1
COMPOSED BY:- MUHAMMAD SALMAN SHERAZI 03337727666/03067856232
MATHEMATICS
1st Year
SHORT TERM PREPARATION
IMPORTANT MCQs
IMPORTANT Short Questions
IMPORTANT Long Questions
EXERCISE WISE
M.SALMAN SHERAZI
03337727666/03067856232
,2
COMPOSED BY:- MUHAMMAD SALMAN SHERAZI 03337727666/03067856232
EXERCISE 1.1
Tick (✔) the correct answer.
1. √𝟐 is a number
(a) Rational (b) ✔ Irrational (c) Prime (d) Natural
𝒑
2. The numbers which can be written in the form of 𝒒 , 𝒑, 𝒒 ∈ 𝒁 , 𝒒 ≠ 𝟎 are :
(a) ✔Rational number (b) Irrational number (c) Complex number (d) Whole number
3. A decimal which has a finite numbers of digits in its decimal part is called______ decimal.
(a) ✔Terminating (b) Non-Terminating (c) Recurring (d) Non recurring
4. 𝟓. 𝟑𝟑𝟑 …. Is
(a) ✔Rational (b) Irrational (c) an integer (d) a prime number
5. 𝝅 is
(a) Rational (b) ✔ Irrational (c) Natural number (d) None
𝟐𝟐
6. 𝟕
is
(a) ✔Rational (b) Irrational (c) an integer (d) a whole number
7. Multiplicative inverse of ′𝟎′ is
(a) 0 (b) any real number (c) ✔ not defined (d) 1
𝒂
8. Golden rule of fraction is that for 𝒌 ≠ 𝟎, 𝒃 =
𝑘𝑎 𝑎𝑏 𝑘𝑎 𝑘𝑏
(a) ✔𝑘𝑏 (b) 𝑙 (c) 𝑏 (d) 𝑏
9. The set {𝟏, −𝟏} possesses closure property 𝒘. 𝒓. 𝒕
(a) ′+′ (b) ✔ ′ × ′ (c) ′ ÷ ′ (d) ′ − ′
10. If 𝒂 < 𝑏 then
1 1 1 1
(a) 𝑎<𝑏 (b) 𝑎 < 𝑏 (c) ✔ 𝑎
>𝑏 (d) 𝑎 − 𝑏 > 0
SHORT QUESTONS
i. Write down the “Closure Property for addition”.
ii. Deos the set {𝟏, −𝟏} possess closure property with respect to
(a) addition (b) multiplication
iii. Name the properties : 𝟏𝟎𝟎𝟎 × 𝟏 = 𝟏𝟎𝟎𝟎 and – 𝟑 < −2 ⇒ 0 < 1
𝟕 𝟓 −𝟐𝟏−𝟏𝟎
iv. Prove that – 𝟏𝟐 − 𝟏𝟖 = 𝟑𝟔
𝟒+𝟏𝟔𝒙
v. Simplify justifying each step :
𝟒
EXERCISE 1.2
Tick (✔) the correct answer.
1. The multiplicative identity of complex number is
(a) (0,0) (b) (0,1) (c) ✔ (1,0) (d) (1,1)
𝟏𝟑
2. 𝒊 equals:
(a) ✔𝑖 (b) – 𝑖 (c) 1 (d) -1
3. The multiplicative inverse of (𝟒, −𝟕) is:
4 7 4 7 4 7 4 7
(a) (− 65 , − 65) (b) (− 65 , 65) (c) (65 , − 65) (d) ✔ (65 , 65)
4. (𝟎, 𝟑)(𝟎, 𝟓) =
(a) 15 (b) ✔ -15 (c) −8𝑖 (d) 8𝑖
𝟐𝟏
−
5. (−𝟏) 𝟐 =
(a) 𝑖 (b) ✔ – 𝑖 (c) 1 (d) -1
𝟏
,3
COMPOSED BY:- MUHAMMAD SALMAN SHERAZI 03337727666/03067856232
8. The multiplicative inverse of (𝟎, 𝟎) is:
(a) (0,1) (b) (1,0) (c) (0,0) (d) ✔ Does not exist
9. The product of any two conjugate complex numbers is
(a) ✔Real number (b) complex number (c) zero (d) 1
10. Identity element of complex number is
(a) (0,1) (b) (0,1) (c) (0,0) (d) ✔(1,0)
SHORT QUESTIONS
𝒂 −𝒃
i. Prove that “multiplicative inverse” of (𝒂, 𝒃) is (𝒂𝟐+𝒃𝟐 , 𝒂𝟐 +𝒃𝟐 ).
ii. Find the “multiplicative inverse” of (𝟏, 𝟎)
iii. Factorize 𝒂𝟐 + 𝟒𝒃𝟐 and 𝟗𝒂𝟐 + 𝟏𝟔𝒃𝟐
iv. Prove that the sum as well as the product of two conjugate complex numbers is a real
number.
