Mathematics for Competitive Exam Success

Mathematics

3D Geometry Applications of Derivatives Binomial Theorem Circles Complex Numbers Conic Sections Coordinate Geometry Differential Equations Differentiation Integration Limits & Continuity Matrices & Determinants Permutations & Combinations Probability Quadratic Equations Sequences & Series Sets & Relations Straight Lines Trigonometry Vector Algebra

Q. If a student is selected at random from a group of 20 students, 12 of whom are girls, what is the probability that the student is a girl?

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

Solution

Probability of selecting a girl = 12/20 = 3/5.

Correct Answer: A — 3/5

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Q. If a student is selected at random from a group of 40 students, 25 of whom are boys, what is the probability that the student is a girl? (2020)

A. 1/2
B. 3/8
C. 3/5
D. 5/8

Solution

There are 40 - 25 = 15 girls. The probability of selecting a girl is 15/40 = 3/8.

Correct Answer: D — 5/8

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Q. If C = [[0, 1], [1, 0]], what is det(C)? (2022)

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

Solution

Determinant of C = (0*0) - (1*1) = 0 - 1 = -1.

Correct Answer: C — -1

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Q. If cos(θ) = 0, what is the value of θ? (2017)

A. 0°
B. 90°
C. 180°
D. 270°

Solution

cos(90°) = 0, so θ = 90°

Correct Answer: B — 90°

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Q. If cos(θ) = 0.5, what is the value of θ in degrees? (2017)

A. 30°
B. 45°
C. 60°
D. 90°

Solution

cos(60°) = 0.5, so θ = 60°

Correct Answer: C — 60°

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Q. If cos(θ) = 0.5, what is the value of θ? (2020)

A. 0°
B. 30°
C. 60°
D. 90°

Solution

cos(60°) = 0.5, so θ = 60°.

Correct Answer: C — 60°

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Q. If cos(θ) = 0.6, what is sin(θ) using Pythagorean identity? (2017)

A. 0.4
B. 0.5
C. 0.6
D. 0.8

Solution

sin²(θ) + cos²(θ) = 1; sin²(θ) = 1 - 0.6² = 0.64; sin(θ) = √0.64 = 0.8.

Correct Answer: A — 0.4

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Q. If cos(θ) = 0.707, what is θ? (2020)

A. 30°
B. 45°
C. 60°
D. 90°

Solution

cos(45°) = √2/2 ≈ 0.707, so θ = 45°.

Correct Answer: B — 45°

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Q. If cos(θ) = 1/2, what is θ in degrees?

A. 30°
B. 45°
C. 60°
D. 90°

Solution

cos(60°) = 1/2, so θ = 60°

Correct Answer: C — 60°

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Q. If E = [[2, 1, 3], [1, 0, 2], [4, 1, 1]], what is det(E)? (2020)

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

Solution

The determinant of E can be calculated using the rule of Sarrus or cofactor expansion, resulting in 0.

Correct Answer: A — -1

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Q. If E = [[a, b], [c, d]], what is the expression for det(E)? (2023)

A. ad - bc
B. ab + cd
C. ac - bd
D. bc - ad

Solution

The determinant of E is calculated as (a*d) - (b*c) = ad - bc.

Correct Answer: A — ad - bc

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Q. If F = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], what is det(F)? (2021)

A. -14
B. 14
C. 0
D. 10

Solution

Det(F) = 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1.

Correct Answer: A — -14

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Q. If F = [[2, 0], [0, 3]], what is det(F)? (2020)

A. 0
B. 6
C. 5
D. 2

Solution

The determinant of F is calculated as (2*3) - (0*0) = 6.

Correct Answer: B — 6

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Q. If F = [[2, 1, 3], [1, 0, 2], [0, 1, 1]], what is det(F)? (2023)

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

Solution

Det(F) = 2(0*1 - 2*1) - 1(1*1 - 2*0) + 3(1*1 - 0*0) = 2(0 - 2) - 1(1) + 3(1) = -4 - 1 + 3 = -2.

Correct Answer: C — 3

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Q. If F = [[2, 1, 3], [1, 0, 2], [3, 1, 1]], find det(F). (2022)

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

Solution

Using the determinant formula, det(F) = 2(0*1 - 2*1) - 1(1*1 - 2*3) + 3(1*1 - 0*3) = 2(0 - 2) - 1(1 - 6) + 3(1 - 0) = -4 + 5 + 3 = 4.

