Q. Find the value of sin(30°) + cos(60°).
Solution
sin(30°) = 1/2 and cos(60°) = 1/2, thus the sum is 1/2 + 1/2 = 1.
Correct Answer: B — 1/2
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Q. Find the value of sin(cos^(-1)(1/2)).
Solution
Let θ = cos^(-1)(1/2). Then cos(θ) = 1/2, which corresponds to θ = π/3. Therefore, sin(θ) = sin(π/3) = √3/2.
Correct Answer: A — √3/2
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Q. Find the value of sin(tan^(-1)(1)).
-
A.
1/√2
-
B.
1/2
-
C.
√2/2
-
D.
√3/2
Solution
sin(tan^(-1)(1)) = 1/√2, using the triangle with opposite and adjacent sides equal.
Correct Answer: C — √2/2
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Q. Find the value of sin(tan^(-1)(x)).
-
A.
x/√(1+x^2)
-
B.
√(1+x^2)/x
-
C.
1/x
-
D.
x
Solution
Using the right triangle definition, sin(tan^(-1)(x)) = opposite/hypotenuse = x/√(1+x^2).
Correct Answer: A — x/√(1+x^2)
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Q. Find the value of sin^(-1)(sin(π/3)).
-
A.
π/3
-
B.
2π/3
-
C.
π/6
-
D.
0
Solution
Since π/3 is in the range of sin^(-1), sin^(-1)(sin(π/3)) = π/3.
Correct Answer: A — π/3
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Q. Find the value of sin^(-1)(sin(π/4)).
-
A.
π/4
-
B.
3π/4
-
C.
π/2
-
D.
0
Solution
Since sin(π/4) = 1/√2, sin^(-1)(1/√2) = π/4.
Correct Answer: A — π/4
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Q. Find the value of sin^(-1)(√(1 - cos^2(θ))).
-
A.
θ
-
B.
π/2 - θ
-
C.
0
-
D.
π/4
Solution
Since sin^2(θ) = 1 - cos^2(θ), we have sin^(-1)(√(1 - cos^2(θ))) = sin^(-1)(sin(θ)) = θ.
Correct Answer: A — θ
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Q. Find the value of sin^(-1)(√3/2) + cos^(-1)(1/2).
-
A.
π/3
-
B.
π/2
-
C.
π/4
-
D.
π/6
Solution
sin^(-1)(√3/2) = π/3 and cos^(-1)(1/2) = π/3. Therefore, the sum is π/3 + π/3 = 2π/3.
Correct Answer: A — π/3
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Q. Find the value of sin^(-1)(√3/2) + sin^(-1)(1/2).
-
A.
π/2
-
B.
π/3
-
C.
π/4
-
D.
π/6
Solution
sin^(-1)(√3/2) = π/3 and sin^(-1)(1/2) = π/6. Therefore, π/3 + π/6 = π/2.
Correct Answer: A — π/2
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Q. Find the value of sin^(-1)(√3/2).
-
A.
π/3
-
B.
π/6
-
C.
π/4
-
D.
2π/3
Solution
sin^(-1)(√3/2) = π/3.
Correct Answer: A — π/3
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Q. Find the value of the coefficient of x^2 in the expansion of (3x - 4)^4.
-
A.
-144
-
B.
-216
-
C.
216
-
D.
144
Solution
The coefficient of x^2 is C(4,2) * (3)^2 * (-4)^2 = 6 * 9 * 16 = 864.
Correct Answer: A — -144
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Q. Find the value of the derivative of f(x) = x^4 - 4x^3 + 6x^2 at x = 1.
Solution
f'(x) = 4x^3 - 12x^2 + 12x. Evaluating at x = 1 gives f'(1) = 4 - 12 + 12 = 4.
Correct Answer: A — 0
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Q. Find the value of the determinant \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) when \( a=1, b=2, c=3, d=4 \).
Solution
The determinant is \( 1*4 - 2*3 = 4 - 6 = -2 \).
