Q. Find the real part of the complex number z = 2e^(iπ/3).
Solution
The real part is 2 * cos(π/3) = 2 * 1/2 = 1.
Correct Answer: B — 2
Learn More →
Q. Find the real part of the complex number z = 3 + 4i.
Solution
The real part of z is 3.
Correct Answer: A — 3
Learn More →
Q. Find the real part of the complex number z = 4 + 3i.
Solution
The real part of z = 4 + 3i is 4.
Correct Answer: A — 4
Learn More →
Q. Find the real part of the complex number z = 4(cos(π/3) + i sin(π/3)).
Solution
The real part is 4 * cos(π/3) = 4 * 1/2 = 2.
Correct Answer: A — 2
Learn More →
Q. Find the real part of the complex number z = 5 - 2i.
Solution
The real part of z = 5 - 2i is 5.
Correct Answer: A — 5
Learn More →
Q. Find the roots of the equation 3x² - 12x + 12 = 0. (2021)
Solution
Dividing by 3 gives x² - 4x + 4 = 0, which factors to (x - 2)² = 0, hence the root is 2.
Correct Answer: A — 2
Learn More →
Q. Find the roots of the equation 4x² - 12x + 9 = 0. (2023)
Solution
This is a perfect square: (2x - 3)² = 0, hence the root is x = 1.5.
Correct Answer: B — 2
Learn More →
Q. Find the roots of the equation x^2 + 4x + 4 = 0.
Solution
The equation factors to (x + 2)^2 = 0, giving a double root x = -2.
Correct Answer: A — -2
Learn More →
Q. Find the roots of the equation x^2 + 5x + 6 = 0.
-
A.
-2, -3
-
B.
-1, -6
-
C.
-3, -2
-
D.
0, -6
Solution
The roots are x = -2 and x = -3.
Correct Answer: C — -3, -2
Learn More →
Q. Find the roots of the equation x² + 2x - 8 = 0. (2022)
-
A.
-4 and 2
-
B.
4 and -2
-
C.
2 and -4
-
D.
0 and 8
Solution
Factoring gives (x + 4)(x - 2) = 0, hence the roots are 4 and -2.
Correct Answer: B — 4 and -2
Learn More →
Q. Find the roots of the quadratic equation x^2 + 4x + 4 = 0.
-
A.
{-2}
-
B.
{2, -2}
-
C.
{-4, 0}
-
D.
{0, 4}
Solution
The equation factors to (x + 2)(x + 2) = 0, giving a double root x = -2.
Correct Answer: A — {-2}
Learn More →
Q. Find the scalar product of A = (1, 2, 3) and B = (4, 5, 6).
Solution
A · B = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.
Correct Answer: B — 30
Learn More →
Q. Find the scalar product of A = 2i + 3j + k and B = i + 2j + 3k. (2020)
Solution
A · B = (2)(1) + (3)(2) + (1)(3) = 2 + 6 + 3 = 11
Correct Answer: A — 14
Learn More →
Q. Find the scalar product of the vectors (3, -2, 5) and (1, 4, -1).
Solution
Scalar product = 3*1 + (-2)*4 + 5*(-1) = 3 - 8 - 5 = -10.
Correct Answer: A — -1
Learn More →
Q. Find the scalar product of the vectors (4, 5) and (1, 2).
Solution
Scalar product = 4*1 + 5*2 = 4 + 10 = 14.
Correct Answer: A — 14
Learn More →
Q. Find the scalar product of the vectors (7, 8, 9) and (0, 1, 2).
Solution
Scalar product = 7*0 + 8*1 + 9*2 = 0 + 8 + 18 = 26.
Correct Answer: A — 26
Learn More →
Q. Find the scalar product of the vectors A = (2, 3) and B = (4, -1).
Solution
A · B = 2*4 + 3*(-1) = 8 - 3 = 5.
Correct Answer: C — 10
Learn More →
Q. Find the scalar product of the vectors A = 5i + 12j and B = 3i - 4j.
Solution
A · B = (5)(3) + (12)(-4) = 15 - 48 = -33.
