Defence Exams Preparation Resources

Defence Exams

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Q. The equation of a parabola with vertex at (0, 0) and focus at (0, 3) is?

A. x^2 = 12y
B. y^2 = 12x
C. x^2 = 6y
D. y^2 = 6x

Solution

The distance from the vertex to the focus is 3, so the equation is x^2 = 12y.

Correct Answer: A — x^2 = 12y

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Q. The equation x^2 - 2x + 1 = 0 has how many distinct roots?

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

Solution

The discriminant is 0, indicating that there is exactly one distinct root.

Correct Answer: B — 1

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Q. The equation x^2 - 2x + k = 0 has roots that are both positive. What is the range of k?

A. k < 0
B. k > 0
C. k > 1
D. k < 1

Solution

For both roots to be positive, k must be greater than 1 (from Vieta's formulas).

Correct Answer: C — k > 1

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Q. The equation x^2 - 4x + k = 0 has equal roots when k is equal to:

A. 4
B. 0
C. 8
D. 16

Solution

For equal roots, the discriminant must be zero: (-4)^2 - 4*1*k = 0 leads to k = 4.

Correct Answer: A — 4

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Q. The equation x^2 - 4x + k = 0 has no real roots if k is:

A. < 4
B. ≥ 4
C. ≤ 4
D. > 4

Solution

The discriminant must be less than zero: (-4)^2 - 4*1*k < 0 leads to k > 4.

Correct Answer: A — < 4

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Q. The equation x^2 - 6x + 9 = 0 has how many distinct roots?

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

Solution

The equation can be factored as (x - 3)(x - 3) = 0, indicating it has one distinct root (a double root).

Correct Answer: B — 1

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Q. The equation x^2 - 7x + 10 = 0 has roots that are:

A. 1 and 10
B. 2 and 5
C. 3 and 4
D. 5 and 2

Solution

Factoring the equation gives (x - 2)(x - 5) = 0, so the roots are 2 and 5.

Correct Answer: C — 3 and 4

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Q. The equation x^3 - 3x^2 + 3x - 1 = 0 has a root at x = 1. What is the multiplicity of this root? (2023)

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

Solution

The polynomial can be expressed as (x - 1)^3, indicating that the root x = 1 has a multiplicity of 3.

Correct Answer: C — 3

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Q. The equation x^3 - 3x^2 + 3x - 1 = 0 has how many distinct real roots? (2022)

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

Solution

The given polynomial can be factored as (x - 1)^3 = 0, which has one distinct real root, x = 1.

Correct Answer: A — 1

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Q. The following data represents the number of books read by students in a month: 1, 2, 2, 3, 3, 3, 4, 4, 5. What is the mode?

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

Solution

The mode is 3, as it appears most frequently (3 times) in the data set.

Correct Answer: C — 3

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Q. The following data represents the number of books read by students in a month: 4, 5, 6, 5, 4, 5, 7, 8, 5. What is the mode?

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

Solution

The mode is 5, as it appears 4 times, more than any other number.

Correct Answer: B — 5

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Q. The following data set has the numbers: 2, 3, 4, 4, 5, 5, 5, 6, 7. What is the mode of this data set? (2020)

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

Solution

The number 5 appears most frequently (3 times), so the mode is 5.

Correct Answer: B — 5

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Q. The following data set represents the ages of participants in a survey: 22, 25, 22, 30, 25, 22, 28. What is the mode?

A. 22
B. 25
C. 28
D. 30

Solution

The mode is 22, as it appears most frequently (three times) in the data set.

Correct Answer: A — 22

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Q. The following data set shows the number of pets owned by families: 2, 3, 2, 4, 3, 2, 5. What is the mode?

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

Solution

The mode is 2, as it appears 3 times, which is more than any other number.

Correct Answer: A — 2

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Q. The following data shows the number of goals scored by a team in 7 matches: 1, 2, 2, 3, 2, 4, 5. What is the mode?

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

Solution

The mode is 2, as it appears 3 times, more than any other number.

