Q. If the lengths of the semi-major and semi-minor axes of an ellipse are 5 and 3 respectively, what is the distance between the foci?
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Solution
The distance between the foci is given by 2c, where c = √(a^2 - b^2). Here, c = √(5^2 - 3^2) = √16 = 4, so the distance is 2c = 8.
Correct Answer: A — 4
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Q. If the lengths of the sides of a rectangle are 4 cm and 6 cm, what is its perimeter? (2022)
A.
20 cm
B.
24 cm
C.
10 cm
D.
12 cm
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Solution
Perimeter = 2 * (length + width) = 2 * (4 + 6) = 20 cm
Correct Answer: A — 20 cm
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Q. If the lengths of the sides of a triangle are 3 cm, 4 cm, and 5 cm, what type of triangle is it? (2023)
A.
Acute
B.
Obtuse
C.
Right
D.
Equilateral
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Solution
A triangle with sides 3 cm, 4 cm, and 5 cm satisfies the Pythagorean theorem (3² + 4² = 5²), thus it is a right triangle.
Correct Answer: C — Right
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Q. If the lengths of the sides of a triangle are 7 cm, 24 cm, and 25 cm, is it a right triangle? (2023)
A.
Yes
B.
No
C.
Cannot be determined
D.
Only if angles are known
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Solution
Using Pythagoras theorem, 7² + 24² = 49 + 576 = 625 = 25², so it is a right triangle.
Correct Answer: A — Yes
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Q. If the lengths of two sides of a triangle are 8 cm and 15 cm, what is the maximum possible length of the third side? (2021)
A.
22 cm
B.
23 cm
C.
24 cm
D.
25 cm
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Solution
The maximum length of the third side is less than the sum of the other two sides, so it can be 22 cm.
Correct Answer: A — 22 cm
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Q. If the lever arm is doubled while keeping the force constant, how does the torque change?
A.
It doubles
B.
It triples
C.
It remains the same
D.
It halves
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Solution
Torque is directly proportional to the lever arm; if the lever arm is doubled, the torque also doubles.
Correct Answer: A — It doubles
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Q. If the line 2x + 3y = 6 intersects the x-axis, what is the point of intersection?
A.
(3, 0)
B.
(0, 2)
C.
(0, 3)
D.
(2, 0)
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Solution
Setting y = 0 gives 2x = 6, thus x = 3. The point of intersection is (3, 0).
Correct Answer: A — (3, 0)
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Q. If the line 2x + 3y = 6 intersects the x-axis, what is the x-coordinate of the intersection point?
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Solution
Setting y = 0 gives 2x = 6, thus x = 3. The x-coordinate of the intersection point is 3.
Correct Answer: B — 2
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Q. If the line 3x + 4y = 12 intersects the x-axis, what is the point of intersection?
A.
(4, 0)
B.
(0, 3)
C.
(0, 4)
D.
(3, 0)
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Solution
Set y = 0 in the equation: 3x = 12 => x = 4. The point is (4, 0).
Correct Answer: A — (4, 0)
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Q. If the line 3x + 4y = 12 intersects the x-axis, what is the x-coordinate of the intersection point?
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Solution
Set y = 0 in the equation: 3x = 12 => x = 4.
Correct Answer: A — 4
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Q. If the line 3x + 4y = 12 intersects the y-axis, what is the point of intersection? (2021)
A.
(0, 3)
B.
(0, 4)
C.
(0, 2)
D.
(0, 1)
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Solution
Setting x = 0 gives 4y = 12, thus y = 3. The point of intersection is (0, 3).
Correct Answer: A — (0, 3)
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Q. If the line 3x + 4y = 12 is transformed to slope-intercept form, what is the slope?
A.
-3/4
B.
3/4
C.
4/3
D.
-4/3
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Solution
Rearranging gives y = -3/4x + 3. The slope is -3/4.
Correct Answer: A — -3/4
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Q. If the line 3x + 4y = 12 is transformed to slope-intercept form, what is the y-intercept?
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Solution
Rearranging gives y = -3/4x + 3. The y-intercept is 3.
Correct Answer: B — 4
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Q. If the line 3x - 4y + 12 = 0 intersects the y-axis, what is the point of intersection?
A.
(0, 3)
B.
(0, -3)
C.
(0, 4)
D.
(0, -4)
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Solution
Setting x = 0 gives -4y + 12 = 0, thus y = -3. The point of intersection is (0, -3).
Correct Answer: D — (0, -4)
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Q. If the line 3x - 4y + 12 = 0 is parallel to another line, what is the slope of the parallel line?
A.
3/4
B.
-3/4
C.
4/3
D.
-4/3
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Solution
Rearranging gives y = (3/4)x + 3. The slope of the line is 3/4, so the slope of the parallel line is -3/4.
Correct Answer: B — -3/4
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Q. If the line 3x - 4y + 12 = 0 is parallel to another line, what is the slope of that line?
