Algebraic expressions

\(8a^2b\) and \(-3a^2b\) are like terms because both have \(a\) raised to 2 and \(b\) raised to 1. Adding their coefficients gives \(8-3=5\), so they combine to form \(5a^2b\). The term \(4ab\) is not like \(a^2b\), since the power of \(a\) is 1, so it remains separate. Hence, the simplified expression is \(5a^2b+4ab\). Exam tip: combine coefficients only when the variables and all their exponents are exactly the same.

The square of \(x\) is \(x^2\), and three times \(x\) is \(3x\). In the difference between these quantities, the second quantity is subtracted from the first, so the expression is \(x^2-3x\). \(3x-x^2\) reverses the order of subtraction. Exam tip: read “the difference between A and B” as \(A-B\).

A term in algebra can be separated into its numerical coefficient and its variable part. In \(12x^2y\), the number 12 tells how many times the variable product is taken, so 12 is the coefficient. The letters and their powers together form the variable part. Hence the variable part is \(x^2y\), making option B correct. The exponent 2 belongs to x, while y has an understood exponent of 1.

To verify this, rewrite the term as \(12\times x^2\times y\). The first factor is numerical, and the remaining factors contain variables, so they form the variable part. Option A names only the coefficient. Option C contains both a number and a variable, so it is not the complete variable part. Option D leaves out \(x^2\). Therefore, separating the term into coefficient and literal or variable factors leads to option B.

A minus sign before a bracket changes the sign of every term inside it: \(4m-(2m-7)=4m-2m+7=2m+7\). Therefore, \(2m+7\) is correct. \(2m-7\) would result if the sign of \(-7\) were not changed. Exam tip: Before removing brackets, check the sign immediately preceding them.

\(x\) and \(2x\) are like terms because both have \(x\) raised to the power 1. Adding their coefficients gives \(x+2x=3x\). The term \(x^2\) is not like \(x\), so it remains separate. Therefore, the expression becomes \(x^2+3x\). Exam tip: combine terms only when both the variable and its exponent are the same.

In \(-4x^2+7\), the \(x^2\)-term is \(-4x^2\), which can be written as \((-4)\times x^2\). Hence, the coefficient of \(x^2\) is \(-4\). In option C, \(4-x^2=4+(-1)x^2\), so its coefficient is \(-1\), not \(-4\). Exam tip: Always include the sign when identifying a coefficient.

The cost of n notebooks is \(12n\) rupees because each notebook costs ₹12. The ₹5 cover charge is paid only once, so the total cost is \(12n+5\). In \(17n\), the cover charge has incorrectly been added for every notebook. Exam tip: multiply the per-item cost by the variable, then add any one-time charge as a constant term.

Combine like terms in the expression: \(2a+5a=7a\) and \(-3b-b=-4b\). Therefore, \(2a-3b+5a-b=7a-4b\), so option B is correct. In option A, the \(b\)-terms have been combined incorrectly, while option D wrongly multiplies \(a\) and \(b\). Exam tip: Add or subtract only terms with the same variable and the same power.

In \(3x^2y+4xy^2-5\), the only distinct variables are \(x\) and \(y\), so there are 2 variables. The exponent 2 in \(x^2\) and \(y^2\) does not create an additional variable, and \(-5\) is a constant. Exam tip: count distinct letters representing variables, not their powers.

Using the distributive property, \(7(z-1)=7z-7\). Then \(7z-7+2=7z-5\), so the correct answer is \(7z-5\). The expression \(7z-9\) would result from incorrectly treating \(+2\) as \(-2\). Exam tip: after expanding brackets, combine only like terms.

In \(5s^2\) and \(5s\), the variable \(s\) is the same, but its powers are 2 and 1 respectively. Their literal parts are therefore different, so they are unlike terms. In contrast, \(4c\) and \(-9c\) are like terms because both contain \(c\) to the power 1. Exam tip: for like terms, every variable and its exponent must match exactly; only the coefficients may differ.

Given k = 4, substitute 4 for k in the expression: 4² − 2(4) + 3 = 16 − 8 + 3 = 11. Therefore, the correct answer is 11. The value 9 may result if the final +3 is mistakenly omitted. Exam tip: substitute the value first, then apply powers, multiplication, and addition/subtraction in order.

The two \(a^2\) terms are like terms, so \(a^2+a^2=2a^2\). The term \(a\) and the constant \(3\) are unlike terms, so they cannot be combined further. Hence, the simplified expression is \(2a^2+a+3\). In particular, \(a^2\) cannot be added to \(a\) to make \(2a\). Exam tip: combine only terms with the same variable and exponent.

Multiplying x by -3 gives -3x. Adding 8 afterwards gives the expression -3x+8. In option D, -3 multiplies both x and 8, producing -3x-24, so it is not correct. Exam tip: words such as “then adding” mean that the added number stays outside the multiplication.

Here, \(2x^2\) and \(5x^2\) are like terms, so their sum is \(7x^2\). Similarly, \(-3x\) and \(-x\) add to \(-4x\). Therefore, the simplified form is \(7x^2-4x\). In option C, the \(x^2\) terms have been added incorrectly. Exam tip: add or subtract only terms with the same variable and the same exponent.

Using the distributive property, multiply \(3a\) by each term inside the bracket: \(3a\times 2a=6a^2\) and \(3a\times(-1)=-3a\). Therefore, the expression becomes \(6a^2-3a\). Option A is incorrect because it does not use \(a\times a=a^2\). Exam tip: When multiplying terms, multiply the coefficients and also combine the powers of the variables correctly.

In 5mn + 2m, both terms, 5mn and 2m, contain variables. Therefore, there is no term consisting only of a number, so it has no constant term. In contrast, -7 is the constant term in 4y - 7. Exam tip: A term with no variable is called a constant term.

Substituting u=1 and v=-2, \(4u+v^2=4(1)+(-2)^2=4+4=8\). Therefore, the correct answer is 8. A common error is getting 6 by treating \((-2)^2\) as -4; the square of a negative number is positive. Exam tip: Always use brackets when squaring a negative number.

First, the difference of \(x\) and \(y\) is \(x-y\). Twice this entire difference is \(2(x-y)\), which equals \(2x-2y\). In \(2x-y\), only \(x\) is doubled, so it is not correct. Exam tip: For “twice the difference,” put the difference in brackets and multiply the whole bracket by 2.

Only like terms can be added or subtracted. Here, 10q^2 and q^2 are like terms, so 10q^2+q^2=11q^2. Similarly, -4q+6q=2q. Therefore, the simplified expression is 11q^2+2q. In 9q^2+2q, the coefficients of the q^2 terms have been combined incorrectly. Exam tip: Group terms according to the power of the variable before simplifying.