Q. Find the solutions of the equation 2sin(x) - 1 = 0 in the interval [0, 2π].
A.
π/6, 5π/6
B.
π/4, 3π/4
C.
π/3, 2π/3
D.
π/2, 3π/2
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Solution
The solutions are x = π/6 and x = 5π/6.
Correct Answer:
A
— π/6, 5π/6
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Q. Find the solutions of the equation 2sin(x) - 1 = 0.
A.
π/6
B.
5π/6
C.
7π/6
D.
11π/6
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Solution
The solutions are x = π/6 and x = 5π/6.
Correct Answer:
A
— π/6
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Q. Find the sum of the first 15 terms of the geometric series where the first term is 2 and the common ratio is 3.
A.
143
B.
145
C.
146
D.
147
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Solution
The sum of the first n terms of a geometric series is S_n = a(1 - r^n) / (1 - r). Here, a = 2, r = 3, n = 15. So, S_15 = 2(1 - 3^15) / (1 - 3) = 2(1 - 14348907) / -2 = 14348906.
Correct Answer:
C
— 146
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Q. Find the sum of the first 5 terms of the series 1, 4, 9, 16, ...
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Solution
The series is the sum of squares: 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55.
Correct Answer:
B
— 31
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Q. Find the sum of the roots of the equation 2x^2 - 3x + 1 = 0.
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Solution
The sum of the roots is given by -b/a = 3/2.
Correct Answer:
A
— 1
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Q. Find the sum of the roots of the equation 3x^2 - 12x + 9 = 0.
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Solution
The sum of the roots is given by -b/a = 12/3 = 4.
Correct Answer:
B
— 4
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Q. Find the surface area of a cone with a radius of 4 cm and a slant height of 5 cm.
A.
25.12 cm²
B.
50.24 cm²
C.
62.83 cm²
D.
78.54 cm²
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Solution
Surface Area = πr(l + r) = π(4)(5 + 4) = π(4)(9) = 36π ≈ 113.10 cm²
Correct Answer:
B
— 50.24 cm²
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Q. Find the term containing x^3 in the expansion of (x + 5)^6.
A.
150
B.
200
C.
250
D.
300
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Solution
The term containing x^3 is C(6,3) * (5)^3 = 20 * 125 = 250.
Correct Answer:
A
— 150
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Q. Find the term containing x^3 in the expansion of (x - 1)^5.
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Solution
The term containing x^3 is C(5,3) * x^3 * (-1)^2 = 10 * x^3 * 1 = 10.
Correct Answer:
C
— -10
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Q. Find the term independent of x in the expansion of (x^2 - 2x + 3)^4. (2022)
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Solution
The term independent of x occurs when the powers of x cancel out. The term is 81.
Correct Answer:
A
— 81
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Q. Find the term independent of x in the expansion of (x^2 - 3x + 1)^5. (2023)
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Solution
The term independent of x occurs when the powers of x cancel out. The term is C(5,2)(-3)^2(1)^3 = 45.
Correct Answer:
A
— -15
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Q. Find the term independent of x in the expansion of (x^2 - 4x + 4)^4. (2020)
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Solution
The expression can be rewritten as (x - 2)^4. The term independent of x occurs when k = 4, which gives us (-2)^4 = 16.
Correct Answer:
C
— 256
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Q. Find the term independent of x in the expansion of (x^2 - 4x + 4)^6. (2020)
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Solution
The expression can be rewritten as (x - 2)^6. The term independent of x occurs when k = 3, which gives us 6C3 * (-2)^3 = 20 * (-8) = -160. The term independent of x is 24.
Correct Answer:
C
— 24
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Q. Find the unit vector in the direction of the vector (3, 4).
A.
(0.6, 0.8)
B.
(0.8, 0.6)
C.
(1, 1)
D.
(0.5, 0.5)
Show solution
Solution
Magnitude = √(3^2 + 4^2) = 5. Unit vector = (3/5, 4/5) = (0.6, 0.8).
Correct Answer:
A
— (0.6, 0.8)
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Q. Find the unit vector in the direction of the vector (3, 4, 0).
A.
(0.6, 0.8, 0)
B.
(0.3, 0.4, 0)
C.
(1, 1, 0)
D.
(0, 0, 1)
Show solution
Solution
Magnitude = √(3^2 + 4^2) = 5. Unit vector = (3/5, 4/5, 0) = (0.6, 0.8, 0).
Correct Answer:
A
— (0.6, 0.8, 0)
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Q. Find the unit vector in the direction of the vector (4, 3).
A.
(4/5, 3/5)
B.
(3/5, 4/5)
C.
(1, 0)
D.
(0, 1)
Show solution
Solution
Unit vector = (4, 3) / √(4^2 + 3^2) = (4, 3) / 5 = (4/5, 3/5).
Correct Answer:
A
— (4/5, 3/5)
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Q. Find the unit vector in the direction of the vector (6, 8).
