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Showing posts with label Maths For Class XI. Show all posts
Showing posts with label Maths For Class XI. Show all posts

Wednesday, 12 June 2024

Unit 10: Trigonometric Identities Of Sum And Difference Of Angles - Mathematics for XI (Science Group) - Multiple Choice Questions (MCQs)

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Mathematics for XI (Science Group)
Unit 10: Trigonometric Identities Of Sum And Difference Of Angles
Multiple Choice Questions (MCQs)



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Unit 9: Linear Programming (LP) - Mathematics for XI (Science Group) - Multiple Choice Questions (MCQs)

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Mathematics for XI (Science Group)
Unit 9: Linear Programming (LP)
Multiple Choice Questions (MCQs)





Unit 8: Functions And Graphs - Mathematics for XI (Science Group) - Multiple Choice Questions (MCQs)

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Mathematics for XI (Science Group)
Unit 8: Functions And Graphs
Multiple Choice Questions (MCQs)



Unit 7: Mathematical Inductions And Binomial Theorem - Mathematics for XI (Science Group) - Multiple Choice Questions (MCQs)

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Mathematics for XI (Science Group)
Unit 7: Mathematical Inductions And Binomial Theorem
Multiple Choice Questions (MCQs)


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Unit 6: Permutation, Combination And Probability - Mathematics for XI (Science Group) - Multiple Choice Questions (MCQs)

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Mathematics for XI (Science Group)
Unit 6: Permutation, Combination And Probability
Multiple Choice Questions (MCQs)



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Special Thanks To Newton's Inn Coaching Center


Unit 5: Miscellaneous Series - Mathematics for XI (Science Group) - Multiple Choice Questions (MCQs)

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Mathematics for XI (Science Group)
Unit 5: Miscellaneous Series
Multiple Choice Questions (MCQs)


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Unit 4: Sequences And Series - Mathematics for XI (Science Group) - Multiple Choice Questions (MCQs)

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Mathematics for XI (Science Group)
Unit 4: Sequences And Series
Multiple Choice Questions (MCQs)



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Special Thanks To Newton's Inn Coaching Center


Unit 3: Vectors - Mathematics for XI (Science Group) - Multiple Choice Questions (MCQs)

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Mathematics for XI (Science Group)
Unit 3: Vectors
Multiple Choice Questions (MCQs)




xxi) Ă® x Ä´ =:
a) Ă®
b) k̂ ✔
c) Ä´
d) -Ă®


Monday, 10 June 2024

Unit 2: Matrices And Determinants - Mathematics for XI (Science Group) - Multiple Choice Questions (MCQs)

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Mathematics for XI (Science Group)
Unit 2: Matrices And Determinants
Multiple Choice Questions (MCQs)

Review Exercise 2

Select correct option:
i. If a matrix A has m rows and n columns, then order of A is:
(a) m x n
(b) n x m
(c) mn
(d) mn

ii. Any matrix of order m x 1 is called:
(a) Row matrix
(b) Column matrix ✔
(c) Square matrix
(d) Zero matrix


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Unit 1: Complex Numbers - Mathematics for XI (Science Group) - Multiple Choice Questions (MCQs)

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Mathematics for XI (Science Group)
Unit 1: Complex Numbers
Multiple Choice Questions (MCQs)


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Special Thanks To Newton's Inn Coaching Center


Friday, 1 March 2024

Unit 12: Graphs Of Trigonometry And Inverse Trigonometric Functions And Solutions Of Trigonometric Equations - Mathematics for XI (Science Group) - Review Exercise 12 -

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Mathematics for XI (Science Group)
Unit 12: Graphs Of Trigonometry And Inverse Trigonometric Functions And Solutions Of Trigonometric Equations
Review Exercise 12



2. Find the maximum and minimum value of the each of the following functions.
  1. y = 5 - 7cos𝛉
  2. y = 4 + 3sin(2𝛉 -5)
Solution:

3. By using graph, find the solution of the following equation.
2cos𝛉 - 𝛉 = 0

Solution:

4. Without using calculator, show that:

Solution:

5. Show that:

Solution:

6. Find the solution set of the equation sec3x = secx.
Solution:



Unit 11: Applications Of Trignometry - Mathematics for XI (Science Group) - Multiple Choice Questions (MCQs) -

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Mathematics for XI (Science Group)
Unit 11: Applications Of Trignometrys
Multiple Choice Questions (MCQs)



21. In equilateral triangle, each angle is equal to:
(a) 30°
(b) 60° ✔
(c) 90°
(d) 45°

22. In equilateral triangle, each side of x units, the area of triangle will be:
(a) 3x / 2
(b) √ 3x / 2
(c) √ 3x / 4
(d) √ 3x2 / 4 ✔


Unit 11: Applications Of Trignometry - Mathematics for XI (Science Group) - Review Exercise 11

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Mathematics for XI (Science Group)
Unit 11: Applications Of Trignometrys
Review Exercise 11





Unit 12: Graphs Of Trigonometry And Inverse Trigonometric Functions And Solutions Of Trigonometric Equations - Mathematics for XI (Science Group) - Multiple Choice Questions (MCQs)

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Mathematics for XI (Science Group)
Unit 12: Graphs Of Trigonometry And Inverse Trigonometric Functions And Solutions Of Trigonometric Equations
Multiple Choice Questions (MCQs)




Wednesday, 24 January 2024

Period Of Trigonometric Functions - Solved Exercise 12.1 - Unit 12: Graphs Of Trigonometry And Inverse Trigonometric Functions And Solutions Of Trigonometric Equations - Mathematics for XI (Science Group)

