Q. In triangle XYZ, if angle X = 45 degrees and angle Y = 45 degrees, what is the ratio of the lengths of sides opposite to these angles? (2021)
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A.
1:1
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B.
1:√2
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C.
√2:1
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D.
2:1
Solution
In an isosceles triangle with angles 45 degrees, the sides opposite these angles are equal, hence the ratio is 1:1.
Correct Answer: A — 1:1
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Q. Is the function f(x) = 1/(x-1) continuous at x = 1?
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A.
Yes
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B.
No
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C.
Only left continuous
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D.
Only right continuous
Solution
The function f(x) = 1/(x-1) is not defined at x = 1, hence it is discontinuous at that point.
Correct Answer: B — No
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Q. Is the function f(x) = sqrt(x) continuous at x = 0?
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A.
Yes
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B.
No
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C.
Only from the right
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D.
Only from the left
Solution
The function f(x) = sqrt(x) is continuous at x = 0 as it is defined and the limit exists.
Correct Answer: A — Yes
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Q. Is the function f(x) = |x| continuous at x = 0?
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A.
Yes
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B.
No
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C.
Only left continuous
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D.
Only right continuous
Solution
The function f(x) = |x| is continuous at x = 0 because the left limit, right limit, and f(0) all equal 0.
Correct Answer: A — Yes
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Q. Solve for x: 3x - 7 = 2x + 5. (2021)
Solution
Subtract 2x from both sides: x - 7 = 5. Then add 7: x = 12.
Correct Answer: A — 12
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Q. Solve the differential equation dy/dx = 2y.
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A.
y = Ce^(2x)
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B.
y = 2Ce^x
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C.
y = Ce^(x/2)
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D.
y = 2x + C
Solution
This is a separable equation. Separating variables and integrating gives ln|y| = 2x + C, hence y = Ce^(2x).
Correct Answer: A — y = Ce^(2x)
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Q. Solve the differential equation dy/dx = 5 - 2y.
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A.
y = 5/2 + Ce^(-2x)
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B.
y = 5/2 - Ce^(-2x)
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C.
y = 2.5 + Ce^(2x)
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D.
y = 2.5 - Ce^(2x)
Solution
Rearranging gives dy/(5 - 2y) = dx. Integrating both sides leads to y = 5/2 + Ce^(-2x).
Correct Answer: A — y = 5/2 + Ce^(-2x)
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Q. Solve the differential equation dy/dx = y^2.
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A.
y = 1/(C - x)
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B.
y = Cx
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C.
y = C + x^2
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D.
y = C - x
Solution
This is a separable equation. Integrating gives y = 1/(C - x), where C is the constant.
Correct Answer: A — y = 1/(C - x)
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Q. The ages of a group of people are: 22, 25, 22, 30, 25, 22, 28. What is the mode of their ages?
Solution
The mode is 22, as it appears 3 times, more than any other age.
Correct Answer: A — 22
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Q. The angle between the vectors A = i + j and B = j + k is:
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A.
45°
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B.
60°
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C.
90°
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D.
30°
Solution
cos(θ) = (A · B) / (|A||B|) = (0) / (√2 * √2) = 0, thus θ = 90°.
Correct Answer: C — 90°
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Q. The average of 10, 20, 30, and x is 25. What is the value of x?
Solution
Mean = (10 + 20 + 30 + x) / 4 = 25. Solving gives x = 50.
Correct Answer: C — 40
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Q. The average of 4, 6, 8, and x is 7. What is the value of x?
Solution
Mean = (4 + 6 + 8 + x) / 4 = 7.\n\n18 + x = 28\n\nx = 28 - 18 = 10.
Correct Answer: C — 10
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Q. The average of five consecutive integers is 12. What is the smallest of these integers?
Solution
Let the integers be x, x+1, x+2, x+3, x+4.\n\nMean = (5x + 10) / 5 = 12\n\n5x + 10 = 60\n\n5x = 50\n\nx = 10.
Correct Answer: B — 11
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Q. The average of five numbers is 20. If one number is excluded, the average of the remaining numbers becomes 18. What is the excluded number?
Solution
Total of five numbers = 5 * 20 = 100. Total of remaining four = 4 * 18 = 72. Excluded number = 100 - 72 = 28.
Correct Answer: A — 22
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Q. The average of three numbers is 15. If one number is increased by 5, what will be the new average?
Solution
Total of three numbers = 3 * 15 = 45.\n\nNew total = 45 + 5 = 50.\n\nNew average = 50 / 3 = 16.67.
Correct Answer: B — 16
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Q. The average of two numbers is 30. If one number is 20, what is the other number?
Solution
Mean = (20 + x) / 2 = 30.\n\n20 + x = 60\n\nx = 60 - 20 = 40.
