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. For the function f(x) = -x^2 + 4x + 1, find the x-coordinate of the vertex. (2023)

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

Solution

The vertex x-coordinate is given by -b/(2a) = -4/(2*-1) = 2.

Correct Answer: A — 2

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Q. For the function f(x) = 2x^3 - 9x^2 + 12x, find the critical points. (2022)

A. (0, 0)
B. (1, 5)
C. (2, 0)
D. (3, 3)

Solution

Set f'(x) = 0. f'(x) = 6x^2 - 18x + 12 = 0. Critical points are x = 2.

Correct Answer: C — (2, 0)

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Q. For the function f(x) = 2x^3 - 9x^2 + 12x, find the intervals of increase. (2022)

A. (-∞, 0)
B. (0, 3)
C. (3, ∞)
D. (0, 2)

Solution

f'(x) = 6x^2 - 18x + 12. The critical points are x = 1 and x = 2. Test intervals show increase in (0, 3).

Correct Answer: B — (0, 3)

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Q. For the function f(x) = 2x^3 - 9x^2 + 12x, find the local maxima. (2023) 2023

A. (1, 5)
B. (2, 6)
C. (3, 3)
D. (0, 0)

Solution

Find f'(x) = 6x^2 - 18x + 12, set to 0. Local maxima at x = 2 gives f(2) = 6.

Correct Answer: B — (2, 6)

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Q. For the function f(x) = 3x^2 - 12x + 7, find the minimum value. (2022)

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

Solution

The vertex occurs at x = 2. f(2) = 3(2^2) - 12(2) + 7 = -5.

Correct Answer: B — -4

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Q. For the function f(x) = 3x^2 - 12x + 7, find the x-coordinate of the vertex. (2022)

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

Solution

The x-coordinate of the vertex is given by x = -b/(2a) = 12/(2*3) = 2.

Correct Answer: B — 2

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Q. For the function f(x) = 3x^2 - 12x + 9, find the vertex. (2021)

A. (2, 3)
B. (3, 0)
C. (0, 9)
D. (1, 6)

Solution

The vertex occurs at x = -b/(2a) = 12/(2*3) = 2. f(2) = 3(2^2) - 12(2) + 9 = 3.

Correct Answer: A — (2, 3)

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Q. For the function f(x) = x^2 + 2x, find the local maximum. (2022)

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

Solution

f'(x) = 2x + 2. Setting f'(x) = 0 gives x = -1. f(-1) = 1.

Correct Answer: A — -1

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Q. For the matrix E = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find det(E). (2021)

A. -24
B. 24
C. 0
D. 12

Solution

Using the determinant formula, det(E) = 1*(1*0 - 4*6) - 2*(0*0 - 4*5) + 3*(0*6 - 1*5) = 1*(-24) - 2*(-20) - 15 = -24 + 40 - 15 = 1.

Correct Answer: A — -24

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Q. For the matrix E = [[1, 2], [2, 4]], what is the determinant? (2021)

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

Solution

Determinant of E = (1*4) - (2*2) = 4 - 4 = 0.

Correct Answer: A — 0

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Q. For the quadratic equation x² + 6x + k = 0 to have real roots, what is the minimum value of k? (2021)

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

Solution

The discriminant must be non-negative: 6² - 4*1*k ≥ 0, thus k ≤ 9. The minimum value of k is -9.

Correct Answer: A — -9

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Q. For which value of m does the equation x² + mx + 9 = 0 have roots that are both negative? (2021)

A. -6
B. -4
C. -2
D. 2

Solution

For both roots to be negative, m must be greater than 0 and m² < 36. Thus, m must be in the range (-6, 0). The suitable value is -4.

Correct Answer: B — -4

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Q. For which value of m does the equation x² - mx + 9 = 0 have roots 3 and 3? (2023)

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

Solution

The sum of the roots is 3 + 3 = 6, hence m = 6.

Correct Answer: A — 6

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Q. For which value of p does the equation x² + px + 4 = 0 have roots 2 and -2? (2022)

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

Solution

Using the sum of roots: 2 + (-2) = -p, hence p = 0.

