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. Determine the coordinates of the centroid of the triangle with vertices A(0, 0, 0), B(6, 0, 0), and C(0, 8, 0). (2023)

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

Solution

Centroid = ((0+6+0)/3, (0+0+8)/3, (0+0+0)/3) = (2, 2.67, 0).

Correct Answer: A — (2, 2, 0)

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Q. Determine the critical points of f(x) = 3x^4 - 8x^3 + 6. (2021)

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

Solution

f'(x) = 12x^3 - 24x^2. Setting f'(x) = 0 gives x = 0, 2. Check f(1) = 1.

Correct Answer: B — (1, 1)

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Q. Determine the critical points of f(x) = e^x - 2x. (2021)

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

Solution

f'(x) = e^x - 2. Setting f'(x) = 0 gives e^x = 2, so x = ln(2).

Correct Answer: B — 1

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Q. Determine the distance between the points (2, 3) and (5, 7). (2020)

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

Solution

Using the distance formula, d = √((5 - 2)² + (7 - 3)²) = √(9 + 16) = √25 = 5.

Correct Answer: A — 5

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Q. Determine the distance from the point (3, 4) to the line 2x + 3y - 12 = 0.

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

Solution

Using the formula for distance from a point to a line, d = |Ax1 + By1 + C| / sqrt(A^2 + B^2), we find d = |2(3) + 3(4) - 12| / sqrt(2^2 + 3^2) = 3.

Correct Answer: B — 3

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Q. Determine the intervals where f(x) = -x^2 + 4x is concave up. (2023)

A. (-∞, 0)
B. (0, 2)
C. (2, ∞)
D. (0, 4)

Solution

f''(x) = -2, which is always negative, indicating concave down everywhere.

Correct Answer: C — (2, ∞)

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Q. Determine the intervals where f(x) = x^4 - 4x^3 has increasing behavior. (2023)

A. (-∞, 0)
B. (0, 2)
C. (2, ∞)
D. (0, 4)

Solution

f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3). f'(x) > 0 for x in (0, 3).

Correct Answer: B — (0, 2)

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Q. Determine the intervals where f(x) = x^4 - 4x^3 has local minima. (2020)

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

Solution

f'(x) = 4x^3 - 12x^2. Setting f'(x) = 0 gives x = 0, 3. Testing intervals shows local minima at (0, 2).

Correct Answer: A — (0, 2)

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Q. Determine the local maxima of f(x) = -x^3 + 3x^2 + 1. (2021)

A. (0, 1)
B. (1, 3)
C. (2, 5)
D. (3, 4)

Solution

f'(x) = -3x^2 + 6x. Setting f'(x) = 0 gives x = 0 or x = 2. f(2) = 5 is a local maximum.

Correct Answer: B — (1, 3)

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Q. Determine the local minima of f(x) = x^4 - 4x^2. (2021)

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

Solution

f'(x) = 4x^3 - 8x. Setting f'(x) = 0 gives x = 0, ±2. f(0) = 0.

Correct Answer: B — 0

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Q. Determine the maximum area of a triangle with a base of 10 units and height as a function of x. (2020)

A. 25
B. 50
C. 30
D. 40

Solution

Area = 1/2 * base * height = 5h. Max area occurs when h is maximized, thus Area = 50 when h = 10.

Correct Answer: B — 50

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Q. Determine the maximum height of the function f(x) = -x^2 + 6x + 5. (2020) 2020

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

Solution

The vertex occurs at x = 3. f(3) = -3^2 + 6*3 + 5 = 8.

Correct Answer: A — 8

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Q. Determine the maximum height of the projectile modeled by h(t) = -16t^2 + 64t + 80. (2020)

A. 80
B. 64
C. 48
D. 96

Solution

The maximum height occurs at t = -b/(2a) = 64/(2*16) = 2. h(2) = -16(2^2) + 64(2) + 80 = 80.

Correct Answer: A — 80

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Q. Determine the maximum value of f(x) = -x^2 + 6x - 8. (2022)

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

Solution

The maximum occurs at x = 3. f(3) = -3^2 + 6(3) - 8 = 6.

Correct Answer: C — 6

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Q. Determine the minimum value of f(x) = x^2 - 4x + 5. (2021)

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

Solution

The vertex form gives the minimum at x = 2. f(2) = 2^2 - 4(2) + 5 = 1.

Correct Answer: A — 1

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Q. Determine the point of inflection for f(x) = x^4 - 4x^3 + 6. (2023)

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

Solution

f''(x) = 12x^2 - 24x. Setting f''(x) = 0 gives x(12x - 24) = 0, so x = 0 or x = 2. Check f(1) = 3.

