Arithmetic Aptitude for Competitive Exam Success

Arithmetic Aptitude

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Q. If y = (2x + 1)^3, find dy/dx at x = 1.

A. 12
B. 18
C. 24
D. 30

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

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

Solution

dy/dx = 6(2x + 1)^2 by the chain rule.

Correct Answer: A — 6(2x + 1)^2

Q. If y = (x^2 + 1)^5, find dy/dx at x = 1.

A. 10
B. 20
C. 30
D. 40

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

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

Solution

dy/dx = 10x(x^2 + 1)^4 by the chain rule.

Correct Answer: A — 10x(x^2 + 1)^4

Q. If y = 2^x, find dy/dx at x = 1.

A. 0.693
B. 1.386
C. 2.718
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

Q. If y = 3x^2 + 2x, find dy/dx at x = 2.

A. 14
B. 18
C. 22
D. 26

Solution

First, find dy/dx = 6x + 2. At x = 2, dy/dx = 6(2) + 2 = 12 + 2 = 14.

Correct Answer: B — 18

Q. If y = 4x^2 + 3x + 2, find dy/dx at x = -1.

A. -5
B. -3
C. 1
D. 5

Solution

dy/dx = 8x + 3. At x = -1, dy/dx = 8(-1) + 3 = -8 + 3 = -5.

Correct Answer: A — -5

Q. If y = 4x^3 - 2x + 1, find dy/dx at x = -1.

A. -10
B. -8
C. -6
D. -4

Solution

dy/dx = 12x^2 - 2. At x = -1, dy/dx = 12(-1)^2 - 2 = 12 - 2 = 10.

Correct Answer: C — -6

Q. If y = 5x^4 + 3x^2 - x, find dy/dx at x = 1.

A. 20
B. 22
C. 24
D. 26

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

Q. If y = 5x^4 - 3x^3 + 2x - 1, find dy/dx at x = 1.

A. 14
B. 16
C. 18
D. 20

Solution

dy/dx = 20x^3 - 9x^2 + 2. At x = 1, dy/dx = 20 - 9 + 2 = 13.

Correct Answer: B — 16

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)

Solution

dy/dx = -10xsin(5x^2) by the chain rule.

Correct Answer: A — -10xsin(5x^2)

Q. If y = e^(3x), find dy/dx at x = 0.

A. 1
B. 3
C. e
D. 3e

Solution

dy/dx = 3e^(3x). At x = 0, dy/dx = 3e^(0) = 3.

Correct Answer: A — 1

Q. If y = e^(3x), find dy/dx.

A. 3e^(3x)
B. e^(3x)
C. 9e^(3x)
D. 6e^(3x)

Solution

dy/dx = 3e^(3x) by the chain rule.

Correct Answer: A — 3e^(3x)

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

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

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)

Solution

dy/dx = (10x)/(5x^2 + 3) by the chain rule.

Correct Answer: A — 10/(5x^2 + 3)

Q. If y = sin(2x), find dy/dx at x = π/4.

A. 0
B. 1
C. √2/2
D. √2

Solution

dy/dx = 2cos(2x). At x = π/4, dy/dx = 2cos(π/2) = 2(0) = 0.

Correct Answer: D — √2

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))

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))

Q. If y = sqrt(x^2 + 1), find dy/dx at x = 0.

A. 0
B. 1
C. 1/2
D. 1/√2

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

Q. If y = tan(3x), find dy/dx at x = π/6.

A. 3√3
B. 3
C. √3
D. 1

Solution

dy/dx = 3sec^2(3x). At x = π/6, dy/dx = 3sec^2(π/2) = 3(∞) = undefined.

Correct Answer: A — 3√3

Q. If y = tan(3x), find dy/dx.

A. 3sec^2(3x)
B. 3tan^2(3x)
C. sec^2(3x)
D. 3tan(3x)

Solution

Using the chain rule, dy/dx = 3sec^2(3x).

Correct Answer: A — 3sec^2(3x)

Q. If y = x^3 * e^x, find dy/dx at x = 0.

A. 0
B. 1
C. 3
D. 6

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

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))

Solution

dy/dx = (4x)/(2√(4x^2 + 1)) = 4x/(√(4x^2 + 1)).

Correct Answer: A — 4x/(√(4x^2 + 1))

Q. If y = √(x^2 + 1), find dy/dx at x = 1.

A. 1/√2
B. 1/2
C. 1
D. 2

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

Q. If you invest $1000 at an interest rate of 8% compounded annually, how much will you have after 2 years?

A. $1166.40
B. $1080.00
C. $1200.00
D. $1150.00

Solution

Total Amount = 1000(1 + 0.08)^2 = 1000(1.1664) = 1166.40

Correct Answer: A — $1166.40

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?

A. 20
B. 30
C. 40
D. 50

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

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?

A. 15
B. 20
C. 25
D. 30

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

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?

A. 15
B. 20
C. 25
D. 30

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

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?

A. 15
B. 20
C. 25
D. 30

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

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?

A. 1/3
B. 2/5
C. 1/2
D. 3/5

Solution

The number of boys is 30 - 18 = 12. The probability is 12/30 = 2/5.

Correct Answer: A — 1/3

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?

A. 0.4
B. 0.6
C. 0.5
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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