Q. If the odds in favor of an event are 3:2, what is the probability of the event occurring?
-
A.
0.6
-
B.
0.4
-
C.
0.5
-
D.
0.3
Solution
Probability = Odds in favor / (Total odds) = 3 / (3 + 2) = 3/5 = 0.6.
Correct Answer: A — 0.6
Learn More →
Q. If the pair of lines represented by ax^2 + 2hxy + by^2 = 0 are perpendicular, then:
-
A.
a + b = 0
-
B.
a - b = 0
-
C.
h = 0
-
D.
a = b
Solution
For the lines to be perpendicular, the condition a + b = 0 must hold.
Correct Answer: A — a + b = 0
Learn More →
Q. If the pair of lines represented by the equation ax^2 + 2hxy + by^2 = 0 are perpendicular, then:
-
A.
a + b = 0
-
B.
a - b = 0
-
C.
h = 0
-
D.
a = b
Solution
For the lines to be perpendicular, the condition a*b + h^2 = 0 must hold.
Correct Answer: A — a + b = 0
Learn More →
Q. If the parabola y = ax^2 + bx + c has its vertex at (1, -2), what is the value of a if it passes through the point (0, 0)?
Solution
Using the vertex form y = a(x - 1)^2 - 2 and substituting (0, 0), we get 0 = a(0 - 1)^2 - 2 => 2 = a => a = 2.
Correct Answer: B — 2
Learn More →
Q. If the parabola y^2 = 16x opens to the right, what is the value of p?
Solution
In the equation y^2 = 4px, we have 4p = 16, thus p = 4.
Correct Answer: B — 4
Learn More →
Q. If the parabola y^2 = 20x opens to the right, what is the value of p?
Solution
In the equation y^2 = 4px, we have 4p = 20, thus p = 20/4 = 5.
Correct Answer: A — 5
Learn More →
Q. If the points (1, 2), (3, 4), and (5, 6) are collinear, what is the slope of the line?
-
A.
1
-
B.
2
-
C.
0
-
D.
undefined
Solution
Slope = (4-2)/(3-1) = 1, and it remains the same for other points.
Correct Answer: A — 1
Learn More →
Q. If the points A(1, 2), B(3, 4), and C(5, 6) are collinear, what is the area of triangle ABC?
Solution
Area = 1/2 | x1(y2-y3) + x2(y3-y1) + x3(y1-y2) | = 0.
Correct Answer: A — 0
Learn More →
Q. If the polynomial P(x) = x^3 - 6x^2 + 11x - 6 has a root at x = 1, what is P(2)?
Solution
P(2) = 2^3 - 6(2^2) + 11(2) - 6 = 8 - 24 + 22 - 6 = 0.
Correct Answer: D — 3
Learn More →
Q. If the position vector of a point is (5, 12), what is its distance from the origin?
Solution
Distance = √(5^2 + 12^2) = √(25 + 144) = √169 = 13
Correct Answer: A — 13
Learn More →
Q. If the position vector of a point is given by r = (2t, 3t, 4t), what is the velocity vector?
-
A.
(2, 3, 4)
-
B.
(4, 6, 8)
-
C.
(2t, 3t, 4t)
-
D.
(0, 0, 0)
Solution
Velocity vector = dr/dt = (2, 3, 4)
Correct Answer: A — (2, 3, 4)
Learn More →
Q. If the position vector of a point P is (2, 3, 4), what is the distance from the origin to point P?
Solution
Distance = √(2^2 + 3^2 + 4^2) = √(4 + 9 + 16) = √29 ≈ 5.385.
Correct Answer: B — 6
Learn More →
Q. If the position vector of a point P is (x, y, z) and the vector a = (1, 2, 3), what is the projection of P onto a?
-
A.
(1, 2, 3)
-
B.
(2, 4, 6)
-
C.
(0, 0, 0)
-
D.
(x, y, z)
Solution
Projection of P onto a = ((P · a) / |a|^2) * a.
Correct Answer: D — (x, y, z)
Learn More →
Q. If the position vector of a point P is given by r = (2t, 3t, 4t), find the coordinates of P when t = 1.
-
A.
(2, 3, 4)
-
B.
(1, 1, 1)
-
C.
(0, 0, 0)
-
D.
(2, 4, 6)
Solution
Substituting t = 1, r = (2*1, 3*1, 4*1) = (2, 3, 4).
Correct Answer: A — (2, 3, 4)
Learn More →
Q. If the position vector of point P is (3, -2) and Q is (1, 4), what is the vector PQ?
-
A.
(-2, 6)
-
B.
(2, -6)
-
C.
(4, -6)
-
D.
(6, 2)
Solution
Vector PQ = Q - P = (1 - 3, 4 - (-2)) = (-2, 6).
Correct Answer: A — (-2, 6)
Learn More →
Q. If the position vector of point P is (3, 4) and Q is (1, 2), what is the vector PQ?
-
A.