Correct Answer: A — (1, 3)

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Q. Determine the product of the roots of the equation x² + 6x + 9 = 0. (2021)

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

Solution

The product of the roots is c/a = 9/1 = 9.

Correct Answer: A — 9

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Q. Determine the roots of the equation x² + 2x - 8 = 0. (2023)

A. -4 and 2
B. 4 and -2
C. 2 and -4
D. 0 and 8

Solution

Factoring gives (x + 4)(x - 2) = 0, hence the roots are -4 and 2.

Correct Answer: A — -4 and 2

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Q. Evaluate the integral ∫ (3x^2 + 2x) dx. (2020)

A. x^3 + x^2 + C
B. x^3 + x^2 + 2C
C. x^3 + x^2 + 1
D. x^3 + 2x + C

Solution

The integral is (3/3)x^3 + (2/2)x^2 + C = x^3 + x^2 + C.

Correct Answer: A — x^3 + x^2 + C

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Q. Evaluate the integral ∫(3x^2 + 2)dx. (2022)

A. x^3 + 2x + C
B. x^3 + 2x^2 + C
C. x^3 + 2x^3 + C
D. 3x^3 + 2x + C

Solution

Integrating term by term, ∫3x^2dx = x^3 and ∫2dx = 2x. Thus, ∫(3x^2 + 2)dx = x^3 + 2x + C.

Correct Answer: A — x^3 + 2x + C

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Q. Evaluate the limit: lim (x -> 0) (tan(x)/x) (2023)

A. 0
B. 1
C. ∞
D. Undefined

Solution

Using the limit property lim (x -> 0) (tan(x)/x) = 1, we find that the limit is 1.

Correct Answer: B — 1

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Q. Evaluate ∫ (2x + 3) dx. (2022)

A. x^2 + 3x + C
B. x^2 + 3 + C
C. x^2 + 3x + 1
D. 2x^2 + 3 + C

Solution

The integral is (2/2)x^2 + 3x + C = x^2 + 3x + C.

Correct Answer: A — x^2 + 3x + C

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Q. Evaluate ∫ (4x^3 - 2x) dx. (2019)

A. x^4 - x^2 + C
B. x^4 - x^2 + 2C
C. x^4 - x + C
D. 4x^4 - 2x^2 + C

Solution

The integral is (4/4)x^4 - (2/2)x^2 + C = x^4 - x^2 + C.

Correct Answer: A — x^4 - x^2 + C

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Q. Evaluate ∫ (5 - 3x) dx. (2022)

A. 5x - (3/2)x^2 + C
B. 5x - (3/3)x^2 + C
C. 5x - (3/4)x^2 + C
D. 5x - (3/5)x^2 + C

Solution

The integral is 5x - (3/2)x^2 + C.

Correct Answer: A — 5x - (3/2)x^2 + C

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Q. Evaluate ∫(2x^2 + 3x + 1)dx. (2021)

A. (2/3)x^3 + (3/2)x^2 + x + C
B. (2/3)x^3 + (3/2)x + C
C. (2/3)x^3 + (3/2)x^2 + C
D. (2/3)x^3 + 3x + C

Solution

Integrating term by term: ∫2x^2dx = (2/3)x^3, ∫3xdx = (3/2)x^2, and ∫1dx = x. Thus, ∫(2x^2 + 3x + 1)dx = (2/3)x^3 + (3/2)x^2 + x + C.

Correct Answer: A — (2/3)x^3 + (3/2)x^2 + x + C

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Q. Evaluate ∫(5x^4)dx. (2020)

A. (5/5)x^5 + C
B. (1/5)x^5 + C
C. (5/4)x^4 + C
D. (1/4)x^4 + C

Solution

The integral of 5x^4 is (5/5)x^5 + C = x^5 + C.

Correct Answer: A — (5/5)x^5 + C

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Q. Find the area of the triangle formed by the points (0, 0), (4, 0), and (0, 3). (2022) 2022

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

Solution

Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.

Correct Answer: A — 6

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Q. Find the area of the triangle formed by the points A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9). (2022)

A. 0
B. 6
C. 12
D. 3

Solution

The points are collinear, hence the area = 0.

Correct Answer: A — 0

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Q. Find the coefficient of x^2 in the expansion of (2x + 3)^6.

A. 540
B. 720
C. 810
D. 960

Solution

The coefficient of x^2 is given by 6C2 * (2)^2 * (3)^4 = 15 * 4 * 81 = 4860.

Correct Answer: A — 540

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Q. Find the coefficient of x^2 in the expansion of (x + 4)^5. (2023)

A. 80
B. 100
C. 120
D. 160

Solution

The coefficient of x^2 is C(5,2)(4)^3 = 10 * 64 = 640.

Correct Answer: A — 80

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Mathematics

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