1. How many edges are there in a graph with 10 vertices each of degree 6?
Solution:
In any graph, the sum of degrees of all vertices is twice the number of edges.
Sum of degrees = 10 × 6 = 60
So, number of edges = 60 / 2 = 301.
2. What is the minimum number of colours required to colour the given graph?
Solution:
The answer given is 3. This is a typical graph colouring problem, where the chromatic number is 31.
3. What is the weight of a minimum cost spanning tree of the given graph?
Solution:
The answer is 23. (This is a direct answer, as the graph is not provided here.)1
4. How many critical points of $ f(x, y) $ are there?
Solution:
The answer is 9. Critical points are where the gradient is zero1.
5. The function $ f(x) = x^3 - 12x $ has a:
Local maximum at $ x = -2 $
Local minimum at $ x = 2 $
Explanation:
Find critical points by setting $ f’(x) = 0 $:
$ f’(x) = 3x^2 - 12 = 0 \implies x^2 = 4 \implies x = -2, 2 $
Second derivative: $ f’’(x) = 6x $
At $ x = -2 $, $ f’’(-2) = -12 < 0 $ ⇒ local maximum
At $ x = 2 $, $ f’’(2) = 12 > 0 $ ⇒ local minimum1.
6. Let $ f $ be differentiable at $ x = 2 $. The tangent at $ (2, 6) $ passes through $ (6, -18) $. What is $ f’(2) $?
Solution:
Slope of the tangent = $ \frac{-18 - 6}{6 - 2} = \frac{-24}{4} = -6 $
So, $ f’(2) = -6 $1.
7. What is the minimum sum of three non-negative numbers whose product is 27?
Solution:
Let the numbers be $ a, b, c \geq 0 $, $ abc = 27 $.
By AM-GM inequality, minimum sum occurs when $ a = b = c $.
So, $ a = b = c = 3 $, sum = $ 3 + 3 + 3 = 9 $1.
8. What is the value of $ f’(2) $ if the tangent at $ (2, 6) $ passes through $ (6, -18) $?
9. Sushant throws a die, then throws as many coins as the number on the die. If the die shows 5, what is the probability of getting exactly 3 heads?
Solution:
Number of coins = 5.
Probability of 3 heads = $ \binom{5}{3} \times (0.5)^3 \times (0.5)^2 = 10 \times 0.125 \times 0.25 = 0.3125 $
So, the answer is approximately 0.3131.
10. Expected number of people shortlisted if for every 5 interviewed, 1 is selected, and 20 people are interviewed?
11. Suppose the lifetime of a radio is uniformly distributed between 100 to 120 weeks. What is the probability it lasts more than 115 weeks given it has already worked for 110 weeks?
12. The number of customers arriving each day at a petrol pump is Poisson with mean 10. What is the probability that on a particular day, more than 3 customers arrive?
Solution:
Probability = $ 1 - P(at most 3) $. For Poisson(10), $ P(X \leq 3) $ is very small, so $ P(X > 3) \approx 1 $. The answer is in the range 0.96 to 11.
13. Find the total numbers greater than 6000 that can be formed using the digits 0, 2, 3, 6, 9 without repetition.
Solution:
Numbers must be 4 or 5 digits, starting with 6 or 9.
For 4-digit:
First digit: 6 or 9 (2 choices), then choose 3 from 4 digits: $ 4 \times 3 \times 2 = 24 $ for each, total 48.
For 5-digit: All digits used, first digit can’t be 0, so 4 choices (2, 3, 6, 9), but only 6 and 9 are >6000, so 2 choices, total $ 4! = 24 $ each, so 48.
Total = 48 + 96 = 1441.
Introduction to Python
14. Which of the following code blocks prints the product of the digits for a given number?
Solution:
The correct code block should:
Initialize a variable (e.g., prod = 1)
Loop through each digit of the number (using while n > 0)
20. If a procedure returns a list of names of customers who have visited shop B, and each customer must be represented exactly once, what is a possible mistake?
Solution:
Incorrect initialization of the set/list to store unique names.
For many questions, the original exam refers to diagrams, code, or data not included here. Where possible, the answer is explained or the formula used is shown.
For MCQs with only options and no context, only the correct answer is provided as per the answer key.
