Q1. What is the value of x in the equation x^2 + 6x + 9 = 0?
Solution:
This is a perfect square trinomial: (x + 3)^2 = 0. Therefore, x + 3 = 0, which gives x = -3.
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Q2. Solve for x: 5x + 3 = 2x + 12.
Solution:
First, subtract 2x from both sides: 3x + 3 = 12. Then subtract 3: 3x = 9. Finally, divide by 3: x = 3.
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Q3. If the first term of an arithmetic progression is 4 and the common difference is 5, what is the 10th term?
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.
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Q4. 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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Q5. 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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Q6. 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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Q7. Which expression represents the sum of the first n terms of an arithmetic series with first term a and common difference d?
Solution:
The formula for the sum of the first n terms is S_n = n/2 * (2a + (n-1)d).
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Q8. If the first term of a geometric progression is 5 and the common ratio is 3, what is the 3rd term?
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.
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Q9. 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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Q10. If the first term of a geometric sequence is 3 and the common ratio is 2, what is the 4th term?
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.
