Intermediate Part FIRST Series Test#1CHAPTER NO.
1,2
MULTIPLE CHOICE QUESTIONS
i. The solution of the equation x +a=b if a> b where a, b ∈ W is
(a) Natural number (b) Whole number (c) Integers (d) Rational number
ii. The property of inequality used in a> band b> c ⇒ a>c is:
(a) Additive property (b) Multiplicative property (c) Transitive pro. (d) Additive inverse
iii. π and e are
(a) Integers (b) Natural numbers (c) Rational numbers (d) Irrational number
iv. If ‘n’ is a prime number then its square root is:
(a) An irrational number (b) A rational number (c) Prime number (d) None of these
v. Every real number is a complex number with zero as its
(a) Real part (b) Imaginary part (c) à & b both (d) None of these
( )
2
b −b
vi. 4 2
, 4 2 is multiplicative inverse of:
b +b b + b
(a) (b3 ,b) ( 1b )
(b) b , (c) (b2 ,b) (d) b - b2
vii. if z=a+ibthen z . z
(a) a+ b (b) (a+ ib)2 (c) a2 + b2 (d) a2 - b2
viii. ¿ ¿ __________
(a) cosnθ – i sin nθ (b) cos (−n ) θ−i sin(−n)θ (c) cosθ +sinθ (d) cosnθ−sin nθ
ix. The figure representing one or more complex numbers on the z-plane is called
(a) Argand diagram (b) Histogram (c) Polygon (d) Histogram
x. Which set does not satisfy the order axioms:
(a) C (b) Q (c) N (d) Q'
xi. If (1−1) correspondence can be established between the sets, then they are
(a) Equal (b) Equivalent (c) Finite (d) Infinite
xii. If A ⊆ B then:
(a) AUB = A (b) A∩B = A (c) AUB = A' (d) A∩B = A’
xiii. The possible number subsets of the set {a ,{b , c }} are
(a) zero (b) 1 (c) 4 (d) 2
xiv. Number of identity elements in any group is/are:
(a) 2 (b) 1 (c) 3 (d) None of these
xv. If A ∩ B=Ø then n(B−A )=¿
(a) n( A) (b) n(B) (c) n(B)−n( A ∩ B) (d) n( A)−n( A ∩ B)
xvi. ( p ⋀ q)=?
(a) p ∨ q (b) p V q (c) ~ (p∧q) (d) None of these
xvii. The inverse of a linear function {(x , y )/ y=mx +c } is
x−c
(a) { y−mx−c=0 } (b) { ( x , y ) / y= } (c) {(x , y )x =my+ c } (d) {(x , y ) y=x }
m
xviii. If a function f → A+ B is such that range f =B then f is called an:
(a) Into function (b) 1 - 1 function (c) Onto function (d) Bijective function
xix. The set Z+∪ {0} is monoid w.r.t multiplication but not
(a) Addition (b) Subtraction (c) Division (d) None
xx. The set N with respect to addition is a:
(a) Group (b) Monoid (c) Null set (d)Semi group
Q No. 2. Write at least Eight (8) short answers of the following questions.
(i) Define recurring decimals.
(ii) Define binary operation.
(iii) Does the set {0 , – 1} satisfy closure w.r.t. addition and multiplication?
(iv) Simplify(5 , – 4)+¿).
(v) Define an Irrational Number.
a c
+
b d
(vi) Simplify by justifying each step.
a c
−
b d
(vii) Find the multiplicative inverse of(−4 , 7).
(viii) Factorize 3x2 + 3y2
2−7 i
(ix) Separate into real and imaginary parts of
4+5 i
−17 5 −21−10
(x) Prove that − =
12 18 36
(xi) Prove that the sum as well as product of any two conjugate complex numbers is a real number.
i
(xii) Separate into real and imaginary parts of
1+ i
Q No. 2. Write at least Eight (8) short answers of the following questions.
2
(i). Simplify by expressing in the form of a + bi
√5+ √−8
(ii). Show that ∀ z ∈ C, ¿ z ) is real number.
( )
3
−1 √ 3
(iii). Simplify + i
2 2
(iv). Simplify (a+ bi)−2
(v). Define complex number.
(vi). Write down the power set of set {a ,{b , c }}
(vii). Define proper subset.
(viii). Exhibit AUB and A∩B by Venn diagrams when A and B are overlapping sets.
(ix). Show that A−B and B− A by Venn diagram when A and B are overlapping sets.
(x). Using Venn diagram, verify that AUB = AU (A' ∩ B').
(xi). Write the converse, inverse and contra positive of p ∧ q
(xii). Define tautology.
Q.4 Write at least Nine (9) short answers of the following questions.
(i). Construct truth table for p ⟶( p ∨q)
(ii). Show that ∼ ( p ⟶ q ) ⟷( p ∧∼ q) is tautology
(iii). If U = {1, 2, 3, 4, 5, ...... 20}, A = {1, 3, 5, ...... 19} verify the following AU A' = U
(iv). For A = {1, 2, 3, 4), find the following relation in A. State the domain and range of {(x , y )∨x + y <5 }
(v). Define Injective function.
(vi). Define semi group.
(vii). Prepare a table of addition of the elements of the set of residue classes modulo.
(viii). Find the inverse of {(x , y )∨ y =2 x +3 , x ∈ R }
(ix). Differentiate between a contingency and an absurdity.
(x). If G is a group under the operation * anda , b ∈ G, find the solution of equationa∗x=b .
(xi). Define monoid
(xii). Prepare a table of multiplication of the elements of the set of residue classes modulo 5.
(xiii). Show that the set of natural numbers is not closed with respect to subtraction
SECTION-II
Note: Attempt any three (3) questions. (10 x 3 = 30)
Q No 5 (a) State and prove De Morgan Laws analytically
(b) Convert into logical form and prove: A ∪ ( B∩ C )= ( A ∪ B ) ∩( A ∪ C )
Q No 6 (a) Use Venn diagrams to verify the A−B= A ∩ Bc
(b) Prove that p ⋁ ( ∽ p ⋀ ∽ q ) ⋁ ( p ⋀ q )= p ⋁ (∽ p ⋀ ∽ q)
Q No 7 (a) Set S= { 1,−1 ,i ,−i } setup its multiplication table and show, it is an abelian group under
multiplication
(b) Prove that all 2 × 2 non – singular matrices form a non- abelian group under multiplication.
Q No 8 (a) Show that the set {1 , ω , ω 2 }, whereω 3=1 , is an Abelian group w.r.t ordinary multiplication
(b) If G be a group then shows that identity element is unique and inverse of each element is unique
Q No 9 (a) State the distributive property of intersection over union and prove it analytically
(b) Prove that the set of numbers of the form a+ √ 3 b (where a , b ∈ Q¿ form group under
addition