𝒊
v. Separate into real and imaginary parts : 𝟏+𝒊
𝟐𝟏
vi. Simplify the following : (−𝟏)− 𝟐 and 𝒊𝟗
−𝟏𝟔
vii. Write in terms of 𝒊: √−𝟓 and √ 𝟐𝟓
viii. Simplify the following (𝟐, 𝟔)(𝟑, 𝟕) and (𝟐, 𝟔) ÷ (𝟑, 𝟕)
EXERCISE 1.3
Tick (✔) the correct answer.
1. If 𝒛1 and 𝒛2 are complex numbers then |𝒛1+𝒛2| is _______
(a) <|𝑧1+𝑧2| (b) ✔ ≤|𝑧1 |+|𝑧2| (c) ≥ |𝑧1+𝑧2| (d) None of these
2. The figure representing one or more complex numbers on the complex plane is called:
(a) Cartesian plane (c) Z-Plane (c) Complex plane (d) ✔Argand diagram
3. 𝒚 − 𝒂𝒙𝒊𝒔 represents
(a) Real numbers (b) ✔ Imaginary numbers (c) natural numbers (d) Rational numbers
4. If 𝒛 = 𝒙 + 𝒊𝒚 then |𝒛| =
(a) 𝑥 2 + 𝑦 2 (b) 𝑥 2 − 𝑦 2 (c) ✔ √𝑥 2 + 𝑦 2 (d) √𝑥 2 − 𝑦 2
5. The moduli of 3 is
(a) ✔3 (b) 4 (c) 5 (d) 6
6. 𝒛𝒛̅ =
(a) 𝑧 2 (b) 𝑧 (c) 𝑧̅ (d) ✔ |𝑧|2
7. (𝒛 − 𝒛̅)𝟐 is
(a) Complex number (b) ✔ Real number (c) both (a) and (b) (d) None of these
𝟏𝟎𝟏
8. 𝒊 =
(a) 1 (b) −1 (c) ✔ 𝑖 (d) – 𝑖
SHORT QUESTIONS
i. Show that 𝒛𝟐 + 𝒛̅𝟐 is a real number.
ii. Prove that 𝒛̅ = 𝒛 𝒊𝒇𝒇 𝒛 is real.
iii. Simplify (−𝒂𝒊)𝟒 , 𝒂 ∈ 𝑹
iv. Simplify the following 𝟓 + 𝟐√−𝟒
, 4
COMPOSED BY:- MUHAMMAD SALMAN SHERAZI 03337727666/03067856232
EXERCISE 2.1
Tick (✔) the correct answer.
1. A set is a collection of objects which are
(a) Well defined (b) ✔ Well defined and distinct (c) identical (d) not defined
2. The set of odd numbers between 1 and 9 are
(a) {1,3,5,7} (b) {3,5,7,9} (c) {1,3,5,7,9} (d) ✔ {3,5,7}
3. There are _______ methods to describe a set.
(a) 2 (b) ✔ 3 (c) 4 (d) 5
4. {1,2,3} and {2,1,3} are sets.
