Trigonometric Equation

Formulae
Condition Solution Condition Solution
π
sin θ = 0 θ = nπ, [n ∈ ℤ] sin θ = −1 θ = 4n − 1 , [n ∈ ℤ]
2
π
cos θ = 0 θ = 2n + 1 , [n ∈ ℤ] cos θ = −1 θ = (2n + 1)π, [n ∈ ℤ]
2
π
θ = nπ − , [n ∈ ℤ]
tan θ = 0 θ = nπ, [n ∈ ℤ] tan θ = −1 4
π
Or θ = 4n − 1 , [n ∈ ℤ]
4
π n
sin θ = 1 θ = 4n + 1 , [n ∈ ℤ] sin θ = sin α θ = nπ + −1 α, [n ∈ ℤ]
2
cos θ = 1 θ = 2nπ, [n ∈ ℤ] cos θ = cos α θ = 2nπ ± α, [n ∈ ℤ]
π
θ = nπ + , [n ∈ ℤ]
tan θ = 1 4 tan θ = tan α θ = nπ + α, [n ∈ ℤ]
π
Or θ = 4n + 1 , [n ∈ ℤ]
4
 Related to Trigonometric Equation Solution

❖ Solve: sin + cos = 2sin2θ

 Related to Trigonometric Equation Solution

❖ Solve: cos 2x + 2 = sin x + cos x [0 ≤ x ≤ π]

 Related to Trigonometric Equation Solution

❖ Solve: cos x + 3 sin x = 2. [BUET’10-11]

 Related to Trigonometric Equation Solution

3 1
❖ Solve: − =4 [BUET’06-07]
sin 2x cos 2x
 Quiz – 01

❖ How many integer value of k are there so that 7 cos x + 5 sin x = 2k + 1 equation
may have at least one solution? [2002, 1M]

(a) 4

(b) 8

(c) 10

(d) 12
 Related to Trigonometric Equation Solution

❖ Solve: cot θ cot 3θ = 1 [DU’18-19]

 Related to Trigonometric Equation Solution

❖ Solve: cotx + cot2x + cot3x = cotx cot2x cot3x. [BUET’14-15]

 Related to Trigonometric Equation Solution

❖ Solve: sec 4θ − sec 2θ = 2

 Related to Trigonometric Equation Solution

❖ Solve: sin θ + sin 3θ + sin 2θ = 1 + cos 2θ + cos θ [0 < θ < π]

[BUET’08-09, RUET’15-16]
 Related to Trigonometric Equation Solution

❖ Solve: 1 + sin 2ϕ + sin 2θ = cos 2ϕ + 2θ [θ ≤ ϕ, θ ≤ 90°] [BUET’05-06]

 Related to Trigonometric Equation Solution

❖ Solve: 4 cos x cos 2x cos 3x = 1 ; 0 < x < π. [CUET’08-09, BUTEX’03-04, KUET’06-07]

 Related to Trigonometric Equation Solution

2
❖ Determine the values of x that satisfies 2
1+ cos x + cos x + ………
= 4 where, x ∈
−π, π . [1984, 2M]
 Related to Use of Maximum / Minimum Value

❖ Solve: sin x cos y = 1 ; x, y ∈ [0,2π]

 Related to Use of Maximum / Minimum Value

❖ Solve: sin x + cos y = 2 sec z

2 2 2
 Related to Use of Maximum / Minimum Value

❖ Solve: cos 50
x − sin 50
x=1
 Quiz – 02

❖ No of solutions of the equation sin(e ) = 5 + 5

x x −x
are- [1991, 2M]

(a) 0

(b) 1

(c) 2

(d) Infinity
 Related to Use of Maximum / Minimum Value

π
❖ Determine the range of the value of k if sin−1 2 −1
x + 2x + 2 + tan 2
x − 3x − k 2
> .
2
 Solution to Pair of Trigonometric Equation with Two Variables

❖ Solve: 5 sin x cos y = 1 and 4 tan x = tan y.

 Related to Proof

❖ If the roots of the equation a tan θ + b sec θ = c are α, β , then prove that
2ac
tan α + β = [BUET’19-20]
a2 −c2
 Matrix &
Determinant
 Discussions Related to Matrix, Order of Matrix & Classification of
Matrix

❖ Symmetric Matrix:

❖ Skew-Symmetric Matrix:
 Discussions Related to Matrix, Order of Matrix & Classification of
Matrix

2 3
❖ If A is a symmetric and B is a skew-symmetric matrix and A + B = , then
5 −1
the value of AB is- [2019 Main, 12 April I]

