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Title: Important questions
Description: These are 1st year some important question which can help you to get t good marks
Description: These are 1st year some important question which can help you to get t good marks
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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
...
1
Tick (✔) the correct answer
...
√𝟐 is a number
(a) Rational
(b) ✔ Irrational
(c) Prime
𝒑
2
...
(a)
4
...
(a)
6
...
✔Terminating
(b) Non-Terminating
(c) Recurring
(d) Non recurring
𝟓
...
(a)
8
...
(a)
10
...
𝒕
′+′
(b) ✔ ′ × ′
(c) ′ ÷ ′
If 𝒂 < 𝑏 then
1
1
𝑎<𝑏
(b) 𝑎 < 𝑏
(c) ✔
(d)
𝑘𝑏
𝑏
(d) ′ − ′
1
𝑎
1
>𝑏
(d) 𝑎 − 𝑏 > 0
SHORT QUESTONS
i
...
iii
...
Write down the “Closure Property for addition”
...
Simplify justifying each step :
𝟒+𝟏𝟔𝒙
𝟒
EXERCISE 1
...
1
...
(a)
3
...
(a)
The multiplicative identity of complex number is
(0,0)
(b) (0,1)
(c) ✔ (1,0)
𝒊𝟏𝟑 equals:
✔𝑖
(b) – 𝑖
(c) 1
The multiplicative inverse of (𝟒, −𝟕) is:
4
7
4 7
4
7
(− 65 , − 65)
(b) (− 65 , 65)
(c) (65 , − 65)
(𝟎, 𝟑)(𝟎, 𝟓) =
15
(b) ✔ -15
(c) −8𝑖
𝟐𝟏
−
𝟐
5
...
√– =
𝟒
(a)
1
2
1
𝑖
√2
𝟐
(b)
1
(c) ✔ 2 𝑖
1
(d) − 2 𝑖
7
...
The multiplicative inverse of (𝟎, 𝟎) is:
(a) (0,1)
(b) (1,0)
9
...
(a)
(d) ✔ Does not exist
(c) (0,0)
The product of any two conjugate complex numbers is
✔Real number
(b) complex number
(c) zero
Identity element of complex number is
(0,1)
(b) (0,1)
(c) (0,0)
(d) 1
(d) ✔(1,0)
SHORT QUESTIONS
𝒂
−𝒃
Prove that “multiplicative inverse” of (𝒂, 𝒃) is (𝒂𝟐+𝒃𝟐 , 𝒂𝟐+𝒃𝟐)
...
𝒊
Separate into real and imaginary parts :
i
...
iii
...
v
...
𝟐𝟏
𝟐
𝟏+𝒊
𝟗
and 𝒊
−𝟏𝟔
√
𝟐𝟓
vii
...
Simplify the following (𝟐, 𝟔)(𝟑, 𝟕) and (𝟐, 𝟔) ÷ (𝟑, 𝟕)
EXERCISE 1
...
1
...
The figure representing one or more complex numbers on the complex plane is called:
(a) Cartesian plane
3
...
If 𝒛 = 𝒙 + 𝒊𝒚 then |𝒛| =
(a) 𝑥 2 + 𝑦 2
5
...
𝒛𝒛̅ =
(a) 𝑧 2
7
...
𝒊
(a) 1
=
SHORT QUESTIONS
i
...
iii
...
Prove that 𝒛̅ = 𝒛 𝒊𝒇𝒇 𝒛 is real
...
v
...
vii
...
Simplify the following
Simplify the following
State "𝑫𝒆 𝑴𝒐𝒊𝒗𝒓𝒆′𝒔 𝒕𝒉𝒆𝒐𝒓𝒆𝒎"
...
1
Tick (✔) the correct answer
...
A set is a collection of objects which are
(a) Well defined
(b) ✔ Well defined and distinct (c) identical
2
...
There are _______ methods to describe a set
...
{1,2,3} and {2,1,3} are sets
...
The sets 𝑵 and 𝑶 are sets
...
Which of the following is true?
(c) Not equal
(d) None of these
(a) 𝑁 ⊂ 𝑅 ⊂ 𝑄 ⊂ 𝑍
(b) 𝑅 ⊂ 𝑍 ⊂ 𝑄 ⊂ 𝑁
7
...
Total number of subsets that can be formed from the set {𝒙, 𝒚, 𝒛} is
(a) 1
(b) ✔ 8
(c) 5
9
...
The set of odd integers between 2 and 4 is
(c) Power set
(d) Subset
(c) ✔ Singleton set
(d) Subset
(a) Null set
(b) Power set
(d)2
SHORT QUESTIONS
i
...
iii
...
v
...
vii
...
Write the following sets in “𝑺𝒆𝒕 − 𝒃𝒖𝒊𝒍𝒅𝒆𝒓 𝒎𝒆𝒕𝒉𝒐𝒅”
(a) {January , June , July } (b) {100, 101, 102,…
...
What is the difference between {𝒂, 𝒃} and {{𝒂, 𝒃}}?
Write down the power set of {𝝋} and {+, −, ×,÷}
...
2
Tick (✔) the correct answer
...
A diagram which represents a set is called______
(a) ✔Venn’s
(b) Argand
(c) Plane
2
...
𝑨 − 𝑼 =
(a) ✔𝜑
4
...
If 𝑨 ⊆ 𝑩 then 𝑨 ∪ 𝑩 =
(a) 𝐴
(b) ✔ 𝐵
(c) 𝐴𝑐
(d) 𝐵𝑐
6
...
If 𝑨 and 𝑩 are disjoint sets then :
(a) ✔𝐴 ∩ 𝐵 = 𝜑
(b) 𝐴 ∩ 𝐵 ≠ 𝜑
(c) 𝐴 ⊂ 𝐵
8
...
If the intersection of two sets is the empty then sets are called
(a) ✔Disjoint sets
(b) Overlapping Sets
(c) Subsets
(d) Power sets
SHORT QUESTIONS
i
...
iii
...
Exhibit 𝑨 ∪ 𝑩 and 𝑨 ∩ 𝑩 by “𝑽𝒆𝒏𝒏 𝑫𝒊𝒂𝒈𝒓𝒂𝒎” in the following case :- 𝑩 ⊆ 𝑨
Under what conditions on 𝑨 and 𝑩 are the following statements true?
(a) 𝑨 ∪ 𝑩 = 𝑩
(b) 𝑼 − 𝑨 = 𝝋
Let 𝑼 = {𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎} , 𝑨 = {𝟐, 𝟒, 𝟔, 𝟖, 𝟏𝟎} find 𝑼𝒄 and 𝑨𝒄
Define Union of Two sets
...
3
Tick (✔) the correct answer
...
Commutative property of intersection is:
(a)
2
...
𝐴∪𝐵 =𝐵∪𝐴
(𝑨 ∪ 𝑩)′ =
(a)
4
...
✔𝐴
(a)
6
...
𝐴
If 𝒙 ∈ 𝑳 ∪ 𝑴 then
𝑥 ∉ 𝐿 𝑜𝑟 𝑥 ∉ 𝑀
If 𝒙 ∈ (𝑨 ∪ 𝑩)′ then
(b) ✔ 𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴 (c) 𝐴 ∩ 𝐵 = 𝐵 ∪ 𝐴
𝐴′ ∪ 𝐵′
(b) ✔ 𝐴′ ∩ 𝐵′
(c) 𝐴′ ∩ 𝐵
Take any set, say 𝑨 = {𝟏, 𝟐, 𝟑, 𝟒, 𝟓} then 𝑨 ∪ ∅ =
(d) 𝐴 ∪ 𝐵 = 𝐵 ∩ 𝐴
(d) 𝐴 ∪ 𝐵′
(b) ∅
(c) 𝑈
(d) None of these
𝑳 ∪ 𝑴 = 𝑳 ∩ 𝑴 then 𝑳 is equal to
✔𝑀
(b) 𝐿
(c) 𝜑
(d) 𝑀′
If 𝑼 = {𝟏, 𝟐, 𝟑, 𝟒, 𝟓, … … , 𝟐𝟎} and 𝑨 = {𝟏, 𝟑, 𝟓, …
...
ii
...
iv
...
Write down “𝑫𝒆 𝑴𝒐𝒓𝒈𝒂𝒏 𝒔 𝑳𝒂𝒘𝒔”
...
Prove that
𝑨∩𝑩=𝑩∩𝑨
Verify “𝑨𝒔𝒔𝒐𝒄𝒊𝒂𝒕𝒊𝒗𝒊𝒕𝒚 𝒐𝒇 𝒊𝒏𝒕𝒆𝒓𝒔𝒆𝒄𝒕𝒊𝒐𝒏” of 𝑵, 𝒁, 𝑸
Let 𝑼 =The set of English alphabet , 𝑨 = {𝒙|𝒙 is a vowel } and 𝑩 = {𝒚|𝒚 is
consonant} Verify (𝑨 ∪ 𝑩)′ = 𝑨′ ∩ 𝑩′
vi
...
If 𝑼 = {𝟏, 𝟐, 𝟑, 𝟒, 𝟓, … … , 𝟐𝟎} and 𝑨 = {𝟏, 𝟑, 𝟓, …
...
EXERCISE 2
...
1
...
For the propositions 𝒑 and 𝒒 , 𝒑 → (𝒑 ∨ 𝒒) is:
(a) ✔Tautology
(b) Absurdity
(c) Contingency
3
...
Truth set of a tautology is
(a) ✔Universal set
(b) 𝜑
(c) True
5
...
𝒑 → ~𝒑 is
(b) ✔ Absurdity
(c) Contingency
(d) None of these
(d) None of these
(d) ∨
(d) False
(d) Contra positive
(a) Tautology
(b) ✔Absurdity
(c) Contingency
(d) Contra positive
7
...
Contra positive of ~𝒑 → ~𝒒 is
(a) 𝑝 → 𝑞
(b) ✔ 𝑞 → 𝑝
9
...
The symbol “∀” is called
(a) ✔Universal quantifier
(d) ~𝑞 → 𝑝
(c) Converse
(d) Inverse
(b) Existential quantifier (c) Converse
(d) Inverse
SHORT QUESTIONS
i
...
iii
...
Write converse , inverse and contra positive of ~𝒑 → 𝒒
Construct the truth table of [(𝒑 → 𝒒) ∧ 𝒑 → 𝒒]
Show that ~(𝒑 → 𝒒) → 𝒑 is tautology
...
LONG QUESTIONS
Prove that 𝒑 ∨ (∼ 𝒑 ∧∼ 𝒒) ∨ (𝒑 ∧ 𝒒) = 𝒑 ∨ (∼ 𝒑 ∧∼ 𝒒)
EXERCISE 2
...
1
...
𝑷 = 𝑸 is the truth set of
(b) 𝑃 ∪ 𝑄
(c) 𝑃 − 𝑄
(d) 𝑃 + 𝑄
(a) 𝑝 = 𝑞
(b) 𝑝 → 𝑞
3
...
7
COMPOSED BY:- MUHAMMAD SALMAN SHERAZI 03337727666/03067856232
EXERCISE 2
...
1
...
An onto function is also called:
(a) Injective
(b) ✔ Surjective
(c) Bijective
(d) Inverse
3
...
If set 𝑨 has 2 elements and 𝑩 has 4 elements , then number of elements in 𝑨 × 𝑩 is :
(a) 6
(b) ✔ 8
(c) 16
(d) None of these
5
...
