10 questionswith explanationsfor RAS, Patwari, VDO and other Rajasthan exams
Consider the following statements about the polynomial x² − 5x − 24: 1. (x − 8) is a factor of it. 2. (x − 3) is a factor of it. Which of the statements given above is/are correct?
Answer: (1) 1 only
Explanation
x² − 5x − 24 = (x − 8)(x + 3). So (x − 8) is a factor, but (x + 3), not (x − 3), is the other one.
The roots of the quadratic equation x² − 7x + k = 0 differ by 3. What is the value of k?
Answer: (2) 10
Explanation
Sum of roots = 7 and difference = 3, so the roots are 5 and 2; k = product = 10.
If log₁₀ 2 = 0.3010 and log₁₀ 3 = 0.4771, what is the value of log₁₀ 72?
Answer: (3) 1.8572
Explanation
72 = 2³ × 3², so log 72 = 3(0.3010) + 2(0.4771) = 0.9030 + 0.9542 = 1.8572.
Which value(s) of x satisfy the equation log₂(x − 1) + log₂(x + 1) = 3?
Answer: (4) 3 only
Explanation
log₂[(x − 1)(x + 1)] = 3 gives x² − 1 = 8, so x = ±3. A logarithm needs a positive argument, so x = −3 is rejected; only x = 3.
For what value of k is the polynomial 2x³ + 5x² + kx − 3 exactly divisible by (x + 3)?
Answer: (2) −4
Explanation
By the factor theorem, (x + 3) divides the polynomial when p(−3) = 0: 2(−27) + 5(9) + k(−3) − 3 = −54 + 45 − 3k − 3 = −12 − 3k = 0, so k = −4. Check by division: 2x³ + 5x² − 4x − 3 = (x + 3)(2x² − x − 1). Taking k = 4 comes from a sign slip, writing −3k as +3k.
Taking log₁₀ 2 = a and log₁₀ 3 = b, log₁₀ 15 expressed in terms of a and b is:
Answer: (3) 1 + b − a
Explanation
Split 15 using 2, 3 and 10: 15 = (3 × 10)/2, so log 15 = log 3 + log 10 − log 2 = b + 1 − a. Check with values: 0.4771 + 1 − 0.3010 = 1.1761 = log 15. The option a + b is log 6, not log 15, because 2 × 3 = 6, and 1 + a − b = log (20/3).
If a + b = 7 and ab = 10, what is the value of a³ + b³?
Answer: (2) 133
Explanation
a³ + b³ = (a + b)³ − 3ab(a + b) = 343 − 3 × 10 × 7 = 343 − 210 = 133 (check: a and b are 2 and 5, and 8 + 125 = 133). The value 343 is only (a + b)³, and 553 adds the term instead of subtracting it.
If one root of the equation x² − 8x + k = 0 is three times the other root, what is the value of k?
Answer: (1) 12
Explanation
Let the roots be a and 3a. Their sum is 4a = 8, so a = 2 and the roots are 2 and 6; the product k = 2 × 6 = 12. The value 16 is the k for equal roots (4 and 4), not for roots in the ratio 1 : 3.
Two numbers x and y differ by 4 (x − y = 4) and their product xy is 21. Find the value of x² + y².
Answer: (1) 58
Explanation
x² + y² = (x − y)² + 2xy = 16 + 42 = 58. Adding 16 and 21 gives 37, and 100 is the value of (x + y)².
Consider the following statements about the equation 2x² − 5x − 12 = 0: 1. The product of its roots is −6. 2. The sum of its roots is −5/2. 3. One of its roots is 4. Which of the statements given above are correct?
Answer: (3) 1 and 3 only
Explanation
For ax² + bx + c = 0, the product of the roots is c/a = −12/2 = −6 and the sum is −b/a = 5/2, not −5/2, so statement 2 is wrong. The equation factorises as (x − 4)(2x + 3) = 0, so the roots are 4 and −3/2.