Q. In triangle ABC, if the lengths of the sides are a = 8, b = 15, and c = 17, what is the value of cos A?
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A.
0.5
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B.
0.6
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C.
0.8
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D.
0.9
Solution
Using the cosine rule, cos A = (b² + c² - a²) / (2bc) = (15² + 17² - 8²) / (2 * 15 * 17) = 0.8.
Correct Answer: C — 0.8
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Q. In triangle ABC, if the lengths of the sides are in the ratio 3:4:5, what type of triangle is it?
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A.
Acute
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B.
Obtuse
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C.
Right
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D.
Equilateral
Solution
Since the sides are in the ratio of a Pythagorean triplet (3, 4, 5), triangle ABC is a right triangle.
Correct Answer: C — Right
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Q. In triangle ABC, if the sides are in the ratio 3:4:5, what is the nature of the triangle?
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A.
Equilateral
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B.
Isosceles
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C.
Right
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D.
Scalene
Solution
The sides satisfy the Pythagorean theorem, hence it is a right triangle.
Correct Answer: C — Right
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Q. In triangle ABC, if the sides are in the ratio 3:4:5, what type of triangle is it?
-
A.
Acute
-
B.
Obtuse
-
C.
Right
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D.
Equilateral
Solution
A triangle with sides in the ratio 3:4:5 is a right triangle, as it satisfies the Pythagorean theorem.
Correct Answer: C — Right
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Q. In triangle MNO, if angle M = 45 degrees and angle N = 45 degrees, what is angle O?
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A.
90 degrees
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B.
45 degrees
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C.
60 degrees
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D.
30 degrees
Solution
Angle O = 180 - (angle M + angle N) = 180 - (45 + 45) = 90 degrees.
Correct Answer: A — 90 degrees
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Q. In triangle PQR, if PQ = 10 cm, QR = 24 cm, and PR = 26 cm, what is the area of the triangle?
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A.
120 cm²
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B.
120√3 cm²
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C.
240 cm²
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D.
48 cm²
Solution
Using Heron's formula, s = (10 + 24 + 26)/2 = 30. Area = √(30(30-10)(30-24)(30-26)) = √(30*20*6*4) = 120 cm².
Correct Answer: A — 120 cm²
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Q. In triangle XYZ, if XY = 8 cm, YZ = 15 cm, and XZ = 17 cm, is it a right triangle?
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A.
Yes
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B.
No
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C.
Cannot be determined
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D.
Only if XY is the hypotenuse
Solution
Since 8^2 + 15^2 = 17^2, triangle XYZ is a right triangle.
Correct Answer: A — Yes
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Q. Is the function f(x) = x^2 - 2x + 1 differentiable at x = 1?
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A.
Yes
-
B.
No
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C.
Only from the left
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D.
Only from the right
Solution
f(x) is a polynomial function, which is differentiable everywhere, including at x = 1.
Correct Answer: A — Yes
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Q. Is the function f(x) = x^2 - 4x + 4 differentiable at x = 2?
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A.
Yes
-
B.
No
-
C.
Only from the left
-
D.
Only from the right
Solution
The function is a polynomial and is differentiable everywhere, hence yes.
Correct Answer: A — Yes
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Q. Is the function f(x) = x^2 - 4x + 4 differentiable everywhere?
-
A.
Yes
-
B.
No
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C.
Only at x = 0
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D.
Only at x = 2
Solution
This is a polynomial function, which is differentiable everywhere on its domain.
Correct Answer: A — Yes
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Q. Is the function f(x) = x^2 sin(1/x) differentiable at x = 0?
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A.
Yes
-
B.
No
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C.
Only from the left
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D.
Only from the right
Solution
Using the limit definition, f'(0) = lim (h -> 0) [(h^2 sin(1/h) - 0)/h] = 0. Thus, f(x) is differentiable at x = 0.
Correct Answer: A — Yes
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Q. Is the function f(x) = x^3 - 3x + 2 differentiable at x = 1?
-
A.
Yes
-
B.
No
-
C.
Only left differentiable
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D.
Only right differentiable
Solution
The function is a polynomial and hence differentiable everywhere, including at x = 1.
Correct Answer: A — Yes
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Q. Is the function f(x) = { e^x, x < 0; ln(x + 1), x >= 0 } continuous at x = 0?
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A.
Yes
-
B.
No
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C.
Only left continuous
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D.
Only right continuous
Solution
Both limits as x approaches 0 from the left and right are equal to 1, hence f(x) is continuous at x = 0.
Correct Answer: A — Yes
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Q. Is the function f(x) = { sin(x), x < 0; x^2, x >= 0 } continuous at x = 0?
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A.
Yes
-
B.
No
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C.
Depends on x
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D.
Not defined
Solution
Both limits as x approaches 0 from the left and right are equal to 0, hence f(x) is continuous at x = 0.
Correct Answer: A — Yes
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Q. Is the function f(x) = { x^3, x < 1; 2x + 1, x >= 1 } continuous at x = 1?
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A.
Yes
-
B.
No
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C.
Only left continuous
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D.
