RAMAIAH COLLEGE OF ARTS, SCIENCE AND COMMERCE

I SEMESTER BCA (DISCRETE MATHEMATICS)

ASSIGNMENT

Assignment -1(Sets, Relations and Functions)

1. If A={3,5,7} B={6,7,8} and C={7,8,9}, then find (A∩B)×(B∩C).
2. If A={2, 3, 4, 8}, B={1, 3, 4} and U={0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Verify 𝐴 −
𝐵 = 𝐴 ∩ 𝐵′.
3. In a group of 50 students, 23 can speak Hindi only, 13 can speak English only.
How many can speak both Hindi and English? How many students speak
Hindi?
4. In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How
many like tennis only and not cricket? How many like tennis?
5. If A={a, b, c, d}, B={c, d} and C={d, e} find 𝐴 − 𝐵, (𝐴 − 𝐵) ∩ (𝐵 − 𝐶),
𝐵 × 𝐶.
6. If f : R → R is defined by f(x)= 2x+5. Prove that f is one-one and onto.
7. Show that the function f : R → R defined by f(x) = 4x+3 is invertible and find
the inverse of f.

Assignment-2(Logic)
8. Construct the truth table for the proposition p ∨ ~q.
9. Prove that p → q ↔ ( ~q → ~p) is a tautology.
10. Prove that 𝑝 ∨ (𝑞 ∧ 𝑟) ↔ [(𝑝 ∨ 𝑞) ∧ (𝑝 ∨ 𝑟)] is a tautology.
11. If (𝑝 → 𝑞) ∧ (𝑝 ∧ 𝑟) is given to be false, find the truth values of p, q, r.
12. Write the converse, inverse and contrapositive for the proposition “If I work
hard then I get a grade”.
13. Define tautology and contradiction and verify whether the following
compound proposition is a tautology or a contradiction: p →(~p ∨ q)
14. Examine whether each of the following are logically equivalent.
i) p → q and ~p ∨ q ii) ~(p → q) and p ↔ ~q
iii) p → q and (~p ∨ q) ∧ (~q ∨ p)
15. If the truth values of the propositions p, q, r are respectively F, F, T. Find the
truth value of each of the following compound propositions.
i)(𝑝 ∨ 𝑞) ∧ 𝑟 ii) (𝑝 ⟶ 𝑞) ∧ 𝑟 iii)(𝑝 ∨ 𝑟) ⟶ ~𝑞
 Assignment-3(Matrices)
4−𝑦 3 −1 𝑧 + 1
16. Find x, y, z if [ ]=[ ].
𝑥 5 1 5
3
17.If 𝐴 = [2] , 𝐵 = [2 3 5] prove that (𝐴𝐵)′ = 𝐵′𝐴′.
3
3 2 3 5
18. If 𝐴 = [ ] and 𝐵 = [ ] find 2A + 3B.
−1 4 −2 4
4 4 7 −3 2 1
19. If 2𝐴 + 𝐵 = [ ] , 𝐴 − 2𝐵 = [ ] then find A and B.
7 3 4 1 −1 2
20. Find the characteristic equation, Eigen value and Eigen vectors of the
following matrices.
3 0 5 −1 4 1 1 2
i)A= [ ] ii) A= [ ] iii) A= [ ] iv) A= [ ]
2 5 4 9 −1 2 5 4
21. Solve the following equations using Cramer’s rule.
i) x + 3y = 5 , 2x – y = 3
ii) 5x + 3y = 1 , 3x + 5y = −9
iii) 3x – y + 2z = 13 , 2x + y – z = 3 , x + 3y – 5z = -8
iv) x + y + z = 7 , 2x + 3y + 2z = 17 , 4x + 9y + z = 37
22. Find the inverse of the following matrices.
3 −3 4
2 −1
i) [ ] ii) [2 −3 4]
3 2
0 −1 1
23. Solve the following equations using matrix method.
i) 5x + 2y = 4 , 7x + 3y = 5
ii) 2x + 5y = 1 , 3x + 2y = 7
iii) 𝑥 + 𝑦 + 𝑧 = 7, 2𝑥 + 3𝑦 + 2𝑧 = 17, 4𝑥 + 9𝑦 + 𝑧 = 37
iv)𝑥 − 𝑦 − 2𝑧 = 3, 2𝑥 + 𝑦 + 𝑧 = 5, 4𝑥 − 𝑦 − 2𝑧 = 11
24. State and verify the Cayley Hamilton Theorem for the following matrices.
3 1 5 4
i) [ ] ii) [ ]
−1 2 1 2
25. For the following matrices find A4 , A−1 , A−2 using Cayley Hamilton Theorem.
−1 3 3 −2
i) [ ] ii) [ ]
−2 4 4 1

