Q. Factor the polynomial x^2 - 5x + 6.
A.
(x - 2)(x - 3)
B.
(x + 2)(x + 3)
C.
(x - 1)(x - 6)
D.
(x + 1)(x + 6)
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Solution
To factor x^2 - 5x + 6, we look for two numbers that multiply to 6 and add to -5. The numbers -2 and -3 work. Thus, the factorization is (x - 2)(x - 3).
Correct Answer:
A
— (x - 2)(x - 3)
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Q. Factor the polynomial x^2 - 9.
A.
(x - 3)(x + 3)
B.
(x - 9)(x + 1)
C.
(x + 3)(x + 3)
D.
(x - 1)(x + 9)
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Solution
The expression x^2 - 9 is a difference of squares, which factors to (x - 3)(x + 3).
Correct Answer:
A
— (x - 3)(x + 3)
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Q. If 5x + 2 = 3x + 10, what is the value of x?
A.
x = 1
B.
x = 2
C.
x = 3
D.
x = 4
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Solution
Subtract 3x from both sides: 2x + 2 = 10. Then subtract 2: 2x = 8. Finally, divide by 2: x = 4.
Correct Answer:
B
— x = 2
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Q. If the first term of a geometric progression is 3 and the common ratio is 2, what is the 4th term?
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Solution
The nth term of a geometric progression is given by a_n = a * r^(n-1). Here, a = 3, r = 2, and n = 4. So, a_4 = 3 * 2^(4-1) = 3 * 2^3 = 3 * 8 = 24.
Correct Answer:
A
— 24
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Q. If the first term of a geometric progression is 5 and the common ratio is 3, what is the 3rd term?
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Solution
The nth term of a geometric progression is given by a_n = a * r^(n-1). Here, a = 5, r = 3, and n = 3. So, a_3 = 5 * 3^(3-1) = 5 * 3^2 = 5 * 9 = 45.
Correct Answer:
C
— 135
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Q. If the first term of a geometric sequence is 3 and the common ratio is 2, what is the 4th term?
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Solution
The nth term of a geometric sequence is a * r^(n-1). Here, a = 3, r = 2, n = 4. So, 3 * 2^(4-1) = 3 * 8 = 24.
Correct Answer:
A
— 24
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Q. If the first term of an arithmetic progression is 4 and the common difference is 5, what is the 10th term?
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Solution
The nth term of an AP is given by a_n = a + (n-1)d. Here, a = 4, d = 5, and n = 10. So, a_10 = 4 + (10-1)5 = 4 + 45 = 49.
Correct Answer:
B
— 54
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Q. If the quadratic equation x^2 - 4x - 5 = 0 is factored, what are the roots?
A.
-1, 5
B.
1, -5
C.
5, -1
D.
5, 1
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Solution
Factoring the quadratic gives (x - 5)(x + 1) = 0. Thus, the roots are x = 5 and x = -1.
Correct Answer:
A
— -1, 5
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Q. Solve for x: 5x + 3 = 2x + 12.
A.
x = 3
B.
x = 4
C.
x = 5
D.
x = 6
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Solution
First, subtract 2x from both sides: 3x + 3 = 12. Then subtract 3: 3x = 9. Finally, divide by 3: x = 3.
Correct Answer:
B
— x = 4
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Q. Solve the inequality 2x + 3 > 7.
A.
x < 2
B.
x > 2
C.
x < 3
D.
x > 3
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Solution
To solve 2x + 3 > 7, subtract 3 from both sides: 2x > 4. Then divide by 2: x > 2.
Correct Answer:
B
— x > 2
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Q. Solve the inequality 3x - 5 < 7.
A.
x < 4
B.
x > 4
C.
x < 2
D.
x > 2
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Solution
To solve 3x - 5 < 7, add 5 to both sides: 3x < 12. Then divide by 3: x < 4.
Correct Answer:
A
— x < 4
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Q. What is the 3rd term of the arithmetic progression with first term 4 and common difference 5?
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Solution
The nth term of an arithmetic progression is given by a_n = a + (n-1)d. Here, a = 4, d = 5, and n = 3. So, a_3 = 4 + (3-1)5 = 4 + 10 = 14.
Correct Answer:
B
— 19
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Q. What is the common difference in the arithmetic progression 7, 10, 13, ...?
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Solution
The common difference d is found by subtracting the first term from the second term: d = 10 - 7 = 3.
Correct Answer:
B
— 3
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Q. What is the sum of the first 5 terms of the arithmetic progression 2, 5, 8, ...?
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Solution
The first term a = 2, common difference d = 3. The sum of the first n terms S_n = n/2 * (2a + (n-1)d). For n = 5, S_5 = 5/2 * (2*2 + 4*3) = 5/2 * (4 + 12) = 5/2 * 16 = 40/2 = 20.
Correct Answer:
C
— 20
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Q. What is the sum of the first 6 terms of the arithmetic progression 1, 4, 7, ...?
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Solution
The first term a = 1, common difference d = 3. The sum of the first n terms S_n = n/2 * (2a + (n-1)d). For n = 6, S_6 = 6/2 * (2*1 + 5*3) = 3 * (2 + 15) = 3 * 17 = 51.
Correct Answer:
B
— 45
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Q. What is the sum of the first 6 terms of the geometric progression 1, 3, 9, ...?
A.
364
B.
364/2
C.
364/3
D.
182
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Solution
The first term a = 1, and the common ratio r = 3. The sum of the first n terms of a GP is S_n = a(1 - r^n) / (1 - r). For n = 6, S_6 = 1(1 - 3^6) / (1 - 3) = (1 - 729) / -2 = -728 / -2 = 364.
Correct Answer:
A
— 364
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Q. What is the value of x in the equation 2x^2 - 8 = 0?
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Solution
First, add 8 to both sides: 2x^2 = 8. Then divide by 2: x^2 = 4. Taking the square root gives x = ±2. The positive solution is x = 2.
Correct Answer:
B
— 4
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Q. What is the value of x in the equation x^2 + 6x + 9 = 0?
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Solution
This is a perfect square trinomial: (x + 3)^2 = 0. Therefore, x + 3 = 0, which gives x = -3.
Correct Answer:
A
— -3
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Q. Which expression represents the difference of squares for x^2 - 16?
A.
(x - 4)(x + 4)
B.
(x - 8)(x + 8)
C.
(x - 2)(x + 2)
D.
(x - 16)(x + 16)
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Solution
The expression can be factored as (x - 4)(x + 4) since 16 is 4^2.
Correct Answer:
A
— (x - 4)(x + 4)
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Q. Which expression represents the sum of the first n terms of an arithmetic series with first term a and common difference d?
A.
n/2 * (2a + (n-1)d)
B.
n * (a + d)
C.
n * a + d
D.
n/2 * (a + d)
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Solution
The formula for the sum of the first n terms is S_n = n/2 * (2a + (n-1)d).
Correct Answer:
A
— n/2 * (2a + (n-1)d)
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