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)
Learn More →
Q. Determine the coordinates of the centroid of the triangle with vertices at (0, 0), (6, 0), and (3, 6).
-
A.
(3, 2)
-
B.
(3, 3)
-
C.
(2, 3)
-
D.
(0, 0)
Solution
Centroid = ((0+6+3)/3, (0+0+6)/3) = (3, 2).
Correct Answer:
B
— (3, 3)
Learn More →
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)
Learn More →
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)
Learn More →
Q. Determine the critical points of f(x) = e^x - 2x. (2021)
Solution
f'(x) = e^x - 2. Setting f'(x) = 0 gives e^x = 2, so x = ln(2).
Correct Answer:
B
— 1
Learn More →
Q. Determine the critical points of f(x) = x^3 - 3x + 2.
-
A.
-1, 1
-
B.
0, 2
-
C.
1, -2
-
D.
2, -1
Solution
Setting f'(x) = 3x^2 - 3 = 0 gives x^2 = 1, so critical points are x = -1 and x = 1.
Correct Answer:
A
— -1, 1
Learn More →
Q. Determine the critical points of f(x) = x^3 - 3x^2 + 4.
-
A.
(0, 4)
-
B.
(1, 2)
-
C.
(2, 1)
-
D.
(3, 0)
Solution
f'(x) = 3x^2 - 6x. Setting f'(x) = 0 gives x = 0 and x = 2. Critical points are (0, 4) and (2, 1).
Correct Answer:
B
— (1, 2)
Learn More →
Q. Determine the critical points of f(x) = x^3 - 6x^2 + 9x.
-
A.
x = 0, 3
-
B.
x = 1, 2
-
C.
x = 2, 3
-
D.
x = 1, 3
Solution
Setting f'(x) = 0 gives critical points at x = 0 and x = 3.
Correct Answer:
A
— x = 0, 3
Learn More →
Q. Determine the critical points of f(x) = x^4 - 4x^3 + 6.
-
A.
x = 0, 3
-
B.
x = 1, 2
-
C.
x = 2, 3
-
D.
x = 1, 3
Solution
Setting f'(x) = 0 gives critical points at x = 1 and x = 2.
Correct Answer:
B
— x = 1, 2
Learn More →
Q. Determine the critical points of f(x) = x^4 - 8x^2 + 16.
-
A.
x = 0, ±2
-
B.
x = ±4
-
C.
x = ±1
-
D.
x = 2
Solution
Setting f'(x) = 0 gives critical points at x = 0, ±2.
Correct Answer:
A
— x = 0, ±2
Learn More →
Q. Determine the critical points of f(x) = x^4 - 8x^2.
-
A.
x = 0, ±2
-
B.
x = ±4
-
C.
x = ±1
-
D.
x = 2
Solution
f'(x) = 4x^3 - 16x = 4x(x^2 - 4). Critical points are x = 0, ±2.
Correct Answer:
A
— x = 0, ±2
Learn More →
Q. Determine the critical points of the function f(x) = x^2 - 4x + 4. (2022)
Solution
f'(x) = 2x - 4; Setting f'(x) = 0 gives x = 2 as the critical point.
Correct Answer:
C
— 2
Learn More →
Q. Determine the critical points of the function f(x) = x^3 - 6x^2 + 9x.
-
A.
(0, 0)
-
B.
(1, 4)
-
C.
(2, 0)
-
D.
(3, 0)
Solution
f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives (x - 1)(x - 3) = 0, so critical points are x = 1 and x = 3.
Correct Answer:
D
— (3, 0)
Learn More →
Q. Determine the derivative of f(x) = 1/x.
-
A.
-1/x^2
-
B.
1/x^2
-
C.
1/x
-
D.
-1/x
Solution
Using the power rule, f'(x) = -1/x^2.
Correct Answer:
A
— -1/x^2
Learn More →
Q. Determine 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) = (1/(x^2 + 1)) * (2x) = 2x/(x^2 + 1).
Correct Answer:
A
— 2x/(x^2 + 1)
Learn More →
Q. Determine the derivative of f(x) = x^2 * e^x.
-
A.
e^x * (x^2 + 2x)
-
B.
e^x * (2x + 1)
-
C.
2x * e^x
-
D.
x^2 * e^x
Solution
Using the product rule, f'(x) = d/dx(x^2 * e^x) = e^x * (x^2 + 2x).
Correct Answer:
A
— e^x * (x^2 + 2x)
Learn More →
Q. Determine the derivative of f(x) = x^3 - 4x + 7. (2023)
-
A.
3x^2 - 4
-
B.
3x^2 + 4
-
C.
x^2 - 4
-
D.