The function 𝒇 = {(𝒙, 𝒚), 𝒚 = 𝒙} is :
(a) ✔Identity function (b) Null function
(c) not a function
(d) similar function
(𝟑,
(𝟒,
(𝟓,
(𝟔,
7
...
The inverse of {(𝒙, 𝒚)|𝒙 + 𝒚 = 𝟗 , |𝒙| ≤ 𝟑, |𝒚| ≤ 𝟑} is
(a) Function
(b) ✔ Not function
(c) (1-1) function
(d) onto function
SHORT QUESTIONS
i
...
iii
...
Find the inverse of {(𝒙, 𝒚)|𝒚 = 𝟐𝒙 + 𝟑 , 𝒙 ∈ 𝑹}
...
How we can find inverse of a function in “𝑺𝒆𝒕 − 𝒃𝒖𝒊𝒍𝒅𝒆𝒓 𝒏𝒐𝒕𝒂𝒕𝒊𝒐𝒏”
...
State domain and range of the
{(𝒙, 𝒚)| 𝒙 + 𝒚 < 5}
relation
...
Tell it is function or not
...
7
Tick (✔) the correct answer
...
Negation of a given number is example of
(a) Binary operation
(b) ✔Unary operation (c) relation
2
...
A binary operation is denoted by
(a) +
4
...
The set {𝟏, −𝟏, 𝒊, −𝒊} where 𝒊 = √−𝟏 is closed 𝒘
...
The set {𝟏, 𝝎, 𝝎 } where 𝒊 = √−𝟏 is closed 𝒘
...
𝑵 is closed 𝒘
...
Inverse and identity of a set 𝑺 under binary operation ∗ is
(a) ✔Unique
(b) Two
(c) Three
SHORT QUESTIONS
(d) Four
8
COMPOSED BY:- MUHAMMAD SALMAN SHERAZI 03337727666/03067856232
i
...
iii
...
Prepare a table of addition of the elements of the set of residue classes modulo 4
...
EXERCISE 2
...
1
...
(a)
3
...
𝒕
✔+
(b) ×
(𝒁,
...
Subtraction is non-commutative and non-associative on
(d) Not closed
(a) ✔𝑁
(b) 𝑅
5
...
For every 𝒂, 𝒃 ∈ 𝑮 , 𝒂 ∗ 𝒃 = 𝒃 ∗ 𝒂 then 𝑮 is called
(d) Not closed
(a) Group
(b) Monoid
7
...
In group (𝒁, +) inverse of 1 is
(c) 𝑥 = 𝑎𝑏
(d) 𝑥𝑎 = 𝑏
(a) 1
(b) ✔ -1
9
...
In a group the inverse is
(a) ✔Unique
(b) two
SHORT QUESTIONS
i
...
iii
...
Prove that (𝒂𝒃)−𝟏 = 𝒃−𝟏 𝒂−𝟏
Define Finite and Infinite group
...
Set up its ‘
...
’
If 𝑮 is a group under the operation ∗ and 𝒂, 𝒃 ∈ 𝑮, find the solutions of the
equation 𝒂 ∗ 𝒙 = 𝒃 and 𝒙 ∗ 𝒂 = 𝒃
LONG QUESTIONS
Show that the set {𝟏, 𝝎, 𝝎𝟐 } , 𝝎𝟑 = 𝟏, is an Abelian group 𝒘
...
Prove that all 𝟐 × 𝟐 non-singular matrices over the real field form a non-abelian
group under multiplication
...
1
Tick (✔) the correct answer
...
A rectangular array of numbers enclosed by a square brackets is called:
(a) ✔Matrix
(b) Row
(c) Column
(d) Determinant
9
COMPOSED BY:- MUHAMMAD SALMAN SHERAZI 03337727666/03067856232
2
...
(a)
4
...
(a)
6
...
(a)
8
...
(a)
10
...
(a)
The horizontal lines of numbers in a matrix are called:
Columns
(b) ✔ Rows
(c) Column matrix
(d) Row matrix
The vertical lines of numbers in a matrix are called:
✔Columns
(b) Rows
(c) Column matrix
(d) Row matrix
If a matrix A has 𝒎 rows and 𝒏 columns , then order of A is :
✔𝑚 × 𝑛
(b) 𝑛 × 𝑚
(c) 𝑚 + 𝑛
(d) 𝑚𝑛
[𝟏 −𝟏 𝟑 𝟒] is an example of
✔Row vector
(b) column vector
(c) Rectangular matrix (d) Square matrix
The matrix 𝑨 is said to be real if its all entries are
Rational
(b) ✔ real
(c) natural
(d) complex
If a matrix A has different number of rows and columns then A is called:
Row vector
(b) ✔ Column vector (c) square matrix (d) Rectangular matrix
For the square matrix 𝑨 = [𝒂𝒊𝒋 ]𝒏×𝒏 then 𝒂𝟏𝟏 , 𝒂𝟐𝟐 , 𝒂𝟑𝟑… 𝒂𝒏𝒏 are:
✔Main diagonal
(b) primary diagonal
(c) proceding diagonal (d) secondary diagonal
For a square matrix 𝑨 = [𝒂𝒊𝒋 ]if all 𝒂𝒊𝒋 = 𝟎, 𝒊 ≠ 𝒋 and all 𝒂𝒊𝒋 = 𝒌(𝒏𝒐𝒏 − 𝒛𝒆𝒓𝒐) for 𝒊 = 𝒋 then A
is called:
Diagonal matrix
(b) ✔ Scalar Matrix (c) Unit matrix
(d) Null matrix
A square matrix 𝑨 is singular if
✔|𝐴| = 0
(b) |𝐴| ≠ 0
(c) 𝐴 = 0
(d) 𝐴 ≠ 0
If order of matrix 𝑨 is 𝒎 × 𝒏 and order of matrix 𝑩 is 𝒏 × 𝒑 then order of matrix 𝑨𝑩 is
𝑚 ×𝑛
(b) 𝑛 × 𝑚
(c) 𝑛 × 𝑝
(d) ✔ 𝑚 × 𝑝
SHORT QUESTIONS
i
...
iii
...
v
...
𝒂 𝒃
𝟎 𝟏
𝟐 𝟏
𝟓 𝟐
Find the matrix 𝑿 if ;
[
]𝑿 = [
]
𝟓 𝟏𝟎
−𝟐 𝟏
𝟑 −𝟕
𝟓 −𝟏
Find the matrix 𝑨 if ;
[𝟎 𝟎 ] 𝑨 = [𝟎 𝟎 ]
𝟕 𝟐
𝟑 𝟏
If 𝑨 = [
𝒊
𝟏
LONG QUESTIONS
𝟐
Find 𝒙 and 𝒚 if [
𝟏
𝟎 𝒙
𝟏 𝒙
] + 𝟐[
𝒚 𝟑
𝟎 𝟐
𝒚
𝟒 −𝟐 𝟑
]=[
]
𝟏 𝟔 𝟏
−𝟏
EXERCISE 3
...
1
...
(a)
3
...
(a)
5
...
R
(b) Q
Which of the following Sets is not a field
...
ii
...
𝟐𝒊 𝒊
[
]
𝒊 −𝒊
If 𝑨 and 𝑩 are square matrices of the same order, then explain why in general
(a) (𝑨 + 𝑩)𝟐 ≠ 𝑨𝟐 + 𝟐𝑨𝑩 + 𝑩𝟐
(b) 𝑨𝟐 − 𝑩𝟐 ≠ (𝑨 − 𝑩)(𝑨 + 𝑩)
𝟐 −𝟏 𝟑 𝟎
If 𝑨 = [ 𝟏
𝟎 𝟒 −𝟐 ] then find 𝑨𝒕 𝑨
−𝟑 𝟓 𝟐 −𝟏
Find the inverse of
LONG QUESTIONS
Solve the following system of linear equations:
Solve the following matrix equation for 𝑨: [
𝟑𝒙 − 𝟓𝒚 = 𝟏 ; −𝟐𝒙 + 𝒚 = −𝟑
𝟒 𝟑
𝟐
]𝑨 − [
𝟐 𝟐
−𝟏
−𝟏 −𝟒
𝟑
]=[
]
𝟑
𝟔
−𝟐
EXERCISE 3
...
1
...
(a)
3
...
(a)
5
...
(a)
7
...
| 𝟎
𝒓𝒔𝒊𝒏𝝋 𝟏 𝒄𝒐𝒔𝝋
(a) 1
(b) 2
𝟏 𝟐 𝟑
9
...
(𝑨 ) =
(a) 𝐴−1
(b) (𝐴−1 )𝑡
(c) ✔ (𝐴𝑡 )−1
SHORT QUESTIONS
i
...
Without expansion show that
iii
...
Find the value of 𝒙 if
𝟏 𝟐 −𝟐
|−𝟏 𝟏 𝟑 |
𝟐 𝟒 −𝟏
𝟏 𝟐 𝟑𝒙
|𝟐 𝟑 𝟔𝒙| = 𝟎
𝟑 𝟓 𝟗𝒙
𝟑 𝟏 𝒙
|−𝟏 𝟑 𝟒| = −𝟑𝟎
𝒙 𝟏 𝟎
(d) 𝐴𝑡
11
COMPOSED BY:- MUHAMMAD SALMAN SHERAZI 03337727666/03067856232
v
...
4
Tick (✔) the correct answer
...
A square matrix A is symmetric if:
𝑡
(a) ✔𝐴𝑡 = 𝐴
(b) 𝐴𝑡 = −𝐴
2
...
A square matrix A is Hermitian if:
(c) (𝐴) = 𝐴
(a) 𝐴𝑡 = 𝐴
(b) 𝐴𝑡 = −𝐴
4
...
(a)
6
...
(a)
8
...
(a)
10
...
ii
...
iv
...
vi
...
] , show that 𝑨 + (𝑨
𝟏
−𝒊
𝟏 𝟐 𝟎
If = [ 𝟑 𝟐 −𝟏] , show that 𝑨 + 𝑨𝒕 is symmetric
...
𝟏
̅ )𝒕
...
Define lower and upper triangular matrix
...
5
Tick (✔) the correct answer
...
In a homogeneous system of linear equations , the solution (0,0,0) is:
(a) ✔Trivial solution
(b) non trivial solution (c) exact solution
(d) anti symmetric
12
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If 𝑨𝑿 = 𝑶 then 𝑿 =
𝐼
(b)✔ 𝑂
(c) 𝐴−1
(d) Not possible
If the system of linear equations have no solution at all, then it is called a/an
Consistent system
(b) ✔ Inconsistent system(c) Trivial System
(d) Non Trivial System
The value of 𝝀 for which the system 𝒙 + 𝟐𝒚 = 𝟒; 𝟐𝒙 + 𝝀𝒚 = −𝟑 does not possess the unique
solution
(a) ✔4
(b) -4
(c) ±4
(d) any real number
5
...
(a)
3
...
LONG QUESTIONS
Use matrices to solve the system
𝒙𝟏 − 𝟐𝒙𝟐 + 𝒙𝟑 = −𝟒 ; 𝟐𝒙𝟏 − 𝟑𝒙𝟐 + 𝟐𝒙𝟑 = −𝟔 ; 𝟐𝒙𝟏 + 𝟐𝒙𝟐 + 𝒙𝟑 = 𝟓
Solve by using Cramer’s Rule
𝟐𝒙 + 𝟐𝒚 + 𝒛 = 𝟑 ; 𝟑𝒙 − 𝟐𝒚 − 𝟐𝒛 = 𝟏 ; 𝟓𝒙 + 𝒚 − 𝟑𝒛 = 𝟐
Find the value of 𝝀 for which the following system does not possess a unique
solution
...
