From a survey of 120 people, the following data was obtained:

90 owned a car, 35 owned a computer, 40 owned a house, 32 owned a car and a house, 21 owned a house and a computer, 26 owned a car and a computer, 17 owned all the three facilities.
i. How many people owned neither of the three.

ii. How many people owned only a car?

iii. How many people owned only a computer? 

How many numbers form 0 to 999 (0 and 999 inclusive) are not divisible by 7 or 11? 

Compute the time complexity of the merge sort algorithm. 

Use generating function to prove the identity

Find the general form of the solution to a linear homogeneous recurrence relation with constant coefficients for which the characteristic roots are 4, -2 and 3 with multiplicities 2, 1 and 3, respectively. The relation also has a non-homogeneous part which is a linear combination of 3ⁿ and 4ⁿ. 

Write the expression  in conjunction normal form and disjunctive normal form. 

For any natural numbers n and k, prove that . 

Write an algorithm that computes the sum of the squares of all the odd integers lying between two given numbers p and q, where . 

A die is rolled twice and the sum of the numbers that appear is observed. What is the probability that the sum is either a perfect square or a perfect cube? 

If 5 points are chosen in a square of side 2cm, show that there will always be two points at a distance of at most cm. 

A bank pays you  interest per year. In addition, you receive Rs. 100 as bonus at the end of the year (after the interest is paid). Find a recurrence for the amount of money after n years if you invest Rs. 2000. 

If the solution of the recurrence relation  is , then determine the values of  and f(n). 

Write down the converse of each of the following statements: 

i. If p is a prime number and a and b are any two natural numbers and if p divides a or b, then p divides ab.

ii. In a triangle , if , then .

What is the probability of selecting 2 red and 3 green balls from a bag containing 4 red balls and 8 green balls? 

If p and q are any statements, show that the statement

 is a tautology. 

Consider the propositions ‘’ and 'The Sun rises in the West'.

i. Write the disjunction of the statements and give its truth value.

ii. Write the conjunction of the statements and give its truth value.

iii. Write the exclusive disjunction of the statements and give its truth value. 

Which of the following statements are true and which are false? Give reasons for your answer. 
(i) ‘x² + y² - 3 is not divisible by 4.’ is a mathematical statement.
(ii) The number of onto functions from  to  is 4!S₆⁴.
(iii) The generating function associated with a sequence can never be a polynomial.
(iv) .
(v) Selection sort is faster than insertion sort in worst case.
(vi) , where n is a power of 2, is a linear recurrence relation.
(vii) The generating function of the sequence  is (1 - z)^-2.
(viii) If g(x) is the generating function for , then (1 - x)g(x) is the generating function for the sequence  where .
(ix) The number of partitions of 10 with no part larger than 5 is Q₁₀⁵.
(x) The number of integer solutions of the equation  in positive integers is .

दिल्ली की चार-पहिया गाड़ियों की संख्या पट्टिका सामान्यतया DL3AXC4126 जैसे दिखती है, जिसके प्रथम दो अक्षर DL नियत होते हैं, तीसरा अक्षर एक अंक होता है, और अगले तीन अक्षर अंग्रेजी के अक्षर हैं, और अंत के चार अक्षर अंक हैं। इस पैटर्न के साथ दिल्ली में कितनी संख्या पट्टिकाएँ पंजीकृत की जा सकती हैं? 

विभाजन  का फेरर ग्राफ बनाइए। साथ ही, इस विभाजन का संयुग्मी विभाजन लिखिए। 

बैल संख्याएँ B₆ और B₇ ज्ञात कीजिए।