3x^2 - 7
Solution
Using the power rule, f'(x) = 3x^2 - 4.
Correct Answer:
A
— 3x^2 - 4
Learn More →
Q. Determine the derivative of f(x) = x^5 - 3x^3 + 2x. (2023)
-
A.
5x^4 - 9x^2 + 2
-
B.
5x^4 - 9x + 2
-
C.
5x^4 - 3x^2 + 2
-
D.
5x^4 - 3x^3
Solution
Using the power rule, f'(x) = 5x^4 - 9x^2 + 2.
Correct Answer:
A
— 5x^4 - 9x^2 + 2
Learn More →
Q. Determine the distance between the points (-1, -1) and (2, 2).
Solution
Using the distance formula: d = √[(2 - (-1))² + (2 - (-1))²] = √[(2 + 1)² + (2 + 1)²] = √[9 + 9] = √18 = 3√2 ≈ 4.24.
Correct Answer:
C
— 5
Learn More →
Q. Determine the distance between the points (0, 0) and (0, 8).
Solution
Using the distance formula: d = √[(0 - 0)² + (8 - 0)²] = √[0 + 64] = √64 = 8.
Correct Answer:
A
— 8
Learn More →
Q. Determine the distance between the points (1, 2) and (4, 6). (2022)
Solution
Using the distance formula: d = √[(4 - 1)² + (6 - 2)²] = √[9 + 16] = √25 = 5.
Correct Answer:
A
— 5
Learn More →
Q. Determine the distance between the points (2, 3) and (2, -1).
Solution
Using the distance formula: d = √[(2 - 2)² + (-1 - 3)²] = √[0 + 16] = √16 = 4.
Correct Answer:
A
— 4
Learn More →
Q. Determine the distance between the points (2, 3) and (5, 7). (2020)
Solution
Using the distance formula, d = √((5 - 2)² + (7 - 3)²) = √(9 + 16) = √25 = 5.
Correct Answer:
A
— 5
Learn More →
Q. Determine the distance from the point (1, 2) to the line 2x + 3y - 6 = 0. (2023)
Solution
Using the formula for distance from a point to a line, the distance is |2(1) + 3(2) - 6| / sqrt(2^2 + 3^2) = 1.
Correct Answer:
B
— 2
Learn More →
Q. Determine the distance from the point (3, 4) to the line 2x + 3y - 12 = 0.
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
Learn More →
Q. Determine the equation of the circle with center (2, -3) and radius 5.
-
A.
(x - 2)² + (y + 3)² = 25
-
B.
(x + 2)² + (y - 3)² = 25
-
C.
(x - 2)² + (y - 3)² = 25
-
D.
(x + 2)² + (y + 3)² = 25
Solution
Equation of circle: (x - h)² + (y - k)² = r² => (x - 2)² + (y + 3)² = 5² = 25.
Correct Answer:
A
— (x - 2)² + (y + 3)² = 25
Learn More →
Q. Determine the equation of the line that passes through the points (0, 0) and (3, 9).
-
A.
y = 3x
-
B.
y = 2x
-
C.
y = 3x + 1
-
D.
y = x + 1
Solution
The slope m = (9 - 0) / (3 - 0) = 3. The equation is y = 3x.
Correct Answer:
A
— y = 3x
Learn More →
Q. Determine the equation of the tangent line to the curve y = x^2 + 2x at the point where x = 1.
-
A.
y = 3x - 2
-
B.
y = 2x + 1
-
C.
y = 2x + 3
-
D.
y = x + 3
Solution
f'(x) = 2x + 2. At x = 1, f'(1) = 4. The point is (1, 4). The tangent line is y - 4 = 4(x - 1) => y = 4x - 4 + 4 => y = 4x - 2.
Correct Answer:
A
— y = 3x - 2
Learn More →
Q. Determine the family of curves represented by the equation x^2 - y^2 = c, where c is a constant.
-
A.
Circles
-
B.
Ellipses
-
C.
Hyperbolas
-
D.
Parabolas
Solution
The equation x^2 - y^2 = c represents a family of hyperbolas with varying values of c.
Correct Answer:
C
— Hyperbolas
Learn More →
Q. Determine the family of curves represented by the equation x^2/a^2 + y^2/b^2 = 1.
-
A.
Circles
-
B.
Ellipses with varying axes
-
C.
Hyperbolas
-
D.
Parabolas
Solution
The equation x^2/a^2 + y^2/b^2 = 1 represents a family of ellipses with varying semi-major and semi-minor axes.
Correct Answer:
B
— Ellipses with varying axes
Learn More →
Showing 3721 to 3750 of 27896 (930 Pages)
