Home / Digital SAT Math Practice Questions – Medium : Equivalent expressions

Digital SAT Math Practice Questions – Medium : Equivalent expressions

Digital SAT Math Practice Questions - Medium : Equivalent expressions - New Syllabus

DSAT MAth Practice questions – all topics

  • Advanced Math Weightage: 35% Questions: 13-15
    • Equivalent expressions
    • Nonlinear equations in one variable and systems of equations in two variables
    • Nonlinear functions

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Question Medium

Which of the following is an equivalent form of \((1.5x – 2.4)^2 – (5.2x^2 – 6.4)\)?

A) \(-2.2x^2 + 16\)

B) \(-2.2x^2 + 11.2\)

C) \(-2.95x^2 – 7.2x + 12.16\)

D) \(-2.95x^2 – 7.2x + 0.64\)

▶️ Answer/Explanation
Solution

Answer: C

Expand \((1.5x – 2.4)^2\): \((1.5x – 2.4)(1.5x – 2.4) = 2.25x^2 – 3.6x – 3.6x + 5.76\).

Subtract \((5.2x^2 – 6.4)\): \((2.25x^2 – 7.2x + 5.76) – (5.2x^2 – 6.4)\).

Distribute and combine: \(2.25x^2 – 7.2x + 5.76 – 5.2x^2 + 6.4\).

Combine like terms: \((-2.95x^2) + (-7.2x) + (12.16) = -2.95x^2 – 7.2x + 12.16\).

Question Medium

\(\sqrt[3]{x^3 y^6}\)

Which of the following expressions is equivalent to the expression above?

A) \(y^2\)

B) \(xy^2\)

C) \(y^3\)

D) \(xy^3\)

▶️ Answer/Explanation
Solution

Answer: B

Use property \(\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}\).

Rewrite: \(\sqrt[3]{x^3} \cdot \sqrt[3]{y^6}\).

Simplify: \(x^1 \cdot y^2 = xy^2\).

Question Medium

Blood volume, \( V_B \), in a human can be determined using the equation \( V_B = \frac{V_P}{1 – H} \), where \( V_P \) is the plasma volume and \( H \) is the hematocrit (the fraction of blood volume that is red blood cells). Which of the following correctly expresses the hematocrit in terms of the blood volume and the plasma volume?

A) \( H = 1 – \frac{V_P}{V_B} \)

B) \( H = \frac{V_B}{V_P} \)

C) \( H = 1 + \frac{V_B}{V_P} \)

D) \( H = V_B – V_P \)

▶️ Answer/Explanation
Solution

Answer: A

Given: \( V_B = \frac{V_P}{1 – H} \).

Rearrange: \( 1 – H = \frac{V_P}{V_B} \).

Solve for \( H \): \( H = 1 – \frac{V_P}{V_B} \).

Question Medium

\[ q = s (r – 1)^2 \]

The given equation relates the positive numbers \(q\), \(r\), and \(s\). Which equation gives \(r\) in terms of \(q\) and \(s\), when \(r > 1\)?

A) \( r = 1 + \sqrt{\frac{q}{s}} \)

B) \( r = 1 + \frac{\sqrt{q}}{s} \)

C) \( r = -1 – \sqrt{\frac{q}{s}} \)

D) \( r = -1 – \frac{\sqrt{q}}{s} \)

▶️ Answer/Explanation
Solution

Answer: A

Given: \( q = s (r – 1)^2 \).

Divide by \( s \): \(\frac{q}{s} = (r – 1)^2\).

Take square root: \(\sqrt{\frac{q}{s}} = |r – 1|\).

Since \( r > 1 \), \( r – 1 > 0 \), so \( r – 1 = \sqrt{\frac{q}{s}} \).

Add 1: \( r = 1 + \sqrt{\frac{q}{s}} \).

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