Subtract 3x from both sides: 2x + 2 = 10. Then subtract 2: 2x = 8. Finally, divide by 2: x = 4.
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Q2. What is the sum of the first 6 terms of the geometric progression 1, 3, 9, ...?
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.
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Q3. What is the common difference in the arithmetic progression 7, 10, 13, ...?
Solution:
The common difference d is found by subtracting the first term from the second term: d = 10 - 7 = 3.
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Q4. Factor the polynomial x^2 - 9.
Solution:
The expression x^2 - 9 is a difference of squares, which factors to (x - 3)(x + 3).
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Q5. Factor the polynomial x^2 - 5x + 6.
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).
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Q6. Solve the inequality 2x + 3 > 7.
Solution:
To solve 2x + 3 > 7, subtract 3 from both sides: 2x > 4. Then divide by 2: x > 2.
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Q7. If the quadratic equation x^2 - 4x - 5 = 0 is factored, what are the roots?
Solution:
Factoring the quadratic gives (x - 5)(x + 1) = 0. Thus, the roots are x = 5 and x = -1.
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Q8. What is the 3rd term of the arithmetic progression with first term 4 and common difference 5?
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.
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Q9. What is the sum of the first 5 terms of the arithmetic progression 2, 5, 8, ...?
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.
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Q10. Solve the inequality 3x - 5 < 7.
Solution:
To solve 3x - 5 < 7, add 5 to both sides: 3x < 12. Then divide by 3: x < 4.
