Algebraic expressions

Correct answer: B. 5c

Explanation: Here, 4c and c are like terms because both contain c to the first power. The coefficient of c is 1, so 4c+c = 4c+1c = 5c. The option 4c^2 is incorrect because adding like terms does not increase the exponent. Exam tip: When adding like terms, add only their coefficients.

Correct answer: C. \(8q\)

Explanation: All the terms contain the same variable \(q\), so they are like terms. Adding their coefficients gives \(10-3+1=8\). Therefore, \(10q-3q+q=8q\). The result \(7q\) would occur if the final \(+q\) were omitted. Exam tip: When simplifying like terms, operate on the coefficients and keep the variable unchanged.

Correct answer: B. Unlike terms

Explanation: The variable parts of 2x and 3y are x and y respectively. Like terms must have exactly the same variables with the same powers; only their coefficients may differ. Since the variables are different, these are unlike terms. A constant term has no variable. Exam tip: Compare the variable parts before deciding whether terms are like terms.

Correct answer: B. Constant expression

Explanation: The expression 15 contains no variable such as x or y, and its value always remains 15. Therefore, it is a constant expression. A binomial and a trinomial have two and three terms respectively, whereas 15 has only one term. Exam tip: An expression with no variable is identified as a constant expression.

Correct answer: C. 6

Explanation: The numerical coefficient is the number that multiplies the variable part. Here, 6x^2 = 6 \(\times\) x^2, so the numerical coefficient is 6. The number 2 is the exponent of x, not the coefficient. Exam tip: In a term, the number left after identifying the variable part is its numerical coefficient.

Correct answer: C. Only \(5p^2-3p+7\) is a polynomial in \(p\).

Explanation: In \(5p^2-3p+7\), the powers of \(p\) are 2, 1, and 0, all non-negative integers, so it is a polynomial. In \(4p^{-1}+2\), the power is \(-1\), so it is not a polynomial. Exam tip: check for negative powers first.

Correct answer: C. \(\frac{k}{2}\)

Explanation: To find half of a quantity, divide it by 2. Therefore, half of k is \(\frac{k}{2}\). The expression \(2k\) represents twice k, not half of it. Exam tip: When you see “half,” divide the quantity by 2.

Correct answer: A. \(2r+9\)

Explanation: Twice \(r\) is \(2r\). Adding 9 to this quantity gives \(2r+9\). In \(2(r+9)\), 9 is added to \(r\) first and then the entire sum is doubled, so it represents a different expression. Exam tip: when a number is added “to twice” a variable, add it outside the doubled term.

Correct answer: B. 4s − 7

Explanation: The governing skill is translating a verbal statement into an algebraic expression while preserving the order indicated by the language. “Four times s” means 4s. The phrase “subtracting 7 from four times s” means that 4s is the starting quantity and 7 is removed from it, giving 4s − 7. Therefore option B is correct. Option A changes subtraction into addition. Option C reverses the order and represents 7 − 4s, which means subtracting 4s from 7. Option D incorrectly changes the multiplication into a different subtraction and does not contain the required term 4s. Careful attention to the word “from” prevents the common reversal error.

Correct answer: A. \(7a-3\)

Explanation: \(2a\) and \(5a\) are like terms because both have the variable \(a\) to the first power. Adding their coefficients gives \(2+5=7\), so \(2a+5a=7a\). The constant term \(-3\) remains unchanged; therefore, the simplified form is \(7a-3\). \(10a-3\) would result from incorrectly multiplying the coefficients. Exam tip: add or subtract coefficients only of like terms.

Correct answer: B. Unlike terms

Explanation: Like terms must have exactly the same variables raised to the same powers. Here, the power of b is 2 in b^2, whereas it is 1 in 2b. Therefore, they are unlike terms. Having the same variable b alone does not make them like terms. Exam tip: Compare variables and their exponents, not just the coefficients.

Correct answer: A. \(3x\) और \(2x\)

Explanation: Like terms have the same variables raised to the same powers. Both \(3x\) and \(2x\) contain \(x\) to the power 1, so they are like terms. The term \(5\) is a constant, so it is not like either term containing \(x\). Exam tip: coefficients may differ; compare the variables and their powers to identify like terms.

Correct answer: C. 3

Explanation: Substituting
(x=1) gives
(x^2+2=1^2+2=1+2=3). Therefore, the correct answer is 3. Option 2 could result from incorrectly treating the squared term
(x^2) as 0. Exam tip: evaluate powers first, then perform addition or subtraction.

