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. What is the length of the diagonal of a rectangle with vertices at (1, 1), (1, 4), (5, 1), and (5, 4)? (2021)

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

Solution

Diagonal length = √[(5-1)² + (4-1)²] = √[16 + 9] = √25 = 5.

Correct Answer: B — 5

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Q. What is the maximum area of a triangle with a base of 10 cm and height as a function of x? (2020)

A. 25
B. 50
C. 75
D. 100

Solution

Area = 1/2 * base * height = 1/2 * 10 * x. Max area occurs when x = 10, giving Area = 50.

Correct Answer: B — 50

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Q. What is the maximum profit if the profit function is P(x) = -x^2 + 10x - 16? (2021)

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

Solution

The maximum profit occurs at x = -b/(2a) = 10/2 = 5. P(5) = -5^2 + 10*5 - 16 = 9.

Correct Answer: C — 8

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Q. What is the maximum value of f(x) = -x^2 + 4x + 1? (2023)

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

Solution

The vertex occurs at x = 2. f(2) = -2^2 + 4(2) + 1 = 5.

Correct Answer: A — 5

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Q. What is the maximum value of f(x) = -x^2 + 6x - 8? (2023)

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

Solution

The vertex occurs at x = 3. f(3) = -9 + 18 - 8 = 1.

Correct Answer: C — 6

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Q. What is the minimum distance from the point (3, 4) to the line 2x + 3y - 6 = 0? (2023)

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

Solution

Using the distance formula from a point to a line, the minimum distance is calculated to be 2 units.

Correct Answer: A — 2

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Q. What is the minimum value of f(x) = 3x^2 - 12x + 9? (2022)

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

Solution

The vertex occurs at x = 2. f(2) = 3(2^2) - 12(2) + 9 = 0.

Correct Answer: B — 1

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Q. What is the minimum value of f(x) = x^2 - 4x + 5? (2020)

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

Solution

The vertex form gives the minimum at x = 2. f(2) = 2.

Correct Answer: A — 1

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Q. What is the minimum value of f(x) = x^2 - 4x + 6? (2022)

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

Solution

The vertex occurs at x = 2. f(2) = 2.

Correct Answer: B — 3

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Q. What is the minimum value of f(x) = x^2 - 6x + 10? (2020)

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

Solution

The vertex form gives the minimum at x = 3. f(3) = 3^2 - 6(3) + 10 = 1.

Correct Answer: A — 4

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Q. What is the minimum value of the function f(x) = 4x^2 - 16x + 20? (2021) 2021

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

Solution

The vertex occurs at x = 2. f(2) = 4(2^2) - 16(2) + 20 = 4.

Correct Answer: B — 5

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Q. What is the modulus of the complex number 3 + 4i? (2023)

A. 5
B. 7
C. 1
D. 4

Solution

The modulus is calculated as √(3^2 + 4^2) = √(9 + 16) = √25 = 5.

Correct Answer: A — 5

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Q. What is the particular solution of dy/dx = 4y with the initial condition y(0) = 2?

A. y = 2e^(4x)
B. y = e^(4x)
C. y = 4e^(4x)
D. y = 2e^(x)

Solution

The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).

Correct Answer: A — y = 2e^(4x)

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Q. What is the point of inflection for the function f(x) = x^3 - 6x^2 + 9x? (2023) 2023

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

Solution

To find inflection points, set f''(x) = 0. f''(x) = 6x - 12 = 0, x = 2. f(2) = 4.

Correct Answer: A — (1, 4)

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Q. What is the point of intersection of the lines x + y = 5 and 2x - y = 1?

A. (2, 3)
B. (3, 2)
C. (1, 4)
D. (4, 1)

Solution

Solving the equations simultaneously, we find x = 2 and y = 3. Thus, the point of intersection is (2, 3).

Correct Answer: A — (2, 3)

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Q. What is the point of intersection of the lines y = 3x + 1 and y = -x + 5?