−4 −2
(a)
−1 4

4 −2
(b)
−1 −4

4 −2
(c)
1 −4

−4 2
(d)
1 4
 Discussions Related to Matrix, Order of Matrix & Classification of
Matrix

a b c
❖ If A = b c a where a, b, c are positive real numbers, abc = 1 and A A = I, then
T

c a b
determine the value of a + b + c .
3 3 3
[2003, 2M]
 Some Properties of Diagonal Matrix

i 0 0
❖ If A = 0 i 0 , then σk=1 A =?
100 k

0 0 i
 Related to Trace of Matrix

2
k 7 5
❖ If A = 4 9k −6 and Trace(A) = 0 , then k =?
−3 2 14
 Related to Matrix Equity

x+y −6 −3 a+b
❖ Given A = and B = , where A + B = I. What are the
T T
ab 7 −9 x+y
values of a and b? [IUT’21-22]

(a) 3, −3

(b) −3, −3

(c) 3, 3

(d) −3, 3
 Related to value Determination

❖ Two matrix will be multipliable if the no. of columns of the first matrix is equal to
the no. of rows of the second matrix. If the order of matrix A is m × n and order of
matrix B is n × p , then AB is multipliable but BA is not multipliable. The order of
matrix AB will be = m × p

❖ If the orders of matrices A, B and C are 4 × 5, 5 × 4 and 4 × 2 then the order of

matrix (A + B)C will be-
T
[BUET’10-11, BUTEX’ 15-16, IUT'18-19]
(a) 5 × 4
(b) 4 × 2
(c) 5 × 2
(d) 2 × 5
 Matrix Multiplication

❖ If A and B are two matrices, AB = B and BA = A, then A + B =?

2 2

(a) I

(b) AB

(c) A − B

(d) A + B
 Related to value Determination

1 n
❖ Given the matrix, A = , what is the resultant matrix A ?
n
[IUT’21-22]
0 1
n
1 n
(a)
0 1
2
(b) 1 n
0 1
n n
(c)
0 n
n n
(d)
0 1
 Related to value Determination

cos α − sin α 0 −1
❖ Say, A = , (α ∈ R) which satisfies A =
32
. Then one of the
sin α cos α 1 0
value of α is- [2019 Main, 8 April I]

π
(a)
32

(b) 0

π
(c)
64

π
(d)
16
 Related to the Entries of Matrices

4 3 10 17
❖ If A = and AB = , then determine the entries of matrix B.
2 1 4 7
[BUET’16-17, RUET'09-10]
 Related to the Entries of Matrices

4 −4 8 4
❖ If 1 × A = −1 2 1 , then determine the matrix A. [BUET’17-18]
3 −3 6 3
 Related to the Entries of Matrices

2 1 −2 1 −2 4
❖ If A = ,C = and ABC = , then B =? [BUET'20-21]
3 2 5 −2 3 −1
 Singular/Non-Singular Matrix

❖ Singular Matrix:
6 −3
Example: A =
−16 8

❖ Non-Singular Matrix:
1 2
Example: A =
5 8
 Singular/Non-Singular Matrix

k k 2
❖ If is a real matrix. For which value of k the matrix won’t be invertible?
2 k
[KUET'12-13, SUST’17-18]
(a) −2
(b) 2
(c) ±2
(d) 2
(e) − 2
 Minor and Cofactor of Determinant

2 3 −1
❖ Determine the minor and cofactor of the (3,2) entry 5 6 0 .
−2 1 4
 Minor and Cofactor of Determinant

❖ What is the cofactor of (−2a) of the determinant below? [KUET'11-12, RUET’14-15]

2 2
1+a −b 2ab −2b
2 2
2ab 1−a +b 2a
2 2
2b −2a 1−a −b

(a) 1 − a 4 2
−b 4−b 2

(b) 2a(1 + a + b )
2 2

3
(c) 1 + a + b
2 2

(d) (1 + a + b )
2 2

(e) −2a(1 + a + b ) 2 2
 Inverse Matrix

1
❖ If A is a Non-Singular square matrix , then A
−1
= adj A
A

0 1 2
❖ If A = 1 2 3 and B = A , then b23 =?
−1
[IUT'17-18]
3 1 1

(a) 1
(b) 2
(c) −1
(d) −2
 Inverse Matrix

❖ Two matrices A and B are given. Determine the relation between AB and BA (if they
have any). Represent B −1
in terms of x and A. [BUET'19-20]

3x −4x 2x x 2x −2x
A = −2x x 0 and B = 2x 5x −4x
−x −x x 3x 7x −5x
 Inverse Matrix

5 1
7 7
❖ If A
−1
= 3 2 , then determine the value of A + 2A.
2
[BUET'18-19]
7 7
 Inverse Matrix

4 5 0 0
3 4 0 0
❖ Find the inverse of A = [IUT’21-22]
0 0 3 2
0 0 4 3
1 1 1
0 0 −5 0 0
4 5 4
1 1 1
0 0 −3 0 0
3 4 4
(a) 1 1 (b) 1
0 0 0 0 −2
3 2 3
1 1 1
0 0 0 0 −4
4 3 3