Correct Answer: A — -2
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Q. Find the value of the determinant \( |D| \) where \( D = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix} \).
-
A.
-12
-
B.
-10
-
C.
-8
-
D.
-6
Solution
The determinant is calculated as 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1.
Correct Answer: A — -12
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Q. Find the value of the determinant: | 2 3 1 | | 1 0 4 | | 0 5 2 |
Solution
Using the determinant formula, we calculate it to be 1.
Correct Answer: B — 1
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Q. Find the value of the determinant: | 2 3 1 | | 1 0 4 | | 5 6 2 |
-
A.
-20
-
B.
-10
-
C.
10
-
D.
20
Solution
Using the determinant formula, we calculate it to be -10.
Correct Answer: B — -10
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Q. Find the value of the determinant: | 2 3 1 | | 1 0 4 | | 5 6 7 |
-
A.
-30
-
B.
-20
-
C.
20
-
D.
30
Solution
Using the determinant formula, we calculate it to be -20.
Correct Answer: B — -20
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Q. Find the value of the determinant: | x 1 2 | | 3 x 4 | | 5 6 x | when x = 1.
Solution
Substituting x = 1 gives the determinant | 1 1 2 | | 3 1 4 | | 5 6 1 | = 6.
Correct Answer: C — 6
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Q. Find the value of the integral ∫(0 to 1) (1 - x^2)dx.
-
A.
1/3
-
B.
1/2
-
C.
2/3
-
D.
1
Solution
The integral evaluates to [x - (1/3)x^3] from 0 to 1 = 1 - 1/3 = 2/3.
Correct Answer: C — 2/3
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Q. Find the value of the integral ∫(0 to 1) (3x^2)dx.
Solution
The integral ∫(3x^2)dx from 0 to 1 = [x^3] from 0 to 1 = 1.
Correct Answer: A — 1
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Q. Find the value of the integral ∫(0 to 1) (x^2 + 2x)dx.
Solution
The integral evaluates to [(1/3)x^3 + x^2] from 0 to 1 = (1/3 + 1) - 0 = 4/3.
Correct Answer: B — 2
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Q. Find the value of x if 3x + 5 = 20.
Solution
Subtracting 5 from both sides gives 3x = 15, thus x = 15/3 = 5.
Correct Answer: A — 5
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Q. Find the value of z if z^2 + 4z + 8 = 0.
-
A.
-2 + 2i
-
B.
-2 - 2i
-
C.
-4 + 0i
-
D.
-4 - 0i
Solution
Using the quadratic formula, z = [-4 ± √(16 - 32)]/2 = -2 ± 2i.
Correct Answer: A — -2 + 2i
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Q. Find the value of z if z^2 = -16.
Solution
Taking square root, z = ±√(-16) = ±4i.
Correct Answer: A — 4i
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Q. Find the value of z^2 if z = 1 + i.
Solution
z^2 = (1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i.
Correct Answer: B — 2
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Q. Find the value of \( k \) for which the determinant \( \begin{vmatrix} 1 & 2 & 3 \\ 4 & k & 6 \\ 7 & 8 & 9 \end{vmatrix} = 0 \)
Solution
Setting the determinant to zero gives \( k = 6 \).
Correct Answer: B — 6
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Q. Find the value of \( \det \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \).
Solution
The determinant of the identity matrix is always 1.
Correct Answer: B — 1
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Q. Find the value of \( \sin(\sin^{-1}(\frac{1}{2})) \).
-
A.
0
-
B.
\( \frac{1}{2} \)
-
C.
1
-
D.
undefined
Solution
By definition, \( \sin(\sin^{-1}(x)) = x \). Therefore, \( \sin(\sin^{-1}(\frac{1}{2})) = \frac{1}{2} \).
Correct Answer: B — \( \frac{1}{2} \)
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Q. Find the value of ∫ 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. Find the value of ∫ 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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