Q. Calculate ∫ from 0 to 1 of (x^2 + 1/x^2) dx.
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Solution
The integral evaluates to [x^3/3 - 1/x] from 0 to 1 = (1/3 - 1) = -2/3.
Correct Answer:
C
— 3
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Q. Calculate ∫ from 0 to 1 of (x^2 + 2x + 1) dx.
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Solution
The integral evaluates to [x^3/3 + x^2 + x] from 0 to 1 = (1/3 + 1 + 1) - (0) = 7/3.
Correct Answer:
C
— 3
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Q. Calculate ∫ from 0 to 1 of (x^2 + 4x + 4) dx.
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Solution
The integral evaluates to [x^3/3 + 2x^2 + 4x] from 0 to 1 = (1/3 + 2 + 4) = 25/3.
Correct Answer:
C
— 5
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Q. Calculate ∫ from 0 to 1 of (x^4 - 2x^2 + 1) dx.
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Solution
The integral evaluates to [x^5/5 - 2x^3/3 + x] from 0 to 1 = (1/5 - 2/3 + 1) = (15/15 - 10/15 + 3/15) = 8/15.
Correct Answer:
D
— 2/3
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Q. Calculate ∫ from 0 to 1 of (x^4 - 2x^3 + x^2) dx.
A.
0
B.
1/5
C.
1/3
D.
1/2
Show solution
Solution
The integral evaluates to [x^5/5 - (2/4)x^4 + (1/3)x^3] from 0 to 1 = (1/5 - 1/2 + 1/3) = 1/30.
Correct Answer:
B
— 1/5
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Q. Calculate ∫ from 0 to 2 of (x^3 - 3x^2 + 4) dx.
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Solution
The integral evaluates to [x^4/4 - x^3 + 4x] from 0 to 2 = (4 - 8 + 8) - 0 = 4.
Correct Answer:
C
— 6
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Q. Calculate ∫ from 0 to π of sin(x) dx.
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Solution
The integral evaluates to [-cos(x)] from 0 to π = [1 - (-1)] = 2.
Correct Answer:
C
— 2
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Q. Calculate ∫ from 0 to π/2 of sin(x) cos(x) dx.
A.
1/2
B.
1
C.
π/4
D.
π/2
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Solution
Using the identity sin(2x) = 2sin(x)cos(x), the integral becomes 1/2 ∫ from 0 to π/2 of sin(2x) dx = 1/2 [-1/2 cos(2x)] from 0 to π/2 = 1/2 [0 - (-1/2)] = 1/4.
Correct Answer:
A
— 1/2
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Q. Calculate ∫ from 0 to π/2 of sin^2(x) dx.
A.
π/4
B.
π/2
C.
π/3
D.
π/6
Show solution
Solution
Using the identity sin^2(x) = (1 - cos(2x))/2, the integral evaluates to π/4.
Correct Answer:
A
— π/4
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Q. Calculate ∫ from 1 to 3 of (2x + 1) dx.
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Solution
The integral evaluates to [x^2 + x] from 1 to 3 = (9 + 3) - (1 + 1) = 10.
Correct Answer:
B
— 6
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Q. Calculate ∫_0^1 (4x^3 - 3x^2 + 2) dx.
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Solution
∫_0^1 (4x^3 - 3x^2 + 2) dx = [x^4 - x^3 + 2x] from 0 to 1 = (1 - 1 + 2) - (0) = 2.
Correct Answer:
B
— 2
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Q. Calculate ∫_0^1 (e^x) dx.
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Solution
∫_0^1 e^x dx = [e^x] from 0 to 1 = e - 1.
Correct Answer:
A
— e - 1
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Q. Calculate ∫_0^1 (x^3 - 2x^2 + x) dx.
A.
-1/12
B.
0
C.
1/12
D.
1/6
Show solution
Solution
The integral evaluates to [x^4/4 - 2x^3/3 + x^2/2] from 0 to 1 = (1/4 - 2/3 + 1/2) = 1/12.
Correct Answer:
C
— 1/12
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Q. Calculate ∫_0^π/2 cos^2(x) dx.
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Solution
∫_0^π/2 cos^2(x) dx = π/4.
Correct Answer:
A
— π/4
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Q. Calculate ∫_1^e (ln(x)) dx.
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Solution
∫_1^e ln(x) dx = [x ln(x) - x] from 1 to e = (e - e) - (1 - 1) = 1.
Correct Answer:
B
— e - 1
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Q. Calculate ∫_1^e (ln(x))^2 dx.
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Solution
Using integration by parts, the integral evaluates to 1.