Correct Answer: B — 2

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Q. The following data shows the number of pets owned by families: 0, 1, 1, 2, 2, 2, 3, 4. What is the mode?

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

Solution

The mode is 2, as it appears most frequently (3 times) in the data set.

Correct Answer: C — 2

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Q. The following data shows the number of pets owned by families: 1, 1, 2, 2, 3, 3, 3, 4. What is the mode?

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

Solution

The mode is 3, as it appears most frequently (3 times) in the data set.

Correct Answer: C — 3

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Q. The following numbers are given: 8, 3, 5, 12, 7. What is the median?

A. 5
B. 7
C. 8
D. 12

Solution

Arrange the numbers: 3, 5, 7, 8, 12. The median is the middle value, which is 7.

Correct Answer: C — 8

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Q. The following numbers represent the number of books read by students: 1, 2, 2, 3, 4, 4, 4, 5. What is the mode?

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

Solution

The mode is 4, as it appears three times, which is more than any other number.

Correct Answer: C — 4

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Q. The following numbers represent the number of books read by students: 5, 3, 5, 2, 5, 4, 3. What is the mode?

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

Solution

The mode is 5, as it appears 4 times, which is more than any other number.

Correct Answer: D — 5

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Q. The following numbers represent the number of games won by a team in a season: 5, 6, 6, 7, 8, 8, 8, 9. What is the mode?

A. 5
B. 6
C. 7
D. 8

Solution

The mode is 8, as it appears most frequently (3 times) in the data set.

Correct Answer: D — 8

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Q. The following numbers represent the number of goals scored by a team in 10 matches: 1, 2, 2, 3, 3, 3, 4, 4, 5. What is the mode?

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

Solution

The mode is 3, as it appears most frequently (3 times) in the data set.

Correct Answer: B — 3

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Q. The following numbers represent the number of hours studied by students: 1, 2, 2, 3, 3, 4, 4, 4, 5. What is the mode?

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

Solution

The mode is 4, as it appears most frequently (3 times) in the data set.

Correct Answer: C — 4

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Q. The following numbers represent the number of hours studied by students: 5, 6, 5, 7, 8, 5, 6. What is the mode?

A. 5
B. 6
C. 7
D. 8

Solution

The mode is 5, as it appears 3 times, more than any other number.

Correct Answer: A — 5

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Q. The following numbers represent the number of hours studied by students: 5, 6, 6, 7, 8, 8, 8, 9, 10. What is the mode?

A. 6
B. 7
C. 8
D. 9

Solution

The mode is 8, as it appears most frequently (3 times) in the data set.

Correct Answer: C — 8

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Q. The following numbers represent the number of languages spoken by individuals: 1, 2, 2, 3, 3, 3, 4, 5. What is the mode?

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

Solution

The mode is 3, as it appears most frequently (3 times) in the data set.

Correct Answer: C — 3

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Q. The following numbers represent the number of pets owned by a group of people: 0, 1, 1, 2, 2, 2, 3. What is the mode?

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

Solution

The number 2 appears 3 times, which is more than any other number, making it the mode.

Correct Answer: C — 2

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Q. The following numbers represent the number of pets owned by families: 0, 1, 1, 2, 2, 2, 3, 4. What is the mode?

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

Solution

The mode is 2, as it appears most frequently (3 times) in the data set.

Correct Answer: C — 2

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Q. The following numbers represent the scores of a game: 20, 30, 20, 40, 30, 30, 50. What is the mode?

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

Solution

The mode is 30, as it appears three times, more than any other score.

Correct Answer: B — 30

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Q. The following numbers represent the scores of a test: 45, 55, 65, 75, 85, 95. What is the median score? (2019)

A. 65
B. 70
C. 75
D. 80

Solution

Arrange the scores: 45, 55, 65, 75, 85, 95. The median is the average of the 3rd and 4th scores: (65 + 75) / 2 = 70.

Correct Answer: C — 75

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Defence Exams

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