𝒙𝟏 + 𝟒𝒙𝟐 + 𝝀𝒙𝟑 = 𝟐 ; 𝟐𝒙𝟏 + 𝒙𝟐 − 𝟐𝒙𝟑 = 𝟏𝟏 ; 𝟑𝒙𝟏 + 𝟐𝒙𝟐 − 𝟐𝒙𝟑 = 𝟏𝟔
EXERCISE 4
...
1
...
(a)
3
...
How many techniques to solve quadratic equations
...
The solution of a quadratic equation are called
(a) ✔Roots
(b) identity
(d) None of these
(d) 4
(c) quadratic equation (d) solution
SHORT QUESTIONS
i
...
iii
...
v
...
Solve by factorization
𝒙𝟐 − 𝒙 = 𝟐
Solve by completing square 𝒙𝟐 − 𝟑𝒙 − 𝟔𝟒𝟖
Solve by quadratic formula 𝟏𝟓𝒙𝟐 + 𝟐𝒂𝒙 − 𝒂𝟐 = 𝟎
Solve the equation 𝒙𝟐 − 𝟕𝒙 + 𝟏𝟎 = 𝟎 by factorization
...
Derive “𝑸𝒖𝒂𝒅𝒓𝒂𝒕𝒊𝒄 𝒇𝒐𝒓𝒎𝒖𝒍𝒂”
...
2
Tick (✔) the correct answer
...
(a)
2
...
(a)
4
...
Solve
ii
...
iv
...
Solve
𝒙𝟓 + 𝟖 = 𝟔𝒙𝟓
Define “𝑹𝒆𝒄𝒊𝒑𝒓𝒐𝒄𝒂𝒍 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏” with an example
...
𝟐
𝟏
LONG QUESTIONS
𝟏
𝟏
𝒙
𝒙𝟐
Solve 𝒙𝟐 + 𝒙 − 𝟒 + +
=𝟎
Solve 𝟒
...
3
Tick (✔) the correct answer
...
The equations involving redical expressions of the variable are called:
(a) Reciprocal equations(b) ✔ Redical equations (c) Quadratic functions (d) exponential
equations
LONG QUESTIONS
Solve
√𝟐𝒙 + 𝟖 + √𝒙 + 𝟓 = 𝟕
Solve
√𝒙 + 𝟕 + √𝒙 + 𝟐 = √𝟔𝒙 + 𝟏𝟑
Solve
√𝒙𝟐 + 𝒙 + 𝟏 − √𝒙𝟐 + 𝒙 − 𝟏 = 𝟏
Solve
(𝒙 + 𝟒)(𝒙 + 𝟏) = √𝒙𝟐 + 𝟐𝒙 − 𝟏𝟓 + 𝟑𝒙 + 𝟑𝟏
EXERCISE 4
...
1
...
(a)
3
...
(a)
5
...
(a)
7
...
(a)
9
...
(a)
✔1,
−1+√3𝑖 −1−√3𝑖
, 2
2
1+√3𝑖 1+√3𝑖
−1+√3𝑖 −1+√3𝑖
(b) 1, 2 , 2
(c) −1, 2 , 2
Sum of all cube roots of 64 is :
✔0
(b) 1
(c) 64
Product of cube roots of -1 is:
0
(b) -1
(c) ✔ 1
𝟏𝟔𝝎𝟖 + 𝟏𝟔𝒘𝟒 =
0
(b) ✔ -16
(c) 16
The sum of all four fourth roots of unity is:
Unity
(b) ✔0
(c) -1
The product of all four fourth roots of unity is:
Unity
(b) 0
(c) ✔ -1
The sum of all four fourth roots of 16 is:
16
(b) -16
(c) ✔ 0
The complex cube roots of unity are………………
...
✔Additive inverse
(b) Equal to
(c) Conjugate
The complex fourth roots of unity are ……
...
✔Additive inverse
(b) equal to
(c) square of
The cube roots of -1 are
{1, 𝜔, 𝜔2 }
(b) {1, −𝜔, 𝜔2 }
(c) ✔ {−1, −𝜔, −𝜔2 }
(d) −1,
1+√3𝑖 1+√3𝑖
, 2
2
(d) -64
(d) None
(d) -1
(d) None
(d) None
(d) 1
(d) None of these
(d) None of these
(d) {−1, 𝜔, 𝜔2 }
SHORT QUESTIONS
i
...
iii
...
v
...
vii
...
𝟐𝒏 factors = 𝟏
If 𝝎 is cube root of 𝒙𝟐 + 𝒙 + 𝟏 = 𝟎, show that its other root is 𝝎𝟐 and prove that
𝝎𝟑 = 𝟏
If 𝝎 is cube root of unity , form an equation whose roots are 𝟐𝝎 and 𝟐𝝎𝟐
...
Solve
𝟓𝒙𝟓 − 𝟓𝒙 = 𝟎
LONG QUESTIONS
𝒙𝟑 +
Show that
𝒚𝟑 + 𝒛𝟑 − 𝟑𝒙𝒚𝒛 = (𝒙 + 𝒚 + 𝒛)(𝒙 + 𝝎𝒚 + 𝝎𝟐 𝒛)(𝒙 + 𝝎𝟐 𝒚 + 𝝎𝒛)
EXERCISE 4
...
𝟏
𝒙
1
...
(a)
3
...
(a)
5
...
(a)
2
(b) 3
(c) 1
(d) ✔
If 𝒇(𝒙) is divided by −𝒂 , then dividend = (Divisor)(……
...
Divisor
(b) Dividend
(c) ✔ Quotient
If 𝒇(𝒙) is divided by 𝒙 − 𝒂 by remainder theorem then remainder is:
✔ 𝑓(𝑎)
(b) 𝑓(−𝑎)
(c) 𝑓(𝑎) + 𝑅
The polynomial (𝒙 − 𝒂) is a factor of 𝒇(𝒙) if and only if
✔ 𝑓(𝑎) = 0
(b) 𝑓(𝑎) = 𝑅
(c) Quotient = 𝑅
𝟐
𝒙 − 𝟐 is a factor of 𝒙 − 𝒌𝒙 + 𝟒, if 𝒌 is:
2
(b) ✔ 4
(c) 8
If 𝒙 = −𝟐 is the root of 𝒌𝒙𝟒 − 𝟏𝟑𝒙𝟐 + 𝟑𝟔 = 𝟎, then 𝒌 =
2
(b) -2
(c) 1
not a polynomial
(d) 𝑓(𝑎)
(d) 𝑥 − 𝑎 = 𝑅
(d) 𝑥 = −𝑎
(d) -4
(d) ✔
-1
15
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7
...
(a)
𝒙 + 𝒂 is a factor of 𝒙𝒏 + 𝒂𝒏 when 𝒏 is
Any integer
(b) any positive integer (c) ✔ any odd integer (d) any real number
𝒙 − 𝒂 is a factor of 𝒙𝒏 − 𝒂𝒏 when 𝒏 is
✔ Any integer
(b) any positive integer (c) any odd integer
(d) any real number
SHORT QUESTIONS
Find the numerical value of 𝒌 if the polynomial 𝒙𝟑 + 𝒌𝒙𝟐 − 𝟕𝒙 + 𝟔 has a
remainder of – 𝟒 , when divided by 𝒙 + 𝟐
...
When the polynomial 𝒙𝟑 + 𝟐𝒙𝟐 + 𝒌𝒙 + 𝟒 is divided by 𝒙 − 𝟐, the remainder is 𝟏𝟒
...
Use synthetic division to find the quotient and the remainder when the polynomial
𝒙𝟒 − 𝟏𝟎𝒙𝟐 − 𝟐𝒙 + 𝟒 is divided by 𝒙 + 𝟑
...
i
...
iii
...
v
...
Find the values of 𝒂 and 𝒃 if – 𝟐 and 𝟐 are the roots of the polynomial 𝒙𝟑 −
𝟒𝒙𝟐 + 𝒂𝒙 + 𝒃
...
6
Tick (✔) the correct answer
...
Sum of roots of 𝒂𝒙𝟐 − 𝒃𝒙 − 𝒄 = 𝟎 is (𝒂 ≠ 𝟎)
𝑏
𝑏
𝑐
(a)
(b) –
(c)
𝑎
2
...
(a)
4
...
(a)
𝑎
𝑎
(d) ✔ –
𝑐
𝑎
Product of roots of 𝒂𝒙𝟐 − 𝒃𝒙 − 𝒄 = 𝟎 is (𝒂 ≠ 𝟎)
𝑏
𝑏
𝑐
𝑐
✔𝑎
(b) – 𝑎
(c) 𝑎
(d) – 𝑎
If 2 and -5 are roots of a quadratic equation , then equation is:
𝑥 2 − 3𝑥 − 10 = 0 (b) 𝑥 2 − 3𝑥 + 10 = 0 (c) ✔ 𝑥 2 + 3𝑥 − 10 = 0 (d) 𝑥 2 + 3𝑥 + 10 = 0
If 𝜶 and 𝜷 are the roots of 𝟑𝒙𝟐 − 𝟐𝒙 + 𝟒 = 𝟎, then the value of 𝜶 + 𝜷 is:
2
2
4
4
✔3
(b) −
(c)
(d) −
3
3
3
The equation whose roots are given is
𝑥 2 + 𝑆𝑥 + 𝑃 = 0
(b) 𝑥 2 − 𝑆𝑥 − 𝑃 = 0 (c) 𝑥 2 + 𝑆𝑥 − 𝑃 = 0 (d) ✔ 𝑥 2 − 𝑆𝑥 + 𝑃 = 0
SHORT QUESTIONS
If 𝜶, 𝜷 are the roots of 𝟑𝒙𝟐 − 𝟐𝒙 + 𝟒 = 𝟎, find the values of
i
...
iii
...
v
...
If 𝜶, 𝜷 are the roots of the equation 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎, form the equations whose
roots are 𝜶𝟑 , 𝜷𝟑
...
𝟏+𝜷
16
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𝒂
𝒃
Find the condition that
+
= 𝟓 may have roots equal in magnitude but
𝒙−𝒂
𝒙−𝒃
opposite in signs
...
7
Tick (✔) the correct answer
...
(a)
2
...
(a)
4
...
(a)
If roots of 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎, (𝒂 ≠ 𝟎) are real , then
✔Disc≥ 0
(b) Disc< 0
(c) Disc≠ 0
(d) Disc≤ 0
If roots of 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎, (𝒂 ≠ 𝟎) are complex , then
Disc≥ 0
(b) ✔ Disc< 0
(c) Disc≠ 0
(d) Disc≤ 0
𝟐
If roots of 𝒂𝒙 + 𝒃𝒙 + 𝒄 = 𝟎, (𝒂 ≠ 𝟎) are equal , then
✔Disc= 0
(b) Disc< 0
(c) Disc≠ 0
(d) None of these
𝟐
The expression 𝒃 − 𝟒𝒂𝒄 is called:
✔Discriminant
(b) Quadratic equation (c) Linear equation
(d) roots
Disc of 𝒙𝟐 + 𝟐𝒙 + 𝟑 = 𝟎 is
16
(b) −16
(c) ✔ -8
(d) −16
SHORT QUESTIONS
i
...
iii
...