Only right continuous
Solution
Both limits as x approaches 1 from the left and right are equal to 2, hence f(x) is continuous at x = 1.
Correct Answer: A — Yes
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Q. Is the function f(x) = |x|/x continuous at x = 0?
-
A.
Yes
-
B.
No
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C.
Depends on direction
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D.
None of the above
Solution
The left limit is -1 and the right limit is 1, which are not equal. Therefore, f(x) is not continuous at x = 0.
Correct Answer: B — No
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Q. Let A = {1, 2, 3, 4} and R be the relation defined by R = {(a, b) | a < b}. How many ordered pairs are in R?
Solution
The pairs are (1,2), (1,3), (1,4), (2,3), (2,4), (3,4). Thus, there are 6 ordered pairs.
Correct Answer: B — 6
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Q. Let A = {1, 2, 3, 4} and R be the relation defined by R = {(x, y) | x < y}. How many ordered pairs are in R?
Solution
The ordered pairs are (1,2), (1,3), (1,4), (2,3), (2,4), (3,4). Thus, there are 6 ordered pairs.
Correct Answer: B — 6
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Q. Let R be a relation on the set of natural numbers defined by R = {(m, n) | m divides n}. Is R a partial order?
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A.
Yes
-
B.
No
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C.
Only reflexive
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D.
Only transitive
Solution
R is reflexive, antisymmetric, and transitive, thus it is a partial order.
Correct Answer: A — Yes
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Q. Solve for x: 3(x - 2) = 2(x + 1).
Solution
Expanding both sides gives 3x - 6 = 2x + 2. Rearranging gives x = 8.
Correct Answer: B — 0
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Q. Solve for x: log_3(x + 1) - log_3(x - 1) = 1.
Solution
Using properties of logarithms, log_3((x + 1)/(x - 1)) = 1 => (x + 1)/(x - 1) = 3 => x + 1 = 3(x - 1) => x = 2.
Correct Answer: A — 2
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Q. Solve for x: log_3(x) = 2.
Solution
log_3(x) = 2 implies x = 3^2 = 9.
Correct Answer: B — 9
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Q. Solve for x: log_5(x + 1) - log_5(x - 1) = 1.
Solution
Using properties of logarithms: log_5((x + 1)/(x - 1)) = 1 => (x + 1)/(x - 1) = 5 => x + 1 = 5(x - 1) => 4x = 6 => x = 2.
Correct Answer: A — 2
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Q. Solve for x: log_5(x) = 2.
Solution
log_5(x) = 2 implies x = 5^2 = 25.
Correct Answer: C — 25
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Q. Solve the differential equation dy/dx + 2y = 4.
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A.
y = 2 - Ce^(-2x)
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B.
y = 2 + Ce^(-2x)
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C.
y = 4 - Ce^(-2x)
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D.
y = 4 + Ce^(2x)
Solution
This is a linear first-order differential equation. The integrating factor is e^(2x). Solving gives y = 2 - Ce^(-2x).
Correct Answer: A — y = 2 - Ce^(-2x)
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Q. Solve the differential equation dy/dx = 3x^2.
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A.
y = x^3 + C
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B.
y = 3x^3 + C
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C.
y = x^2 + C
-
D.
y = 3x + C
Solution
Integrating both sides gives y = x^3 + C.
Correct Answer: A — y = x^3 + C
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Q. Solve the differential equation dy/dx = x^2 + y^2.
-
A.
y = x^3/3 + C
-
B.
y = x^2 + C
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C.
y = x^2 + x + C
-
D.
y = Cx^2 + C
Solution
This is a non-linear differential equation. The solution can be found using substitution methods.
Correct Answer: A — y = x^3/3 + C
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Q. Solve the differential equation y' = 3y + 6.
-
A.
y = Ce^(3x) - 2
-
B.
y = Ce^(3x) + 2
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C.
y = 2e^(3x)
-
D.
y = 3e^(3x) + 2
Solution
Using the integrating factor method, we find y = Ce^(3x) + 2.
Correct Answer: B — y = Ce^(3x) + 2
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Q. Solve the differential equation y'' + 4y = 0.
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A.
y = C1 cos(2x) + C2 sin(2x)
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B.
y = C1 e^(2x) + C2 e^(-2x)
-
C.
y = C1 cos(x) + C2 sin(x)
-
D.
y = C1 e^(x) + C2 e^(-x)
Solution
The characteristic equation is r^2 + 4 = 0, giving complex roots. The solution is y = C1 cos(2x) + C2 sin(2x).
Correct Answer: A — y = C1 cos(2x) + C2 sin(2x)
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Q. Solve the differential equation y'' - 5y' + 6y = 0.
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A.
y = C1 e^(2x) + C2 e^(3x)
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B.
y = C1 e^(3x) + C2 e^(2x)
-
C.
y = C1 e^(x) + C2 e^(2x)
-
D.
y = C1 e^(2x) + C2 e^(x)
Solution
The characteristic equation is r^2 - 5r + 6 = 0, which factors to (r - 2)(r - 3) = 0, giving the solution y = C1 e^(2x) + C2 e^(3x).
Correct Answer: B — y = C1 e^(3x) + C2 e^(2x)
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