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Mathematics for XI (Science Group)
Unit 12: Graphs Of Trigonometry And Inverse Trigonometric Functions And Solutions Of Trigonometric Equations
Solved Exercise 12.1

Special Thanks To Sir Adnan Ullah Khan
1.Find the domain and range of each of the following functions:
(i) 2sin 3x
SOLUTION:
Here, y = 2sin 3x
∵ Given function is defined for all real numbers
∴ function of 2sin 3x is defined for all real numbers.
So,
Domain of 2sin 3x is same as the Domain of sin x = R OR (-∞0, +∞0)
As Range of sin 3x is same as the range of sin x = {y | y ∈ R ⋀ -1 ≤ y ≤ 1} OR [-1, +1]
So, the Range of 2sin 3x = {y | y ∈ R ⋀ -2 ≤ y ≤ 2} OR [-2, +2]
Ans: Domain = R, Range = -2 ≤ y ≤ 2 ∀ y ∈ R or [-2, +2]

(ii) 5cos 4x
SOLUTION:
Here, y = 5cos 4x
∵ Given function is defined for all real numbers
∴ function of 5cos 4x is defined for all real numbers.
So,
Domain of cos 4x is same as the Domain of cos x = R OR (-∞0, +∞0)
As Range of cos 4x is same as the range of cos x = {y | y ∈ R ⋀ -1 ≤ y ≤ 1} OR [-1, +1]
So, the Range of 5cos 4x = {y | y ∈ R ⋀ -5 ≤ y ≤ 5} OR [-5, +5]
Ans: Domain = R, Range = -5 ≤ y ≤ 5 ∀ y ∈ R OR [-5, +5]

(ii) 8tan 2x
SOLUTION:
Here y = 8tan 2x
∵ Given function is defined for all real numbers
∴ function of 5cos 4x is defined for all real numbers.
So,
Domain of cos 4x is same as the Domain of cos x = R OR (-∞0, +∞0)
As Range of cos 4x is same as the range of cos x = {y | y ∈ R ⋀ -1 ≤ y ≤ 1} OR [-1, +1]
So, the Range of 5cos 4x = {y | y ∈ R ⋀ -5 ≤ y ≤ 5} OR [-5, +5]
Ans: Domain = R, Range = -5 ≤ y ≤ 5 ∀ y ∈ R OR [-5, +5]




Q.2: Determine whether the following trigonometric functions are even, odd or neither.
i. f(x) = sinx cosx
SOLUTION:
Replacing x by -x
we get,
f(-x) = sin(-x) cos(-x)
f(-x) = [-sin(x)][cos(x)] {∵ sin(-x) = -sinx and cos(-x) = cox(x)
f(-x) = -sinx cosx
f(-x) = -f(x) {∴ f(x) = sinx cos x}
Hence f(x) is an odd function
Ans: sinx cosx is an odd function.

ii. k(x) = x3(sinx + cosx)
SOLUTION:
Replacing x by -x
we get,
k(-x) = (-x)3 sin(-x) + cos(-x)
k(-x) = -x3[(-sinx) + cosx] {∵ sin(-x) = -sinx and cos(-x) = +cox(x)
k(-x) = -x3(-sinx + cosx)  
≠ - k(x) or k(x)
Hence k(x) is neither
Ans: k(x) = x3(sinx + cosx) is neither Even nor Odd.


Q.3: Find the period of the following functions:




Q.4: Find the maximum and minimum values of each of the following:
i) y = 4 + 3 sin𝛉
SOLUTION:
y = 4 + 3 sin𝛉
⇒Here a = 4, b = 3
Maximum value of y = a + |b|
= 4 + |3|
= 4 + 3
= 7
Minimum value of y = a -|b|
= 4 - |3|
= 4 - 3
=1
Ans: Maximum value of y = 7 and Minimum value of y = 1.


iv) y = 8 + 5cos(𝛉 - 25)
SOLUTION:
y = 8 + 5cos(𝛉 - 25)
⇒Here a = 8, b = 5
Maximum value of y = a + |b|
= 8 + |5|
= 4 + 5
= 13
Minimum value of y = a -|b|
= 8 - |5|
= 8 - 5
= 3
Ans: Maximum value of y = 13 and Minimum value of y = 3.

SOLUTION:
y = 1 / 25 - 12 sin(3𝛉 - 2)
⇒Here a = 25, b = -12
Maximum value of y =  a + |b|
M = 25 + |-12|
M = 25 + 12
M =  37
Minimum value of y =  a -|b|
m = 25 - |-12|
m = 25 - 12
m = 13
Let M' and m' represents the maximum and minimum value of the reciprocal of the functions respectively
∵ M > 0 and  m > 0 then
∴ M' = 1 /m and
m' = 1/M
Ans: Maximum value of y = 1 / 13 and
Minimum value of y = 1 /37.


SOLUTION:
y = 1 / 1 + 16cos(5𝛉 - 4
⇒Here a = 1, b = 6
Maximum value of y = a + |b|
M = 1 + |6|
M = 1 + 6
M = 7
Minimum value of y = a -|b|
m = 1 - |6|
m = 1 - 6
m = -5
Let M' and m' represents the maximum and minimum value of the reciprocal of the functions respectively
∵ M > 0 and m > 0  then
∴ M' = 1/M and
m' = 1/m
Ans: Maximum value of y = 1/7 and
Minimum value of y = 1/-5


EXAMPLES

DOMAIN AND RANGE




The Domain and Range for the Trigonometric Functions sinθ, cosθ, tanθ, cosecθ, secθ and cotθ

EVEN, ODD & NEITHER FUNCTIONS


PERIODICITY OF TRIGONOMETRIC FUNCTIONS


MAXIMUM & MINIMUM VALUES OF FUNCTIONS