Correct answer: C. 6

Explanation: Given \(y=2\), \(y^2+y=2^2+2=4+2=6\). Therefore, the correct answer is 6. Option 4 is only the value of \(y^2\); the \(+y\) term must also be added. Exam tip: after substituting a variable's value, evaluate powers before addition or subtraction.

Correct answer: B. \(6u-2\)

Explanation: \(5u\) and \(u\) are like terms, so their coefficients are added: \(5u+u=6u\). The constant term \(-2\) remains unchanged. Therefore, the simplified expression is \(6u-2\). In \(6u+2\), the sign of the constant term is incorrect. Exam tip: Combine only terms with the same variable and the same power.

Correct answer: C. 2(x+y)

Explanation: The sum of (x) and (y) is (x+y). Twice this entire sum is 2(x+y). In 2x+y, only x is multiplied by 2, not y. Exam tip: Use brackets when words such as “sum of” or “entire sum” occur.

Correct answer: C. v

Explanation: In the expression \(7-3v\), \(v\) is the variable because its value can change. Here, \(7\) is the constant term and \(-3\) is the coefficient of \(v\), so \(-3\) is not a variable. Exam tip: Identify the letter or symbol whose value can change.

Correct answer: A. (a+b) is a sum and (ab) is a product

Explanation: The governing concept is recognising algebraic operations from notation. The plus sign in a + b explicitly indicates addition, so a + b is a sum. When two variables are written together as ab without an operation sign, multiplication is implied; thus ab is the product of a and b. Option A is therefore correct. The expressions are not always equal: if a = 2 and b = 3, then a + b = 5 while ab = 6. Option C reverses the meanings of addition and multiplication. Option D is also incorrect because a and b are generally variables, not necessarily fixed constants.

Correct answer: A. (x^2+3)

Explanation: Direct answer: Option A, \(x^2+3\). The phrase “the square of \(x\)” means multiplying \(x\) by itself, written \(x^2\). The phrase “adding 3 to” means place 3 after the square and use plus: \(x^2+3\). Step by step: identify the number being squared, which is \(x\); write its square as \(x^2\); add 3; obtain \(x^2+3\). Option A is correct. Option B, \(2x+3\), doubles \(x\) and then adds 3; it does not contain the square of \(x\). Option C, \(x+3^2\), squares 3 instead of squaring \(x\), and would equal \(x+9\). Option D, \(3x^2\), means multiplying the square by 3, not adding 3. A plus sign and a multiplication sign express different operations. Memory cue: “square of x, then add 3” means square first and add later: \(x^2+3\).

Correct answer: C. 2

Explanation: A constant term is a term with no variable. In 4x^2+3x+2, both 4x^2 and 3x contain x, whereas 2 contains no variable. Therefore, 2 is the constant term. Although x^0 equals 1, it is not a term in the given expression. Exam tip: identify the term that has no variable before selecting the constant term.

Correct answer: C. 11

Explanation: Substituting p=3 gives 4p-1=4×3-1=12-1=11. Therefore, the correct answer is 11. Option 12 is a close distractor if the subtraction of 1 is missed. Exam tip: After substitution, perform multiplication before addition or subtraction.

Correct answer: A. 9x

Explanation: Here, 2x, 3x, and 4x are like terms because each term contains x to the power 1. So, add their coefficients: 2+3+4=9. Hence, the simplified expression is 9x. The option 24x comes from multiplying the coefficients, which is not required here. Exam tip: Only terms with the same variable and the same exponent can be added.

Correct answer: B. \(x\)

Explanation: Zero is the additive identity: adding 0 to any number or variable does not change its value. Therefore, \(x+0=x\), so \(x\) is correct. \(1\) and \(x^0\) generally represent 1, so they are not the simplified form of \(x+0\). Exam tip: In addition, a term added with 0 can be removed.

Correct answer: C. x

Explanation: 1 is the multiplicative identity. Hence, multiplying any variable by 1 leaves it unchanged: \(1\cdot x=x\). The close distractor \(x+1\) is wrong because it adds 1 rather than multiplying by 1. Exam tip: multiplying an expression by 1 does not change it.

Correct answer: D. 6

Explanation: The product of 0 and any number or variable is 0. Therefore, \(0\cdot y=0\), and \(0\cdot y+6=0+6=6\). The option \(y+6\) is incorrect because the term containing \(y\) becomes 0 when multiplied by 0. Exam tip: simplify multiplication first, then perform addition or subtraction.