Algebraic expressions

Correct answer: A. 3

Explanation: The equations are \(x+y=11\) and \(x-y=5\). Subtract the second equation from the first: \((x+y)-(x-y)=11-5\), so \(2y=6\). Hence, \(y=3\). Option 5 is the constant in \(x-y=5\), not the value of \(y\). Exam tip: While adding or subtracting simultaneous equations, carefully track the signs to eliminate the required variable.

Correct answer: A. 7p-25q

Explanation: Expand the brackets: 5(2p-3q)=10p-15q and -3(p+4q)=-3p-12q. Thus, the expression becomes 10p-15q-3p-12q+2q. Combining like terms gives (10p-3p)+(−15q−12q+2q)=7p−25q. Therefore, option A is correct. Remember that the minus sign before 3(p+4q) affects both terms inside the bracket.

Correct answer: C. 44

Explanation: Given \(m=-1\) and \(n=4\), we have \(m^2=(-1)^2=1\) and \(n^2=4^2=16\). Thus, \(3m^2n=3\times1\times4=12\), while \(-2mn^2=-2\times(-1)\times16=32\). Therefore, the value of the expression is \(12+32=44\). Option 32 is only the value of the second term, not of the complete expression. Exam tip: squaring a negative number gives a positive result.

Correct answer: A. -2

Explanation: Only like terms can be added or subtracted. The terms containing x^2y are 7x^2y and -9x^2y, so their sum is (7-9)x^2y=-2x^2y. Therefore, the coefficient of x^2y is -2. The terms 3xy^2 and 5xy^2 are not like x^2y; they add to 8xy^2 instead. Exam tip: like terms must have the same variables raised to the same powers.

Correct answer: C. \(8x-8\)

Explanation: First simplify the innermost bracket: \(3x-(5x-8)=3x-5x+8=-2x+8\). Then \(6x-(-2x+8)=6x+2x-8=8x-8\). Therefore, \(8x-8\) is the correct option. In \(8x+8\), the sign of the final constant is incorrect because the outer minus changes the sign of every term inside the bracket. Exam tip: When a bracket is preceded by a minus sign, change the signs of all terms inside it.

Correct answer: B. 3

Explanation: The expression has value 9 when x=3. Substituting gives \(k(3)^2-4(3)-6=9\), or \(9k-12-6=9\). Thus, \(9k-18=9\), so \(9k=27\) and \(k=3\). For example, if k=1, the expression does not evaluate to 9. Exam tip: Substitute the given value of the variable carefully into every term of the expression.

Correct answer: A. (4p^2q^3), (-6p^2q^3), (p^2q^3)

Explanation: Like terms have exactly the same variables and powers. In the first option, every term has variable part (p^2q^3).

Correct answer: D. 20

Explanation: Given \(a+b=8\) and \(ab=12\), substitute these values into \(4(a+b)-ab\): \(4\times 8-12=32-12=20\). Hence, the correct answer is 20. The value 32 is only \(4(a+b)\); subtracting \(ab\) is still necessary. Exam tip: substitute the given combined expressions such as \(a+b\) and \(ab\) directly.

Correct answer: A. \(7r^2-10r\)

Explanation: Apply the distributive property: \(4r(r-1)=4r^2-4r\). Also, \(-3r(2-r)=-6r+3r^2\), since \(-3r\times(-r)=+3r^2\). Combining like terms gives \(4r^2+3r^2-4r-6r=7r^2-10r\). Therefore, option A is correct. In option C, the linear terms have been combined incorrectly. Exam tip: when expanding an expression with a negative sign, check the sign of every product separately.

Correct answer: B. 3

Explanation: Combining like terms gives \(8p^2q-5p^2q=3p^2q\) and \(-3pq+6pq=3pq\). Thus, the simplified expression is \(3p^2q+3pq+4\), which has 3 terms: \(3p^2q\), \(3pq\), and \(4\). Choosing 2 would be incorrect because the constant \(4\) is also a separate term. Exam tip: Combine only terms with exactly the same variables and exponents.

Correct answer: B. u^3v^2

Explanation: In u^3v^2, the exponent of u is 3 and the exponent of v is 2, so option B is correct. In option A, the exponents are reversed: u has exponent 2 and v has exponent 3. Exam tip: read the exponent written on each variable separately.

Correct answer: A. 6

Explanation: Substituting the given values, \(x^2y=(-3)^2\times2=9\times2=18\) and \(xy^2=(-3)\times2^2=(-3)\times4=-12\). Hence, \(x^2y+xy^2=18+(-12)=6\). The distractor \(-6\) may result from handling the sign of the negative term incorrectly. Exam tip: write a negative number in brackets before squaring it; \((-3)^2=9\).

Correct answer: B. \(4x^2+4x+7\)

Explanation: While adding polynomials, combine only like terms: \(7x^2+(-3x^2)=4x^2\), \(-4x+8x=4x\), and \(9+(-2)=7\). Therefore, the sum is \(4x^2+4x+7\). Option C incorrectly adds the linear terms. Exam tip: group terms with the same power of the variable before adding.

