Q. What is the value of sin^2(θ) + cos^2(θ)?
A.
1
B.
0
C.
sin(θ)
D.
cos(θ)
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Solution
By the Pythagorean identity, sin^2(θ) + cos^2(θ) = 1.
Correct Answer: A — 1
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Q. What is the value of tan(30°)?
A.
√3/3
B.
1/√3
C.
√3
D.
3
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Solution
tan(30°) = sin(30°)/cos(30°) = (1/2)/(√3/2) = 1/√3.
Correct Answer: A — √3/3
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Q. What is the value of tan(45° + x)?
A.
tan x
B.
1 + tan x
C.
(1 - tan x)/(1 + tan x)
D.
1 - tan x
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Solution
Using the formula tan(A + B) = (tan A + tan B) / (1 - tan A tan B), we have tan(45° + x) = (1 + tan x) / (1 - 1*tan x) = (1 + tan x).
Correct Answer: B — 1 + tan x
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Q. What is the value of tan^(-1)(1) + tan^(-1)(1)?
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Solution
tan^(-1)(1) = π/4, thus tan^(-1)(1) + tan^(-1)(1) = π/4 + π/4 = π/2.
Correct Answer: A — π/2
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Q. What is the value of tan^(-1)(1) + tan^(-1)(2)?
A.
π/4
B.
π/3
C.
π/2
D.
π/6
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Solution
Using the formula tan^(-1)(a) + tan^(-1)(b) = tan^(-1)((a+b)/(1-ab)), we have tan^(-1)(1) + tan^(-1)(2) = tan^(-1)((1+2)/(1-1*2)) = tan^(-1)(3/-1) = π - tan^(-1)(3) = π/4.
Correct Answer: A — π/4
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Q. What is the value of tan^(-1)(1)?
A.
π/4
B.
π/3
C.
π/2
D.
0
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Solution
tan^(-1)(1) = π/4, since tan(π/4) = 1.
Correct Answer: A — π/4
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Q. What is the value of tan^(-1)(√3)?
A.
π/3
B.
π/4
C.
π/6
D.
π/2
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Solution
tan^(-1)(√3) corresponds to the angle π/3.
Correct Answer: A — π/3
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Q. What is the value of the 5th term in the expansion of (x + 2)^6?
A.
80
B.
120
C.
160
D.
240
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Solution
The 5th term is given by C(6,4) * (x)^4 * (2)^2 = 15 * x^4 * 4 = 240.
Correct Answer: B — 120
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Q. What is the value of the coefficient of x^3 in the expansion of (x - 1)^6?
A.
-20
B.
-30
C.
-40
D.
-10
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Solution
The coefficient of x^3 is C(6,3) * (-1)^3 = 20 * (-1) = -20.
Correct Answer: B — -30
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Q. What is the value of the coefficient of x^5 in the expansion of (x + 3)^7?
A.
21
B.
63
C.
126
D.
189
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Solution
The coefficient of x^5 is C(7,5) * (3)^2 = 21 * 9 = 189.
Correct Answer: C — 126
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Q. What is the value of the coefficient of x^5 in the expansion of (x + 3)^8?
A.
1680
B.
168
C.
840
D.
280
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Solution
The coefficient of x^5 is C(8, 5) * (3)^3 = 56 * 27 = 1512.
Correct Answer: A — 1680
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Q. What is the value of the derivative of f(x) = ln(x^2 + 1) at x = 1?
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Solution
f'(x) = (2x)/(x^2 + 1). At x = 1, f'(1) = 2/(1 + 1) = 1.
Correct Answer: B — 1/2
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Q. What is the value of the determinant of the matrix [[1, 2], [3, 4]]?
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Solution
The determinant is calculated as (1*4) - (2*3) = 4 - 6 = -2.
Correct Answer: A — -2
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Q. What is the value of the determinant of the matrix \( A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \)?
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Solution
The determinant of matrix A is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. What is the value of the determinant of the matrix \( \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \)?
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Solution
The determinant of the matrix is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. What is the value of the determinant \( \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{vmatrix} \)?
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Solution
The determinant is 0 because the first column is repeated.
Correct Answer: A — 0
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Q. What is the value of the determinant \( \begin{vmatrix} a & b \\ c & d \end{vmatrix} \) when \( a = 1, b = 2, c = 3, d = 4 \)?
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Solution
The determinant is \( 1*4 - 2*3 = 4 - 6 = -2 \).
Correct Answer: A — -2
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Q. What is the value of the determinant | 1 2 3 | | 0 1 4 | | 5 6 0 |?
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Solution
Calculating the determinant gives -12.
Correct Answer: A — -12
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Q. What is the value of the determinant | a b | | c d | when a = 2, b = 3, c = 4, d = 5?
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Solution
det = (2)(5) - (3)(4) = 10 - 12 = -2.
Correct Answer: C — 2
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Q. What is the value of the integral ∫(0 to 1) (3x^2 + 2)dx?
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Solution
Integral = [x^3 + 2x] from 0 to 1 = (1 + 2) - (0) = 3.
Correct Answer: A — 5/3
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Q. What is the value of the integral ∫(0 to 1) (3x^2)dx?
A.
1
B.
1/3
C.
1/2
D.
3/4
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Solution
The integral evaluates to [x^3] from 0 to 1 = 1^3 - 0^3 = 1.
Correct Answer: A — 1
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Q. What is the value of the integral ∫(0 to 1) x^2 dx?
A.
1/3
B.
1/2
C.
1/4
D.
1
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Solution
The integral ∫(0 to 1) x^2 dx = [x^3/3] from 0 to 1 = 1/3.
Correct Answer: A — 1/3
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Q. What is the value of the integral ∫(1 to 2) (3x^2 - 2)dx?
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Solution
The integral evaluates to [x^3 - 2x] from 1 to 2 = (8 - 4) - (1 - 2) = 3.
Correct Answer: B — 4
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Q. What is the value of the integral ∫(1 to 2) (x^2 + 2x)dx?
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Solution
The integral evaluates to 7.
Correct Answer: A — 7
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Q. What is the value of the integral ∫(1/(x^2 + 1))dx?
A.
tan^-1(x) + C
B.
sin^-1(x) + C
C.
cos^-1(x) + C
D.
ln(x) + C
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Solution
The integral evaluates to tan^-1(x) + C.
Correct Answer: A — tan^-1(x) + C
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Q. What is the value of the integral ∫(2x)dx from 0 to 2?
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Solution
∫(2x)dx = x^2 from 0 to 2 = 2^2 - 0^2 = 4.
Correct Answer: A — 4
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Q. What is the value of the integral ∫(2x^3 - 3x^2 + 4)dx from 1 to 3?
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Solution
The integral evaluates to 20.
Correct Answer: B — 20
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Q. What is the value of the integral ∫_0^1 (3x^2 + 2x) dx?
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Solution
Evaluating the integral gives [x^3 + x^2]_0^1 = (1 + 1) - (0 + 0) = 2.
Correct Answer: B — 2
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Q. What is the value of the limit lim (x -> 1) (x^2 - 1)/(x - 1)?
A.
0
B.
1
C.
2
D.
Infinity
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Solution
Using L'Hôpital's Rule, the limit evaluates to 2.
Correct Answer: C — 2
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Q. What is the value of the limit lim(x->0) (sin(x)/x)?
A.
0
B.
1
C.
∞
D.
undefined
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Solution
The limit lim(x->0) (sin(x)/x) = 1.
Correct Answer: B — 1
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