Q. The function f(x) = x^3 - 6x^2 + 9x has how many local extrema?
Solution
Finding f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives x = 1 and x = 3. Checking the second derivative shows one local maximum and one local minimum.
Correct Answer:
B
— 1
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Q. The function f(x) = { 1/x, x != 0; 0, x = 0 } is continuous at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only from the right
-
D.
Only from the left
Solution
The limit as x approaches 0 does not equal f(0) = 0, hence it is not continuous at x = 0.
Correct Answer:
B
— No
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Q. The function f(x) = { 1/x, x ≠ 0; 0, x = 0 } is:
-
A.
Continuous at x = 0
-
B.
Not continuous at x = 0
-
C.
Continuous everywhere
-
D.
None of the above
Solution
The function is not continuous at x = 0 since the limit does not equal f(0).
Correct Answer:
B
— Not continuous at x = 0
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Q. The function f(x) = { 2x + 3, x < 1; x^2 + 1, x >= 1 } is continuous at x = ?
Solution
To check continuity at x = 1, we find the left limit (5) and the right limit (2). They are not equal, hence f(x) is not continuous at x = 1.
Correct Answer:
B
— 1
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Q. The function f(x) = { 3x + 1, x < 1; 2, x = 1; x^2, x > 1 } is continuous at x = 1 if which condition holds?
-
A.
3 = 2
-
B.
1 = 2
-
C.
2 = 1
-
D.
2 = 4
Solution
For continuity at x = 1, the left limit (3) must equal f(1) (2), which is not true.
Correct Answer:
A
— 3 = 2
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Q. The function f(x) = { 3x + 1, x < 1; 2x + 3, x >= 1 } is continuous at x = 1 if:
Solution
For continuity at x = 1, both pieces must equal 4, hence the function is continuous.
Correct Answer:
A
— 3
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Q. The function f(x) = { x + 1, x < 1; 2, x = 1; x^2, x > 1 } is continuous at x = 1?
-
A.
Yes
-
B.
No
-
C.
Only left continuous
-
D.
Only right continuous
Solution
Left limit as x approaches 1 is 2, right limit is 1, but f(1) = 2. Hence, it is discontinuous at x = 1.
Correct Answer:
B
— No
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Q. The function f(x) = { x + 2, x < 1; 3, x = 1; x^2, x > 1 } is continuous at x = ?
Solution
To check continuity at x = 1, we find the left limit (3) and the right limit (3). Both equal 3, hence f(x) is continuous at x = 1.
Correct Answer:
B
— 1
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Q. The function f(x) = { x^2, x < 0; 0, x = 0; x + 1, x > 0 } is:
-
A.
Continuous
-
B.
Not continuous
-
C.
Continuous from the left
-
D.
Continuous from the right
Solution
The left limit as x approaches 0 is 0, but the right limit is 1. Hence, it is not continuous at x = 0.
Correct Answer:
B
— Not continuous
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Q. The function f(x) = { x^2, x < 0; 1, x = 0; x + 1, x > 0 } is continuous at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only from the right
-
D.
Only from the left
Solution
Limit as x approaches 0 from left is 0, and f(0) = 1, hence it is not continuous at x = 0.
Correct Answer:
A
— Yes
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Q. The function f(x) = { x^2, x < 0; 2, x = 0; x + 1, x > 0 } is continuous at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only left continuous
-
D.
Only right continuous
Solution
At x = 0, lim x→0- f(x) = 0 and lim x→0+ f(x) = 1, hence it is discontinuous at x = 0.
Correct Answer:
B
— No
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Q. The function f(x) = { x^2, x < 0; 2x + 1, x >= 0 } is continuous at which point?
-
A.
x = -1
-
B.
x = 0
-
C.
x = 1
-
D.
x = 2
Solution
To check continuity at x = 0, we find f(0) = 1 and limit as x approaches 0 is also 1.
Correct Answer:
B
— x = 0
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Q. The function f(x) = { x^2, x < 1; 2, x = 1; x + 1, x > 1 } is:
-
A.
Continuous everywhere
-
B.
Continuous at x = 1
-
C.
Not continuous at x = 1
-
D.
Continuous for x < 1
Solution
The function is not continuous at x = 1 because the left-hand limit does not equal the function value.
Correct Answer:
C
— Not continuous at x = 1
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Q. The function f(x) = { x^2, x < 1; 2x - 1, x >= 1 } is continuous at which point?
-
A.
x = 0
-
B.
x = 1
-
C.
x = 2
-
D.
x = -1
Solution
To check continuity at x = 1, we find f(1) = 1, limit as x approaches 1 from left is 1, and from right is also 1.
Correct Answer:
B
— x = 1
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Q. The function f(x) = { x^2, x < 1; 2x - 1, x >= 1 } is continuous at x = ?
Solution
To check continuity at x = 1, we find the limit from both sides. Both limits equal 1, hence f(x) is continuous at x = 1.
