Q. The pair of straight lines represented by the equation x^2 - 4xy + y^2 = 0 are:
-
A.
Parallel
-
B.
Perpendicular
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C.
Coincident
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D.
Intersecting at a point
Solution
The given equation can be factored as (x - 2y)(x - 2y) = 0, indicating that the lines are perpendicular.
Correct Answer: B — Perpendicular
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Q. The parabola y = -3(x - 2)^2 + 5 opens in which direction?
-
A.
Upwards
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B.
Downwards
-
C.
Left
-
D.
Right
Solution
Since the coefficient of (x - 2)^2 is negative, the parabola opens downwards.
Correct Answer: B — Downwards
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Q. The product of the roots of the equation x^2 + 7x + 10 = 0 is:
Solution
The product of the roots is given by c/a = 10/1 = 10.
Correct Answer: A — 10
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Q. The product of the roots of the equation x^2 - 7x + k = 0 is 10. What is the value of k?
Solution
Using Vieta's formulas, the product of the roots is k = 10. Thus, k = 17.
Correct Answer: B — 17
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Q. The product of two complex numbers z1 = 1 + i and z2 = 2 - i is?
-
A.
3 + i
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B.
3 - i
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C.
2 + 3i
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D.
2 - 3i
Solution
z1 * z2 = (1 + i)(2 - i) = 2 - i + 2i - i^2 = 2 + 1 + i = 3 + i.
Correct Answer: A — 3 + i
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Q. The quadratic equation x^2 + 4x + 4 = 0 has:
-
A.
Two distinct real roots
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B.
One real root
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C.
No real roots
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D.
Infinitely many roots
Solution
The discriminant is 0, indicating one real root (a repeated root).
Correct Answer: B — One real root
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Q. The quadratic equation x^2 + 6x + 9 = 0 has roots that are:
-
A.
Real and equal
-
B.
Real and distinct
-
C.
Complex
-
D.
None of these
Solution
The discriminant is 0, hence the roots are real and equal.
Correct Answer: A — Real and equal
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Q. The quadratic equation x^2 + kx + 16 = 0 has equal roots. What is the value of k?
Solution
For equal roots, the discriminant must be zero: k^2 - 4*1*16 = 0, solving gives k = -8.
Correct Answer: A — -8
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Q. The quadratic equation x^2 + px + q = 0 has roots 3 and -2. What is the value of p?
Solution
Using the sum of roots: p = -(3 + (-2)) = -1.
Correct Answer: B — 5
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Q. The quadratic equation x^2 - 3x + 2 = 0 can be factored as?
-
A.
(x-1)(x-2)
-
B.
(x-2)(x-1)
-
C.
(x+1)(x+2)
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D.
(x-3)(x+2)
Solution
The equation factors to (x-1)(x-2) = 0.
Correct Answer: A — (x-1)(x-2)
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Q. The quadratic equation x^2 - 4x + 4 = 0 has how many distinct real roots?
Solution
The discriminant is 0, indicating one distinct real root.
Correct Answer: B — 1
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Q. The quadratic equation x^2 - 6x + 9 = 0 has how many distinct real roots?
-
A.
0
-
B.
1
-
C.
2
-
D.
Infinite
Solution
The discriminant is 0, indicating that there is exactly one distinct real root.
Correct Answer: B — 1
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Q. The quadratic equation x^2 - 6x + k = 0 has roots that differ by 2. What is the value of k?
Solution
Let the roots be r and r+2. Then, r + (r+2) = 6 and r(r+2) = k. Solving gives k = 10.
Correct Answer: B — 10
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Q. The range of sin^(-1)(x) is:
-
A.
[-π/2, π/2]
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B.
[0, π]
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C.
[-1, 1]
-
D.
[0, 1]
Solution
The range of sin^(-1)(x) is [-π/2, π/2].
Correct Answer: A — [-π/2, π/2]
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Q. The range of the data set 1, 3, 5, 7, 9 is:
Solution
Range = Maximum - Minimum = 9 - 1 = 8.
Correct Answer: A — 8
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Q. The range of the data set {10, 15, 20, 25, 30} is?
Solution
Range = Maximum value - Minimum value = 30 - 10 = 20.
