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 integral of cos(2x) dx? (2021)

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

Solution

The integral of cos(2x) is (1/2)sin(2x) + C.

Correct Answer: A — (1/2)sin(2x) + C

Learn More →

Q. What is the integral of e^x with respect to x? (2023)

A. e^x + C
B. xe^x + C
C. e^x/x + C
D. x^2e^x + C

Solution

The integral of e^x is e^x + C.

Correct Answer: A — e^x + C

Learn More →

Q. What is the integral of e^x? (2023)

A. e^x + C
B. xe^x + C
C. e^x/x + C
D. x^2e^x + C

Solution

The integral of e^x is e^x + C.

Correct Answer: A — e^x + C

Learn More →

Q. What is the integral of sin(x) dx? (2021)

A. -cos(x) + C
B. cos(x) + C
C. sin(x) + C
D. -sin(x) + C

Solution

The integral of sin(x) is -cos(x) + C.

Correct Answer: A — -cos(x) + C

Learn More →

Q. What is the integral of tan(x) dx? (2023)

A. -ln
B. cos(x)
C. + C
D. ln
. sin(x)
. + C
. ln
. tan(x)
. + C
. -ln
. sin(x)
. + C

Solution

The integral of tan(x) is -ln|cos(x)| + C.

Correct Answer: A — -ln

Learn More →

Q. What is the integral of tan(x)? (2021)

A. -ln
B. cos(x)
C. + C
D. ln
. sin(x)
. + C
. ln
. tan(x)
. + C
. -ln
. sin(x)
. + C

Solution

The integral of tan(x) is -ln|cos(x)| + C.

Correct Answer: A — -ln

Learn More →

Q. What is the integral of x^2 with respect to x? (2021)

A. (1/3)x^3 + C
B. (1/2)x^2 + C
C. (1/4)x^4 + C
D. (1/5)x^5 + C

Solution

The integral of x^n is (1/(n+1))x^(n+1) + C. Here, n=2, so the integral is (1/3)x^3 + C.

Correct Answer: A — (1/3)x^3 + C

Learn More →

Q. What is the integral of x^3 + 2x? (2021)

A. (1/4)x^4 + x^2 + C
B. (1/3)x^3 + x^2 + C
C. (1/4)x^4 + (1/2)x^2 + C
D. (1/5)x^5 + (1/2)x^2 + C

Solution

Integrating term by term, ∫x^3dx = (1/4)x^4 and ∫2xdx = x^2. Thus, ∫(x^3 + 2x)dx = (1/4)x^4 + (1/2)x^2 + C.

Correct Answer: C — (1/4)x^4 + (1/2)x^2 + C

Learn More →

Q. What is the integral of x^n where n ≠ -1? (2021)

A. (1/n)x^(n+1) + C
B. (1/(n+1))x^n + C
C. (1/(n+1))x^(n+1) + C
D. nx^(n-1) + C

Solution

The integral of x^n is (1/(n+1))x^(n+1) + C, provided n ≠ -1.

Correct Answer: C — (1/(n+1))x^(n+1) + C

Learn More →

Q. What is the integrating factor for the equation dy/dx + (1/x)y = 2?

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

Solution

The integrating factor is e^(∫(1/x)dx) = e^(ln(x)) = x.

Correct Answer: C — e^(ln(x^2))

Learn More →

Q. What is the integrating factor for the equation dy/dx + 2y = 3?

A. e^(2x)
B. e^(-2x)
C. e^(3x)
D. e^(-3x)

Solution

The integrating factor is e^(∫2dx) = e^(2x).

Correct Answer: A — e^(2x)

Learn More →

Q. What is the integrating factor for the equation dy/dx + 2y = 6?

A. e^(2x)
B. e^(-2x)
C. e^(6x)
D. e^(-6x)

Solution

The integrating factor is e^(∫2dx) = e^(2x).

Correct Answer: A — e^(2x)

Learn More →

Q. What is the length of a chord that is 6 cm away from the center of a circle with a radius of 10 cm? (2015)

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

Solution

Using Pythagoras: (radius)² = (distance from center)² + (half chord)²; 10² = 6² + (half chord)²; half chord = 8 cm, so full chord = 16 cm.