Assignment-4(Logarithms)
1
26.Prove that log 3𝑎 2𝑎 ∙ log 4𝑎2 3𝑎 = .
2
27.Prove that log 𝑏 𝑎 ∙ log 𝑐 𝑏 ∙ log 𝑎 𝑐 = 1.
28.If 𝑙𝑜𝑔7 𝑥 + 𝑙𝑜𝑔7 𝑥 2 + 𝑙𝑜𝑔7 𝑥 3 = 6, find x.
 6 1 81 27
29.If log 𝑥 − 2 log = log − log , find x.
7 2 16 196
𝑎−𝑏 1
30.If 𝑙𝑜𝑔 (
5
) = 2 (log 𝑎 + log 𝑏), show that 𝑎2 + 𝑏 2 = 27𝑎𝑏.
𝑎+𝑏 1
31.If 𝑙𝑜𝑔 ( ) = 2 (log 𝑎 + log 𝑏), show that 𝑎 = 𝑏.
2
32.If 𝑥 = log 2𝑎 𝑎 , 𝑦 = log 3𝑎 2𝑎 , 𝑧 = log 4𝑎 3𝑎 , show that 𝑥𝑦𝑧 + 1 = 2𝑦𝑧.
33. Find the value of each of the following using Logarithmic tables:
i) 6.45 × 981.4 ii) 0.0064 × 1.507
1⁄
(7.41)2 × 38.9 3
34.Evaluate ( (0.251)3
)

Assignment-5(Permutation and Combination)

35.How many 5 digit telephone numbers can be constructed using the digits 0 to 9
if each number starts
with 67 and no digit appears more than once.
36.If 𝑛 𝐶30 = 𝑛 𝐶5 , find n.
37.If 2𝑛 𝐶3 : 𝑛 𝐶2 = 44 ∶ 3 find n.
38. Find the number of three digit even numbers that can be formed using 2, 3, 4,
5, 6 repetitions being not allowed.
39.In how many ways the letters of the word “EVALUATE” be arranged so that
all vowels are together.
40.In how many ways can the letters of the word “ASSASSINATION” be
arranged so that all the S’s are together.
41.How many different words can be formed with the letters of the word
BHARAT
i) In how many of these B and H are never together
ii) How many of these begin with B and end with T
42.A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways
can this be done when the committee consist of i) exactly 3 girls ii) atleast 3
girls iii) atmost 3 girls.
43. How many arrangements can be made with the letters of the word
MATHEMATICS?
i) In how many of them vowels are together
ii) How many of them begin with C
iii) How many of them begin with T
 Assignment-6(Groups)
44. Define group.
𝑎𝑏
45.On the set of integers Z, the binary operation ∗ is defined by 𝑎 ∗ 𝑏 = ,
3
∀ 𝑎, 𝑏 ∈ 𝑍. Find identity element.
46.Define order of a group.
47.Prove that H={0, 2, 4} is a subgroup of G={0, 1, 2, 3, 4, 5} under addition
modulo 6.
48.If 𝐺 = {3𝑛 : 𝑛 ∈ 𝑍} prove that G is an abelian group under multiplication.
49.Prove that G={1, 5, 7, 11} is a group under multiplication modulo 12.
50.Prove that the set G= {3𝑛 |𝑛 ∈ 𝑍} is an abelian group w.r.t addition.
51.Prove that the set 𝐺 = {1, −1, 𝑖, −𝑖} is a group under multiplication.
52.Prove that G = {2, 4, 6, 8} forms an abelian group under ×10 .