A. (1, 4)
B. (2, 7)
C. (3, 10)
D. (4, 13)

Solution

Setting 3x + 1 = -x + 5 gives 4x = 4, thus x = 1. Substituting x back gives y = 4. The intersection point is (1, 4).

Correct Answer: A — (1, 4)

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Q. What is the probability of drawing a king from a standard deck of cards? (2023)

A. 1/13
B. 1/52
C. 1/4
D. 1/12

Solution

Number of kings = 4. Probability = 4/52 = 1/13.

Correct Answer: A — 1/13

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Q. What is the probability of drawing a queen from a standard deck of cards?

A. 1/13
B. 1/26
C. 1/52
D. 1/4

Solution

Number of queens = 4. Probability = 4/52 = 1/13.

Correct Answer: A — 1/13

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Q. What is the probability of flipping a coin and getting heads? (2023)

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

Solution

There are 2 outcomes: heads or tails. Probability of heads = 1/2.

Correct Answer: A — 1/2

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Q. What is the probability of rolling a sum of 7 with two dice?

A. 1/6
B. 1/12
C. 1/36
D. 5/36

Solution

Possible combinations for sum of 7 = (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6. Total outcomes = 36. Probability = 6/36 = 1/6.

Correct Answer: D — 5/36

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Q. What is the product of 9 and 6? (2016)

A. 54
B. 60
C. 48
D. 72

Solution

9 × 6 = 54.

Correct Answer: A — 54

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Q. What is the product of the complex numbers (1 + i) and (1 - i)? (2023) 2023

A. 2
B. 0
C. 1
D. 1 + 0i

Solution

(1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 2.

Correct Answer: A — 2

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Q. What is the product of the complex numbers 2 + 3i and 4 - i? (2021)

A. 11 + 10i
B. 14 + 5i
C. 5 + 10i
D. 10 + 5i

Solution

(2 + 3i)(4 - i) = 8 - 2i + 12i - 3i^2 = 8 + 10i + 3 = 11 + 10i.

Correct Answer: A — 11 + 10i

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Q. What is the product of the roots of the equation 3x² - 12x + 9 = 0? (2022)

A. 1
B. 3
C. 4
D. 9

Solution

The product of the roots is given by c/a = 9/3 = 3.

Correct Answer: B — 3

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Q. What is the product of the roots of the equation x² + 5x + 6 = 0? (2022)

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

Solution

The product of the roots is given by c/a = 6/1 = 6.

Correct Answer: A — 6

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Q. What is the product of the roots of the equation x² - 10x + 24 = 0? (2021)

A. 24
B. 10
C. 16
D. 14

Solution

The product of the roots is given by c/a = 24/1 = 24.

Correct Answer: A — 24

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Q. What is the product of the roots of the equation x² - 8x + 15 = 0? (2022)

A. 15
B. 8
C. 7
D. 6

Solution

The product of the roots is given by c/a = 15/1 = 15.

Correct Answer: A — 15

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Q. What is the projection of vector A = 3i + 4j onto vector B = i + j? (2020)

A. 3
B. 4
C. 5
D. 7

Solution

proj_B(A) = (A · B / |B|^2)B = ((3*1 + 4*1) / (1^2 + 1^2))(i + j) = (7/2)(i + j) = 3.5.

Correct Answer: C — 5

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Q. What is the radius of a circle if the area is 154 square units? (2023)

A. 7 units
B. 14 units
C. 11 units
D. 10 units

Solution

Area = πr²; 154 = 22/7 * r²; r² = 154 * 7/22 = 49; r = 7 units.

Correct Answer: A — 7 units

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Q. What is the radius of a circle if the circumference is 31.4 m? (2020)

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

Solution

Circumference = 2πr; 31.4 = 2πr; r = 31.4/(2π) ≈ 5 m.

Correct Answer: A — 5 m

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Mathematics

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