(2, 2)
-
B.
(4, 6)
-
C.
(2, 4)
-
D.
(1, 1)
Solution
Vector PQ = Q - P = (1 - 3, 2 - 4) = (-2, -2).
Correct Answer: A — (2, 2)
Learn More →
Q. If the probability of an event A is 0.3, what is the probability of the event not occurring?
-
A.
0.7
-
B.
0.3
-
C.
0.5
-
D.
0.2
Solution
Probability of not A = 1 - P(A) = 1 - 0.3 = 0.7.
Correct Answer: A — 0.7
Learn More →
Q. If the probability of an event A is 0.7, what is the probability of the event not occurring?
-
A.
0.3
-
B.
0.7
-
C.
0.5
-
D.
0.1
Solution
Probability of not A = 1 - P(A) = 1 - 0.7 = 0.3.
Correct Answer: A — 0.3
Learn More →
Q. If the probability of an event is 0.7, what is the probability of its complement?
-
A.
0.3
-
B.
0.7
-
C.
1
-
D.
0.5
Solution
Probability of complement = 1 - P(event) = 1 - 0.7 = 0.3.
Correct Answer: A — 0.3
Learn More →
Q. If the probability of an event is 0.7, what is the probability of the event not occurring?
-
A.
0.3
-
B.
0.7
-
C.
0.5
-
D.
0.1
Solution
Probability of not occurring = 1 - P(event) = 1 - 0.7 = 0.3.
Correct Answer: A — 0.3
Learn More →
Q. If the probability of event A is 0.4 and event B is 0.5, what is the probability of both events occurring if they are independent?
-
A.
0.2
-
B.
0.4
-
C.
0.5
-
D.
0.6
Solution
P(A and B) = P(A) * P(B) = 0.4 * 0.5 = 0.2.
Correct Answer: A — 0.2
Learn More →
Q. If the quadratic equation ax^2 + bx + c = 0 has roots 3 and -2, what is the value of a?
Solution
Using the fact that the product of the roots is c/a and the sum is -b/a, we can set a = 1, b = -1, c = -6.
Correct Answer: A — 1
Learn More →
Q. If the quadratic equation x^2 + 2px + p^2 - 4 = 0 has real roots, what is the condition on p?
-
A.
p > 2
-
B.
p < 2
-
C.
p = 2
-
D.
p >= 2
Solution
The discriminant must be non-negative: (2p)^2 - 4(1)(p^2 - 4) >= 0 => 4p^2 - 4p^2 + 16 >= 0, which is always true. Thus, p can be any real number.
Correct Answer: D — p >= 2
Learn More →
Q. If the quadratic equation x^2 + 2px + p^2 - 4 = 0 has roots that are equal, what is the value of p?
Solution
Setting the discriminant to zero: (2p)^2 - 4(1)(p^2 - 4) = 0 leads to p = ±2.
Correct Answer: C — -2
Learn More →
Q. If the quadratic equation x^2 + 2x + k = 0 has equal roots, what is the value of k?
Solution
For equal roots, the discriminant must be zero: 2^2 - 4*1*k = 0, leading to k = 1.
Correct Answer: C — -1
Learn More →
Q. If the quadratic equation x^2 + 2x + k = 0 has no real roots, what is the condition for k?
-
A.
k < 0
-
B.
k > 0
-
C.
k >= 0
-
D.
k <= 0
Solution
For no real roots, the discriminant must be less than zero: 2^2 - 4*1*k < 0 => 4 - 4k < 0 => k > 1.
Correct Answer: A — k < 0
Learn More →
Q. If the quadratic equation x^2 + 2x + k = 0 has no real roots, what is the condition on k?
-
A.
k < 0
-
B.
k > 0
-
C.
k >= 0
-
D.
k <= 0
Solution
For no real roots, the discriminant must be less than zero: 2^2 - 4*1*k < 0, hence k > 1.
Correct Answer: A — k < 0
Learn More →
Q. If the quadratic equation x^2 + 2x + k = 0 has roots that are equal, what is the value of k?
Solution
For equal roots, the discriminant must be zero: 2^2 - 4*1*k = 0 leads to k = -1.
Correct Answer: D — -2
Learn More →
Q. If the quadratic equation x^2 + 4x + c = 0 has one root equal to -2, what is the value of c?
Solution
If one root is -2, then substituting x = -2 gives: (-2)^2 + 4(-2) + c = 0 => 4 - 8 + c = 0 => c = 4.
Correct Answer: A — 0
Learn More →
Q. If the quadratic equation x^2 + 4x + k = 0 has roots -2 and -2, what is the value of k?
Solution
Using the formula for roots, k = (-2)^2 - 4*(-2) = 4 + 8 = 12.
Correct Answer: B — 4
Learn More →
Showing 1501 to 1530 of 2847 (95 Pages)