LONG QUESTIONS
𝟐
Show that the roots of 𝒙 + (𝒎𝒙 + 𝒄)𝟐 = 𝒂𝟐 will be equal
if 𝒄𝟐 = 𝒂𝟐 (𝟏 + 𝒎𝟐 )
Show that the roots of the equation
(𝒂𝟐 − 𝒃𝒄)𝒙𝟐 + 𝟐(𝒃𝟐 − 𝒄𝒂)𝒙 + 𝒄𝟐 − 𝒂𝒃 = 𝟎 will be equal, if either
𝒂𝟑 + 𝒃𝟑 + 𝒄𝟑 = 𝟑𝒂𝒃𝒄 or 𝒃 = 𝟎
...
8
LONG QUESTIONS
Solve the following systems of equations
...
𝒙+𝒚=𝟓
;
𝒙𝟐 + 𝟐𝒚𝟐 = 𝟏𝟕
Solve the following systems of equations
...
9
LONG QUESTIONS
Solve the following systems of equations
...
𝒙𝟐 − 𝒚𝟐 = 𝟏𝟔
𝒙𝒚 = 𝟏𝟓
;
Solve the following systems of equations
...
10
LONG QUESTIONS
The sum of a positive number and its reciprocal is
𝟐𝟔
𝟓
...
Find two consecutive numbers, whose product is 𝟏𝟑𝟐
...
Find the number
...
1
Tick (✔) the correct answer
...
(a)
2
...
(a)
An open sentence formed by using sign of “ = ” is called a/an
✔Equation
(b) Formula
(c) Rational fraction
If an equation is true for all values of the variable, then it is called:
a conditional equation (b) ✔ an identity
(c) proper rational fraction
𝟐
(𝒙 + 𝟑)(𝒙 + 𝟒) = 𝒙 + 𝟕𝒙 + 𝟏𝟐 is a/an:
Conditional equation (b) ✔an identity
(c) proper rational fraction
4
...
A
𝑷(𝒙)
fraction 𝑸(𝒙) , 𝑸(𝒙)
𝑷(𝒙)
, 𝑸(𝒙)
𝑸(𝒙)
(d) Theorem
(d) All of these
(d) a formula
≠ 𝟎 is called :
(b) Irrational fraction
(c) Partial fraction
(d) Proper fraction
≠ 𝟎 is called proper fraction if :
(a) ✔Degree of 𝑃(𝑥) < Degree of 𝑄(𝑥) (b) Degree of 𝑃(𝑥) = Degree of 𝑄(𝑥)
(c) Degree of 𝑃(𝑥) > Degree of 𝑄(𝑥)
(d) Degree of 𝑃(𝑥) ≥ Degree of 𝑄(𝑥)
𝑷(𝒙)
6
...
A mixed form of fraction is :
(a) An integer+ improper fraction (b) a polynomial+improper fraction
(c) ✔ a polynomial+proper fraction (d) a polynomial+rational fraction
8
...
Resolve
ii
...
into Partial fraction
...
into Partial fraction
...
2
Tick (✔) the correct answer
...
The number of Partial fraction of 𝒙(𝒙+𝟏)(𝒙𝟐−𝟏)are:
(a) 2
(c) ✔ 4
(b) 3
2
...
𝟗𝒙𝟐
is
𝒙𝟑 −𝟏
(b) 3
(d) None of these
𝒙𝟓
are:
𝒙(𝒙+𝟏)(𝒙𝟐 −𝟒)
(c) 4
(d) ✔ 6
an
(b) ✔ Proper fraction (c) Polynomial
(a) Improper fraction
(d) equation
LONG QUESTIONS
Resolve
Resolve
Resolve
𝟏
(𝒙+𝟏)𝟐 (𝒙𝟐 −𝟏)
𝒙𝟐
into Partial fraction
...
(𝒙−𝟐)(𝒙−𝟏)𝟐
𝟏
into Partial fraction
...
3
Tick (✔) the correct answer
...
Which is a reducible factor:
(a) 𝑥 3 − 6𝑥 2 + 8𝑥
(b) 𝑥 2 + 16𝑥
(c) 𝑥 2 + 5𝑥 − 6
(d) ✔ all of these
2
...
(𝒙𝟐 +𝟏)(𝒙+𝟏)
𝒙𝟐 +𝟏
into Partial fraction
...
(𝒙𝟐 +𝟒)(𝒙+𝟑)
EXERCISE 5
...
into Partial fraction
...
19
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EXERCISE 6
...
1
...
(a)
3
...
(a)
5
...
(a)
7
...
(a)
9
...
, set of real numbers is called:
✔Real sequence
(b) Imaginary sequence (c) Natural sequence
𝒏
If 𝒂𝒏 = {𝒏 + (−𝟏) }, then 𝒂𝟏𝟎 =
10
(b) 11
(c) 12
(d) 13
The last term of an infinite sequence is called :
𝑛𝑡ℎ term
(b) 𝑎𝑛
(c) last term
The next term of the sequence −𝟏, 𝟐, 𝟏𝟐, 𝟒𝟎, … is
✔112
(b) 120
(c) 124
𝒏+𝟏
For 𝒂𝒏 = (−𝟏)
, 𝒂𝟐𝟔 =
1
(b) ✔ -1
(c) 0
The next two terms of the sequence 𝟏, −𝟑, 𝟓, −𝟕, 𝟗, −𝟏𝟏, … are
13,15
(b) −13, −15
(c) ✔ 13, −15
10
...
ii
...
iv
...
vi
...
Write first two , 21st and 26th terms of the sequence whose general term is
(−𝟏)𝒏+𝟏
...
EXERCISE 6
...
1
...
(a)
3
...
P
(b) G
...
P
(d) None of these
𝒏𝒕𝒉 term of an A
...
ii
...
iv
...
Define “𝑨𝒓𝒊𝒕𝒉𝒎𝒆𝒕𝒊𝒄 𝑷𝒓𝒐𝒈𝒓𝒆𝒔𝒔𝒊𝒐𝒏”
...
Find the 𝟏𝟑th term of the sequence
𝒙, 𝟏, 𝟐 − 𝒙, 𝟑 − 𝟐𝒙, …
𝟏 𝟏 𝟏
𝟐𝒂𝒄
If 𝒂 , 𝒃 , 𝒄 are in A
...
, show that 𝒃 = 𝒂+𝒄
...
vii
...
P
...
P
...
P
...
3
Tick (✔) the correct answer
...
(a)
2
...
P, then 𝒂𝒏 is
✔A
...
M
Arithmetic mean between 𝒄 and 𝒅 is:
𝑐+𝑑
𝑐+𝑑
✔
(b)
2
2𝑐𝑑
(c) H
...
The arithmetic mean between √𝟐 and 𝟑√𝟐 is:
4
(a) 4√2
(b) 2
(c) ✔ 2√2
4
...
M between 𝒂 and 𝒃 if
(a) ✔𝑛 = 1
(b) 𝑛 = 0
(c) 𝑛 > 1
(d) 𝑛 < 1
SHORT QUESTIONS
Find A
...
between 𝟏 − 𝒙 + 𝒙𝟐 and 𝟏 + 𝒙 + 𝒙𝟐
If 𝟓, 𝟖 are two A
...
Define “𝑨𝒓𝒊𝒕𝒉𝒎𝒆𝒕𝒊𝒄 𝒎𝒆𝒂𝒏”
...
ii
...
LONG QUESTIONS
Find 6 A
...
between 2 and 5
...
M
...
Show that the sum of 𝒏 A
...
M
...
4
Tick (✔) the correct answer
...
(a)
2
...
(a)
The sum of terms of a sequence is called:
Partial sum
(b) ✔ Series
(c) Finite sum
𝟐
Forth partial sum of the sequence {𝒏 } is called:
16
(b) ✔ 1+4+9+16
(c) 8
Sum of 𝒏 −term of an Arithmetic series 𝑺𝒏 is equal to:
𝑛
𝑛
𝑛
✔ [2𝑎 + (𝑛 − 1)𝑑] (b) [𝑎 + (𝑛 − 1)𝑑] (c) [2𝑎 + (𝑛 + 1)𝑑]
2
2
2
(d) none of these
(d) 1+2+3+4
𝑛
(d) 2 [2𝑎 + 𝑙]
SHORT QUESTIONS
iii
...
iv
...
ii
...
Find the sum of 20 terms of the series whose 𝒓th term is 𝟑𝒓 + 𝟏
...
The ratio of the sums of 𝒏 terms of two series in A
...
is 𝟑𝒏 + 𝟐: 𝒏 + 𝟏
...
If 𝑺𝟐 , 𝑺𝟑 , 𝑺𝟓 are the sums of 𝟐𝒏, 𝟑𝒏, 𝟓𝒏 terms of an A
...
, show that 𝑺𝟓 =
𝟓(𝑺𝟑 − 𝑺𝟐 )
Find four numbers in A
...
whose sum is 32 and the sum of whose squares is
276
...
P
...
P
...
5
LONG QUESTIONS
A clock strikes once when its hour hand is at one , twice when it is at two and
so on
...
The sum of interior angles of polygons having sides 3,4,5,… etc form an A
...
Find the sum of interior angles for a 16 sided polygon
...
6
Tick (✔) the correct answer
...
For any 𝑮
...
(a)
3
...
, is:
✔0
(b) 1
The general term of a 𝑮
...
ii
...
P
...
Which term of the sequence: 𝒙𝟐 − 𝒚𝟐 , 𝒙 + 𝒚, 𝒙−𝒚 , … is (𝒙−𝒚)𝟗 ?
iv
...
P, prove that 𝒂 − 𝒃, 𝒃 − 𝒄, 𝒄 − 𝒅 are in G
...
v
...
P
...
𝒙+𝒚
𝟏 𝟏
𝟏
𝒙+𝒚
𝒂
LONG QUESTIONS
Find three, consecutive numbers in G
...
If three consecutive numbers in A
...
are increased by 𝟏, 𝟒, 𝟏𝟓 respectively,
22
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the resulting numbers are in G
...
Find the original numbers if their sum is 6
...
7
Tick (✔) the correct answer
...
Geometric mean between 4 and 16 is
(a) ±2
(b) ±4
2
...
ii
...
iv
...
vi
...
P
...
P
...
M
...
M
...
M is 4
...
The A
...
of two positive integral numbers exceeds their (positive) G
...
Insert four real geometric means between 𝟑 and 𝟗𝟔
...
For what value 𝒏,
𝒂𝒏 +𝒃𝒏
𝒂𝒏−𝟏 +𝒃𝒏−𝟏
LONG QUESTIONS
is the positive geometric mean between 𝒂 and 𝒃?
The A
...
between two numbers is 5 and their (Positive) G
...
is 4
...
EXERCISE 6
...
1
...
For the series 𝟏 + 𝟓 + 𝟐𝟓 + 𝟏𝟐𝟓 + ⋯ + ∞ , the sum is
(a)
3
...
(a)
5
...
(a)
7
...
Sum to 𝒏 terms of the series : 𝒓 + (𝟏 + 𝒌)𝒓𝟐 + (𝟏 + 𝒌 + 𝒌𝟐 )𝒓𝟑 + ⋯
ii
...
iii
...
𝟎𝟐𝟓 + ⋯
𝟏
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Find vulgar fractions equivalent to the recurring decimal : 𝟎
...
v
...
𝒚
)
EXERCISE 6
...
What will be its population after 3
years if it increases geometrically at the rate of 4% annually
...
