Q. Evaluate the limit: lim (x -> 0) (x^2 * sin(1/x))
-
A.
0
-
B.
1
-
C.
Infinity
-
D.
Undefined
Solution
Since |sin(1/x)| <= 1, we have |x^2 * sin(1/x)| <= x^2, and thus lim (x -> 0) x^2 * sin(1/x) = 0.
Correct Answer: A — 0
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Q. Evaluate the limit: lim (x -> 0) (x^2)/(sin(x))
-
A.
0
-
B.
1
-
C.
Infinity
-
D.
Undefined
Solution
As x approaches 0, sin(x) approaches x, thus lim (x -> 0) (x^2)/(sin(x)) = 0.
Correct Answer: A — 0
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Q. Evaluate the limit: lim (x -> 1) (x^2 - 1)/(x - 1)
-
A.
2
-
B.
0
-
C.
1
-
D.
Infinity
Solution
Using L'Hôpital's Rule, the limit evaluates to 2.
Correct Answer: A — 2
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Q. Evaluate the limit: lim (x -> ∞) (2x^2 + 3)/(5x^2 - 4x + 1)
-
A.
2/5
-
B.
3/5
-
C.
1/2
-
D.
Infinity
Solution
Divide numerator and denominator by x^2. The limit becomes lim (x -> ∞) (2 + 3/x^2)/(5 - 4/x + 1/x^2) = 2/5.
Correct Answer: A — 2/5
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Q. Evaluate the limit: lim (x -> ∞) (2x^3 - 3x)/(4x^3 + 5)
Solution
Dividing numerator and denominator by x^3 gives lim (x -> ∞) (2 - 3/x^2)/(4 + 5/x^3) = 2/4 = 1/2.
Correct Answer: B — 1/2
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Q. Evaluate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4)
-
A.
3/5
-
B.
0
-
C.
1
-
D.
Infinity
Solution
Dividing numerator and denominator by x^2 gives lim (x -> ∞) (3 + 0)/(5 - 0) = 3/5.
Correct Answer: A — 3/5
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Q. Evaluate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1)
Solution
Dividing numerator and denominator by x^2 gives lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x + 1/x^2) = 3/5.
Correct Answer: A — 3/5
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Q. Evaluate the limit: lim(x->1) (x^2 - 1)/(x - 1)^2
-
A.
1
-
B.
2
-
C.
0
-
D.
Undefined
Solution
This is an indeterminate form (0/0). Factor the numerator: (x-1)(x+1)/(x-1)^2 = (x+1)/(x-1). Thus, lim(x->1) = 2.
Correct Answer: B — 2
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Q. Evaluate the limit: lim(x->infinity) (2x^3 - 3x)/(4x^3 + 5)
-
A.
1/2
-
B.
0
-
C.
1
-
D.
Infinity
Solution
Divide numerator and denominator by x^3: lim(x->infinity) (2 - 3/x^2)/(4 + 5/x^3) = 2/4 = 1/2.
Correct Answer: A — 1/2
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Q. Evaluate the limit: lim(x->infinity) (3x^2 + 2)/(5x^2 - 4)
-
A.
3/5
-
B.
0
-
C.
1
-
D.
Infinity
Solution
Divide numerator and denominator by x^2: lim(x->infinity) (3 + 2/x^2)/(5 - 4/x^2) = 3/5.
Correct Answer: A — 3/5
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Q. Evaluate \( \begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix} \)
Solution
The determinant of the identity matrix is 1.
Correct Answer: B — 1
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Q. Evaluate \( \begin{vmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 3 & 1 \end{vmatrix} \)
Solution
The determinant is calculated as \( 1(1*1 - 0*3) - 2(0*1 - 0*2) + 1(0*3 - 1*2) = 1 - 0 - 2 = -1 \).
Correct Answer: B — 2
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Q. Evaluate \( \begin{vmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{vmatrix} \)
Solution
The determinant is calculated as \( 1(1*0 - 4*6) - 2(0 - 4*5) + 3(0 - 1*5) = -24 + 40 - 15 = 1 \).
Correct Answer: A — -12
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Q. Evaluate \( \begin{vmatrix} x & 1 \\ 1 & y \end{vmatrix} \) when \( x = 2 \) and \( y = 3 \).
Solution
The determinant is \( 2*3 - 1*1 = 6 \).
Correct Answer: B — 6
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Q. Evaluate \( \cos(\cos^{-1}(\frac{3}{5})) \).
-
A.
0
-
B.
