Algebraic expressions

Correct answer: B. 6

Explanation: Both terms contain the same variable \(x\), so add their coefficients: \((k-2)x+5x=(k-2+5)x=(k+3)x\). Since the simplified expression is \(9x\), we get \(k+3=9\). Therefore, \(k=6\). If 9 were chosen, the coefficient would become 12, not 9. Exam tip: while combining like terms, add only their coefficients.

Correct answer: C. 11

Explanation: Add the two given equations: \((x-y)+(x+y)=8+14\). The terms \(-y\) and \(+y\) cancel, giving \(2x=22\). Hence, \(x=11\). Option 22 is the value of \(2x\), not of \(x\). Exam tip: In a pair of linear equations, add equations when one variable has opposite signs so that it is eliminated.

Correct answer: A. \(2p-3q\)

Explanation: Using the distributive property, \(4(3p-2q)=12p-8q\) and \(-2(5p+q)=-10p-2q\). Thus, the expression becomes \(12p-8q-10p-2q+7q\). Combining like terms gives \(12p-10p=2p\) and \(-8q-2q+7q=-3q\), so the result is \(2p-3q\). In \(2p+3q\), the sign of the \(q\)-term is incorrect. Exam tip: when a negative multiplier is outside brackets, apply it to every term inside the brackets.

Correct answer: B. -6

Explanation: Substituting m=2 and n=-3, we get 2m²n = 2×(2²)×(-3) = -24 and mn² = 2×(-3)² = 18. Therefore, 2m²n + mn² = -24 + 18 = -6. The nearby option 6 can result from incorrectly handling the sign of the negative term. Exam tip: the square of a negative number is positive, so (-3)² = 9.

Correct answer: C. \((a+b)^2-3ab\)

Explanation: “The square of the sum of (a) and (b)” is \((a+b)^2\). Subtracting \(3ab\) from it gives \((a+b)^2-3ab\), so option C is correct. In option D, the order of subtraction is reversed, so its value is generally different. Exam tip: In phrases such as “subtract X from Y,” write Y first and then subtract X.

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

Explanation: In \(3x^4-2x+7\), the powers of x are 4, 1 and 0, all non-negative integers. Since \(1/x=x^{-1}\), option B is not a polynomial. Exam tip: reject expressions with negative or fractional powers.

Correct answer: C. \(7x+3\)

Explanation: First simplify the inner bracket: \(2x-(4x+3)=2x-4x-3=-2x-3\). Now, \(5x-(-2x-3)=5x+2x+3=7x+3\). Therefore, the correct answer is \(7x+3\). In \(7x-3\), the sign of the constant term has been changed incorrectly. Exam tip: When a bracket is preceded by a minus sign, change the signs of all terms inside it.

Correct answer: C. 4

Explanation: At x=2, the expression is equal to 9. So, k(2)^2-5(2)+3=9, or 4k-10+3=9. Hence, 4k-7=9, giving 4k=16 and k=4. If k=3, the expression evaluates to 5, not 9. Exam tip: In value-based questions, substitute the given variable value carefully, especially in terms with exponents.

Correct answer: A. (3x^2yz), (-7x^2yz), (11x^2yz)

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

Correct answer: C. 60

Explanation: Given \(x+y=9\) and \(xy=14\), substitute these values directly: \(2(x+y)+3xy=2(9)+3(14)=18+42=60\). Therefore, option C is correct. \(42\) is only the value of \(3xy\); the term \(2(x+y)\) must also be added. Exam tip: When \(x+y\) and \(xy\) are given directly, substitute them into the expression without finding \(x\) and \(y\) separately.

Correct answer: A. \(5r^2-2r\)

Explanation: Expand each bracket first: \(3r(r+2)=3r^2+6r\). Also, \(-2r(4-r)=-8r+2r^2\), since \(-2r\times(-r)=+2r^2\). Combining like terms gives \(3r^2+2r^2+6r-8r=5r^2-2r\). Hence, \(5r^2-2r\) is correct. The option \(5r^2+14r\) results from handling the negative sign incorrectly. Exam tip: carefully track signs while expanding brackets.

Correct answer: B. 3

Explanation: Combine like terms: \(7p^2q-4p^2q=3p^2q\) and \(-2pq+5pq=3pq\). Thus, the expression becomes \(3p^2q+3pq+9\), which has three terms: \(3p^2q\), \(3pq\), and \(9\). Note that \(p^2q\) and \(pq\) are not like terms because the exponent of \(p\) is different. Exam tip: combine only terms with identical variables and exponents.