If we
𝟐
start with a colony of A bacteria, how many bacteria will have in 𝒏 hours?
EXERCISE 6
...
1
...
, then it is called:
(a) ✔𝐻
...
𝑃
(c) 𝐴
...
The 𝒏𝒕𝒉 term of 𝟐 , 𝟓 , 𝟖 , … is
(a)
3
...
(a)
5
...
(a)
7
...
𝑃
(c) 𝐴 > 𝐺 > 𝐻 (d) all of these
If 𝑨, 𝑮 and 𝑯 are Arithmetic , Geometric and Harmonic means between two negative numbers
then
𝐺 2 = 𝐴𝐻
(b) 𝐴, 𝐺, 𝐻 𝑎𝑟𝑒 𝑖𝑛 𝐺
...
If
𝒂𝒏+𝟏 +𝒃𝒏+𝟏
𝒂𝒏 +𝒃𝒏
(a) ✔0
is H
...
Find the 𝒏th and 𝟖th term of 𝟐 , 𝟓 , 𝟖 , …
ii
...
iii
...
v
...
P
...
P
...
M between 𝒂 and 𝒃
...
M
...
M
...
𝟏𝟔
𝟓
, find the
EXERCISE 6
...
1
...
∑𝒌=𝟏 𝒌 =
𝑛(𝑛+1)
(a)
2
3
...
EXERCISE 7
...
1
...
(a)
3
...
𝟖!
𝟕!
=
✔8
𝟎! =
0
𝒏! =
𝑛(𝑛 − 1)
𝟗!
𝟔!𝟑!
5
...
𝟔
is
𝟑
...
Evaluate
ii
...
iv
...
vi
...
Write in factorial form:
Prove that 𝟎! = 𝟏
Write in factorial form:
𝟗!
𝟐!𝟒!𝟓!
and 𝟒! 𝟎! 𝟏!
𝟐!(𝟗−𝟐)!
𝟐𝟎
...
𝟏
𝒏(𝒏 − 𝟏)(𝒏 − 𝟐) …
...
2
Tick (✔) the correct answer
...
(a)
2
...
(a)
4
...
(a)
6
...
(a)
𝑛!
(c) ✔ 6840
(d) 6880
(c) 6
(d) 10
(c) ✔ 𝑛!
(d) (𝑛 − 1)!
𝑛!
(c) ✔ (𝑛−𝑟)!
(b) 𝑟!
___________ of 𝒏 different objects is called permutation
...
ii
...
𝒏!
Prove that 𝒏𝑷𝒓 = (𝒏−𝒓)!
iii
...
v
...
Find the value of 𝟏𝟏𝑷𝒏 = 𝟏𝟏
...
𝒏 − 𝟏𝑷𝒓−𝟏
How many words can be formed from the letters of “𝑭𝑨𝑺𝑻𝑰𝑵𝑮” using all letters
when no letter is to be repeated
...
𝒏 − 𝟏𝑷𝒓−𝟏
Find the numbers greater than 23000 that can be formed from the
digits 1,2,3,5,6, without repeating any digit
...
EXERCISE 7
...
1
...
The permutation of things which can be represented by the points on a circle are called
(a) Combinations
(b) ✔ Circular permutation (c) Probability
(d) factorial
3
...
ii
...
iv
...
How many arrangements of the letters of “𝑷𝑨𝑲𝑰𝑺𝑻𝑨𝑵” taken all together , can
be made
...
In how many ways can
they be seated at a round table?
How many necklace can be made from 6 beads of different colors ?
LONG QUESTIONS
How many numbers greater than 𝟏𝟎𝟎𝟎, 𝟎𝟎𝟎 can be formed from the digits
𝟎, 𝟐, 𝟐, 𝟑, 𝟒, 𝟒
...
4
Tick (✔) the correct answer
...
(a)
2
...
(a)
4
...
(a)
6
...
(a)
8
...
ii
...
iv
...
Show that : 𝟏𝟔𝑪𝟏𝟏 + 𝟏𝟔𝑪𝟏𝟎 = 𝟏𝟕𝑪𝟏𝟏
Find the values of 𝒏 and 𝒓, when 𝒏𝑪𝒓 = 𝟑𝟓 and 𝒏𝑷𝒓 = 𝟐𝟏𝟎
How many (a) diagonals and (b) triangles can be formed by joining the vertices of
the polygon having 8 sides
...
LONG QUESTIONS
Prove that 𝒏 − 𝟏𝑪𝒓 + 𝒏 − 𝟏𝑪𝒓−𝟏 = 𝒏𝑪𝒓
Prove that 𝒏𝑪𝒓 + 𝒏𝑪𝒓−𝟏 = 𝒏 + 𝟏𝑪𝒓
EXERCISE 7
...
1
...
(a)
3
...
(a)
5
...
(a)
7
...
Non occurrence of an event E is denoted by:
∼𝐸
(b) ✔ 𝐸
(c) 𝐸 𝑐
(d) All of these
A card is drawn from a deck of 52 playing cards
...
Sample space for tossing a coin is:
(a) {𝐻}
(b) {𝑇}
(c) {𝐻, 𝐻}
(d) ✔ {𝐻, 𝑇}
SHORT QUESTIONS
i
...
iii
...
v
...
, 10?
A die is rolled
...
Define “ 𝑬𝒒𝒖𝒂𝒍𝒍𝒚 𝒍𝒊𝒌𝒆𝒍𝒚 𝒆𝒗𝒆𝒏𝒕𝒔 𝒂𝒏𝒅 𝑴𝒖𝒕𝒖𝒂𝒍𝒍𝒚 𝒆𝒙𝒄𝒍𝒖𝒔𝒊𝒗𝒆 𝒆𝒗𝒆𝒏𝒕𝒔”
...
6
Tick (✔) the correct answer
...
If 𝑷(𝑬) = 𝟏𝟐, 𝒏(𝑺) = 𝟖𝟒𝟎𝟎 , 𝒏(𝑬) =
(a) 108
2
...
7
Tick (✔) the correct answer
...
For independent events 𝑷(𝑨 ∪ 𝑩) =
𝑃(𝑆)
𝑛(𝐸)
(d) 14400
(d)
𝑛(𝐸)
𝑃(𝑆)
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𝟏
𝑃(𝐴)
(b) ✔ 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵) (c) 𝑃(𝐴)
...
If 𝑷(𝑨) = 𝟐 , 𝑷(𝑩) = 𝟐 and 𝑷(𝑨 ∩ 𝑩) = 𝟑 then =𝑷(𝑨 ∪ 𝑩) =
(a)
1
2
2
1
(b) ✔ 3
(c) 3
1
(d) 4
SHORT QUESTIONS
If sample space 𝑺 = {𝟏, 𝟐, 𝟑, …
...
A natural number is chosen out of the first fifty natural numbers
...
What is the probability that the sum of number of dots
appearing on them is 4 or 6?
i
...
iii
...
8
Tick (✔) the correct answer
...
For independent events 𝑷(𝑨 ∩ 𝑩) =
(a) 𝑃(𝐴) + 𝑃(𝐵)
(b) 𝑃(𝐴) − 𝑃(𝐵)
(c) ✔ 𝑃(𝐴)
...
If an event 𝑨 can occur in 𝒑 ways and 𝑩 can occur 𝒒 ways , then number of ways that both
events occur is:
(a) 𝑝 + 𝑞
(b) ✔ 𝑝
...
If 𝑷(𝑨) = 𝟎
...
𝟕𝟓 then 𝑷(𝑨 ∩ 𝑩) =
(b) ✔ 0
...
5
(c) 0
...
9
SHORT QUESTIONS
i
...
Define “𝑰𝒏𝒅𝒆𝒑𝒆𝒏𝒅𝒆𝒏𝒕 𝒆𝒗𝒆𝒏𝒕𝒔”
...
EXERCISE 8
...
1
...
(a)
3
...
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Use the principle of extended mathematical induction to prove that
𝟏 + 𝒏𝒙 ≤ (𝟏 + 𝒙)𝒏 𝒇𝒐𝒓 𝒏 ≥ 𝟐 𝒂𝒏𝒅 𝒙 > −1
Prove by mathematical induction that all positive integral values of 𝒏
𝟏
𝟏
𝟏
𝟒
𝟐𝒏−𝟏
𝟏 + + + ⋯+
𝟐
= 𝟐 [𝟏 −
𝟏
𝟐𝒏
]
Prove by mathematical induction that all positive integral values of 𝒏
𝒓𝟐 + 𝒓𝟑 + ⋯ + 𝒓𝒏 =
𝒓(𝟏−𝒓𝒏 )
𝟏−𝒓
𝒓+
,𝒓 ≠ 𝟏
EXERCISE 8
...
4
...
(a)
6
...
(a)
8
...
(a)
General term in the expansion of (𝒂 + 𝒃)𝒏 is:
𝑛
𝑛
(b)✔(𝑟−1)𝑎𝑛−𝑟 𝑥𝑟
(c) (𝑟+1
(𝑛+1
)𝑎𝑛−𝑟 𝑥^𝑟
)𝑎𝑛−𝑟 𝑥 𝑟
𝑟
The number of terms in the expansion of (𝒂 + 𝒃)𝒏 are:
𝑛
(b) ✔ 𝑛 + 1
(c) 2𝑛
𝟏𝟒
Middle term/s in the expansion of (𝒂 − 𝟑𝒙) is/are :
𝑇7
(b) ✔ 𝑇8
(c) 𝑇6 &𝑇7
The coefficient of the last term in the expansion of (𝟐 − 𝒙)𝟕 is :
1
(b) ✔ −1
(c) 7
𝟐𝒏
𝟐𝒏
𝟐𝒏
+
+
+
⋯
+
is
equal
to:
(𝟐𝒏
)
(
)
(
)
(
)
𝟎
𝟏
𝟐
𝟐𝒏
2𝑛
(b) ✔ 22𝑛
(c) 22𝑛−1
𝟏 + 𝒙 + 𝒙𝟐 + 𝒙𝟑 + ⋯
(1 + 𝑥)−1
(b) ✔(1 − 𝑥)−1
(c) (1 + 𝑥)−2
10
...
𝟎𝟐)𝟒
Expand and simplify (𝟐 + 𝒊)𝟓 − (𝟐 − 𝒊)𝟓
iv
...
Determine the middle term in (𝟐𝒙 − 𝟐𝒙)
𝟐 𝟏𝟑
𝟏 𝟐𝒎+𝟏
LONG QUESTIONS
𝟏
𝟏
...
3
Tick (✔) the correct answer
...
The expansion (𝟏 − 𝟒𝒙)−𝟐 is valid if:
(d) −7
(d) 22𝑛+1
(d) (1 − 𝑥)−2
(d) 19
i
...
iii
...
The number of terms in the expansion of (𝒂 + 𝒃)𝟐𝟎 is:
(a) 18
(b) ✔20
(c) 21
Show that the middle term of (𝟏 + 𝒙)𝟐𝒏 is
(d) (𝑛𝑟)𝑎𝑛−𝑟 𝑥 𝑟
𝟐𝒏 𝒙𝒏
𝟐𝒏+𝟏 −𝟏
𝒏+𝟏
30
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1
1
(a) ✔|𝑥| < 4
(b) |𝑥| > 4
(c) −1 < 𝑥 < 1
(d) |𝑥| < −1
SHORT QUESTIONS
Expand (𝟏 + 𝟐𝒙)−𝟏 upto 4 terms, taking the values of 𝒙 such that the expansion is
valid
...
i
...
