Q. A car travels 60 km in 1 hour. If it increases its speed by 20%, how far will it travel in 1.5 hours?
A.
90 km
B.
100 km
C.
110 km
D.
120 km
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Solution
New speed = 60 + (20% of 60) = 60 + 12 = 72 km/h. Distance in 1.5 hours = 72 * 1.5 = 108 km.
Correct Answer:
C
— 110 km
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Q. A man sells two articles for $120 each. If he makes a profit of 20% on the first and a loss of 20% on the second, what is his overall profit or loss percentage?
A.
0%
B.
5%
C.
10%
D.
20%
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Solution
Cost price of first article = 120 / 1.2 = $100. Cost price of second article = 120 / 0.8 = $150. Total cost price = 100 + 150 = $250. Total selling price = 120 + 120 = $240. Overall loss = 250 - 240 = $10. Loss percentage = (10 / 250) * 100 = 4%.
Correct Answer:
B
— 5%
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Q. A person can complete a work in 12 days. If he works for 4 days, what fraction of the work is left?
A.
1/3
B.
1/4
C.
1/2
D.
2/3
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Solution
Work done in 4 days = 4/12 = 1/3. Work left = 1 - 1/3 = 2/3.
Correct Answer:
D
— 2/3
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Q. A shopkeeper bought an article for $200 and sold it for $240. What is the profit percentage?
A.
15%
B.
20%
C.
25%
D.
30%
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Solution
Profit = Selling Price - Cost Price = 240 - 200 = $40. Profit percentage = (40 / 200) * 100 = 20%.
Correct Answer:
B
— 20%
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Q. A sum of money is invested at a rate of 10% per annum compound interest. How much will it amount to in 2 years if the principal is $1000?
A.
$1100
B.
$1200
C.
$1210
D.
$1300
Show solution
Solution
Amount = P(1 + r)^n = 1000(1 + 0.10)^2 = 1000(1.21) = $1210.
Correct Answer:
C
— $1210
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Q. If a man can complete a work in 10 days and another man can complete the same work in 15 days, how long will it take for both to complete the work together?
A.
6 days
B.
8 days
C.
10 days
D.
12 days
Show solution
Solution
Work done by first man in one day = 1/10. Work done by second man in one day = 1/15. Together they do = 1/10 + 1/15 = 1/6. Therefore, they will complete the work in 6 days.
Correct Answer:
A
— 6 days
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Q. If a man can walk 5 km in 1 hour, how long will it take him to walk 20 km?
A.
3 hours
B.
4 hours
C.
5 hours
D.
6 hours
Show solution
Solution
Time = Distance / Speed = 20 km / 5 km/h = 4 hours.
Correct Answer:
B
— 4 hours
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Q. If y = (2x + 1)^3, find dy/dx at x = 1.
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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.
A.
6(2x + 1)^2
B.
3(2x + 1)^2
C.
2(2x + 1)^2
D.
12(2x + 1)^2
Show solution
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.
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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.
A.
10x(x^2 + 1)^4
B.
5(x^2 + 1)^4
C.
5x(x^2 + 1)^4
D.
20x(x^2 + 1)^3
Show solution
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.
A.
0.693
B.
1.386
C.
2.718
D.
3.141
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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.
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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.
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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.
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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.
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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.
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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.
A.
-10xsin(5x^2)
B.
-5xsin(5x^2)
C.
-25xsin(5x^2)
D.
-2xsin(5x^2)
Show solution
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.
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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.
A.
3e^(3x)
B.
e^(3x)
C.
9e^(3x)
D.
6e^(3x)
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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.
A.
5/8
B.
3/8
C.
1/8
D.
1/5
Show solution
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.
A.
10/(5x^2 + 3)
B.
5/(5x^2 + 3)
C.
2/(5x^2 + 3)
D.
15/(5x^2 + 3)
Show solution
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.
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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.
A.
(4x)/(sqrt(4x^2 + 1))
B.
(2x)/(sqrt(4x^2 + 1))
C.
(8x)/(sqrt(4x^2 + 1))
D.
(2)/(sqrt(4x^2 + 1))
Show solution
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.
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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.
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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.
A.
3sec^2(3x)
B.
3tan^2(3x)
C.
sec^2(3x)
D.
3tan(3x)
Show solution
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.
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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.
A.
4x/(√(4x^2 + 1))
B.
2x/(√(4x^2 + 1))
C.
2/(√(4x^2 + 1))
D.
8x/(√(4x^2 + 1))
Show solution
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.
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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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