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 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 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, 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 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 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 (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 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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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 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 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 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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Mathematics

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