𝟏
iii
...
Use Binomial theorem find the value of (
...
Find the coefficient of 𝒙𝒏 in the expansion of 𝟏−𝒙𝟐
vi
...
If 𝒙 is very nearly equal 1, then prove that 𝒑𝒙𝒑 − 𝒒𝒙𝒒 ≈ (𝒑 − 𝒒)𝒙𝒑+𝒒
𝟐
𝟏
...
𝟓 𝟐 𝟑
𝟑!
(𝟓) + ⋯, then prove that 𝒚𝟐 + 𝟐𝒚 − 𝟒 = 𝟎
EXERCISE 9
...
1
...
(a)
3
...
(a)
5
...
(a)
7
...
(a)
9
...
3 radian is:
✔171
...
ii
...
Define “𝑹𝒂𝒅𝒊𝒂𝒏”
...
𝟐𝟓𝝅
𝝅
Convert 𝟑𝟔 and 𝟔 into sexagesimal system
...
v
...
Find 𝒓, when 𝒍 = 𝟓 𝒄𝒎 , 𝜽 = 𝟐 𝒓𝒂𝒅𝒊𝒂𝒏
What is circular measure of the angle between the hands of a watch at 4 O’clock?
Find the radius of the circle, in which the arms of a central angle of measure 1
radian cut off an arc of length 35cm
...
vii
...
2
Tick (✔) the correct answer
...
(a)
2
...
(a)
4
...
(a)
6
...
(a)
8
...
(a)
10
...
I
(b) ✔ II
(c) III
(d) IV
If 𝒔𝒆𝒄𝜽 < 0 and 𝒔𝒊𝒏𝜽 < 0 then the terminal arm of angle lies in ___________ Quad
...
ii
...
𝟏𝟐
If 𝒔𝒊𝒏𝜽 = 𝟏𝟑 and the terminal arm of the angle is in quadrant 𝐈st
...
For 𝑪𝒐𝒕𝜽 > 0 , 𝑎𝑛𝑑 𝑆𝑖𝑛𝜃 < 0 In when Quadrant 𝜽 lies ?
𝟏𝟐
Find 𝑪𝒐𝒔 𝜽 𝒂𝒏𝒅 𝑻𝒂𝒏 𝜽 if 𝑺𝒊𝒏 𝜽 = 𝟏𝟑 and the terminal arm of the angle is in quad
1st
...
𝟓
If 𝒄𝒐𝒕𝜽 = and the terminal arm of the angle is in 𝐈 quad
...
EXERCISE 9
...
Tick (✔) the correct answer
...
(a)
2
...
(a)
4
...
(a)
(5)
In right angle triangle, the measure of the side opposite to 𝟑𝟎° is:
✔Half of Hypotenuse (b) Half of Base
(c) Double of base
The point (𝟎, 𝟏) lies on the terminal side of angle:
0°
(b) ✔ 90°
(c) 180°
The point (−𝟏, 𝟎) lies on the terminal side of angle:
0°
(b) 90°
(c) ✔ 180°
The point (𝟎, −𝟏) lies on the terminal side of angle:
0°
(b) 90°
(c) 180°
𝟏
𝟐𝑺𝒊𝒏𝟒𝟓° + 𝑪𝒐𝒔𝒆𝒄𝟒𝟓° =
𝟐
2
√
(b)
3
✔
3
√2
(d) None of these
(d) 270°
(d) 270°
(d) ✔ 270°
(c) −1
(d) 1
SHORT QUESTIONS
𝝅
𝝅
𝝅
𝝅
𝟔
𝟒
𝟑
𝟐𝒕𝒂𝒏𝜽
𝟐
i
...
iii
...
v
...
Verify
𝒕𝒂𝒏 𝟐𝜽 = 𝟏−𝒕𝒂𝒏𝟐 𝜽 when 𝜽 = 𝟒𝟓°
Find 𝒙, if 𝒕𝒂𝒏𝟐 𝟒𝟓° − 𝒄𝒐𝒔𝟐 𝟔𝟎° = 𝒙𝒔𝒊𝒏𝟐 𝟒𝟓°𝒄𝒐𝒔𝟒𝟓°𝒕𝒂𝒏𝟒𝟓°
𝟓
Find the values of the trigonometric functions of
𝝅
𝟐
Find the values of the trigonometric functions of
𝟕𝟔𝟓°
Define “𝑸𝒖𝒂𝒅𝒓𝒂𝒏𝒕𝒂𝒍 𝒂𝒏𝒈𝒍𝒆𝒔”
...
Evaluate
𝝅
𝟑
𝝅
𝟏+𝒕𝒂𝒏𝟐
𝟑
𝟏−𝒕𝒂𝒏𝟐
EXERCISE 9
...
1
...
Domain of 𝒄𝒐𝒔𝜽 is:
(c) 𝜃 ∈ 𝑅 𝑏𝑢𝑡 𝜃 ≠
(a) ✔𝑅
(b) 𝜃 ∈ 𝑅 𝑏𝑢𝑡 𝜃 ≠ 𝑛𝜋, 𝑛 ∈ 𝑍
3
...
Domain of 𝒔𝒆𝒄𝜽 is:
(c) 𝜃 ∈ 𝑅 𝑏𝑢𝑡 𝜃 ≠
(a)
5
...
(a)
7
...
Show that 𝒄𝒐𝒕𝟒 𝜽 + 𝒄𝒐𝒕𝟐 𝜽 = 𝒄𝒐𝒔𝒆𝒄𝟒 𝜽 − 𝒄𝒐𝒔𝒆𝒄𝟐 𝜽, where 𝜽 is not an integral
𝝅
multiple of 𝟐
...
iii
...
v
...
Prove that 𝒄𝒐𝒔𝜽 + 𝒕𝒂𝒏𝜽𝒔𝒊𝒏𝜽 = 𝒔𝒆𝒄𝜽
Prove that (𝒔𝒆𝒄𝜽 − 𝒕𝒂𝒏𝜽)(𝒔𝒆𝒄𝜽 + 𝒕𝒂𝒏𝜽) = 𝟏
𝒄𝒐𝒔𝜽−𝒔𝒊𝒏𝜽
𝒄𝒐𝒕𝜽−𝟏
Prove that
= 𝒄𝒐𝒕+𝟏
𝒄𝒐𝒔𝜽+𝒔𝒊𝒏𝜽
Prove that
𝟐𝒄𝒐𝒔𝟐 𝜽 − 𝟏 = 𝟏 − 𝟐𝒔𝒊𝒏𝟐 𝜽
𝟏−𝒔𝒊𝒏𝜽
(𝒔𝒆𝒄𝜽 − 𝒕𝒂𝒏𝜽)𝟐 =
Prove that
𝟏+𝒔𝒊𝒏𝜽
33
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vii
...
1
Tick (✔) the correct answer
...
Fundamental law of trigonometry is , 𝒄𝒐𝒔(𝜶 − 𝜷)
(a) ✔𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 + 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽
(b) 𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 − 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽
(c) 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽 + 𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛽
(d) 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽 − 𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛽
2
...
𝒄𝒐𝒔 ( 𝟐 − 𝜷) =
(a)
4
...
(a)
𝑐𝑜𝑠𝛽
𝒔𝒊𝒏(𝟐𝝅 − 𝜽) =
𝑐𝑜𝑠𝜃
𝒕𝒂𝒏(𝜶 − 𝜷) =
(b) – 𝑐𝑜𝑠𝛽
(c) ✔ 𝑠𝑖𝑛𝛽
(d) – 𝑠𝑖𝑛𝛽
(b) – 𝑐𝑜𝑠𝜃
(c) ✔ 𝑠𝑖𝑛𝜃
(d) – 𝑠𝑖𝑛𝜃
𝑡𝑎𝑛𝛼−𝑡𝑎𝑛𝛽
𝑡𝑎𝑛𝛼+𝑡𝑎𝑛𝛽
✔1+𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽
𝑡𝑎𝑛𝛼−𝑡𝑎𝑛𝛽
(b) 1−𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽
𝑡𝑎𝑛𝛼+𝑡𝑎𝑛𝛽
(c) 1−𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽
(d) 1−𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽
6
...
𝒔𝒊𝒏 (
𝜽) =
(a) 𝑠𝑖𝑛𝜃
(b) 𝑐𝑜𝑠𝜃
8
...
𝒔𝒊𝒏(𝟏𝟖𝟎° + 𝜶)𝒔𝒊𝒏(𝟗𝟎° − 𝜶) =
(a) ✔ 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛼
(b) – 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛼
10
...
Which is the allied angle
(a) ✔ 90° + 𝜃
(d) ✔ – 𝑐𝑜𝑠𝜃
(c) −𝑠𝑖𝑛𝜃
1
√2
(d)
(c) 𝑐𝑜𝑠𝛾
𝜶+𝜷
ABC then 𝒄𝒐𝒔 ( 𝟐 )
𝛾
(c) cos 2
(b) 60° + 𝜃
√3
2
(d) – 𝑐𝑜𝑠𝛾
=
𝛾
(d) – cos 2
(c) 45° + 𝜃
(d) 30° + 𝜃
SHORT QUESTIONS
i
...
Without using Calculator
...
iv
...
If 𝜶, 𝜷, 𝜸 are angles of triangle ABC, then prove that 𝑪𝒐𝒔 (
Prove that 𝑪𝒐𝒔𝟑𝟑𝟎°𝑺𝒊𝒏𝟔𝟎𝟎° + 𝑪𝒐𝒔𝟏𝟐𝟎°𝑺𝒊𝒏𝟏𝟓𝟎° = −𝟏
State “Distance formula”
...
2
Tick (✔) the correct answer
...
𝒄𝒐𝒔𝟏𝟏°−𝒔𝒊𝒏𝟏𝟏° =
(a) ✔ 𝑡𝑎𝑛56°
(b) 𝑡𝑎𝑛34°
(c) 𝑐𝑜𝑡56°
(d) 𝑐𝑜𝑡34°
2
...
(𝒔𝒊𝒏𝜶 + 𝒔𝒊𝒏𝜷)(𝒔𝒊𝒏𝜶 − 𝒔𝒊𝒏𝜷) =
(a) ✔𝑠𝑖𝑛2 𝛼 − 𝑠𝑖𝑛2 𝛽 (b) 𝑠𝑖𝑛2 𝛼 − 𝑐𝑜𝑠 2 𝛽
SHORT QUESTIONS
i
...
iii
...
Show that
v
...
If 𝜶, 𝜷, 𝜸 are angles of a triangle 𝑨𝑩𝑪 , Show that
𝟏−𝒕𝒂𝒏𝜽𝒕𝒂𝒏𝝋
𝒄𝒐𝒔𝟖°−𝒔𝒊𝒏𝟖°
𝒄𝒐𝒔𝟖°+𝒔𝒊𝒏𝟖°
𝒄𝒐𝒕
vii
...
LONG QUESTIONS
If 𝒄𝒐𝒔𝜶 = −
𝟐𝟒
𝟐𝟓
, 𝒕𝒂𝒏 𝜷 =
𝟗
𝟒𝟎
, then terminal side of the angle of measure of 𝜶
in the II quadrant and that of 𝜷 is in the III quadtant, find the value of
𝒄𝒐𝒔(𝜶 + 𝜷)
...
3
Tick (✔) the correct answer
...