Correct Answer: C — -4

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Q. For which value of p does the equation x² + px + 4 = 0 have roots that are both negative? (2022)

A. -8
B. -6
C. -4
D. -2

Solution

For both roots to be negative, p must be greater than 0 and p² > 16. Thus, p < -4.

Correct Answer: C — -4

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Q. For which value of p does the equation x² + px + 9 = 0 have roots that are both negative? (2021)

A. -6
B. -4
C. -3
D. -2

Solution

For both roots to be negative, p must be positive and p² > 4*9. Thus, p > 6, so p = -4 is valid.

Correct Answer: B — -4

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Q. How many different 4-digit PINs can be formed using the digits 0-9 if digits cannot be repeated?

A. 5040
B. 10000
C. 9000
D. 7200

Solution

The first digit can be any of 10 digits, the second can be any of 9, the third can be any of 8, and the fourth can be any of 7. Total = 10 * 9 * 8 * 7 = 5040.

Correct Answer: A — 5040

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Q. How many different 4-digit PINs can be formed using the digits 0-9 without repetition?

A. 5040
B. 9000
C. 10000
D. 1000

Solution

The number of 4-digit PINs is P(10, 4) = 5040.

Correct Answer: A — 5040

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Q. How many different ways can 3 boys and 2 girls be seated in a row?

A. 30
B. 60
C. 120
D. 180

Solution

The number of ways to arrange 5 people is 5! = 120.

Correct Answer: C — 120

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Q. How many different ways can the letters of the word 'MATH' be arranged?

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

Solution

The number of arrangements of the letters in 'MATH' is 4! = 24.

Correct Answer: B — 24

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Q. How many ways can 2 boys and 3 girls be selected from 5 boys and 6 girls?

A. 100
B. 150
C. 200
D. 250

Solution

The number of ways is 5C2 * 6C3 = 10 * 20 = 200.

Correct Answer: B — 150

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Q. How many ways can 2 students be selected from a group of 5? (2019)

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

Solution

The number of ways to select 2 students from 5 is given by 5C2 = 10.

Correct Answer: A — 10

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Q. How many ways can 3 books be selected from a shelf of 10 books? (2022)

A. 120
B. 210
C. 100
D. 30

Solution

The number of ways to select 3 books from 10 is given by 10C3 = 120.

Correct Answer: B — 210

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Q. How many ways can 3 boys and 2 girls be seated in a row?

A. 30
B. 60
C. 120
D. 180

Solution

The number of ways to arrange 5 people is 5! = 120.

Correct Answer: C — 120

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Q. How many ways can 3 different colored balls be arranged in a row? (2022)

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

Solution

The number of arrangements of 3 distinct balls is 3! = 6.

Correct Answer: A — 6

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Q. How many ways can 3 different gifts be given to 5 children if each child can receive more than one gift? (2022)

A. 125
B. 243
C. 60
D. 30

Solution

Each gift can go to any of the 5 children, so the total ways = 5^3 = 125.

Correct Answer: B — 243

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Q. How many ways can 3 different trophies be awarded to 5 students? (2022)

A. 60
B. 100
C. 120
D. 30

Solution

The number of ways to award 3 trophies to 5 students is given by P(5,3) = 5! / (5-3)! = 60.

Correct Answer: C — 120

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Q. How many ways can 3 out of 8 different colored balls be chosen? (2022)

A. 56
B. 24
C. 8
D. 12

Solution

The number of ways to choose 3 balls from 8 is given by 8C3 = 56.

Correct Answer: A — 56

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Q. How many ways can 3 prizes be awarded to 10 students if no student can win more than one prize?

A. 720
B. 1000
C. 1200
D. 100

Solution

The number of ways to award 3 prizes is P(10, 3) = 720.

Correct Answer: A — 720

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Q. How many ways can 3 students be selected from a group of 10?

A. 120
B. 90
C. 100
D. 80

Solution

The number of ways to select 3 students from 10 is given by 10C3 = 120.

Correct Answer: A — 120

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Mathematics

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