Correct Answer: A — 15
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Q. The range of the function f(x) = |x - 1| is:
-
A.
(-∞, 1)
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B.
[0, ∞)
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C.
(-1, 1)
-
D.
[1, ∞)
Solution
The absolute value function has a minimum value of 0, hence the range is [0, ∞).
Correct Answer: B — [0, ∞)
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Q. The range of the function y = sin^(-1)(x) is:
-
A.
(0, π)
-
B.
[-π/2, π/2]
-
C.
[-1, 1]
-
D.
[0, 1]
Solution
The range of y = sin^(-1)(x) is [-π/2, π/2].
Correct Answer: B — [-π/2, π/2]
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Q. The real part of the complex number z = 4 - 3i is?
Solution
The real part of z = 4 - 3i is 4.
Correct Answer: A — 4
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Q. The roots of the equation 2x^2 - 4x + 1 = 0 are:
-
A.
1
-
B.
2
-
C.
1/2
-
D.
None of these
Solution
Using the quadratic formula, x = [4 ± √(16 - 8)] / 4 = [4 ± 2√2] / 4 = 1 ± √2/2. Hence, the roots are not simple fractions.
Correct Answer: C — 1/2
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Q. The roots of the equation 5x^2 - 20x + 15 = 0 are:
Solution
Using the quadratic formula, the roots are x = [20 ± √(400 - 300)] / 10 = [20 ± 10] / 10 = 3 and 1.
Correct Answer: B — 2
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Q. The roots of the equation x^2 + 2x + 1 = 0 are:
Solution
The equation can be factored as (x + 1)^2 = 0, giving a double root at x = -1.
Correct Answer: A — -1
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Q. The roots of the equation x^2 - 3x + 2 = 0 are:
-
A.
1 and 2
-
B.
2 and 3
-
C.
0 and 1
-
D.
None of these
Solution
Factoring gives (x-1)(x-2) = 0, so the roots are 1 and 2.
Correct Answer: A — 1 and 2
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Q. The scalar product of two unit vectors is 0. What can be said about these vectors?
-
A.
They are parallel
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B.
They are orthogonal
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C.
They are collinear
-
D.
They are equal
Solution
If the scalar product is 0, the vectors are orthogonal.
Correct Answer: B — They are orthogonal
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Q. The scalar product of vectors A = (a, b, c) and B = (1, 2, 3) is 14. If a = 2, find b and c.
-
A.
3, 4
-
B.
4, 3
-
C.
5, 2
-
D.
2, 5
Solution
A · B = 2*1 + b*2 + c*3 = 14. Thus, 2 + 2b + 3c = 14, leading to 2b + 3c = 12.
Correct Answer: B — 4, 3
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Q. The scalar product of vectors A = (a, b, c) and B = (1, 2, 3) is 14. If a = 2, what is the value of b + c?
Solution
A · B = 2*1 + b*2 + c*3 = 14. Thus, 2 + 2b + 3c = 14, leading to 2b + 3c = 12. Solving gives b + c = 6.
Correct Answer: C — 6
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Q. The slope of the line represented by the equation 2x - 3y + 6 = 0 is:
-
A.
2/3
-
B.
-2/3
-
C.
3/2
-
D.
-3/2
Solution
Rearranging gives y = (2/3)x + 2, so slope = 2/3.
Correct Answer: B — -2/3
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Q. The slope of the line represented by the equation 3x - 4y + 12 = 0 is:
-
A.
3/4
-
B.
4/3
-
C.
-3/4
-
D.
-4/3
Solution
Rearranging gives y = (3/4)x + 3. Slope = 3/4.
Correct Answer: C — -3/4
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Q. The slope of the tangent to the curve y = sin(x) at x = π/4 is:
-
A.
1
-
B.
√2/2
-
C.
√3/3
-
D.
√2
Solution
The derivative f'(x) = cos(x). At x = π/4, f'(π/4) = cos(π/4) = √2/2.
Correct Answer: B — √2/2
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Q. The slope of the tangent to the curve y = x^3 - 3x at x = 1 is:
Solution
The derivative f'(x) = 3x^2 - 3. At x = 1, f'(1) = 3(1)^2 - 3 = 0, so the slope is 0.
Correct Answer: B — 1
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