Correct answer: C. 35

Explanation: Given \(5x-3=22\), add 3 to both sides to get \(5x=25\), so \(x=5\). Now substitute in the expression: \(x^2+2x=5^2+2(5)=25+10=35\). Hence, 35 is the correct answer. The value 30 may result from incorrectly evaluating \(x^2\). Exam tip: first find the value of the variable from the equation, then substitute it carefully into the expression.

Correct answer: B. \(-3m^2n^4\)

Explanation: Like terms have exactly the same variables raised to the same powers; only their coefficients may differ. In \(-3m^2n^4\), the variable part is \(m^2n^4\), so it is a like term of \(8m^2n^4\). In \(5m^4n^2\), the powers of \(m\) and \(n\) are different, so it is not a like term. Exam tip: Ignore the coefficients and compare the variables and their exponents.

Correct answer: A. \(4x^2-9xy+4y^2\)

Explanation: A minus sign precedes the second bracket, so the sign of every term inside it changes: \(6x^2-4xy+y^2-2x^2-5xy+3y^2\). Combining like terms gives \((6-2)x^2=4x^2\), \((-4-5)xy=-9xy\), and \((1+3)y^2=4y^2\). Hence, the simplified expression is \(4x^2-9xy+4y^2\). In option D, the sign of the \(y^2\) term is incorrect. Exam tip: when subtracting a bracket, change the sign of every term in that bracket before combining like terms.

Correct answer: D. 12

Explanation: Add the two given equations:

a-b+a+b=7+17.

The terms involving b cancel, so

2a=24.

Hence,

a=12.

Option 10 may result from incorrectly treating the result of subtracting the equations as a; subtraction actually gives 2b. Exam tip: add equations when a variable has opposite signs so that it cancels out.

Correct answer: B. -7

Explanation: The term containing x^2 is -7x^2. The numerical factor multiplying the variable part is called its coefficient, so the coefficient of x^2 is -7. Here, 4, 2, and -11 belong respectively to the x^3 term, the x term, and the constant term. Exam tip: identify the term with the required power first, then write its numerical coefficient.

Correct answer: B. 7

Explanation: Expanding the expression gives \(2(5x+3y)-7(x-y)+4x=10x+6y-7x+7y+4x\). Combining the \(x\)-terms, \(10x-7x+4x=7x\). Therefore, the coefficient of \(x\) is \(7\). The distractor \(11\) can result from mishandling the negative sign in \(-7(x-y)\). Exam tip: when a negative sign is outside parentheses, distribute it to every term inside.

Correct answer: A. \(x^3y^4\)

Explanation: In \(x^3y^4\), the powers of \(x\) and \(y\) are 3 and 4 respectively. Therefore, its total degree is \(3+4=7\), so option A is correct. \(x^2y^4\) has total degree 6; similarly, \(x^5y\) and \(xy^5\) also have total degree 6. Exam tip: for a multivariable monomial, add the exponents of all variables to find its total degree.

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

Explanation: In a polynomial, every exponent of \(x\) must be a non-negative integer. In \(7x^4-3x+2\), the highest exponent is 4, so it is a polynomial of degree 4. Option B has highest exponent 5. Option C contains \(\sqrt{x}=x^{1/2}\), and option D contains an \(x^{-1}\) term, so neither is a polynomial. Exam tip: first check that all exponents are non-negative integers, then use the highest exponent to find the degree.

Correct answer: C. \(12u^2v+3uv-10\)

Explanation: Only like terms can be combined. The terms \(9u^2v\) and \(3u^2v\) are like terms, so their sum is \(12u^2v\). Similarly, \(-5uv+8uv=3uv\), while the constant term \(-10\) remains unchanged. Therefore, the simplified expression is \(12u^2v+3uv-10\). In option A, the coefficients of the \(uv\) terms have been combined incorrectly. Exam tip: before combining terms, check that both the variables and their powers are exactly the same.

Correct answer: B. 22

Explanation: On substituting \(x=-4\), \(x^3=(-4)^3=-64\) and \(x^2=(-4)^2=16\). Hence the numerator is \(-64-2(16)+8=-64-32+8=-88\). Therefore, \(\frac{-88}{-4}=22\), so option B is correct. A value such as \(26\) can result from mishandling the negative sign in \(-2x^2\). Exam tip: the square of a negative number is positive, whereas its cube is negative.

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

Explanation: Combine like terms while adding: \(3x^2-x^2=2x^2\), \(-2x+6x=4x\), and \(5-9=-4\). Hence, the resulting expression is \(2x^2+4x-4\). In option D, the coefficients of the \(x^2\) terms have been added incorrectly. Exam tip: add the \(x^2\), \(x\), and constant terms separately.

Correct answer: A. \(5x^2+4x-6\)

Explanation: While subtracting the second polynomial, the sign of every term in it changes: \((12x^2-5x+4)-(7x^2-9x+10)=12x^2-5x+4-7x^2+9x-10\). Combining like terms gives \(5x^2+4x-6\). In option C, the \(-9x\) term has not been correctly changed to \(+9x\) during subtraction. Exam tip: when a minus sign precedes brackets, reverse every sign inside the brackets.