(a) ✔Equal (b) Equivalent (c) Not equal (d) None of these
5. The sets 𝑵 and 𝑶 are sets.
(a) Equal (b) ✔Equivalent (c) Not equal (d) None of these
6. Which of the following is true?
(a) 𝑁 ⊂ 𝑅 ⊂ 𝑄 ⊂ 𝑍 (b) 𝑅 ⊂ 𝑍 ⊂ 𝑄 ⊂ 𝑁 (c) 𝑍 ⊂ 𝑁 ⊂ 𝑄 ⊂ 𝑅 (d) ✔ 𝑁 ⊂ 𝑍 ⊂ 𝑄 ⊂ 𝑅
7. The empty set is a subset of
(a) Empty set (b) ✔Every set (c) Natural set (d) Whole set
8. Total number of subsets that can be formed from the set {𝒙, 𝒚, 𝒛} is
(a) 1 (b) ✔ 8 (c) 5 (d)2
9. A set having only one element is called
(a) Empty set (b) ✔Singleton set (c) Power set (d) Subset
10. The set of odd integers between 2 and 4 is
(a) Null set (b) Power set (c) ✔ Singleton set (d) Subset
SHORT QUESTIONS
i. If 𝑨 = {𝒂, 𝒃}, then find 𝑷(𝑨).
ii. Write the following sets in “𝑺𝒆𝒕 − 𝒃𝒖𝒊𝒍𝒅𝒆𝒓 𝒎𝒆𝒕𝒉𝒐𝒅”
(a) {January , June , July } (b) {100, 101, 102,….., 400}
iii. Write the following set in “𝒅𝒆𝒔𝒄𝒓𝒊𝒑𝒕𝒊𝒗𝒆 𝒎𝒆𝒕𝒉𝒐𝒅” and “ 𝒕𝒂𝒃𝒖𝒍𝒂𝒓 𝒇𝒐𝒓𝒎”
(a) {𝒙|𝒙 ∈ 𝑵 ∧ 𝟒 < 𝑥 < 12} (b) {𝒙|𝒙 ∈ 𝑹 ∧ 𝒙 = 𝒙}
iv. Write two power subsets of 𝑵 and {𝒂, 𝒃, 𝒄}
v. Is there any set which has no proper sub set? If so name that set.
vi. What is the difference between {𝒂, 𝒃} and {{𝒂, 𝒃}}?
vii. Write down the power set of {𝝋} and {+, −, ×,÷}.
EXERCISE 2.2
Tick (✔) the correct answer.
1. A diagram which represents a set is called______
(a) ✔Venn’s (b) Argand (c) Plane (d) None of these
2. 𝑨 ∪ 𝝋 =
(a) 𝜑 (b) 𝑈 (c) ✔ 𝐴 (d) 𝑈 − 𝐴
COMPOSED BY:- MUHAMMAD SALMAN SHERAZI 03337727666/03067856232
MATHEMATICS
1st Year
SHORT TERM PREPARATION
IMPORTANT MCQs
IMPORTANT Short Questions
IMPORTANT Long Questions
EXERCISE WISE
M.SALMAN SHERAZI
03337727666/03067856232
,2
COMPOSED BY:- MUHAMMAD SALMAN SHERAZI 03337727666/03067856232
EXERCISE 1.1
Tick (✔) the correct answer.
1. √𝟐 is a number
(a) Rational (b) ✔ Irrational (c) Prime (d) Natural
𝒑
2. The numbers which can be written in the form of 𝒒 , 𝒑, 𝒒 ∈ 𝒁 , 𝒒 ≠ 𝟎 are :
(a) ✔Rational number (b) Irrational number (c) Complex number (d) Whole number
3. A decimal which has a finite numbers of digits in its decimal part is called______ decimal.
(a) ✔Terminating (b) Non-Terminating (c) Recurring (d) Non recurring
4. 𝟓. 𝟑𝟑𝟑 …. Is
(a) ✔Rational (b) Irrational (c) an integer (d) a prime number
5. 𝝅 is
(a) Rational (b) ✔ Irrational (c) Natural number (d) None
𝟐𝟐
6. 𝟕
is
(a) ✔Rational (b) Irrational (c) an integer (d) a whole number
7. Multiplicative inverse of ′𝟎′ is
(a) 0 (b) any real number (c) ✔ not defined (d) 1
𝒂
8. Golden rule of fraction is that for 𝒌 ≠ 𝟎, 𝒃 =
𝑘𝑎 𝑎𝑏 𝑘𝑎 𝑘𝑏
(a) ✔𝑘𝑏 (b) 𝑙 (c) 𝑏 (d) 𝑏
9. The set {𝟏, −𝟏} possesses closure property 𝒘. 𝒓. 𝒕
(a) ′+′ (b) ✔ ′ × ′ (c) ′ ÷ ′ (d) ′ − ′
10. If 𝒂 < 𝑏 then
1 1 1 1
(a) 𝑎<𝑏 (b) 𝑎 < 𝑏 (c) ✔ 𝑎
>𝑏 (d) 𝑎 − 𝑏 > 0
SHORT QUESTONS
i. Write down the “Closure Property for addition”.