Correct Answer:
B
— 1
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Q. The function f(x) = { x^2, x < 1; 2x - 1, x ≥ 1 } is differentiable at x = 1 if which condition holds?
-
A.
f(1) = 1
-
B.
f'(1) = 1
-
C.
f'(1) = 2
-
D.
f(1) = 2
Solution
For differentiability, the left and right derivatives must equal at x = 1, hence f'(1) = 1.
Correct Answer:
B
— f'(1) = 1
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Q. The function f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 } is continuous at x = 2 if:
-
A.
f(2) = 4
-
B.
lim x->2 f(x) = 4
-
C.
Both a and b
-
D.
None of the above
Solution
Both conditions must hold true for continuity at x = 2.
Correct Answer:
C
— Both a and b
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Q. The function f(x) = { x^2, x < 2; k, x = 2; 3x - 4, x > 2 } is continuous at x = 2 for which value of k?
Solution
To be continuous at x = 2, k must equal f(2) = 2^2 = 4.
Correct Answer:
C
— 4
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Q. The function f(x) = |x - 3| is continuous at which of the following points?
-
A.
x = 1
-
B.
x = 2
-
C.
x = 3
-
D.
x = 4
Solution
The function |x - 3| is continuous everywhere, including at x = 3.
Correct Answer:
C
— x = 3
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Q. The function f(x) = |x| is differentiable at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only from the right
-
D.
Only from the left
Solution
f(x) = |x| is not differentiable at x = 0 because the left-hand and right-hand derivatives do not match.
Correct Answer:
B
— No
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Q. The Fundamental Duties were added to the Constitution by which amendment? (2023)
-
A.
42nd Amendment
-
B.
44th Amendment
-
C.
61st Amendment
-
D.
73rd Amendment
Solution
The Fundamental Duties were added to the Constitution by the 42nd Amendment.
Correct Answer:
A
— 42nd Amendment
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Q. The fusion of male and female gametes in plants is known as: (2021)
-
A.
Fertilization
-
B.
Pollination
-
C.
Germination
-
D.
Mitosis
Solution
Fertilization is the process where the male gamete (sperm) fuses with the female gamete (egg) to form a zygote.
Correct Answer:
A
— Fertilization
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Q. The game was won by the players. What is the active voice of this sentence? (2023)
-
A.
The players win the game.
-
B.
The players are winning the game.
-
C.
The players won the game.
-
D.
The game is won by the players.
Solution
In active voice, we state that 'The players won the game.' focusing on the players as the doers.
Correct Answer:
C
— The players won the game.
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Q. The garden was _____ with flowers of every color.
-
A.
empty
-
B.
full
-
C.
barren
-
D.
dry
Solution
The word 'full' suggests a vibrant and colorful garden.
Correct Answer:
B
— full
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Q. The general form of the family of curves for circles is given by:
-
A.
(x - h)^2 + (y - k)^2 = r^2
-
B.
x^2 + y^2 = r^2
-
C.
x^2 + y^2 + Dx + Ey + F = 0
-
D.
y = mx + b
Solution
The equation x^2 + y^2 + Dx + Ey + F = 0 represents a family of circles.
Correct Answer:
C
— x^2 + y^2 + Dx + Ey + F = 0
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Q. The general form of the family of curves y^2 = 4ax is known as:
-
A.
Circle
-
B.
Ellipse
-
C.
Parabola
-
D.
Hyperbola
Solution
The equation y^2 = 4ax represents a parabola.
Correct Answer:
C
— Parabola
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Q. The general form of the family of curves y^2 = 4ax represents:
-
A.
Ellipses
-
B.
Hyperbolas
-
C.
Parabolas
-
D.
Circles
Solution
The equation y^2 = 4ax represents a parabola that opens to the right.
Correct Answer:
C
— Parabolas
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Q. The general form of the family of exponential curves is given by:
-
A.
y = a^x
-
B.
y = ax^2 + bx + c
-
C.
y = mx + c
-
D.
y = log(x)
Solution
The equation y = a^x represents an exponential function where a is a constant.
Correct Answer:
A
— y = a^x
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Q. The government announced a new initiative to promote renewable energy sources. What can be inferred about the government's stance on energy?
-
A.
The government believes renewable energy is the only solution.
-
B.
The government is concerned about environmental issues.
-
C.
All energy sources are equally important to the government.
-
D.
The initiative will be implemented immediately.
Solution
The promotion of renewable energy suggests that the government is concerned about environmental issues.
Correct Answer:
B
— The government is concerned about environmental issues.
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Q. The government claims that increasing taxes on sugary drinks will reduce obesity rates. Which of the following can be inferred?
-
A.
All sugary drinks are unhealthy.
-
B.
Higher taxes will definitely lead to lower obesity rates.
-
C.
Reducing sugary drink consumption may help lower obesity rates.
-
D.
Obesity is solely caused by sugary drink consumption.
Solution
The claim suggests that reducing consumption of sugary drinks could help lower obesity rates, but does not guarantee it.
Correct Answer:
C
— Reducing sugary drink consumption may help lower obesity rates.
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