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

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

Solution

The coefficient of x^4 is C(6,4) * (3)^4 * (-2)^2 = 15 * 81 * 4 = 4860.

Correct Answer: A — 540

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Q. Calculate the coefficient of x^4 in the expansion of (x + 1/2)^6. (2021)

A. 15/8
B. 45/8
C. 5/8
D. 1/8

Solution

The coefficient of x^4 is C(6,4)(1/2)^2 = 15 * 1/4 = 15/4.

Correct Answer: B — 45/8

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Q. Calculate the coefficient of x^4 in the expansion of (x + 3)^6. (2021)

A. 54
B. 81
C. 108
D. 729

Solution

The coefficient of x^4 is C(6,4)(3)^2 = 15 * 9 = 135.

Correct Answer: C — 108

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Q. Calculate the derivative of f(x) = 5x^5. (2016)

A. 25x^4
B. 5x^4
C. 20x^4
D. 10x^4

Solution

The derivative f'(x) = d/dx(5x^5) = 25x^4.

Correct Answer: A — 25x^4

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Q. Calculate the derivative of f(x) = ln(x^2 + 1).

A. 2x/(x^2 + 1)
B. 1/(x^2 + 1)
C. 2/(x^2 + 1)
D. x/(x^2 + 1)

Solution

Using the chain rule, f'(x) = d/dx(ln(x^2 + 1)) = (2x)/(x^2 + 1).

Correct Answer: A — 2x/(x^2 + 1)

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Q. Calculate the derivative of f(x) = x^2 * e^x. (2023) 2023

A. e^x(2x + 1)
B. e^x(2x - 1)
C. 2xe^x
D. x^2e^x

Solution

Using the product rule, f'(x) = d/dx(x^2 * e^x) = e^x(2x + 1).

Correct Answer: A — e^x(2x + 1)

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Q. Calculate the determinant of D = [[2, 3, 1], [1, 0, 2], [4, 1, 0]]. (2020)

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

Solution

Det(D) = 2(0*0 - 2*1) - 3(1*0 - 2*4) + 1(1*1 - 0*4) = 2(0 - 2) - 3(0 - 8) + 1(1) = -4 + 24 + 1 = 21.

Correct Answer: A — -10

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Q. Calculate the determinant of D = [[3, 2, 1], [1, 0, 2], [0, 1, 3]]. (2023)

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

Solution

Det(D) = 3(0*3 - 2*1) - 2(1*3 - 0*2) + 1(1*1 - 0*0) = 3(0 - 2) - 2(3) + 1(1) = -6 - 6 + 1 = -11.

Correct Answer: A — 1

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Q. Calculate the determinant of D = [[3, 2, 1], [1, 0, 2], [2, 1, 3]]. (2020)

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

Solution

Det(D) = 3(0*3 - 2*1) - 2(1*3 - 2*2) + 1(1*1 - 0*2) = 3(0 - 2) - 2(3 - 4) + 1(1) = -6 + 2 + 1 = -3.

Correct Answer: A — 1

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Q. Calculate the determinant of D = [[4, 2], [1, 3]]. (2020)

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

Solution

Determinant of D = (4*3) - (2*1) = 12 - 2 = 10.

Correct Answer: A — 10

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Q. Calculate the determinant of D = [[4, 2], [3, 1]]. (2020)

A. -2
B. 2
C. 10
D. 12

Solution

Determinant of D = (4*1) - (2*3) = 4 - 6 = -2.

Correct Answer: A — -2

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Q. Calculate the determinant of F = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]. (2023)

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

Solution

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

Correct Answer: A — -14

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Q. Calculate the determinant of G = [[1, 1, 1], [1, 2, 3], [1, 3, 6]]. (2022)

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

Solution

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

Correct Answer: A — 0

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Q. Calculate the determinant of G = [[1, 2, 1], [0, 1, 0], [2, 3, 1]]. (2023)

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

Solution

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

Correct Answer: A — -1

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Q. Calculate the determinant of G = [[1, 2, 1], [0, 1, 4], [1, 0, 0]]. (2021)

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

Solution

Using the determinant formula for 3x3 matrices, det(G) = 1(1*0 - 4*0) - 2(0*0 - 4*1) + 1(0*0 - 1*1) = 0 + 8 - 1 = 7.

Correct Answer: A — -2

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Q. Calculate the determinant of H = [[1, 2, 1], [0, 1, 3], [1, 0, 1]]. (2023)

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

Solution

Det(H) = 1(1*1 - 3*0) - 2(0*1 - 3*1) + 1(0*0 - 1*1) = 1(1) - 2(-3) + 1(-1) = 1 + 6 - 1 = 6.

Correct Answer: A — -1

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Q. Calculate the determinant of H = [[1, 2, 1], [0, 1, 3], [2, 1, 0]]. (2020)

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

Solution

Det(H) = 1(1*0 - 3*1) - 2(0*0 - 3*2) + 1(0*1 - 1*2) = 1(0 - 3) - 2(0 - 6) + 1(0 - 2) = -3 + 12 - 2 = 7.

Correct Answer: A — -5

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Q. Calculate the determinant of H = [[5, 4], [2, 3]]. (2021)

A. 7
B. 8
C. 9
D. 10

Solution

Det(H) = (5*3) - (4*2) = 15 - 8 = 7.

Correct Answer: B — 8

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Q. Calculate the determinant of J = [[1, 2, 1], [0, 1, 2], [1, 0, 1]]. (2023)

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

Solution

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

Correct Answer: C — 2

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Q. Calculate the determinant of the matrix \( C = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \). (2020)

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

Solution

The determinant is \( 5*8 - 6*7 = 40 - 42 = -2 \).

Correct Answer: A — -2

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Q. Calculate the determinant of the matrix \( H = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{pmatrix} \). (2020)

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

Solution

The determinant of an upper triangular matrix is the product of its diagonal elements: \( 1*1*1 = 1 \).

Correct Answer: A — 1

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Q. Calculate the distance from the point P(1, 2, 3) to the origin O(0, 0, 0). (2023)

A. 3
B. √14
C. √6
D. √9

Solution

Distance = √[(1-0)² + (2-0)² + (3-0)²] = √[1 + 4 + 9] = √14.

Correct Answer: B — √14

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Q. Calculate the limit: lim (x -> ∞) (5x^2 + 3)/(2x^2 + 1) (2023)

A. 0
B. 5/2
C. 2/5
D. ∞

Solution

Dividing the numerator and denominator by x^2, we get lim (x -> ∞) (5 + 3/x^2)/(2 + 1/x^2) = 5/2.

Correct Answer: B — 5/2

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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 (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. 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(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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Mathematics

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