Correct Answer: A — -4

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Q. If F = [[2, 1, 3], [1, 0, 2], [3, 4, 1]], find det(F). (2022)

A. -10
B. 10
C. 0
D. 5

Solution

Using the determinant formula, det(F) = 2*(0*1 - 2*4) - 1*(1*1 - 2*3) + 3*(1*4 - 0*3) = 2*(-8) - 1*(-5) + 3*4 = -16 + 5 + 12 = 1.

Correct Answer: A — -10

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Q. If F = [[2, 1], [1, 3]], what is the value of det(F)? (2022)

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

Solution

The determinant of F is (2*3) - (1*1) = 6 - 1 = 5.

Correct Answer: A — 5

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Q. If f(x) = x^3 - 3x^2 + 4, find the critical points. (2022)

A. 1, 2
B. 0, 3
C. 2, 4
D. 1, 3

Solution

f'(x) = 3x^2 - 6x. Setting f'(x) = 0 gives x(3x - 6) = 0, so x = 0 or x = 2.

Correct Answer: A — 1, 2

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Q. If H = [[1, 2, 1], [0, 1, 0], [2, 1, 1]], find det(H). (2021)

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

Solution

The determinant of H is calculated as 1(1*1 - 0*1) - 2(0*1 - 0*2) + 1(0*1 - 1*2) = 1 - 0 - 2 = -1.

Correct Answer: B — 1

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Q. If H = [[1, 2], [2, 4]], what is det(H)? (2020)

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

Solution

The determinant of H is (1*4) - (2*2) = 4 - 4 = 0.

Correct Answer: A — 0

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Q. If H = [[2, 3], [4, 5]], find det(H). (2022)

A. -2
B. 1
C. 2
D. 7

Solution

Det(H) = (2*5) - (3*4) = 10 - 12 = -2.

Correct Answer: D — 7

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Q. If I = [[1, 0, 2], [0, 1, 3], [1, 0, 4]], find det(I). (2021)

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

Solution

Using cofactor expansion, det(I) = 1(1*4 - 3*0) - 0 + 2(0*0 - 1*1) = 4 - 2 = 2.

Correct Answer: B — 2

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Q. If I = [[1, 0, 2], [0, 1, 3], [1, 1, 0]], find det(I). (2023)

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

Solution

Using the determinant formula for 3x3 matrices, det(I) = 1(1*0 - 3*1) - 0(0 - 3*1) + 2(0 - 1*1) = 0 - 0 - 2 = -2.

Correct Answer: A — -1

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Q. If I = [[1, 2], [2, 4]], what is det(I)? (2021)

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

Solution

The determinant of I is 0 because the rows are linearly dependent.

Correct Answer: A — 0

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Q. If J = [[1, 1], [1, 1]], what is det(J)? (2019)

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

Solution

Det(J) = (1*1) - (1*1) = 1 - 1 = 0.

Correct Answer: A — 0

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Q. If J = [[1, 2, 1], [0, 1, 3], [2, 1, 0]], calculate det(J). (2023)

A. -4
B. 4
C. 0
D. 2

Solution

Using the determinant formula, det(J) = 1*(1*0 - 3*1) - 2*(0*0 - 3*2) + 1*(0*1 - 1*2) = 1*(-3) - 2*(-6) + 1*(-2) = -3 + 12 - 2 = 7.

Correct Answer: A — -4

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Q. If J = [[1, 2], [2, 4]], what is det(J)? (2022)

A. 0
B. 1
C. 2
D. 4

Solution

Det(J) = (1*4) - (2*2) = 4 - 4 = 0.

Correct Answer: A — 0

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Q. If one root of the equation x² - 6x + k = 0 is 2, find k. (2022)

A. 8
B. 10
C. 12
D. 6

Solution

Using the root 2 in the equation: 2² - 6*2 + k = 0, we find k = 10.

Correct Answer: B — 10

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Q. If one root of the equation x² - 7x + k = 0 is 3, find k. (2023)

A. 10
B. 12
C. 15
D. 9

Solution

Using the root, substitute x = 3: 3² - 7*3 + k = 0, which gives k = 10.

Correct Answer: A — 10

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Q. If one root of the equation x² - 7x + k = 0 is 3, what is the value of k? (2020)

A. 10
B. 12
C. 15
D. 9

Solution

Using the root, substitute x = 3: 3² - 7*3 + k = 0, which gives k = 9.

Correct Answer: D — 9

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Mathematics

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