mathematics beginner 10 sample questions

Algebraic Expressions MCQ Practice Test

Mathematical phrases with variables

Q1. Given the algebraic expression 3(x^2 + 2x - 3) - 2(x^2 - 4x - 7), what is the resulting expression after combining like terms?

A. x^2 + 14x + 5 ✓
B. x^2 + 14x - 5
C. x^2 + 2x + 5
D. 5x^2 + 14x + 5

Explanation: Distribute: 3(x^2 + 2x - 3) = 3x^2 + 6x - 9 and -2(x^2 - 4x - 7) = -2x^2 + 8x + 14. Combine like terms: (3-2)x^2 + (6+8)x + (-9+14) = x^2 + 14x + 5.

Q2. If x² - 3x + 1 = 0, what is the value of x + 1/x?

A. 3 ✓
B. 1/3
C. -3
D. 2

Explanation: Divide the equation by x (x ≠ 0): x - 3 + 1/x = 0, so x + 1/x = 3 — true for both roots, since the equation was built so the expression is single-valued. The old stem paired x² + 5x - 6 = 0 (roots 1 and -6, giving 2 or -37/6) with options that matched neither, and its LaTeX was broken.

Q3. If f(x) = sqrt(16x^2 + 9), what is the value of f(-1)?

A. 5 ✓
B. -5
C. 7
D. 25

Explanation: Substitute x = -1: f(-1) = sqrt(16(-1)^2 + 9) = sqrt(16 + 9) = sqrt(25) = 5. Because the radical symbol denotes the principal (non-negative) root, the value is 5, not -5.

Q4. Given the algebraic expression −2x² + 5x + 3, what is the value of the coefficient of the linear term?

A. the coefficient of the quadratic term
B. the constant term
C. the coefficient of the linear term ✓
D. the coefficient of the constant term

Explanation: In the given expression −2x² + 5x + 3, the linear term is 5x. The coefficient of a term is the numerical value multiplied by the variable. In this case, the coefficient of the linear term is 5.

Q5. If \(x^2 + 2x - 6 = 0\), what is the value of \((x + 1)^2\) when \(x\) is expressed in terms of its roots?

A. \(x = 3 - 2\sqrt{2}\) and \(x = 3 + 2\sqrt{2}\)\, then \((x + 1)^2 = 16\) ✓
B. \(x = 3 - 2\sqrt{2}\) and \(x = 3 + 2\sqrt{2}\)\, then \((x + 1)^2 = 4\)
C. \(x = 3 - 2\sqrt{2}\) and \(x = 3 + 2\sqrt{2}\)\, then \((x + 1)^2 = 12\)
D. \(x = 3 - 2\sqrt{2}\) and \(x = 3 + 2\sqrt{2}\)\, then \((x + 1)^2 = 20\]

Explanation: To find the value of \((x + 1)^2\) when \(x\) is expressed in terms of its roots, we first need to find the roots of the quadratic equation \(x^2 + 2x - 6 = 0\). Factoring the quadratic equation, we get \((x + 3)(x - 2) = 0\), which gives us the roots \(x = -3\) and \(x = 2\). However, we are given the roots as \(x = 3 - 2\sqrt{2}\) and \(x = 3 + 2\sqrt{2}\), which can be obtained by using the quadratic formula. Now, we can substitute the value of \(x\) in the expression \((x + 1)^2\) to find its value.

Q6. Given the expression 2(x² + 2x) + 3(x² + 2x) − 2(x − 3), what is the result after combining like terms?

A. 5x² + 8x + 6 ✓
B. 5x² + 12x + 6
C. 4x² + 8x + 6
D. 5x² + 8x + 9

Explanation: Distribute first: 2(x² + 2x) = 2x² + 4x and 3(x² + 2x) = 3x² + 6x, while −2(x − 3) = −2x + 6. Grouping like terms: (2x² + 3x²) + (4x + 6x − 2x) + 6 = 5x² + 8x + 6.

Q7. What is the simplified form of the algebraic expression ⚑(3x² + 2x) - 2(2x² + 3x) + x²?

A. ⚑(3x² + 2x) - 2(2x² + 3x) + x² = -3x² - 2x
B. ⚑(3x² + 2x) - 2(2x² + 3x) + x² = x² - 7x ✓
C. ⚑(3x² + 2x) - 2(2x² + 3x) + x² = 2x² - 9x
D. ⚑(3x² + 2x) - 2(2x² + 3x) + x² = 4x² + 5x

Explanation: To simplify the expression, we need to distribute the coefficients inside the parentheses, then combine like terms. Distributing the coefficients, we get: 3x² + 2x - 4x² - 6x + x². Combining like terms, we get: -4x² + 3x² + 2x - 6x + x² = x² - 7x

Q8. If x² + 6x + 11 = (x + a)² + b, what are the values of a and b?

A. a = 3, b = 2 ✓
B. a = -3, b = 2
C. a = 6, b = 11
D. a = 3, b = 11

Explanation: (x + 3)² = x² + 6x + 9, so x² + 6x + 11 = (x + 3)² + 2 — that is, a = 3 and b = 2. The old stem equated the square root of a quadratic with a completed square, which is algebraically impossible, and its equations did not resolve.

Q9. What is the simplified form of the expression (x² + 2x − 3) + 2(x² − 4x + 2)?

A. 3x² − 6x + 1 ✓
B. 3x² + 6x − 1
C. 3x² − 6x − 1
D. 3x² + 6x + 1

Explanation: Expanding: x² + 2x − 3 + 2x² − 8x + 4 = 3x² − 6x + 1. The old stem carried a stray '✎' glyph in place of a coefficient, and none of the old options matched the true simplification.

Q10. If \( x^2 + 5x + 6 = 0 \), what is the value of \( rac{x}{x+2} \) when \( x \) is expressed in terms of \( x+2 \)?

A. x
B. x+2 ✓
C. 2
D. 3

Explanation: To find the value of \( \frac{x}{x+2} \) when \( x \) is expressed in terms of \( x+2 \), we need to solve the quadratic equation \( x^2 + 5x + 6 = 0 \) to find the value of \( x \). The quadratic equation can be factored as \( (x+2)(x+3) = 0 \), so \( x = -2 \) or \( x = -3 \). Since \( x \) is expressed in terms of \( x+2 \), we can substitute \( x = -2 \) into the expression \( \frac{x}{x+2} \) to get \( \frac{-2}{-2+2} = \frac{-2}{0} \), which is undefined. However, if we substitute \( x = -3 \) into the expression \( \frac{x}{x+2} \), we get \( \frac{-3}{-3+2} = \frac{-3}{-1} = 3 \). Therefore, the correct answer is \( 3 \).

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