General Aptitude
Q. If y = (2x + 1)^3, find dy/dx at x = 1.
Solution
dy/dx = 6(2x + 1)^2. At x = 1, dy/dx = 6(2(1) + 1)^2 = 6(3)^2 = 54.
Correct Answer: A — 12
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Q. If y = (2x + 1)^3, find dy/dx.
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A.
6(2x + 1)^2
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B.
3(2x + 1)^2
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C.
2(2x + 1)^2
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D.
12(2x + 1)^2
Solution
dy/dx = 6(2x + 1)^2 by the chain rule.
Correct Answer: A — 6(2x + 1)^2
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Q. If y = (x^2 + 1)^5, find dy/dx at x = 1.
Solution
dy/dx = 5(x^2 + 1)^4(2x). At x = 1, dy/dx = 5(2)^4(2) = 5(16)(2) = 160.
Correct Answer: B — 20
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Q. If y = (x^2 + 1)^5, find dy/dx.
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A.
10x(x^2 + 1)^4
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B.
5(x^2 + 1)^4
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C.
5x(x^2 + 1)^4
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D.
20x(x^2 + 1)^3
Solution
dy/dx = 10x(x^2 + 1)^4 by the chain rule.
Correct Answer: A — 10x(x^2 + 1)^4
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Q. If y = 2^x, find dy/dx at x = 1.
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A.
0.693
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B.
1.386
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C.
2.718
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D.
3.141
Solution
dy/dx = 2^x * ln(2). At x = 1, dy/dx = 2^1 * ln(2) = 2 * 0.693 = 1.386.
Correct Answer: A — 0.693
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Q. If y = 3x^2 + 2x, find dy/dx at x = 2.
Solution
First, find dy/dx = 6x + 2. At x = 2, dy/dx = 6(2) + 2 = 12 + 2 = 14.
Correct Answer: B — 18
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Q. If y = 4x^2 + 3x + 2, find dy/dx at x = -1.
Solution
dy/dx = 8x + 3. At x = -1, dy/dx = 8(-1) + 3 = -8 + 3 = -5.
Correct Answer: A — -5
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Q. If y = 4x^3 - 2x + 1, find dy/dx at x = -1.
Solution
dy/dx = 12x^2 - 2. At x = -1, dy/dx = 12(-1)^2 - 2 = 12 - 2 = 10.
Correct Answer: C — -6
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Q. If y = 5x^4 + 3x^2 - x, find dy/dx at x = 1.
Solution
dy/dx = 20x^3 + 6x - 1. At x = 1, dy/dx = 20(1)^3 + 6(1) - 1 = 20 + 6 - 1 = 25.
Correct Answer: B — 22
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Q. If y = 5x^4 - 3x^3 + 2x - 1, find dy/dx at x = 1.
Solution
dy/dx = 20x^3 - 9x^2 + 2. At x = 1, dy/dx = 20 - 9 + 2 = 13.
Correct Answer: B — 16
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Q. If y = cos(5x^2), find dy/dx.
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A.
-10xsin(5x^2)
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B.
-5xsin(5x^2)
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C.
-25xsin(5x^2)
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D.
-2xsin(5x^2)
Solution
dy/dx = -10xsin(5x^2) by the chain rule.
Correct Answer: A — -10xsin(5x^2)
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Q. If y = e^(3x), find dy/dx at x = 0.
Solution
dy/dx = 3e^(3x). At x = 0, dy/dx = 3e^(0) = 3.
Correct Answer: A — 1
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Q. If y = e^(3x), find dy/dx.
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A.
3e^(3x)
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B.
e^(3x)
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C.
9e^(3x)
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D.
6e^(3x)
Solution
dy/dx = 3e^(3x) by the chain rule.
Correct Answer: A — 3e^(3x)
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Q. If y = ln(5x^2 + 3), find dy/dx at x = 1.
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A.
5/8
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B.
3/8
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C.
1/8
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D.
1/5
Solution
dy/dx = (10x)/(5x^2 + 3). At x = 1, dy/dx = (10)/(5(1) + 3) = 10/8 = 5/4.
Correct Answer: A — 5/8
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Q. If y = ln(5x^2 + 3), find dy/dx.
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A.
10/(5x^2 + 3)
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B.
5/(5x^2 + 3)
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C.
2/(5x^2 + 3)
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D.
15/(5x^2 + 3)
Solution
dy/dx = (10x)/(5x^2 + 3) by the chain rule.
Correct Answer: A — 10/(5x^2 + 3)
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Q. If y = sin(2x), find dy/dx at x = π/4.
