Q. Is the function f(x) = x^2 - 4x + 4 differentiable at x = 2?
-
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
-
C.
Only at x = 0
-
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?
-
A.
Yes
-
B.
No
-
C.
Only from the left
-
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
-
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?
-
A.
Yes
-
B.
No
-
C.
Only left continuous
-
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?
-
A.
Yes
-
B.
No
-
C.
Depends on x
-
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?
-
A.
Yes
-
B.
No
-
C.
Only left continuous
-
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| continuous at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only left continuous
-
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. Is the function f(x) = |x|/x continuous at x = 0?
-
A.
Yes
-
B.
No
-
C.
Depends on direction
-
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. J and K invest $10000 and $15000 respectively. If they make a profit of $5000, what is J's share?
-
A.
$2000
-
B.
$1500
-
C.
$2500
-
D.
$3000
Solution
Total investment = 10000 + 15000 = 25000. J's share = (10000/25000) * 5000 = $2000.
Correct Answer:
A
— $2000
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Q. J and K invest $6000 and $9000 respectively. If they earn a profit of $7500, how much does K receive?
-
A.
$3000
-
B.
$4500
-
C.
$4000
-
D.
$3500
Solution
Total investment = 6000 + 9000 = 15000. K's share = (9000/15000) * 7500 = $4500.
Correct Answer:
B
— $4500
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Q. J and K invest $7000 and $3000 respectively. If the profit is $5000, what is K's share?
-
A.
$1500
-
B.
$2000
-
C.
$2500
-
D.
$3000
Solution
Total investment = 7000 + 3000 = 10000. K's share = (3000/10000) * 5000 = $1500.
Correct Answer:
B
— $2000
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Q. J and K invest $7000 and $3000 respectively. If they earn a profit of $5000, how much does K receive?
-
A.
$1500
-
B.
$2000
-
C.
$2500
-
D.
$3000
Solution
Total investment = 7000 + 3000 = 10000. K's share = (3000/10000) * 5000 = $1500.
Correct Answer:
B
— $2000
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Q. J and K invest in a business with J investing $20,000 and K $30,000. If the profit is $10,000, how much does J receive?
-
A.
$4,000
-
B.
$4,500
-
C.
$5,000
-
D.
$6,000
Solution
J's share = (20,000 / (20,000 + 30,000)) * 10,000 = (20,000 / 50,000) * 10,000 = $4,000.
Correct Answer:
C
— $5,000
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Q. J and K invest in a business with J investing $6000 and K $9000. If the profit is $9000, what is J's share?
-
A.
$3000
-
B.
$4000
-
C.
$5000
-
D.
$6000
Solution
Total investment = 6000 + 9000 = 15000. J's share = (6000/15000) * 9000 = $3600.
Correct Answer:
B
— $4000
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Q. John starts from point A and walks 30 meters East, then 40 meters North. How far is he from point A? (2023)
-
A.
50 meters
-
B.
70 meters
-
C.
30 meters
-
D.
40 meters
Solution
Using the Pythagorean theorem, the distance from point A is √(30^2 + 40^2) = 50 meters.
Correct Answer:
A
— 50 meters
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Q. K and L invest in a business with K investing $15,000 and L investing $25,000. If they make a profit of $40,000, how much does L get?
-
A.
$15,000
-
B.
$20,000
-
C.
$25,000
-
D.
$30,000
Solution
Total investment = 15000 + 25000 = 40000. L's share = (25000 / 40000) * 40000 = $25,000.
Correct Answer:
D
— $30,000
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Q. Keyboard is to Type as Mouse is to?
-
A.
Click
-
B.
Scroll
-
C.
Drag
-
D.
Select
Solution
You type with a keyboard, just as you click with a mouse.
Correct Answer:
A
— Click
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Q. L and M invest in a business in the ratio 5:3. If L's profit is $1,250, what is M's profit?
-
A.
$750
-
B.
$500
-
C.
$600
-
D.
$800
Solution
If L's profit is $1,250 and the ratio is 5:3, then M's profit is (3/5) * 1,250 = $750.
Correct Answer:
A
— $750
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Q. L and M invest in a business with L investing $6000 and M investing $9000. If the profit is $4500, how much does M earn?
-
A.
$2000
-
B.
$3000
-
C.
$2500
-
D.
$1500
Solution
Total investment = 6000 + 9000 = 15000. M's share = (9000/15000) * 4500 = $2700, rounded to $3000.
Correct Answer:
B
— $3000
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Q. L, M, and N invest in a business in the ratio 2:3:5. If the total profit is $10000, how much does M receive?
-
A.
$3000
-
B.
$4000
-
C.
$5000
-
D.
$6000
Solution
Total parts = 2 + 3 + 5 = 10. M's share = (3/10) * 10000 = $3000.
Correct Answer:
B
— $4000
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Q. L, M, and N invest in a business with amounts of $8000, $6000, and $4000 respectively. If the profit is $3000, what is M's share?
-
A.
$1200
-
B.
$900
-
C.
$600
-
D.
$1500
Solution
Total investment = 8000 + 6000 + 4000 = 18000. M's share = (6000/18000) * 3000 = $1000.
Correct Answer:
A
— $1200
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Q. L, M, and N invest in a business with investments of $8000, $6000, and $4000 respectively. If the profit is $6000, how much does M receive?
-
A.
$2400
-
B.
$1800
-
C.
$2000
-
D.
$3000
Solution
Total investment = 8000 + 6000 + 4000 = 18000. M's share = (6000/18000) * 6000 = $2000.
Correct Answer:
A
— $2400
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Q. L, M, and N invest in a business with L investing $10000, M $15000, and N $5000. If the total profit is $12000, how much does M receive?
-
A.
$6000
-
B.
$8000
-
C.
$4000
-
D.
$5000
Solution
Total investment = 10000 + 15000 + 5000 = 30000. M's share = (15000/30000) * 12000 = $6000.
Correct Answer:
B
— $8000
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Q. Lenz's law states that the direction of induced current is such that it opposes what?
-
A.
The change in magnetic flux
-
B.
The flow of electric current
-
C.
The resistance in the circuit
-
D.
The applied voltage
Solution
Lenz's law states that the direction of induced current will oppose the change in magnetic flux that produced it.
Correct Answer:
A
— The change in magnetic flux
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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?
-
A.
Yes
-
B.
No
-
C.
Only reflexive
-
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. M and N invest in a business in the ratio 3:5. If N's profit is $32,000, what is the total profit?
-
A.
$48,000
-
B.
$56,000
-
C.
$64,000
-
D.
$72,000
Solution
N's share = 5 parts. Total parts = 3 + 5 = 8. Total profit = (8/5) * 32000 = $51,200.
Correct Answer:
C
— $64,000
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Q. N and O invest $15,000 and $25,000 respectively. If the total profit is $8,000, how much does O receive?
-
A.
$3,200
-
B.
$4,000
-
C.
$4,800
-
D.
$5,000
Solution
O's share = (25,000 / (15,000 + 25,000)) * 8,000 = (25,000 / 40,000) * 8,000 = $5,000.
Correct Answer:
B
— $4,000
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