LEPT Practice Questions: Math Specialization with Answers

Answer: A. By the Remainder Theorem, the remainder when f(x) is divided by (x − a) equals f(a). Here, a = 2. Compute f(2) = 2(2)³ − 5(2)² + 3(2) − 7 = 2(8) − 5(4) + 6 − 7 = 16 − 20 + 6 − 7 = −5. The Remainder Theorem is a quick alternative to performing polynomial long division — simply substitute the value of a into the polynomial to find the remainder.

Answer: A. Factor the quadratic: x² − 6x + 8 = (x − 2)(x − 4) = 0. Setting each factor equal to zero gives x = 2 and x = 4. You can verify by substituting: (2)² − 6(2) + 8 = 4 − 12 + 8 = 0, and (4)² − 6(4) + 8 = 16 − 24 + 8 = 0. Both solutions check out.

Answer: A. When a circle is inscribed in a square, the diameter of the circle equals the side length of the square. So the radius r = 10/2 = 5 cm. The area of the square is 10² = 100 cm². The area of the circle is πr² = π(5)² = 25π cm². The area inside the square but outside the circle is 100 − 25π ≈ 100 − 78.54 = 21.46 cm².

Answer: B. Use the distance formula: d = √[(x&sub2; − x&sub1;)² + (y&sub2; − y&sub1;)²]. Substituting the coordinates: d = √[(−2 − 3)² + (5 − (−1))²] = √[(−5)² + (6)²] = √[25 + 36] = √61. The distance between points A and B is √61 units, which is approximately 7.81 units.

Answer: C. The angle 150° lies in Quadrant II, where sine is positive. Its reference angle is 180° − 150° = 30°. Therefore, sin(150°) = sin(30°) = 1/2. Remember: in Quadrant II, sine is positive while cosine and tangent are negative (the "S" in the mnemonic "All Students Take Calculus").

Answer: D. Start with the Pythagorean identity: sin²(θ) + cos²(θ) = 1. So the expression becomes 1 + tan²(θ). By another Pythagorean identity, 1 + tan²(θ) = sec²(θ). Therefore, the expression equals both 1 + tan²(θ) and sec²(θ), making both A and B correct.

Answer: A. Apply the power rule: d/dx[x&sup n;] = nx&sup n;−¹. For each term: d/dx[3x&sup4;] = 12x³, d/dx[−2x³] = −6x², d/dx[5x] = 5, and d/dx[−1] = 0. Combining these gives f'(x) = 12x³ − 6x² + 5. Option B is incorrect because it adds an extra x to the constant term. Option D incorrectly divides each coefficient by the original exponent instead of multiplying.

Answer: B. First, find the antiderivative: ∫(3x² + 2x) dx = x³ + x² + C. Then evaluate from 0 to 2 using the Fundamental Theorem of Calculus: [x³ + x²]&sub0;² = (2³ + 2²) − (0³ + 0²) = (8 + 4) − (0) = 12. The power rule for integration states that ∫x&sup n; dx = x&sup n;+¹/(n+1) + C, which is the reverse of differentiation.

Answer: B. The median is the middle value in an ordered data set. Since there are 7 values (an odd number), the median is the 4th value when arranged in ascending order. The ordered data is: 78, 82, 85, 85, 90, 92, 96. The 4th value is 85. Note that option C (86.86) represents the mean, not the median. The mean = (78 + 82 + 85 + 85 + 90 + 92 + 96) / 7 = 608 / 7 ≈ 86.86.

Answer: B. There are 10 marbles total. The probability of drawing a red marble first is 5/10 = 1/2. After removing one red marble, there are 4 red marbles left out of 9 total. The probability of drawing a second red marble is 4/9. Multiply the two probabilities: P(both red) = (5/10) × (4/9) = 20/90 = 2/9. Alternatively, use combinations: C(5,2) / C(10,2) = 10/45 = 2/9.