Correct Answer: A — -33
Learn More →
Q. Find the scalar product of the vectors G = (2, -3, 1) and H = (4, 0, -2).
Solution
G · H = 2*4 + (-3)*0 + 1*(-2) = 8 + 0 - 2 = 6.
Correct Answer: A — -2
Learn More →
Q. Find the scalar product of the vectors G = (5, -3, 2) and H = (1, 1, 1).
Solution
G · H = 5*1 + (-3)*1 + 2*1 = 5 - 3 + 2 = 4.
Correct Answer: D — 3
Learn More →
Q. Find the scalar projection of vector A = (3, 4) onto vector B = (1, 0).
Solution
Scalar projection = (A · B) / |B| = (3*1 + 4*0) / 1 = 3.
Correct Answer: A — 3
Learn More →
Q. Find the scalar triple product of vectors A = (1, 2, 3), B = (4, 5, 6), and C = (7, 8, 9).
Solution
Scalar triple product = A · (B × C) = 0, as vectors are coplanar.
Correct Answer: A — 0
Learn More →
Q. Find the second derivative of f(x) = 4x^4 - 2x^3 + x. (2019)
-
A.
48x^2 - 12x + 1
-
B.
48x^3 - 6
-
C.
12x^2 - 6
-
D.
12x^3 - 6x
Solution
First derivative f'(x) = 16x^3 - 6x^2 + 1. Second derivative f''(x) = 48x^2 - 12x.
Correct Answer: A — 48x^2 - 12x + 1
Learn More →
Q. Find the second derivative of f(x) = e^x at x = 0.
Solution
f''(x) = e^x, thus f''(0) = e^0 = 1.
Correct Answer: B — 1
Learn More →
Q. Find the second derivative of f(x) = ln(x^2 + 1).
-
A.
-2/(x^2 + 1)^2
-
B.
2/(x^2 + 1)^2
-
C.
0
-
D.
-1/(x^2 + 1)
Solution
First derivative f'(x) = (2x)/(x^2 + 1). Second derivative f''(x) = (2(x^2 + 1) - 4x^2)/(x^2 + 1)^2 = -2/(x^2 + 1)^2.
Correct Answer: A — -2/(x^2 + 1)^2
Learn More →
Q. Find the second derivative of f(x) = x^3 - 3x^2 + 4. (2020)
-
A.
6x - 6
-
B.
6x + 6
-
C.
3x^2 - 6
-
D.
3x^2 + 6
Solution
First derivative f'(x) = 3x^2 - 6x; second derivative f''(x) = 6x - 6.
Correct Answer: A — 6x - 6
Learn More →
Q. Find the second derivative of f(x) = x^3 - 6x^2 + 9x.
Solution
f'(x) = 3x^2 - 12x + 9; f''(x) = 6x - 12. At x = 2, f''(2) = 6(2) - 12 = 0.
Correct Answer: A — 6
Learn More →
Q. Find the second derivative of f(x) = x^4 - 4x^3 + 6x^2.
-
A.
12x - 24
-
B.
12x^2 - 24
-
C.
24x - 12
-
D.
24x^2 - 12
Solution
First derivative f'(x) = 4x^3 - 12x^2 + 12. Second derivative f''(x) = 12x^2 - 24.
Correct Answer: A — 12x - 24
Learn More →
Q. Find the second derivative of the function f(x) = x^3 - 3x^2 + 4. (2020)
-
A.
6x - 6
-
B.
6x + 6
-
C.
3x^2 - 6
-
D.
3x^2 + 6
Solution
First derivative f'(x) = 3x^2 - 6x; Second derivative f''(x) = 6x - 6.
Correct Answer: A — 6x - 6
Learn More →
Q. Find the slope of the line passing through the points (2, 3) and (4, 7).
Solution
The slope m is given by (y2 - y1) / (x2 - x1) = (7 - 3) / (4 - 2) = 4 / 2 = 2.
Correct Answer: A — 2
Learn More →
Showing 3451 to 3480 of 21651 (722 Pages)
