Q. Evaluate the integral ∫_0^1 (x^2 + 2x) dx.
Solution
∫_0^1 (x^2 + 2x) dx = [x^3/3 + x^2] from 0 to 1 = (1/3 + 1) - (0) = 4/3.
Correct Answer: B — 2
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Q. Evaluate the integral ∫_0^1 (x^3 - 3x^2 + 3x - 1) dx.
Solution
∫_0^1 (x^3 - 3x^2 + 3x - 1) dx = [x^4/4 - x^3 + (3/2)x^2 - x] from 0 to 1 = 0.
Correct Answer: A — 0
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Q. Evaluate the integral ∫_0^π/2 cos^2(x) dx.
Solution
∫_0^π/2 cos^2(x) dx = π/4.
Correct Answer: A — π/4
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Q. Evaluate the integral ∫_1^2 (3x^2 - 2) dx.
Solution
∫_1^2 (3x^2 - 2) dx = [x^3 - 2x] from 1 to 2 = (8 - 4) - (1 - 2) = 3.
Correct Answer: A — 1
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Q. Evaluate the integral: ∫ (1/(x^2 + 1)) dx
-
A.
tan^(-1)(x) + C
-
B.
sin^(-1)(x) + C
-
C.
ln
-
D.
x
-
.
+ C
-
.
cos^(-1)(x) + C
Solution
The integral of 1/(x^2 + 1) is tan^(-1)(x) + C.
Correct Answer: A — tan^(-1)(x) + C
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Q. Evaluate the integral: ∫ (2x^3 - 3x^2 + 4) dx
-
A.
(1/2)x^4 - x^3 + 4x + C
-
B.
(1/4)x^4 - (1/3)x^3 + 4x + C
-
C.
(1/2)x^4 - (1/3)x^3 + 4x + C
-
D.
(1/4)x^4 - x^3 + 4x + C
Solution
Integrating term by term gives (1/4)x^4 - (1/3)x^3 + 4x + C.
Correct Answer: A — (1/2)x^4 - x^3 + 4x + C
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Q. Evaluate the limit lim x->1 (x^3 - 1)/(x - 1).
Solution
Factoring gives (x-1)(x^2 + x + 1)/(x - 1). Canceling (x - 1) gives lim x->1 (x^2 + x + 1) = 3.
Correct Answer: C — 2
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Q. Evaluate the limit lim x->1 of (x^3 - 1)/(x - 1).
Solution
Factoring gives (x-1)(x^2 + x + 1)/(x-1) = x^2 + x + 1, thus limit is 3.
Correct Answer: C — 3
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Q. Evaluate the limit lim x->2 (x^2 - 4)/(x - 2).
Solution
Factoring gives (x-2)(x+2)/(x-2). Canceling gives lim x->2 (x + 2) = 4.
Correct Answer: C — 2
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Q. Evaluate the limit lim x->2 of (x^2 - 4)/(x - 2).
-
A.
0
-
B.
2
-
C.
4
-
D.
undefined
Solution
Factoring gives (x-2)(x+2)/(x-2) = x + 2, thus limit is 4.
Correct Answer: C — 4
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Q. Evaluate the limit lim(x→∞) (3x^2 + 2)/(5x^2 - 4).
Solution
Dividing by x^2, lim(x→∞) (3 + 2/x^2)/(5 - 4/x^2) = 3/5.
Correct Answer: A — 3/5
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Q. Evaluate the limit: lim (x -> 0) (1 - cos(x))/(x^2)
-
A.
0
-
B.
1/2
-
C.
1
-
D.
Undefined
Solution
Using the identity 1 - cos(x) = 2sin^2(x/2), we have lim (x -> 0) (2sin^2(x/2))/(x^2) = 1/2.
Correct Answer: B — 1/2
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Q. Evaluate the limit: lim (x -> 0) (e^x - 1)/x
-
A.
0
-
B.
1
-
C.
e
-
D.
Infinity
Solution
Using L'Hôpital's Rule, we differentiate the numerator and denominator: lim (x -> 0) (e^x)/(1) = e^0 = 1.
Correct Answer: B — 1
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Q. Evaluate the limit: lim (x -> 0) (ln(1 + x)/x)
-
A.
0
-
B.
1
-
C.
Infinity
-
D.
Undefined
Solution
Using L'Hôpital's Rule, differentiate the numerator and denominator: lim (x -> 0) (1/(1 + x))/(1) = 1.
Correct Answer: B — 1
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Q. Evaluate the limit: lim (x -> 0) (sin(5x)/x)
-
A.
0
-
B.
5
-
C.
1
-
D.
Infinity
Solution
Using the standard limit lim (x -> 0) (sin(x)/x) = 1, we have lim (x -> 0) (sin(5x)/x) = 5 * lim (x -> 0) (sin(5x)/(5x)) = 5 * 1 = 5.
Correct Answer: B — 5
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Q. Evaluate the limit: lim (x -> 0) (tan(3x)/x)
-
A.
0
-
B.
3
-
C.
1
-
D.
Infinity
Solution
Using the standard limit lim (x -> 0) (tan(x)/x) = 1, we have lim (x -> 0) (tan(3x)/x) = 3 * lim (x -> 0) (tan(3x)/(3x)) = 3 * 1 = 3.
Correct Answer: B — 3
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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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