𝒔𝒊𝒏𝟐𝜶 is equal to:
(a) cos2 𝛼 − sin2 𝛼
(b) 1 + 𝑐𝑜𝑠2𝛼
2
...
𝒕𝒂𝒏𝟐𝜶 =
2𝑡𝑎𝑛𝛼
(a) 1+𝑡𝑎𝑛2 𝛼
4
...
𝟏𝟐
Find the value of 𝒄𝒐𝒔𝟐𝜶, when 𝒔𝒊𝒏𝜶 = 𝟏𝟑 where 𝟎 < 𝛼 <
𝟏+𝒔𝒊𝒏𝜶
Prove that
√
iii
...
Prove that
Prove that
𝟏 + 𝒕𝒂𝒏𝜶𝒕𝒂𝒏𝟐𝜶 = 𝒔𝒆𝒄𝟐𝜶
𝒔𝒊𝒏𝟑𝜽
𝒄𝒐𝒔𝟑𝜽
+ 𝒔𝒊𝒏𝜽 = 𝟐𝒄𝒐𝒕𝟐𝜽
𝒄𝒐𝒔𝜽
v
...
𝟏−𝒔𝒊𝒏𝜶
𝝅
= 𝒔𝒆𝒄𝜽
LONG QUESTIONS
Reduce 𝒄𝒐𝒔𝟒 𝜽 to an expression involving only function of multiples of raised to
the first power
...
4
Tick (✔) the correct answer
...
𝒔𝒊𝒏𝜶 + 𝒔𝒊𝒏𝜷 is equal to:
𝛼+𝛽
𝛼−𝛽
) cos ( 2 )
2
𝛼+𝛽
𝛼−𝛽
(c) −2 sin ( 2 ) sin ( 2 )
𝛼+𝛽
𝛼−𝛽
) sin ( 2 )
2
𝛼+𝛽
𝛼−𝛽
(d) 2 cos ( 2 ) cos ( 2 )
(a) ✔ 2 sin (
(b) 2 cos (
2
...
𝒄𝒐𝒔𝜶 + 𝒄𝒐𝒔𝜷 is equal to:
𝛼+𝛽
𝛼−𝛽
) cos (
)
2
2
𝛼+𝛽
𝛼−𝛽
(c) -2 sin ( 2 ) sin ( 2 )
(c) 2 sin (
𝛼+𝛽
𝛼−𝛽
) sin (
)
2
2
𝛼+𝛽
𝛼−𝛽
✔ 2 cos ( 2 ) cos ( 2 )
(b) 2 cos (
(d)
4
...
𝟐𝒔𝒊𝒏𝟕𝜽𝒄𝒐𝒔𝟑𝜽 =
(a) ✔ 𝑠𝑖𝑛10𝜃 + 𝑠𝑖𝑛4𝜃
(b) 𝑠𝑖𝑛5𝜃 − 𝑠𝑖𝑛2𝜃 (c) 𝑐𝑜𝑠10𝜃 + 𝑐𝑜𝑠4𝜃 (d) 𝑐𝑜𝑠5𝜃 − 𝑐𝑜𝑠2𝜃
6
...
ii
...
iv
...
Express 𝑺𝒊𝒏𝟓𝒙 + 𝑺𝒊𝒏𝟕𝒙 as product
...
𝑺𝒊𝒏𝟖𝒙+𝑪𝒐𝒔𝟑𝒙
Prove that
= 𝒕𝒂𝒏𝟓𝒙
𝑪𝒐𝒔𝟓𝒙+𝑪𝒐𝒔𝟐𝒙
Express 𝒔𝒊𝒏(𝒙 + 𝟒𝟓°)𝒔𝒊𝒏(𝒙 − 𝟒𝟓°) as sum or difference
...
vi
...
𝒔𝒊𝒏𝜶−𝒔𝒊𝒏𝜷
𝝅
𝝅
𝜶−𝜷
𝟐
𝟏
𝒄𝒐𝒕
𝜶+𝜷
𝟐
Prove that 𝒔𝒊𝒏 ( 𝟒 − 𝜽) 𝒔𝒊𝒏 (𝟒 + 𝜽) = 𝟐 𝒄𝒐𝒔𝟐𝜽
LONG QUESTIONS
Prove that 𝑺𝒊𝒏𝟏𝟎°𝑺𝒊𝒏𝟑𝟎°𝑺𝒊𝒏𝟓𝟎°𝑺𝒊𝒏𝟕𝟎° =
𝟏
𝟏𝟔
36
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Prove that
𝑺𝒊𝒏
𝝅
𝟗
𝑺𝒊𝒏
𝟐𝝅
𝟗
𝝅
𝟒𝝅
𝟑
𝟗
𝑺𝒊𝒏 𝑺𝒊𝒏
=
𝟑
𝟏𝟔
EXERCISE 11
...
1
...
(a)
3
...
(a)
5
...
(a)
7
...
(a)
Range of 𝒚 = 𝒔𝒆𝒄𝒙 is
𝑅
(b) ✔ 𝑦 ≥ 1𝑜𝑟 𝑦 ≤ −1
(c) −1 ≤ 𝑦 ≤ 1
(d) 𝑅 − [−1,1]
Range of 𝒚 = 𝒄𝒐𝒔𝒆𝒄𝒙 is
𝑅
(b) ✔ 𝑦 ≥ 1𝑜𝑟 𝑦 ≤ −1
(c) −1 ≤ 𝑦 ≤ 1
(d) 𝑅 − [−1,1]
Smallest +𝒊𝒗𝒆 number which when added to the original circular measure of the angle
gives the same value of the function is called:
Domain
(b) Range
(c) Co domain
(d) ✔ Period
Domain of 𝒚 = 𝒄𝒐𝒔𝒙 is
✔−∞ < 𝑥 < ∞ (b) −1 ≤ 𝑥 ≤ 1 (c) −∞ < 𝑥 < ∞ , 𝑥 ≠ 𝑛𝜋 , 𝑛 ∈ 𝑍 (d) 𝑥 ≥ 1, 𝑥 ≤ −1
Domain of 𝒚 = 𝒕𝒂𝒏𝒙 is
2𝑛+1
−∞ < 𝑥 < ∞ (b) −1 ≤ 𝑥 ≤ 1 (c)✔ −∞ < 𝑥 < ∞ , 𝑥 ≠ 2 𝜋 , 𝑛 ∈ 𝑍 (d) 𝑥 ≥ 1, 𝑥 ≤ −1
Period of 𝒄𝒐𝒔𝜽 is
𝜋
𝜋
(b) ✔ 2𝜋
(c) −2𝜋
(d) 2
Period of 𝒕𝒂𝒏𝟒𝒙 is
𝜋
𝜋
(b) 2𝜋
(c) −2𝜋
(d) ✔
4
Period of 𝒄𝒐𝒕𝟑𝒙 is
𝜋
𝜋
𝜋
(b)✔ 3
(c) −2𝜋
(d) 4
9
...
The graph of 𝒚 = 𝒄𝒐𝒔𝒙 lies between the horizontal line 𝒚 = −𝟏 and
(a) ✔ +1
(b) 0
(c) 2
(d) -2
(a)
10
...
SHORT QUESTIONS
i
...
iii
...
v
...
𝒙
𝒙
Find the period of (a) 𝑪𝒐𝒔𝒆𝒄 𝟒
(b) 𝒕𝒂𝒏 𝟕
Find the domain and range of 𝒚 = 𝑺𝒊𝒏𝒙 and 𝒚 = 𝑪𝒐𝒔𝒙
Find the period of 𝟑𝑺𝒊𝒏 𝒙
Find the period of 𝑪𝒐𝒕 𝟖𝒙
𝒙
(c) 3𝑪𝒐𝒔 𝟓
EXERCISE 12
...
1
...
(a)
3
...
(a)
5
...
2611
(b) 0
...
6211
(d) 0
...
𝟓𝟏𝟎𝟎 then 𝒙 =
✔30°40′
(b) 35°40′
(c) 40°40′
(d) 44°44′
37
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SHORT QUESTIONS
i
...
Find the values of 𝒄𝒐𝒔𝟑𝟔°𝟐𝟎′ and 𝒄𝒐𝒕𝟖𝟗°𝟗′
Find 𝜽, if 𝒔𝒊𝒏𝜽 = 𝟎
...
𝟕𝟎𝟓
EXERCISE 12
...
Find the unknown angles and sides of the given triangles
...
iii
...
𝟕𝟒
Solve the right triangle 𝑨𝑩𝑪, in which 𝜸 = 𝟗𝟎° , 𝜷 = 𝟓𝟎°𝟏𝟎′ , 𝒄 = 𝟎
...
3
Tick (✔) the correct answer
...
(a)
2
...
What is the angle of
elevation of the sun at that moment?
What the angle between the ground and the sun is 𝟑𝟎°, flag pole casts a
shadow of 𝟒𝟎𝒎 long
...
EXERCISE 12
...
1
...
To solve an oblique triangle we use:
(a) Law of Sine
3
...
(a)
5
...
In any triangle 𝑨𝑩𝑪, law of tangent is :
(a)
𝑎−𝑏
𝑎+𝑏
=
tan(𝛼−𝛽)
tan(𝛼+𝛽)
𝑎+𝑏
(b) 𝑎−𝑏
=
(𝑺−𝒂)(𝒔−𝒃)
7
...
In any triangle 𝑨𝑩𝑪, √
𝛼
𝒂𝒄
=
𝛾
𝑎−𝑏
(d) 𝑎+𝑏
=
𝛼+𝛽
2
𝛼−𝛽
tan
2
tan
𝛼
(d) cos 2
=
𝛾
𝛼
(c) sin 2
(d) cos 2
=
𝛽
(b) ✔ sin 2
(a) sin 2
(c) ✔
𝛼−𝛽
2
𝛼+𝛽
tan
2
tan
(c) ✔ sin 2
(b) sin 2
8
...
In any triangle 𝑨𝑩𝑪, 𝒄𝒐𝒔 =
𝑠(𝑠−𝑎)
𝑎𝑏
𝑠(𝑠−𝑏)
𝑎𝑐
(a) √
(c) ✔ √
(b) √
𝑠(𝑠−𝑎)
𝑏𝑐
(d) √
𝑠(𝑠−𝑐)
𝑎𝑏
𝑠(𝑠−𝑐)
𝑎𝑏
𝜷
11
...
In any triangle 𝑨𝑩𝑪, 𝒄𝒐𝒔 𝟐 =
𝑠(𝑠−𝑎)
𝑎𝑏
𝑠(𝑠−𝑏)
𝑎𝑐
(a) √
(b) √
13
...
In any triangle 𝑨𝑩𝑪, √(𝒔−𝒂)(𝒔−𝒃) =
𝛾
𝛾
(a) sin 2
(𝒔−𝒂)(𝒔−𝒃)
𝒔(𝒔−𝒄)
𝛾
(b) cos 2
15
...
ii
...
𝟏
Solve the triangle 𝑨𝑩𝑪, if 𝒂 = 𝟓𝟑
, 𝜶 = 𝟒𝟐°𝟒𝟓′
, 𝜷 = 𝟖𝟖°𝟑𝟔′
, 𝜸 = 𝟕𝟒°𝟑𝟐′
, 𝜸 = 𝟑𝟏°𝟓𝟒′
LONG QUESTIONS
State and Prove “𝑳𝒂𝒘 𝒐𝒇 𝑺𝒊𝒏𝒆”
...