Correct Answer: A — 8 cm

Learn More →

Q. What is the length of a chord that is 6 cm from the center of a circle with a radius of 10 cm? (2019)

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

Solution

Using Pythagoras: chord length = 2√(r² - d²) = 2√(10² - 6²) = 2√(100 - 36) = 2√64 = 16 cm.

Correct Answer: A — 8 cm

Learn More →

Q. What is the length of a diameter of a circle with a radius of 7 cm? (2022)

A. 14 cm
B. 21 cm
C. 7 cm
D. 28 cm

Solution

Diameter = 2 × radius; Diameter = 2 × 7 cm = 14 cm.

Correct Answer: A — 14 cm

Learn More →

Q. What is the length of an arc of a circle with a radius of 10 cm and a central angle of 60 degrees? (Use π = 3.14) (2023)

A. 10.47 cm
B. 15.71 cm
C. 20.94 cm
D. 25.13 cm

Solution

Arc length = (θ/360) * 2πr = (60/360) * 2 * 3.14 * 10 = 10.47 cm.

Correct Answer: A — 10.47 cm

Learn More →

Q. What is the length of an arc of a circle with a radius of 10 cm and a central angle of 60 degrees? (2021)

A. 10.47 cm
B. 12.57 cm
C. 15.71 cm
D. 20.94 cm

Solution

Arc length = (θ/360) * 2πr = (60/360) * 2 * 3.14 * 10 = 10.47 cm.

Correct Answer: B — 12.57 cm

Learn More →

Q. What is the length of an arc of a circle with a radius of 10 cm and a central angle of 60 degrees? (2021) 2021

A. 10.47 cm
B. 15.71 cm
C. 20.94 cm
D. 25.13 cm

Solution

Arc length = (θ/360) * 2πr = (60/360) * 2 * 3.14 * 10 = 10.47 cm.

Correct Answer: A — 10.47 cm

Learn More →

Q. What is the length of an arc of a circle with a radius of 4 cm and a central angle of 90 degrees? (2021)

A. 2π cm
B. 4π cm
C. π cm
D. 8 cm

Solution

Arc length = (θ/360) * 2πr = (90/360) * 2π * 4 = 2π cm.

Correct Answer: A — 2π cm

Learn More →

Q. What is the length of an arc of a circle with a radius of 5 cm and a central angle of 60 degrees? (2020)

A. 5.24 cm
B. 3.14 cm
C. 5.00 cm
D. 10.47 cm

Solution

Arc length = (θ/360) * 2πr = (60/360) * 2π(5) = (1/6) * 10π ≈ 5.24 cm.

Correct Answer: A — 5.24 cm

Learn More →

Q. What is the length of an arc of a circle with radius 5 cm and angle 60 degrees? (2020)

A. 5.24 cm
B. 3.14 cm
C. 5.00 cm
D. 6.00 cm

Solution

Arc length = (θ/360) * 2πr = (60/360) * 2 * π * 5 = 5.24 cm.

Correct Answer: A — 5.24 cm

Learn More →

Q. What is the length of the diagonal of a cuboid with dimensions 2, 3, and 6? (2022)

A. √49
B. √45
C. √36
D. √50

Solution

Diagonal = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7.

Correct Answer: A — √49

Learn More →

Q. What is the length of the diagonal of a rectangle with vertices at (0, 0), (0, 3), (4, 0), and (4, 3)? (2022)

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

Solution

Diagonal = √[(4-0)² + (3-0)²] = √[16 + 9] = √25 = 5.

Correct Answer: A — 5

Learn More →

Q. What is the length of the diagonal of a rectangle with vertices at (0, 0), (4, 0), (4, 3), and (0, 3)? (2021)

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

Solution

Length of diagonal = √[(4-0)² + (3-0)²] = √[16 + 9] = √25 = 5.

Correct Answer: A — 5

Learn More →

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

Learn More →

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

Learn More →

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

Learn More →

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

Learn More →

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

Learn More →

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

Learn More →

Showing 541 to 570 of 694 (24 Pages)

Mathematics

On a scale of 0–10, how likely are you to recommend The Soulshift Academy?