Correct Answer:
B
— 2
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Q. Change the following sentence to active voice: 'The homework was completed by the students.' (2023)
A.
The students completed the homework.
B.
The students are completing the homework.
C.
The homework is completed by the students.
D.
The students have completed the homework.
Show solution
Solution
In active voice, the subject 'students' performs the action, thus 'The students completed the homework.' is correct.
Correct Answer:
A
— The students completed the homework.
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Q. Change to active voice: 'The song was sung by the children.' (2023)
A.
The children sing the song.
B.
The children sang the song.
C.
The children have sung the song.
D.
The children are singing the song.
Show solution
Solution
In active voice, the subject 'children' performs the action, thus 'The children sang the song.' is correct.
Correct Answer:
B
— The children sang the song.
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Q. Choose the appropriate linking word: 'He loves to travel. _____, he has visited over 20 countries.' (2023)
A.
However
B.
Therefore
C.
Moreover
D.
Although
Show solution
Solution
'Moreover' adds to the information about his love for travel.
Correct Answer:
C
— Moreover
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Q. Choose the best linking word: 'He is very talented; _____, he still practices every day.'
A.
but
B.
and
C.
so
D.
although
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Solution
'And' is correct as it adds information about his practice.
Correct Answer:
B
— and
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Q. Choose the best linking word: 'I enjoy reading; _____, I don't have much time for it.'
A.
but
B.
and
C.
so
D.
because
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Solution
'But' is the correct choice as it introduces a contrast to the first statement.
Correct Answer:
A
— but
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Q. Choose the best rephrasing for: 'The results were inconclusive.'
A.
The results were clear.
B.
The results did not provide a definite answer.
C.
The results were very informative.
D.
The results were obvious.
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Solution
The second option correctly conveys that the results did not lead to a clear conclusion.
Correct Answer:
B
— The results did not provide a definite answer.
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Q. Choose the best rephrasing of 'The results were inconclusive.' (2020)
A.
The results were clear.
B.
The results did not provide clear answers.
C.
The results were very conclusive.
D.
The results were obvious.
Show solution
Solution
The term 'inconclusive' means that the results did not provide clear answers, making option 1 correct.
Correct Answer:
B
— The results did not provide clear answers.
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Q. Choose the best way to combine the following sentences: 'The storm caused significant damage. Many homes were destroyed.' (2023)
A.
The storm caused significant damage, and many homes were destroyed.
B.
Significant damage was caused by the storm, many homes were destroyed.
C.
The storm caused significant damage; many homes were destroyed.
D.
Many homes were destroyed due to the storm causing significant damage.
Show solution
Solution
The use of a semicolon effectively connects the two independent clauses.
Correct Answer:
C
— The storm caused significant damage; many homes were destroyed.
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Q. Choose the best word to complete the sentence: 'Despite the __________ of the evidence, the jury remained unconvinced.' (2023)
A.
abundance
B.
scarcity
C.
ambiguity
D.
clarity
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Solution
The word 'abundance' fits best, indicating that there was a lot of evidence, yet the jury was still unconvinced.
Correct Answer:
A
— abundance
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Q. Choose the best word to fill in the blank: 'The scientist's findings were ________ by the subsequent research, confirming the initial hypothesis.'
A.
contradicted
B.
supported
C.
ignored
D.
overlooked
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Solution
The context suggests that the subsequent research confirmed the findings, thus 'supported' is the best fit.
Correct Answer:
B
— supported
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Q. Choose the correct active voice form of the sentence: 'The homework was completed by the students.'
A.
The students completed the homework.
B.
The homework is completed by the students.
C.
The students are completing the homework.
D.
The homework has been completed by the students.
Show solution
Solution
The active voice requires the subject to perform the action, which is correctly represented by 'The students completed the homework.'
Correct Answer:
A
— The students completed the homework.
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Q. Choose the correct article: 'She is _____ honest person.' (2022)
A.
a
B.
an
C.
the
D.
no article
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Solution
The correct article is 'an' because 'honest' starts with a vowel sound.
Correct Answer:
B
— an
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Q. Choose the correct article: 'She wants to be ___ doctor.'
A.
a
B.
an
C.
the
D.
none
Show solution
Solution
'An' is used before words that begin with a vowel sound, so 'an doctor' is correct.
Correct Answer:
B
— an
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Q. Choose the correct article: She adopted ___ cat from the shelter.
A.
a
B.
an
C.
the
D.
none
Show solution
Solution
'a' is the correct article to use before a consonant sound in 'cat'.
Correct Answer:
A
— a
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