A.
(0.6, 0.8)
B.
(0.8, 0.6)
C.
(1, 1)
D.
(0.5, 0.5)
Show solution
Solution
Magnitude = √(6^2 + 8^2) = √(36 + 64) = √100 = 10. Unit vector = (6/10, 8/10) = (0.6, 0.8).
Correct Answer:
A
— (0.6, 0.8)
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Q. Find the unit vector in the direction of the vector v = (4, -3).
A.
(4/5, -3/5)
B.
(3/5, 4/5)
C.
(4/3, -3/4)
D.
(3/4, 4/3)
Show solution
Solution
Magnitude |v| = √(4^2 + (-3)^2) = √(16 + 9) = 5. Unit vector = (4/5, -3/5).
Correct Answer:
A
— (4/5, -3/5)
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Q. Find the unit vector in the direction of vector A = 6i - 8j.
A.
3/5 i - 4/5 j
B.
6/10 i - 8/10 j
C.
1/5 i - 2/5 j
D.
2/5 i - 3/5 j
Show solution
Solution
Magnitude |A| = √(6^2 + (-8)^2) = √(36 + 64) = 10. Unit vector = (6/10)i + (-8/10)j = (3/5)i - (4/5)j.
Correct Answer:
A
— 3/5 i - 4/5 j
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Q. Find the unit vector in the direction of vector D = -3i + 4j.
A.
-0.6i + 0.8j
B.
0.6i - 0.8j
C.
0.8i + 0.6j
D.
-0.8i + 0.6j
Show solution
Solution
Magnitude |D| = √((-3)^2 + 4^2) = √(9 + 16) = √25 = 5. Unit vector = D/|D| = (-3/5)i + (4/5)j = -0.6i + 0.8j.
Correct Answer:
A
— -0.6i + 0.8j
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Q. Find the value of (1 + 2)^4 using the binomial theorem.
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Solution
Using the binomial theorem, (1 + 2)^4 = C(4,0) * 1^4 * 2^0 + C(4,1) * 1^3 * 2^1 + C(4,2) * 1^2 * 2^2 + C(4,3) * 1^1 * 2^3 + C(4,4) * 1^0 * 2^4 = 1 + 8 + 24 + 32 + 16 = 81.
Correct Answer:
A
— 16
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Q. Find the value of (1 + i)^2.
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Solution
(1 + i)^2 = 1^2 + 2(1)(i) + i^2 = 1 + 2i - 1 = 2i.
Correct Answer:
B
— 2
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Q. Find the value of (1 + i)^4.
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Solution
(1 + i)^4 = (√2 e^(iπ/4))^4 = 4 e^(iπ) = 4(-1) = -4.
Correct Answer:
C
— 8
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Q. Find the value of (1 + i)².
A.
2i
B.
2
C.
0
D.
1 + 2i
Show solution
Solution
(1 + i)² = 1² + 2(1)(i) + i² = 1 + 2i - 1 = 2i.
Correct Answer:
B
— 2
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Q. Find the value of (1 + x)^10 at x = 1. (2048)
A.
10
B.
11
C.
1024
D.
2048
Show solution
Solution
Using the binomial theorem, (1 + 1)^10 = 2^10 = 1024.
Correct Answer:
C
— 1024
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Q. Find the value of (1 + x)^10 at x = 2.
A.
1024
B.
2048
C.
512
D.
256
Show solution
Solution
Using the binomial theorem, (1 + 2)^10 = 3^10 = 59049.
Correct Answer:
B
— 2048
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Q. Find the value of (1 + x)^6 when x = 2.
A.
64
B.
128
C.
256
D.
512
Show solution
Solution
Using the binomial theorem, (1 + 2)^6 = 3^6 = 729.
Correct Answer:
C
— 256
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Q. Find the value of (3 + 2)^3 using the binomial theorem.
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Solution
Using the binomial theorem, (3 + 2)^3 = C(3,0) * 3^3 * 2^0 + C(3,1) * 3^2 * 2^1 + C(3,2) * 3^1 * 2^2 + C(3,3) * 3^0 * 2^3 = 27 + 54 + 36 + 8 = 125.
Correct Answer:
B
— 27
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Q. Find the value of (a + b)^4 when a = 2 and b = 3.
A.
81
B.
125
C.
625
D.
256
Show solution
Solution
Using the binomial theorem, (a + b)^4 = C(4, 0)a^4b^0 + C(4, 1)a^3b^1 + C(4, 2)a^2b^2 + C(4, 3)a^1b^3 + C(4, 4)a^0b^4. Substituting a = 2 and b = 3 gives 16 + 4*6 + 6*9 + 4*27 + 81 = 81.
Correct Answer:
A
— 81
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Q. Find the value of 3^3 - 2^3. (2020)
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Solution
3^3 = 27 and 2^3 = 8, so 27 - 8 = 19.
Correct Answer:
A
— 19
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