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

Explanation: In \(m^2n^3\), the exponent of \(m\) is 2 and the exponent of \(n\) is 3, so option B is correct. In \(m^3n^2\), the exponents are interchanged, making it a close but incorrect option. Exam tip: Read the exponent written on each variable separately.

Correct answer: A. 48

Explanation: Given \(x=-2\) and \(y=4\), \(x^2y=(-2)^2\times4=4\times4=16\). Also, \(xy^2=(-2)\times4^2=(-2)\times16=-32\). Therefore, \(x^2y-xy^2=16-(-32)=16+32=48\). The option \(-48\) may result from incorrectly handling the subtraction of a negative number. Exam tip: evaluate powers first, then multiply, and carefully track signs.

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

Explanation: Add terms with the same power of \(x\): \(6x^2-2x^2=4x^2\), \(-3x+7x=4x\), and \(2-9=-7\). Hence, the sum is \(4x^2+4x-7\). In option D, the coefficients of \(x^2\) have been added incorrectly: \(6+(-2)=4\), not 8. Exam tip: While adding polynomials, group like terms first and carefully retain negative signs.

Correct answer: B. 24

Explanation: Given 3x+2=17, subtracting 2 from both sides gives 3x=15, so x=5. Hence, x^2-1=5^2-1=25-1=24. Option 25 is only the value of x^2; 1 still has to be subtracted. Exam tip: first find the value of the variable, then substitute it into the complete expression.

Correct answer: B. -5a^3b^2

Explanation: Like terms have exactly the same variables raised to the same powers; only their coefficients may differ. The terms 4a^3b^2 and -5a^3b^2 have the same variable part, a^3b^2, so option B is correct. In option A, the powers of a and b are interchanged, so it is not a like term. Exam tip: Ignore the coefficient and compare the variables and their exponents.

Correct answer: A. \(3x^2-8xy+8y^2\)

Explanation: A minus sign before the second bracket changes the sign of every term inside it: \(4x^2-5xy+2y^2-x^2-3xy+6y^2\). Combining like terms gives \((4-1)x^2+(-5-3)xy+(2+6)y^2=3x^2-8xy+8y^2\). Option D results from incorrectly keeping the sign of the \(xy\) term positive. Exam tip: When a bracket is preceded by ‘−’, change every sign inside it before combining like terms.

Correct answer: A. 4

Explanation: Subtract the first equation from the second: \((a+b)-(a-b)=13-5\). This gives \(2b=8\), so \(b=4\). Option 5 is the given value of \(a-b\), not the value of \(b\). Exam tip: Add or subtract a pair of equations to eliminate one variable quickly.

Correct answer: B. -5

Explanation: The term containing x^2 is -5x^2. The number multiplying x^2 is its coefficient, so the coefficient is -5. The number 2 is the coefficient of the x^3 term, 2x^3, not of x^2. Exam tip: first locate the term with the required power, then identify the number multiplying it.

Correct answer: C. 11

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

Explanation: In \(x^2y^4\), the exponent of \(x\) is 2 and that of \(y\) is 4. Hence, its total degree is \(2+4=6\), so option A is correct. The total degree of \(x^3y^2\) is \(3+2=5\), so it is not correct. Exam tip: For a monomial, add the exponents of all its variables to find the total degree.

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

Explanation: In \(4x^3-2x+7\), the powers of \(x\) are \(3\), \(1\), and \(0\). All are non-negative integers, so it is a polynomial in \(x\). In option A, \(\frac{3}{x}=3x^{-1}\); option B has \(\sqrt{x}=x^{1/2}\); and option D has \(x^{-2}\). These contain negative or fractional powers, so they are not polynomials. Exam tip: in a polynomial, variable exponents can only be \(0,1,2,\ldots\).

Correct answer: A. \(7u^2v+3uv-4\)

Explanation: Only like terms can be added or subtracted. \(5u^2v\) and \(2u^2v\) are like terms, so their sum is \(7u^2v\). Similarly, \(-3uv+6uv=3uv\). Therefore, the simplified expression is \(7u^2v+3uv-4\). In option B, \(5+2\) has incorrectly been taken as \(3\). Exam tip: before combining terms, check that both the variables and their exponents are identical.

Correct answer: A. 5

Explanation: On substituting \(x=-3\), the numerator is \((-3)^3+2(-3)^2-6=-27+18-6=-15\). Hence, \(\frac{-15}{-3}=5\). Therefore, option A is correct. An answer such as \(7\) can result from an error with the power of a negative number or the subtraction sign. Exam tip: always use brackets when substituting a negative value, especially in powers.