ii. Deos the set {𝟏, −𝟏} possess closure property with respect to
(a) addition (b) multiplication
iii. Name the properties : 𝟏𝟎𝟎𝟎 × 𝟏 = 𝟏𝟎𝟎𝟎 and – 𝟑 < −2 ⇒ 0 < 1
𝟕 𝟓 −𝟐𝟏−𝟏𝟎
iv. Prove that – 𝟏𝟐 − 𝟏𝟖 = 𝟑𝟔
𝟒+𝟏𝟔𝒙
v. Simplify justifying each step :
𝟒
EXERCISE 1.2
Tick (✔) the correct answer.
1. The multiplicative identity of complex number is
(a) (0,0) (b) (0,1) (c) ✔ (1,0) (d) (1,1)
𝟏𝟑
2. 𝒊 equals:
(a) ✔𝑖 (b) – 𝑖 (c) 1 (d) -1
3. The multiplicative inverse of (𝟒, −𝟕) is:
4 7 4 7 4 7 4 7
(a) (− 65 , − 65) (b) (− 65 , 65) (c) (65 , − 65) (d) ✔ (65 , 65)
4. (𝟎, 𝟑)(𝟎, 𝟓) =
(a) 15 (b) ✔ -15 (c) −8𝑖 (d) 8𝑖
𝟐𝟏
−
5. (−𝟏) 𝟐 =
(a) 𝑖 (b) ✔ – 𝑖 (c) 1 (d) -1
𝟏
,3
COMPOSED BY:- MUHAMMAD SALMAN SHERAZI 03337727666/03067856232
8. The multiplicative inverse of (𝟎, 𝟎) is:
(a) (0,1) (b) (1,0) (c) (0,0) (d) ✔ Does not exist
9. The product of any two conjugate complex numbers is
(a) ✔Real number (b) complex number (c) zero (d) 1
10. Identity element of complex number is
(a) (0,1) (b) (0,1) (c) (0,0) (d) ✔(1,0)
SHORT QUESTIONS
𝒂 −𝒃
i. Prove that “multiplicative inverse” of (𝒂, 𝒃) is (𝒂𝟐+𝒃𝟐 , 𝒂𝟐 +𝒃𝟐 ).
ii. Find the “multiplicative inverse” of (𝟏, 𝟎)
iii. Factorize 𝒂𝟐 + 𝟒𝒃𝟐 and 𝟗𝒂𝟐 + 𝟏𝟔𝒃𝟐
iv. Prove that the sum as well as the product of two conjugate complex numbers is a real
number.
𝒊
v. Separate into real and imaginary parts : 𝟏+𝒊
𝟐𝟏
vi. Simplify the following : (−𝟏)− 𝟐 and 𝒊𝟗
−𝟏𝟔
vii. Write in terms of 𝒊: √−𝟓 and √ 𝟐𝟓
viii. Simplify the following (𝟐, 𝟔)(𝟑, 𝟕) and (𝟐, 𝟔) ÷ (𝟑, 𝟕)
EXERCISE 1.3
Tick (✔) the correct answer.
1. If 𝒛1 and 𝒛2 are complex numbers then |𝒛1+𝒛2| is _______
(a) <|𝑧1+𝑧2| (b) ✔ ≤|𝑧1 |+|𝑧2| (c) ≥ |𝑧1+𝑧2| (d) None of these
2. The figure representing one or more complex numbers on the complex plane is called:
(a) Cartesian plane (c) Z-Plane (c) Complex plane (d) ✔Argand diagram
3. 𝒚 − 𝒂𝒙𝒊𝒔 represents
(a) Real numbers (b) ✔ Imaginary numbers (c) natural numbers (d) Rational numbers
4. If 𝒛 = 𝒙 + 𝒊𝒚 then |𝒛| =
(a) 𝑥 2 + 𝑦 2 (b) 𝑥 2 − 𝑦 2 (c) ✔ √𝑥 2 + 𝑦 2 (d) √𝑥 2 − 𝑦 2
5. The moduli of 3 is
(a) ✔3 (b) 4 (c) 5 (d) 6
6. 𝒛𝒛̅ =
(a) 𝑧 2 (b) 𝑧 (c) 𝑧̅ (d) ✔ |𝑧|2
7. (𝒛 − 𝒛̅)𝟐 is
(a) Complex number (b) ✔ Real number (c) both (a) and (b) (d) None of these
𝟏𝟎𝟏
8. 𝒊 =
(a) 1 (b) −1 (c) ✔ 𝑖 (d) – 𝑖
SHORT QUESTIONS
i. Show that 𝒛𝟐 + 𝒛̅𝟐 is a real number.
