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. Calculate the term containing x^3 in the expansion of (2x + 5)^6. (2000)

A. 1500
B. 1800
C. 2000
D. 2500

Solution

The term containing x^3 is C(6,3) * (2)^3 * (5)^(6-3) = 20 * 8 * 125 = 20000.

Correct Answer: B — 1800

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Q. Calculate the term containing x^3 in the expansion of (x + 2)^7.

A. 56
B. 84
C. 112
D. 128

Solution

The term containing x^3 is C(7,3) * (2)^4 = 35 * 16 = 560.

Correct Answer: B — 84

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Q. Calculate the term independent of x in the expansion of (2x - 3)^5.

A. -243
B. 0
C. 243
D. 81

Solution

The term independent of x is C(5,5) * (2x)^0 * (-3)^5 = 1 * 1 * (-243) = -243.

Correct Answer: A — -243

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Q. Calculate the term independent of x in the expansion of (2x^2 - 3x + 4)^3.

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

Solution

The term independent of x occurs when the powers of x cancel out. The term is C(3,0)(4)^3 + C(3,1)(-3x)(4)^2 + C(3,2)(-3x)^2(4) + C(3,3)(-3x)^3 = 64 - 36 + 0 + 0 = 28.

Correct Answer: B — 24

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Q. Calculate the term independent of x in the expansion of (2x^2 - 3x + 4)^5.

A. 80
B. 120
C. 160
D. 200

Solution

The term independent of x occurs when the powers of x cancel out. This can be calculated using the binomial expansion.

Correct Answer: C — 160

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Q. Calculate the term independent of x in the expansion of (2x^2 - 3x)^4.

A. -81
B. 108
C. -108
D. 81

Solution

The term independent of x occurs when 2k = 4, k = 2. The term is C(4,2) * (2x^2)^2 * (-3x)^2 = 6 * 4 * 9 = -216.

Correct Answer: C — -108

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Q. Calculate the term independent of x in the expansion of (x/2 - 3)^6.

A. 729
B. 729/64
C. 729/32
D. 729/16

Solution

The term independent of x occurs when k = 3, which gives C(6,3) * (x/2)^3 * (-3)^3 = 20 * (1/8) * (-27) = -67.5.

Correct Answer: B — 729/64

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Q. Calculate the term independent of x in the expansion of (x/2 - 3)^8.

A. -3
B. -8
C. 0
D. 256

Solution

The term independent of x occurs when k = 4. Thus, the term is C(8,4) * (x/2)^4 * (-3)^4 = 70 * (1/16) * 81 = 256.

Correct Answer: D — 256

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Q. Calculate the term independent of x in the expansion of (x^2 - 3x + 2)^4.

A. 8
B. 12
C. 16
D. 20

Solution

The term independent of x occurs when the powers of x cancel out. The term is C(4,2) * (2)^2 * (-3)^2 = 6 * 4 * 9 = 216.

Correct Answer: B — 12

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Q. Calculate the value of (1 + 3)^5 using the binomial theorem.

A. 81
B. 243
C. 125
D. 256

Solution

(1 + 3)^5 = 4^5 = 1024.

Correct Answer: B — 243

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Q. Determine the angle between the lines y = 2x + 3 and y = -1/2x + 1.

A. 90 degrees
B. 60 degrees
C. 45 degrees
D. 30 degrees

Solution

The slopes are m1 = 2 and m2 = -1/2. The angle θ = tan^(-1) |(m1 - m2)/(1 + m1*m2)| = tan^(-1)(5/4) which is approximately 60 degrees.

Correct Answer: B — 60 degrees

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Q. Determine the coefficient of x^4 in the expansion of (2x - 3)^6.

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

Solution

The coefficient of x^4 is given by 6C4 * (2)^4 * (-3)^2 = 15 * 16 * 9 = 2160.

Correct Answer: B — 720

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Q. Determine the coordinates of the centroid of the triangle with vertices A(0, 0, 0), B(0, 4, 0), and C(3, 0, 0). (2021)

A. (1, 1.33, 0)
B. (1, 2, 0)
C. (0, 1.33, 0)
D. (0, 2, 0)

Solution

Centroid = ((0+0+3)/3, (0+4+0)/3, (0+0+0)/3) = (1, 1.33, 0).

Correct Answer: B — (1, 2, 0)

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Q. Determine the coordinates of the centroid of the triangle with vertices A(0, 0, 0), B(4, 0, 0), C(0, 3, 0). (2023)

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

Solution

Centroid G = ((0+4+0)/3, (0+0+3)/3, (0+0+0)/3) = (4/3, 1, 0).

Correct Answer: B — (2, 1, 0)

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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 coordinates of the centroid of the triangle with vertices A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9). (2021)

A. (4, 5, 6)
B. (3, 4, 5)
C. (5, 6, 7)
D. (6, 7, 8)

Solution

Centroid G = ((1+4+7)/3, (2+5+8)/3, (3+6+9)/3) = (4, 5, 6).

Correct Answer: B — (3, 4, 5)

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Q. Determine the coordinates of the foot of the perpendicular from the point (1, 2, 3) to the plane x + 2y + 3z = 14. (2023)

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

Solution

Using the formula for the foot of the perpendicular, we find the coordinates to be (1, 2, 4).

Correct Answer: B — (1, 2, 4)

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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^3 - 3x is increasing. (2021)

A. (-∞, -1)
B. (-1, 1)
C. (1, ∞)
D. (-∞, 1)

Solution

f'(x) = 3x^2 - 3. Setting f'(x) = 0 gives x = -1, 1. f'(x) > 0 for x > 1.

Correct Answer: C — (1, ∞)

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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 limit: lim (x -> 0) (tan(5x)/x) (2022)

A. 0
B. 1
C. 5
D. Undefined

Solution

Using the standard limit lim (x -> 0) (tan(kx)/x) = k, we have k = 5. Thus, lim (x -> 0) (tan(5x)/x) = 5.

Correct Answer: C — 5

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Q. Determine the limit: lim (x -> 1) (x^3 - 1)/(x - 1) (2020)

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

Solution

Factoring gives (x - 1)(x^2 + x + 1)/(x - 1). For x ≠ 1, this simplifies to x^2 + x + 1. Evaluating at x = 1 gives 3.

Correct Answer: C — 3

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Q. Determine the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2021)

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

Solution

Factoring gives (x - 1)(x^3 + x^2 + x + 1)/(x - 1). For x ≠ 1, this simplifies to x^3 + x^2 + x + 1. Evaluating at x = 1 gives 4.

Correct Answer: D — Undefined

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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 maxima of f(x) = x^4 - 8x^2 + 16. (2021)

A. (0, 16)
B. (2, 12)
C. (4, 0)
D. (1, 9)

Solution

Find f'(x) = 4x^3 - 16x. Setting f'(x) = 0 gives x = 0, ±2. f(2) = 12 is a local maximum.

Correct Answer: B — (2, 12)

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Mathematics

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