A.
3/4
B.
-3/4
C.
4/3
D.
-4/3
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Solution
The slope of the line is given by -A/B = -3/-4 = 3/4. Parallel lines have the same slope.
Correct Answer: B — -3/4
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Q. If the line 3x - 4y + 12 = 0 is parallel to which of the following lines?
A.
6x - 8y + 24 = 0
B.
3x + 4y - 12 = 0
C.
x + 2y - 5 = 0
D.
2x - 3y + 6 = 0
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Solution
Parallel lines have the same slope. The slope of the given line is 3/4, which is the same as the slope of 6x - 8y + 24 = 0.
Correct Answer: A — 6x - 8y + 24 = 0
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Q. If the line 3x - 4y + 12 = 0 is transformed to slope-intercept form, what is the slope?
A.
3/4
B.
-3/4
C.
4/3
D.
-4/3
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Solution
Rearranging gives y = (3/4)x + 3, so the slope is -3/4.
Correct Answer: B — -3/4
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Q. If the line 5x + 12y = 60 is transformed to slope-intercept form, what is the slope?
A.
-5/12
B.
5/12
C.
12/5
D.
-12/5
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Solution
Rearranging gives y = -5/12 x + 5, so the slope is -5/12.
Correct Answer: A — -5/12
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Q. If the line 5x + 2y = 10 intersects the y-axis, what is the y-coordinate of the intersection point?
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Solution
Setting x = 0 in the equation gives 2y = 10, hence y = 5.
Correct Answer: B — 2
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Q. If the line 5x - 2y + 10 = 0 is reflected about the x-axis, what is the new equation?
A.
5x + 2y + 10 = 0
B.
5x - 2y - 10 = 0
C.
5x + 2y - 10 = 0
D.
5x - 2y + 10 = 0
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Solution
Reflecting about the x-axis changes the sign of y-coefficient: 5x + 2y + 10 = 0.
Correct Answer: A — 5x + 2y + 10 = 0
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Q. If the line 7x + 2y = 14 is transformed to slope-intercept form, what is the slope?
A.
-7/2
B.
7/2
C.
2/7
D.
-2/7
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Solution
Rearranging to y = -7/2x + 7 shows that the slope is -7/2.
Correct Answer: A — -7/2
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Q. If the line graph indicates a consistent decline in sales for Product C over several months, what strategy might the company consider?
A.
Increase marketing efforts
B.
Reduce prices
C.
Introduce new features
D.
All of the above
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Solution
To counteract the decline, the company might consider all these strategies.
Correct Answer: D — All of the above
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Q. If the line graph indicates a steady decline in the sales of Product C from January to June, what can be inferred about consumer preferences?
A.
Consumers prefer Product A
B.
Product C is being discontinued
C.
Product C has a seasonal demand
D.
None of the above
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Solution
The decline suggests that consumers are shifting their preferences towards Product A.
Correct Answer: A — Consumers prefer Product A
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Q. If the line graph indicates a steady decline in the sales of Product C from January to June, what can be inferred about the market trend for that product?
A.
Increased competition
B.
Seasonal demand
C.
Improved marketing
D.
Consumer preference shift
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Solution
The steady decline suggests increased competition affecting Product C's sales.
Correct Answer: A — Increased competition
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Q. If the line graph indicates that Product A's sales are consistently higher than Product C's, what can be concluded?
A.
Product A is more popular
B.
Product C is being discontinued
C.
Sales data is misleading
D.
Product A has better marketing
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Solution
Consistently higher sales suggest that Product A is more popular among consumers.
Correct Answer: A — Product A is more popular
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Q. If the line graph indicates that Product A's sales were consistently higher than Product B's throughout the year, what can be inferred?
A.
Product A is more popular
B.
Product B is being phased out
C.
Sales strategies are ineffective
D.
None of the above
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Solution
Consistently higher sales suggest that Product A is more popular among consumers.
Correct Answer: A — Product A is more popular
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Q. If the line graph shows a downward trend for Product C but an upward trend for Product A, what could be a potential market implication?
A.
Product C is losing market share
B.
Product A is gaining popularity
C.
Both products are affected by external factors
D.
All of the above
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Solution
The trends suggest that Product C is losing market share while Product A is gaining popularity.
Correct Answer: D — All of the above
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Q. If the line graph shows a downward trend for Product C over several months, what strategy might the company consider?
A.
Increase advertising
B.
Reduce prices
C.
Enhance product features
D.
All of the above
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Solution
All these strategies could help reverse the downward trend in sales.
Correct Answer: D — All of the above
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Q. If the line graph shows a plateau in sales for Product A after a period of growth, what could be a possible reason?
A.
Increased competition
B.
Market saturation
C.
Seasonal fluctuations
D.
All of the above
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Solution
A plateau can result from various factors including increased competition and market saturation.
Correct Answer: D — All of the above
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