Approximation & Simplification for Competitive Exams

Approximation & Simplification

Q. If 2x - 3 = 5, what is the value of x?

A. 1
B. 2
C. 4
D. 3

Solution

Adding 3 to both sides gives 2x = 8, so x = 4.

Correct Answer: B — 2

Q. If 5x - 3 = 12, what is the value of x?

A. 2
B. 3
C. 4
D. 5

Solution

Add 3 to both sides to get 5x = 15, then divide by 5 to find x = 3.

Correct Answer: B — 3

Q. If 5x - 3 = 2x + 12, what is the value of x?

A. 3
B. 4
C. 5
D. 6

Solution

Rearranging gives 5x - 2x = 12 + 3, so 3x = 15, thus x = 5.

Correct Answer: B — 4

Q. If 5x = 20, what is the value of x?

A. 2
B. 4
C. 5
D. 10

Solution

Dividing both sides by 5 gives x = 4.

Correct Answer: B — 4

Q. If a number is divisible by both 2 and 3, what is it also divisible by?

A. 4
B. 5
C. 6
D. 7

Solution

A number divisible by both 2 and 3 is also divisible by their least common multiple, which is 6.

Correct Answer: C — 6

Q. If a number is divisible by both 2 and 3, what is the smallest number it could be?

A. 6
B. 12
C. 3
D. 9

Solution

The least common multiple of 2 and 3 is 6.

Correct Answer: A — 6

Q. If a number is divisible by both 2 and 3, what is the smallest possible value?

A. 2
B. 3
C. 6
D. 12

Solution

The least common multiple of 2 and 3 is 6.

Correct Answer: C — 6

Q. If a number is divisible by both 4 and 6, what is the smallest positive integer that is also divisible by 12?

A. 12
B. 24
C. 36
D. 48

Solution

The least common multiple of 4 and 6 is 12.

Correct Answer: A — 12

Q. If a sequence is defined by a_n = 2n + 1, what is a_4?

A. 7
B. 8
C. 9
D. 10

Solution

a_4 = 2(4) + 1 = 8 + 1 = 9.

Correct Answer: A — 7

Q. If x + 5 = 12, what is the value of x?

A. 5
B. 7
C. 12
D. 17

Solution

Subtracting 5 from both sides gives x = 7.

Correct Answer: B — 7

Q. If x = 3, what is the value of 2x^2 - 5?

A. 1
B. 7
C. 10
D. 11

Solution

Substituting x = 3 gives 2(3^2) - 5 = 18 - 5 = 13.

Correct Answer: B — 7

Q. What is 12 mod 7?

A. 5
B. 4
C. 3
D. 2

Solution

12 divided by 7 is 1 with a remainder of 5, so 12 mod 7 = 5.

Correct Answer: A — 5

Q. What is 15 divided by 4?

A. 3.75
B. 4
C. 3
D. 3.5

Solution

15 divided by 4 equals 3.75.

Correct Answer: A — 3.75

Q. What is the next number in the sequence 2, 4, 8, 16?

A. 20
B. 24
C. 32
D. 36

Solution

The sequence doubles each time, so the next number is 32.

Correct Answer: C — 32

Q. What is the next term in the sequence 2, 4, 8, 16?

A. 20
B. 24
C. 32
D. 36

Solution

The sequence doubles each time, so the next term is 32.

Correct Answer: C — 32

Q. What is the remainder when 57 is divided by 5?

A. 2
B. 3
C. 1
D. 4

Solution

57 divided by 5 is 11 with a remainder of 2.

Correct Answer: B — 3

Q. What is the result of 15 mod 4?

A. 3
B. 1
C. 2
D. 0

Solution

15 divided by 4 gives a remainder of 3, so 15 mod 4 = 3.

Correct Answer: A — 3

Q. What is the value of 10 - 3(2 + 1)?

A. 1
B. 4
C. 7
D. 10

Solution

Calculating gives 10 - 3(3) = 10 - 9 = 1.

Correct Answer: B — 4

Q. What is the value of 5^2 + 3^2?

A. 25
B. 34
C. 35
D. 36

Solution

5^2 = 25 and 3^2 = 9, so 25 + 9 = 34.

Correct Answer: B — 34

Q. What is the value of 7 mod 3?

A. 1
B. 2
C. 0
D. 3

Solution

7 divided by 3 gives a remainder of 1.

Correct Answer: B — 2

Q. What is the value of 9 - 3(2 + 1)?

A. 0
B. 3
C. 6
D. 9

Solution

Calculating gives 9 - 3*3 = 9 - 9 = 0.

Correct Answer: C — 6

Q. What is the value of 9 mod 5?

A. 4
B. 3
C. 2
D. 1

Solution

9 divided by 5 gives a remainder of 4.

Correct Answer: A — 4

Q. What is the value of 9^2?

A. 81
B. 72
C. 90
D. 99

Solution

9 squared is 81.

Correct Answer: A — 81

Q. What is the value of x in the equation 3x + 5 = 20?

A. 5
B. 10
C. 15
D. 20

Solution

Subtract 5 from both sides to get 3x = 15, then divide by 3 to find x = 5.

Correct Answer: A — 5

Q. Which of the following is a factor of 30?

A. 5
B. 7
C. 11
D. 13

Solution

5 divides 30 evenly, so it is a factor.

Correct Answer: A — 5

Q. Which of the following is a prime number?

A. 4
B. 6
C. 9
D. 11

Solution

11 has no divisors other than 1 and itself.

Correct Answer: D — 11

Q. Which of the following is a solution to the equation 2x + 3 = 11?

A. 2
B. 3
C. 4
D. 5

Solution

Solving for x gives 2x = 8, so x = 4.

Correct Answer: C — 4

Q. Which of the following is a valid proof for the statement: If n is an even integer, then n^2 is even?

A. n = 2k for some integer k
B. n = 2k + 1
C. n = k^2
D. n = 2k - 1

Solution

If n = 2k, then n^2 = (2k)^2 = 4k^2, which is even.

Correct Answer: A — n = 2k for some integer k

Q. Which of the following is equivalent to 3(2x + 4)?

A. 6x + 12
B. 6x + 4
C. 6x + 8
D. 2x + 12

Solution

Distributing gives 6x + 12.

Correct Answer: A — 6x + 12

Q. Which of the following numbers is divisible by 6?

A. 14
B. 18
C. 25
D. 30

Solution

18 is divisible by both 2 and 3, hence by 6.

Correct Answer: B — 18

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