Correct Answer: A — 40
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Q. The circumference of a circle is 31.4 cm. What is the radius of the circle? (Use π = 3.14) (2019)
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A.
5 cm
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B.
10 cm
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C.
7 cm
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D.
15 cm
Solution
Circumference = 2πr => r = circumference / (2π) = 31.4 / (2 * 3.14) = 5 cm
Correct Answer: A — 5 cm
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Q. The coordinates of the centroid of a triangle with vertices at (1, 2), (3, 4), and (5, 6) are:
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A.
(3, 4)
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B.
(2, 3)
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C.
(4, 5)
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D.
(3, 5)
Solution
Centroid = ((1+3+5)/3, (2+4+6)/3) = (3, 4).
Correct Answer: A — (3, 4)
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Q. The coordinates of the centroid of a triangle with vertices at (2, 3), (4, 5), and (6, 7) are:
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A.
(4, 5)
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B.
(3, 4)
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C.
(5, 6)
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D.
(6, 5)
Solution
Centroid = ((2+4+6)/3, (3+5+7)/3) = (4, 5).
Correct Answer: B — (3, 4)
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Q. The coordinates of the foot of the perpendicular from the point (1, 2) to the line 2x + 3y = 6 are:
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A.
(2, 0)
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B.
(0, 2)
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C.
(1, 1)
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D.
(0, 0)
Solution
Using the formula for foot of perpendicular, we find the coordinates to be (2, 0).
Correct Answer: A — (2, 0)
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Q. The coordinates of the foot of the perpendicular from the point (1, 2) to the line 3x + 4y = 12 are:
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A.
(2, 1)
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B.
(1, 2)
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C.
(0, 3)
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D.
(3, 0)
Solution
Using the formula for foot of perpendicular, we find the coordinates to be (2, 1).
Correct Answer: A — (2, 1)
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Q. The coordinates of the foot of the perpendicular from the point (3, 4) to the line 2x + 3y = 6 are:
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A.
(2, 0)
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B.
(1, 2)
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C.
(0, 2)
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D.
(2, 2)
Solution
Using the formula for foot of perpendicular, we find the coordinates to be (2, 0).
Correct Answer: A — (2, 0)
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Q. The coordinates of the foot of the perpendicular from the point (3, 4) to the line 2x + 3y - 6 = 0 are:
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A.
(2, 0)
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B.
(0, 2)
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C.
(1, 1)
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D.
(2, 2)
Solution
Using the formula for foot of perpendicular, we find the coordinates to be (2, 0).
Correct Answer: A — (2, 0)
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Q. The equation of a line with slope 2 passing through the point (1, 3) is?
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A.
y = 2x + 1
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B.
y = 2x + 2
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C.
y = 2x + 3
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D.
y = 2x - 1
Solution
Using point-slope form: y - 3 = 2(x - 1) => y = 2x + 1.
Correct Answer: C — y = 2x + 3
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Q. The equation of a parabola with vertex at (0, 0) and directrix y = -3 is?
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A.
x^2 = -12y
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B.
y^2 = -12x
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C.
x^2 = 12y
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D.
y^2 = 12x
Solution
The distance from the vertex to the directrix is 3, so the equation is x^2 = -12y.
Correct Answer: A — x^2 = -12y
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Q. The equation of a parabola with vertex at (0, 0) and focus at (0, 3) is?
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A.
x^2 = 12y
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B.
y^2 = 12x
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C.
x^2 = 6y
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D.
y^2 = 6x
Solution
The distance from the vertex to the focus is 3, so the equation is x^2 = 12y.
Correct Answer: A — x^2 = 12y
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Q. The equation x^2 - 2x + k = 0 has roots that are both positive. What is the range of k?
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A.
k < 0
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B.
k > 0
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C.
k > 1
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D.
k < 1
Solution
For both roots to be positive, k must be greater than 1 (from Vieta's formulas).
Correct Answer: C — k > 1
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Q. The equation x^2 - 4x + k = 0 has equal roots when k is equal to:
Solution
For equal roots, the discriminant must be zero: (-4)^2 - 4*1*k = 0 leads to k = 4.
Correct Answer: A — 4
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Q. The equation x^2 - 4x + k = 0 has no real roots if k is:
-
A.
< 4
-
B.
≥ 4
-
C.
≤ 4
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D.
> 4
Solution
The discriminant must be less than zero: (-4)^2 - 4*1*k < 0 leads to k > 4.
Correct Answer: A — < 4
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Q. The equation x^2 - 7x + 10 = 0 has roots that are:
-
A.
1 and 10
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B.
2 and 5
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C.
3 and 4
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D.
5 and 2
Solution
Factoring the equation gives (x - 2)(x - 5) = 0, so the roots are 2 and 5.
Correct Answer: C — 3 and 4
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