\( \frac{3}{5} \)
-
C.
1
-
D.
undefined
Solution
By definition, \( \cos(\cos^{-1}(x)) = x \). Therefore, \( \cos(\cos^{-1}(\frac{3}{5})) = \frac{3}{5} \).
Correct Answer: B — \( \frac{3}{5} \)
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Q. Evaluate ∫ from 0 to 1 of (1 - x^2) dx.
-
A.
1/3
-
B.
1/2
-
C.
2/3
-
D.
1
Solution
The integral evaluates to [x - x^3/3] from 0 to 1 = (1 - 1/3) = 2/3.
Correct Answer: C — 2/3
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Q. Evaluate ∫ from 0 to 1 of (4x^3 - 3x^2 + 2) dx.
Solution
The integral evaluates to [x^4 - x^3 + 2x] from 0 to 1 = (1 - 1 + 2) - (0) = 2.
Correct Answer: C — 3
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Q. Evaluate ∫ from 0 to 1 of (x^2 + 3x + 2) dx.
Solution
The integral evaluates to [x^3/3 + (3/2)x^2 + 2x] from 0 to 1 = (1/3 + 3/2 + 2) = (1/3 + 3/2 + 6/3) = 27/6 = 4.5.
Correct Answer: C — 3
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Q. Evaluate ∫ from 0 to 1 of (x^3 + 3x^2 + 3x + 1) dx.
Solution
The integral evaluates to [x^4/4 + x^3 + (3/2)x^2 + x] from 0 to 1 = (1/4 + 1 + 3/2 + 1) = 4.
Correct Answer: D — 4
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Q. Evaluate ∫ from 0 to 1 of (x^4 + 2x^3) dx.
-
A.
1/5
-
B.
1/4
-
C.
1/3
-
D.
1/2
Solution
The integral evaluates to [x^5/5 + x^4/2] from 0 to 1 = (1/5 + 1/2) = 7/10.
Correct Answer: C — 1/3
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Q. Evaluate ∫ from 0 to 1 of (x^4) dx.
-
A.
1/5
-
B.
1/4
-
C.
1/3
-
D.
1/2
Solution
The integral evaluates to [x^5/5] from 0 to 1 = 1/5.
Correct Answer: A — 1/5
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Q. Evaluate ∫ from 0 to 1 of e^x dx.
Solution
The integral evaluates to [e^x] from 0 to 1 = e - 1.
Correct Answer: A — e - 1
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Q. Evaluate ∫ from 0 to 2 of (x^2 + 2x + 1) dx.
Solution
The integral evaluates to [x^3/3 + x^2 + x] from 0 to 2 = (8/3 + 4 + 2) = 6.
Correct Answer: C — 6
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Q. Evaluate ∫ from 0 to 2 of (x^3 - 3x^2 + 4) dx.
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. Evaluate ∫ from 1 to 2 of (x^4 - 4x^3 + 6x^2 - 4x + 1) dx.
Solution
The integral evaluates to [x^5/5 - x^4 + 2x^3 - 2x^2 + x] from 1 to 2 = 0.
Correct Answer: A — 0
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Q. Evaluate ∫ from 1 to 3 of (2x + 1) dx.
Solution
The integral evaluates to [x^2 + x] from 1 to 3 = (9 + 3) - (1 + 1) = 10.
Correct Answer: B — 10
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Q. Evaluate ∫ from 1 to 3 of (x^2 - 4) dx.
Solution
The integral evaluates to [x^3/3 - 4x] from 1 to 3 = (27/3 - 12) - (1/3 - 4) = 2.
Correct Answer: C — 2
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Q. Evaluate ∫_0^1 (1 - x^2) dx.
-
A.
1/3
-
B.
1/2
-
C.
2/3
-
D.
1
Solution
∫_0^1 (1 - x^2) dx = [x - x^3/3] from 0 to 1 = (1 - 1/3) = 2/3.
Correct Answer: B — 1/2
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Q. Evaluate ∫_0^1 (e^x) dx.
Solution
∫_0^1 e^x dx = [e^x] from 0 to 1 = e - 1.
Correct Answer: A — e - 1
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Q. Evaluate ∫_0^1 (x^3 + 2x^2) dx.
-
A.
1/4
-
B.
1/3
-
C.
1/2
-
D.
1
Solution
∫_0^1 (x^3 + 2x^2) dx = [x^4/4 + 2x^3/3] from 0 to 1 = (1/4 + 2/3) = 11/12.
Correct Answer: C — 1/2
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