5
LONG QUESTIONS
Solve the triangle 𝑨𝑩𝑪 in which :
𝒃 = 𝟑,
𝒄 = 𝟔 and 𝜷 = 𝟑𝟔°𝟐𝟎′
Solve the triangle 𝑨𝑩𝑪 in which :
𝒂 = √𝟑 − 𝟏 , 𝒃 = √𝟑 + 𝟏 and 𝜸 = 𝟔𝟎°
Solve the triangle using first law of tangents and then law of sines:
(a) 𝒃 = 𝟏𝟒
...
𝟏
𝒂 = 𝟑𝟐
EXERCISE 12
...
1
...
𝟑𝟒
, 𝒃 = 𝟑
...
ii
...
iv
...
𝟎𝟔
Find the measure of the greatest angle , if sides of the angle are 𝟏𝟔, 𝟐𝟎, 𝟑𝟑
...
Prove that the greatest
angle of the triangle is 𝟏𝟐𝟎°
...
7
Tick (✔) the correct answer
...
(a)
2
...
ii
...
Find the area of the triangle 𝑨𝑩𝑪, in which 𝒃 = 𝟐𝟏
...
𝟒 , 𝜸 = 𝟑𝟔°𝟒𝟏′ and 𝜶 = 𝟒𝟓°𝟏𝟕′
Find the area of the triangle 𝑨𝑩𝑪, given three sides :
𝒂 = 𝟑𝟐
...
If 𝒂 = 𝟕𝟗 and 𝒄 = 𝟗𝟕, then find angle 𝜷
...
8
Tick (✔) the correct answer
...
The circle passing through the thee vertices of a triangle is called:
(a) ✔Circum circle
(b) in-circle
(c) ex-centre
(d) escribed circle
2
...
In any triangle 𝑨𝑩𝑪, with usual notations, 𝟐𝒔𝒊𝒏𝜶 =
(a) 𝑟
(b) 𝑟1
(c)✔ 𝑅
𝒂
4
...
(a)
6
...
In any triangle 𝑨𝑩𝑪, with usual notations, 𝒔−𝒂 =
(a) 𝑟
(b) 𝑅
8
...
In any triangle 𝑨𝑩𝑪, with usual notations, 𝒔−𝒄 =
(a)
10
...
(a)
12
...
𝜷
Show that
𝒓𝟐 = 𝒔 𝒕𝒂𝒏 𝟐
Prove that
𝒓𝒓𝟏 𝒓𝟐 𝒓𝟑 = ∆𝟐
Prove that
𝒓𝟏 𝒓𝟐 𝒓𝟑 = 𝒓𝒔𝟐
𝟏
𝟏
𝟏
𝟏
Show that
= 𝒂𝒃 + 𝒃𝒄 + 𝒄𝒂
𝟐𝒓𝑹
i
...
iii
...
v
...
LONG QUESTIONS
Prove that
𝒓𝟏 𝒓𝟐 + 𝒓𝟐 𝒓𝟑 + 𝒓𝟑 𝒓𝟏 = 𝒔𝟐
Prove that 𝒓𝟏 + 𝒓𝟐 + 𝒓𝟑 − 𝒓 = 𝟒𝑹
Prove that in an equilateral triangle, 𝒓: 𝑹: 𝒓𝟏 : 𝒓𝟐 : 𝒓𝟑 = 𝟏: 𝟐: 𝟑: 𝟑: 𝟑
Prove that 𝒓 = 𝒔 𝒕𝒂𝒏
𝜶
𝟐
𝒕𝒂𝒏
𝜷
𝟐
𝒕𝒂𝒏
𝜸
𝟐
EXERCISE 13
...
1
...
(a)
Inverse of a function exist only if it is:
Trigonometric function (b) ✔ (1 − 1) function (c) onto function
𝑺𝒊𝒏−𝟏 𝒙 =
𝜋
𝜋
𝜋
✔ − cos−1 𝑥
(b) − sin−1 𝑥
(c) + cos−1 𝑥
2
2
2
3
...
𝑺𝒆𝒄−𝟏 𝒙 =
𝜋
(a) 2 − sec −1 𝑥
(b)
𝜋
2
− sin−1 𝑥
(c) 2 + sec −1 𝑥
(b)
𝜋
2
− sin−1 𝑥
(c)✔
𝜋
5
...
𝑪𝒐𝒕−𝟏 𝒙 =
𝜋
(a) − sec −1 𝑥
2
7
...
𝑻𝒂𝒏−𝟏 (√𝟑) =
𝜋
(a)
6
𝟏
𝟐
(b) –
𝜋
6
(c) –
𝜋
3
(d)✔
𝜋
3
9
...
Find the value of 𝒔𝒊𝒏−𝟏
ii
...
Evaluate 𝒄𝒐𝒔−𝟏 (𝟐)
iv
...
Find the value of 𝒔𝒊𝒏 (𝒕𝒂𝒏−𝟏 (−𝟏))
vi
...
2
Tick (✔) the correct answer
...
𝑺𝒊𝒏−𝟏 𝑨 − 𝑺𝒊𝒏−𝟏 𝑩 =
(a) ✔ 𝑆𝑖𝑛−1 (𝐴√1 − 𝐵2 + 𝐵√1 − 𝐴2 )
(c)
(b) 𝑆𝑖𝑛−1 (𝐴√1 − 𝐴2 − 𝐵√1 − 𝐵2 )
𝑆𝑖𝑛−1 (𝐵√1 − 𝐴2 + 𝐴√1 − 𝐵2 )
(d) 𝑆𝑖𝑛−1 (𝐴𝐵√(1 − 𝐴2 )(1 − 𝐵2 ))
2
...
𝑻𝒂𝒏−𝟏 𝑨 + 𝑻𝒂𝒏−𝟏 𝑩 =
𝐴−𝐵
(a) ✔𝑇𝑎𝑛−1 (1+𝐴𝐵)
𝐴+𝐵
)
1+𝐴𝐵
𝐴−𝐵
)
1−𝐴𝐵
𝐴+𝐵
)
1+𝐴𝐵
(b) 𝑇𝑎𝑛−1 (
(c) 𝑇𝑎𝑛−1 (
(d) 𝑇𝑎𝑛−1 (
(b) 𝑆𝑖𝑛−1 𝑥
(c) 𝜋 − 𝑆𝑖𝑛−1 𝑥
(d) 𝜋 − 𝑆𝑖𝑛𝑥
(b) 𝐶𝑜𝑠 −1 𝑥
(c) ✔ 𝜋 − 𝐶𝑜𝑠 −1 𝑥
(d) 𝜋 − 𝐶𝑜𝑠𝑥
(b) 𝑇𝑎𝑛−1 𝑥
(c) 𝜋 − 𝑇𝑎𝑛−1 𝑥
(d) 𝜋 − 𝑇𝑎𝑛𝑥
(b) 𝐶𝑜𝑠𝑒𝑐 −1 𝑥
(c) 𝜋 − 𝐶𝑜𝑠𝑒𝑐 −1 𝑥
(d) 𝜋 − 𝐶𝑜𝑠𝑒𝑐𝑥
(b) 𝑆𝑒𝑐 −1 𝑥
(c)✔ 𝜋 − 𝑆𝑒𝑐 −1 𝑥
(d) 𝜋 − 𝑆𝑒𝑐𝑥
(b) 𝐶𝑜𝑡 −1 𝑥
(c) ✔ 𝜋 − 𝐶𝑜𝑡 −1 𝑥
(d) 𝜋 − 𝐶𝑜𝑡𝑥
5𝑺𝒊𝒏−𝟏 (−𝒙) =
(a)
6
...
(a)
8
...
𝑺𝒆𝒄−𝟏 (−𝒙) =
(a) – 𝑆𝑒𝑐 −1 𝑥
10
...
ii
...
iv
...
General solution of 𝒕𝒂𝒏𝒙 = 𝟏 is:
𝜋
5𝜋
4
4
(a) ✔{ + 𝑛𝜋,
𝜋
+ 𝑛𝜋} (b) { 4 + 2𝑛𝜋,
5𝜋
4
𝜋
+ 2𝑛𝜋} (c) { 4 + 𝑛𝜋,
3𝜋
4
𝜋
+ 𝑛𝜋} (d) { 4 + 2𝑛𝜋,
3𝜋
4
+ 2𝑛𝜋}
(a)
7
...
(a)
9
...
(a)
If 𝒕𝒂𝒏𝟐𝒙 = −𝟏, then solution in the interval [𝟎, 𝝅]is:
𝜋
𝜋
3𝜋
3𝜋
✔8
(b)
(c)
(d)
4
8
4
If 𝒔𝒊𝒏𝒙 + 𝒄𝒐𝒔𝒙 = 𝟎 then value of 𝒙 ∈ [𝟎, 𝟐𝝅]
𝜋 3𝜋
𝜋 7𝜋
3𝜋 7𝜋
𝜋 −𝜋
{4 , 4 }
(b) { 4 , 4 }
(c) ✔ { 4 , 4 }
(d) { 4 , 4 }
General solution of 𝟒𝒔𝒊𝒏𝒙 − 𝟖 = 𝟎 is:
{𝜋 + 2𝑛𝜋}
(b) {𝜋 + 𝑛𝜋}
(c) {−𝜋 + 𝑛𝜋}
(d) ✔not possible
General solution of 𝟏 + 𝒄𝒐𝒔𝒙 = 𝟎 is:
✔{𝜋 + 2𝑛𝜋}
(b) {𝜋 + 𝑛𝜋}
(c) {−𝜋 + 𝑛𝜋}
(d) not possible
For the general solution , we first find the solution in the interval whose length is equal to
its:
Range
(b) domain
(c) co-domain
(d) ✔ period
All trigonometric functions are ………………
...
✔Periodic
(b) continues
(c) injective
(d) bijective
General solution of every trigonometric equation consists of :
One solution only (b) two solutions (c)✔ infinitely many solutions (d) no real solution
Solution of the equation 𝟐𝒔𝒊𝒏𝒙 + √𝟑 = 𝟎 in the 4th quadrant is:
𝜋
−𝜋
−𝜋
11𝜋
(b) ✔ 3
(c) 6
(d) 6
2
If 𝒔𝒊𝒏𝒙 = 𝒄𝒐𝒔𝒙, then general solution is:
𝜋
𝜋
{ 4 + 𝑛𝜋, 𝑛 ∈ 𝑍}
(b) { 4 + 2𝑛𝜋, 𝑛 ∈ 𝑍}
11
...
(a)
(c)✔ { 4 + 𝑛𝜋, 4 + 𝑛𝜋}
(d){ 4 + 𝑛𝜋, 4 + 𝑛𝜋}
In which quadrant is the solution of the equation 𝒔𝒊𝒏𝒙 + 𝟏 = 𝟎
1st and 2nd
(b) 2nd and 3rd
(c) ✔ 3rd and 4th
If 𝒔𝒊𝒏𝒙 = 𝟎 then 𝒙 =
𝑛𝜋
✔𝑛𝜋 , 𝑛 ∈ 𝑍
(b) 2 , 𝑛 ∈ 𝑍
(c) 0
2
...
(a)
4
...
(a)
6
...
ii
...
iv
...
𝟏
Solve 𝒕𝒂𝒏𝒙 =
v
...
SALMAN SHERAZI
03337727666/03067856232
Title: Important questions
Description: These are 1st year some important question which can help you to get t good marks
Description: These are 1st year some important question which can help you to get t good marks