Assignment-6(Vectors)
53.If 𝑎⃗ = 2î − 3 ̂j + 4 k̂ , 𝑏⃗⃗ = ̂i − ̂j + 2 k̂ find unit vector along 𝑎⃗ − 𝑏⃗⃗.
54.If 𝑎⃗ = 2î + 3 ̂j + 4 k̂ , 𝑏⃗⃗ = 2î − 7 ̂j + 5 k̂, find |2𝑎⃗ + 𝑏⃗⃗| .
55.If 𝑎⃗ = 2î + ̂j + 4 k̂ , 𝑏⃗⃗ = 3î − ̂j + 2 k̂ and 𝑐⃗ = 3î + ̂j + 4 k̂ find 𝑎⃗ ∙ (𝑏⃗⃗ × 𝑐⃗) .
56.Find the area of the triangle whose vertices are A(3, 2, 1), B(4, −1, 2) and
C(−1, 3, 2) using vector method.
57.Find the value of m if 𝑎⃗ = 𝑚î − 3 ̂j + 4 k̂, 𝑏⃗⃗ = ̂i + 3 ̂j + k̂ and 𝑐⃗ = 2î + ̂j + k̂
are coplanar.
58.If 𝑎⃗ = ̂i − ̂j + 2 k̂ , 𝑏⃗⃗ = 2î + 3 ̂j − k̂ find (𝑎⃗ + 2𝑏⃗⃗) ∙ (2𝑎⃗ − 𝑏⃗⃗).
59.Show that the points A(1, 2, 3), B(2, 3, 1) and C(3, 1, 2) are vertices of an
equilateral triangle.
60.Find the value of 𝜆 for which the vectors 𝑎⃗ = 3î + ̂j − 2 k̂ and
𝑏⃗⃗ = ̂i + 𝜆 ̂j − 3 k̂ are perpendicular to each other.
61.Prove that 𝑎⃗ × (𝑏⃗⃗ × 𝑐⃗) + 𝑏⃗⃗ × (𝑐⃗ × 𝑎⃗) + 𝑐⃗ × (𝑎⃗ × 𝑏⃗⃗) = 0.
62.Using vector method show that the points A(2, −1, 3), B(4, 3, 1) and C(3, 1, 2)
are collinear.

Assignment-7(Analytical Geometry)
63. Find the distance between the points A(2,−3) and B(4, 5).
64.Find the equation of the line passing through (2, 5) and having slope 4.
65.Find the midpoint of line joining (−2, 8) and (1, −2).
66.Prove that the points (6, 4), (7, −2), (5, 1), (4, 7) form vertices of a
parallelogram.
67.Prove that the points A(3, −4), B(4, 2), C(5, −4) and D(4, −10) form vertices
of a rhombus.
68.If a vertex of triangle is (1, 1) and the mid-point points of two sides through the
vertex are (−1, 2) and (3, 2) then find the centroid of the triangle.
𝜋 1
69.The angle between two lines is and the slope of one line is . Find the slope of
4 2
the other line.
70.Find the equation of the locus of a point which moves such that its distance
from X-axis is twice its distance from Y-axis.
71.Derive the equation of the straight line whose x-intercept is ‘a’ and y-intercept
is ‘b’.
72.Find ‘k’ for which the lines 2𝑥 − 𝑘𝑦 + 1 = 0 and 𝑥 + (𝑘 + 1)𝑦 − 1 = 0 are
perpendicular.
73.Find the point of intersection of the straight lines 3𝑥 − 4𝑦 − 1 = 0 and 5𝑥 −
7𝑦 − 1 = 0.
74.Find the equation of the line passing through the point of intersection of 2𝑥 +
3𝑦 − 1 = 0 and 3𝑥 + 4𝑦 − 6 = 0 and parallel to the line 5𝑥 − 𝑦 = 0.
75.Find the ratio in which the X-axis divides the line segment joining the points
(7, −3) and (5, 2).