=3×16=48 and 2r=2×4=8. Subtract: 48−8=40. Finally divide the complete numerator by 2: 40÷2=20, or \\(\frac{3(4)^2-2(4)}{2}=\frac{48-8}{2}=20\\). A is wrong because 16 is only r², not the value of the full expression. B is wrong because it does not result from the stated substitution and simplification. D is wrong because the correct numerator gives 40, which becomes 20 after division. C is correct. Memory cue: substitute first, simplify the whole numerator, then divide.

Correct answer: A. \(7a^2b-5ab^2\)

Explanation: \(4a^2b\) and \(3a^2b\) are like terms, so their sum is \(7a^2b\). Similarly, \(-7ab^2\) and \(2ab^2\) add to \(-5ab^2\). Therefore, the simplified expression is \(7a^2b-5ab^2\). In option C, the coefficients of \(a^2b\) have not been added correctly. Exam tip: combine only terms that have exactly the same variables with the same powers.

Correct answer: B. \(x^2-7\)

Explanation: In \(x^2-7\), the \(x^2\) term is present, but there is no first-degree \(x\) term. Therefore, the coefficient of \(x\) is \(0\). In \(2x^2+x\), the coefficient of \(x\) is \(1\), so it is not correct. Exam tip: If a term of a particular degree is absent, its coefficient is \(0\).

Correct answer: A. \(5x-5y\)

Explanation: \(2(x-y)-3(y-x)=2x-2y-3y+3x\), because \(-3(y-x)=-3y+3x\). Combining like terms gives \(5x-5y\). \(5y-5x\) is the negative of the required expression, so it is not correct. Exam tip: When a bracket is preceded by a minus sign, change the sign of every term inside it.

Correct answer: C. 25

Explanation: Given u=-3 and v=2, we have u²=9, uv=(-3)(2)=-6, and v²=4. Hence, u²-2uv+v²=9-2(-6)+4=9+12+4=25. The value 13 can result from handling the sign of the negative product uv incorrectly. Exam tip: determine the sign of uv first, then evaluate the complete term -2uv.

Correct answer: B. \(6x-4\)

Explanation: The perimeter of a rectangle is \(2(\text{length}+\text{breadth})\). Thus, \(2[(2x+3)+(x-5)] = 2(3x-2) = 6x-4\). Therefore, \(6x-4\) is correct. \(3x-2\) is only the sum of the length and breadth, not the perimeter. Exam tip: Always multiply the sum of length and breadth by 2 for a rectangle’s perimeter.

Correct answer: B. The middle terms cancel, but \(3\times(-3)=-9\); therefore, the area is \(x^2-9\).

Explanation: Using distribution, \((x+3)(x-3)=x^2-3x+3x-9=x^2-9\). Rima used the wrong sign for \(3\times(-3)\). Exam tip: apply \((a+b)(a-b)=a^2-b^2\) to avoid expansion errors.

Correct answer: A. \(5a-6b\)

Explanation: On expanding, \(4(2a-3b)=8a-12b\) and \(-3(a-2b)=-3a+6b\). Therefore, \(8a-12b-3a+6b=5a-6b\). In option C, \(-3a\) has incorrectly been added instead of subtracted. Exam tip: When a negative multiplier is outside brackets, apply it to every term inside the brackets.

Correct answer: B. 2

Explanation: For x=1, we have x^3=1 and x^2=1. Therefore, 5x^3-4x^2+3x-2 = 5(1)-4(1)+3(1)-2 = 5-4+3-2 = 2. Hence, the correct answer is 2. Getting 4 may result from an error in addition or subtraction. Exam tip: After substitution, write each term with its sign carefully, especially the negative signs.

Correct answer: A. ((x+y)^2-xy)

Explanation: Direct answer: Option A, \((x+y)^2-xy\). The phrase “the square of the sum of x and y” must first be translated as \((x+y)^2\), because the whole sum is inside the brackets and then squared. The phrase “subtract xy from it” means take away \(xy\), so the complete expression is \((x+y)^2-xy\). Option A follows both instructions in the correct order. Option B is \(x^2+y-xy\), which does not square the complete sum and does not even square y. Option C is \(x+y^2-xy\), which squares only y, not the sum. Option D reverses the subtraction and gives \(xy-(x+y)^2\), the negative of the required expression. The brackets are essential: \((x+y)^2\) is different from \(x+y^2\). Memory cue: translate words step by step—sum, square, then subtract.

Correct answer: A. 2x-5

Explanation: First simplify the innermost bracket: 2x-(x-5)=2x-x+5=x+5. Then, 3x-(x+5)=3x-x-5=2x-5. Hence, the correct answer is 2x-5. In 2x+5, the negative sign before the outer bracket has not been applied to every term inside it. Exam tip: When a bracket is preceded by a minus sign, change the sign of each term inside while removing the bracket.

Correct answer: A. -3

Explanation: Substituting t=-1 gives 2(-1)^3-3(-1)^2+4(-1)+6. Thus, -2-3-4+6=-3, so the correct answer is -3. The option 3 can result from incorrectly handling the negative signs in (-1)^3 or 4(-1). Exam tip: an odd power of a negative number is negative, while an even power is positive.

Correct answer: A. 11x − 8

Explanation: The governing concept is simplifying an algebraic expression by applying the distributive property and combining like terms. First simplify the innermost product: 4(1 − x) = 4 − 4x. Thus the square bracket becomes 3x + 4 − 4x = 4 − x. The original expression is now 9x − 2(4 − x). Distribute −2 across the parentheses: −2(4 − x) = −8 + 2x. Finally combine like terms: 9x + 2x − 8 = 11x − 8. Therefore, option A is correct. A common error is to lose the negative sign before 2, which can produce 7x − 8 or a positive constant. The brackets must be simplified from the inside outward.

Correct answer: B. 4

Explanation: Given a+b=7 and ab=10, substitute these directly into 2(a+b)-ab: 2(7)-10=14-10=4. Therefore, the correct option is 4. The value 14 is only the value of 2(a+b); ab must still be subtracted. In exams, treat a grouped expression such as (a+b) as one unit before substituting.

Correct answer: A. \(3x^4-2x^2+7\)

Explanation: \(3x^4-2x^2+7\) has three terms, and the highest exponent of x is 4, so it is a trinomial polynomial of degree 4. Option B has four terms, while D has a negative exponent and is not a polynomial. Exam tip: check term count and highest exponent separately.

Correct answer: C. 19

Explanation: Substituting the given values, \,\((x+y)^2-xy=(2+3)^2-(2\times3)=5^2-6=25-6=19\). Therefore, the correct answer is 19. The value 25 is only \,\((x+y)^2\); \,\(xy=6\) must also be subtracted. Exam tip: Evaluate the squared term first, then perform multiplication and subtraction in order.

Correct answer: C. -6x^2+5x-1

Explanation: In
-6x^2+5x-1, the term
-6x^2 can be written as
(-6)x^2. Therefore, the coefficient of
x^2 is
-6. In option A, the coefficient is 6, while in option D the coefficient of
x^2 is 1;
-6 is the coefficient of x there. Exam tip: To identify a coefficient, include the sign and consider the complete numerical factor multiplying the term.

Correct answer: B. (\frac{5x}{4})

Explanation: Write (\frac{x}{2}) as (\frac{2x}{4}), then the sum is (\frac{5x}{4}). Make common denominators for fraction terms.