JEE Main 2023 · 1 February · morning · Single correct · +4 / −1
In a binomial distribution B(n,p), the sum and the product of the mean and the variance are 5 and 6 respectively, then 6(n+p-q) is equal to :
- (a) 52
- (b) 50
- (c) 51
- (d) 53
Answer and solution
Answer (a) 52
Solution
Maths sir · 4 steps
Mean and variance have a known sum and product, so they are the two roots of one quadratic.
- Set up a quadratic. Mean = np and variance = npq. Their sum is 5 and their product is 6, so they are the roots of t² - 5t + 6 = 0, which are 2 and 3.
- Pick the order. Variance = npq is the mean multiplied by q < 1, so the variance is always smaller than the mean. That gives mean np = 3 and variance npq = 2.
- Find q, p, n. Dividing, q = 2/3, so p = 1/3. Then n = 3/p = 9.
6(n + p - q) = 6(9 + 1/3 - 2/3) = 54 - 2 = 52
- So. 6(n + p - q) = 52, which is option (a).
The idea to keep. Whenever you know the mean and variance of a binomial, divide them: variance over mean is q. And since q < 1, the variance is always the smaller of the two.
Answer from the NTA final answer key. Solution by PrepWiser.