ii. Prove that 𝒛̅ = 𝒛 𝒊𝒇𝒇 𝒛 is real.
iii. Simplify (−𝒂𝒊)𝟒 , 𝒂 ∈ 𝑹
iv. Simplify the following 𝟓 + 𝟐√−𝟒
, 4
COMPOSED BY:- MUHAMMAD SALMAN SHERAZI 03337727666/03067856232
EXERCISE 2.1
Tick (✔) the correct answer.
1. A set is a collection of objects which are
(a) Well defined (b) ✔ Well defined and distinct (c) identical (d) not defined
2. The set of odd numbers between 1 and 9 are
(a) {1,3,5,7} (b) {3,5,7,9} (c) {1,3,5,7,9} (d) ✔ {3,5,7}
3. There are _______ methods to describe a set.
(a) 2 (b) ✔ 3 (c) 4 (d) 5
4. {1,2,3} and {2,1,3} are sets.
(a) ✔Equal (b) Equivalent (c) Not equal (d) None of these
5. The sets 𝑵 and 𝑶 are sets.
(a) Equal (b) ✔Equivalent (c) Not equal (d) None of these
6. Which of the following is true?
(a) 𝑁 ⊂ 𝑅 ⊂ 𝑄 ⊂ 𝑍 (b) 𝑅 ⊂ 𝑍 ⊂ 𝑄 ⊂ 𝑁 (c) 𝑍 ⊂ 𝑁 ⊂ 𝑄 ⊂ 𝑅 (d) ✔ 𝑁 ⊂ 𝑍 ⊂ 𝑄 ⊂ 𝑅
7. The empty set is a subset of
(a) Empty set (b) ✔Every set (c) Natural set (d) Whole set
8. Total number of subsets that can be formed from the set {𝒙, 𝒚, 𝒛} is
(a) 1 (b) ✔ 8 (c) 5 (d)2
9. A set having only one element is called
(a) Empty set (b) ✔Singleton set (c) Power set (d) Subset
10. The set of odd integers between 2 and 4 is
(a) Null set (b) Power set (c) ✔ Singleton set (d) Subset
SHORT QUESTIONS
i. If 𝑨 = {𝒂, 𝒃}, then find 𝑷(𝑨).
ii. Write the following sets in “𝑺𝒆𝒕 − 𝒃𝒖𝒊𝒍𝒅𝒆𝒓 𝒎𝒆𝒕𝒉𝒐𝒅”
(a) {January , June , July } (b) {100, 101, 102,….., 400}
iii. Write the following set in “𝒅𝒆𝒔𝒄𝒓𝒊𝒑𝒕𝒊𝒗𝒆 𝒎𝒆𝒕𝒉𝒐𝒅” and “ 𝒕𝒂𝒃𝒖𝒍𝒂𝒓 𝒇𝒐𝒓𝒎”
(a) {𝒙|𝒙 ∈ 𝑵 ∧ 𝟒 < 𝑥 < 12} (b) {𝒙|𝒙 ∈ 𝑹 ∧ 𝒙 = 𝒙}
iv. Write two power subsets of 𝑵 and {𝒂, 𝒃, 𝒄}
v. Is there any set which has no proper sub set? If so name that set.
vi. What is the difference between {𝒂, 𝒃} and {{𝒂, 𝒃}}?
vii. Write down the power set of {𝝋} and {+, −, ×,÷}.
EXERCISE 2.2
Tick (✔) the correct answer.
1. A diagram which represents a set is called______
(a) ✔Venn’s (b) Argand (c) Plane (d) None of these
2. 𝑨 ∪ 𝝋 =
(a) 𝜑 (b) 𝑈 (c) ✔ 𝐴 (d) 𝑈 − 𝐴