Solution
dy/dx = 2cos(2x). At x = π/4, dy/dx = 2cos(π/2) = 2(0) = 0.
Correct Answer: D — √2
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Q. If y = sqrt(4x^2 + 1), find dy/dx.
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A.
(4x)/(sqrt(4x^2 + 1))
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B.
(2x)/(sqrt(4x^2 + 1))
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C.
(8x)/(sqrt(4x^2 + 1))
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D.
(2)/(sqrt(4x^2 + 1))
Solution
Using the chain rule, dy/dx = (4x)/(2sqrt(4x^2 + 1)) = (4x)/(sqrt(4x^2 + 1)).
Correct Answer: A — (4x)/(sqrt(4x^2 + 1))
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Q. If y = sqrt(x^2 + 1), find dy/dx at x = 0.
Solution
dy/dx = (1/2)(x^2 + 1)^(-1/2)(2x). At x = 0, dy/dx = (1/2)(1)^(-1/2)(0) = 0.
Correct Answer: B — 1
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Q. If y = tan(3x), find dy/dx at x = π/6.
Solution
dy/dx = 3sec^2(3x). At x = π/6, dy/dx = 3sec^2(π/2) = 3(∞) = undefined.
Correct Answer: A — 3√3
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Q. If y = tan(3x), find dy/dx.
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A.
3sec^2(3x)
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B.
3tan^2(3x)
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C.
sec^2(3x)
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D.
3tan(3x)
Solution
Using the chain rule, dy/dx = 3sec^2(3x).
Correct Answer: A — 3sec^2(3x)
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Q. If y = x^3 * e^x, find dy/dx at x = 0.
Solution
Using product rule, dy/dx = 3x^2 * e^x + x^3 * e^x. At x = 0, dy/dx = 0 + 0 = 0.
Correct Answer: A — 0
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Q. If y = √(4x^2 + 1), find dy/dx.
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A.
4x/(√(4x^2 + 1))
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B.
2x/(√(4x^2 + 1))
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C.
2/(√(4x^2 + 1))
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D.
8x/(√(4x^2 + 1))
Solution
dy/dx = (4x)/(2√(4x^2 + 1)) = 4x/(√(4x^2 + 1)).
Correct Answer: A — 4x/(√(4x^2 + 1))
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Q. If y = √(x^2 + 1), find dy/dx at x = 1.
Solution
dy/dx = (1/2)(x^2 + 1)^(-1/2)(2x). At x = 1, dy/dx = (1/2)(2)^(-1/2)(2) = 1/√2.
Correct Answer: A — 1/√2
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Q. If you invest $1000 at an interest rate of 8% compounded annually, how much will you have after 2 years?
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A.
$1166.40
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B.
$1080.00
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C.
$1200.00
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D.
$1150.00
Solution
Total Amount = 1000(1 + 0.08)^2 = 1000(1.1664) = 1166.40
Correct Answer: A — $1166.40
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Q. In 10 years, a father will be twice as old as his son. If the son is currently 10 years old, how old is the father now?
Solution
Let the father's current age be x. In 10 years, x + 10 = 2(10 + 10). Solving gives x = 30.
Correct Answer: B — 30
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Q. In 5 years, a man will be 3 times as old as his son. If the son is currently 5 years old, how old is the man now?
Solution
Let the man's current age be x. In 5 years, x + 5 = 3 * (5 + 5). Solving gives x = 25.
Correct Answer: C — 25
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Q. In 5 years, A will be 3 times as old as B. If B is currently 10 years old, how old will A be in 5 years?
Solution
In 5 years, B will be 10 + 5 = 15. A will be 3 * 15 = 45. So, A's current age is 45 - 5 = 40.
Correct Answer: C — 25
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Q. In 5 years, A will be 3 times as old as B. If B is currently 10 years old, how old is A now?
Solution
In 5 years, B will be 15. Therefore, A will be 3 * 15 = 45. So, A is currently 45 - 5 = 40 years old.
Correct Answer: B — 20
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Q. In a class of 30 students, 18 are girls. If a student is chosen at random, what is the probability that the student is a boy?
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A.
1/3
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B.
2/5
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C.
1/2
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D.
3/5
Solution
The number of boys is 30 - 18 = 12. The probability is 12/30 = 2/5.
Correct Answer: A — 1/3
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Q. In a class of 30 students, 18 are girls. If a student is selected at random, what is the probability that the student is a boy?
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A.
0.4
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B.
0.6
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C.
0.5
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D.
0.3
Solution
The number of boys is 30 - 18 = 12. The probability is 12/30 = 0.4.
Correct Answer: B — 0.6
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