4 −3 0 0 4 −5 0 0
−5 4 0 0 −3 4 0 0
(c) (d)
0 0 3 −4 0 0 3 −2
0 0 −2 3 0 0 −4 3
 Inverse Matrix

1 1 1 2 1 3 1 n−1 1 78
❖ If ⋅ ⋅ ……… = then what is the inverse
0 1 0 1 0 1 0 1 0 1

1 n
matrix of ? [2019 Main, 0 April I]
0 1

1 0
(a)
12 1

1 −13
(b)
0 1

1 0
(c)
13 1

1 −12
(d)
0 1
 Condition for Value of Determinant to be Zero

1 0 10 2 −5 18
❖ −1 0 8 =? ❖ 2 −5 −9 =?
2 0 11 2 −5 27

1 5 10 2 6 18
❖ −1 3 8 =? ❖ −1 −3 −9 =?
2 6 11 3 9 27
 Determination of the Value of the Determinants

3 0 0 −2 4
0 2 0 0 0
❖ Determine the value of: 0 −1 0 5 −3 [RUET’15-16]
−4 0 1 0 6
0 −1 0 3 2
 Determination of the Value of the Determinants
0 0 1
0 1
❖ If J1 = 1 ; J2 = = −1; J3 = 0 1 0 = −1, then what is the value of
1 0
1 0 0
J100 =? [IUT’21-22]
 Determination of the Value of the Determinant

x−4 2x 2x
❖ If 2x x−4 2x = A + Bx x − A 2
then value of (A, B) is-
2x 2x x−4
[BUET’22-23 (Collected), JEE Main 2018]

(a) (−4, −5)

(b) (−4, 3)

(c) (−4, 5)

(d) (4,5)
 Determination of the Value of the Determinant

2 2 2ab 2b
1+a −b
❖ Prove that: 2ab 2
1−a +b 2
−2a 2 2 3
= (1 + a + b )
2 2
−2b 2a 1−a −b
[KUET’03-04, 04-05, 11-12]
 Related to the Determinant of a Square Matrix

❖ If A is a 3 × 3 matrix and A = −7 then what is the value of 2A −1

?
(a) pA =p A n

1
(b) pA −1
=
pn A

p n
(c) −1
pA =
A

−1 A
(d) pA −1
=
pn

1
[Note: A −1
= ]
A
 Related to the Determinant of a Square Matrix

3 −4
❖ If A = then what is the value of det 2A ?
−1
[DU’19-20]
2 −3
(a) pA =p A n

1
(b) pA −1
=
pn A

p n
(c) −1
pA =
A

−1 A
(d) pA −1
=
pn

1
[Note: A −1
= ]
A
 Related to the Determinant of a Square Matrix

❖ If A is matrix of order 4 × 4 and T

A A A = A P
then what is the value of p?
 Quiz – 03

α 2
❖ If A = and A = 125 then the value of α is-
3
[2004, 1M]
2 α

(a) ±1

(b) ±2

(c) ±3

(d) ±5
 Related to the Determinant of a Square Matrix

❖ Say P = aij is a 3 × 3 matrix and Q = bij where bij = 2i+j

aij (1 ≤ i, j ≤ 3) if
det(P) = 2 , then det Q =? [JEE Main 2012]

(a) 2
10

(b) 2 11

(c) 2
12

(d) 2 13
 Related to the Determinant of a Square Matrix

7 3 1
❖ If A = 5 2 −4 , then i adj A =? and ii adj adj A =?
12 5 −2
 Related to the Solution of Equations Containing Determinants

x+4 3 3
❖ Solve: 3 x+4 5 = 0 [RUET’04-05, KUET’04-05, CUET’13-14, BUET’01-02,13-14]
5 5 x+1
 Solution to Equations Using the Concept of Inverse Matrix

❖ Concept of Solution Using Inverse Matrix:

a1 x + b1 y + c1 z = d1 a1 b1 c1 x d1
❖ a2 x + b2 y + c2 z = d2 Say, A = a2 b2 c2 ; x = y ; B = d2
a3 x + b3 y + c3 z = d3 a3 b3 c3 z d3

Here, AX = B ⇒ A −1 −1 −1 −1
AX = A B ⇒ A A X = A B ⇒ IX = A B ⇒ X = A B −1 −1

2x + y + 3z = 13
❖ x − y + 6z = 17 Solve the equations. (Home Work)
5x + 10y − 6z = 7
 Solution to Equations Using Determinants (Cramer’s Rule)

a1 x + b1 y + c1 z = d1
❖ Solve the equations: a2 x + b2 y + c2 z = d2
a3 x + b3 y + c3 z = d3

a1 b1 c1 d1 b1 c1 a1 d1 c1 a1 b1 d1
D = a2 b2 c2 , Dx = d2 b2 c2 , Dy = a2 d2 c2 , Dz = a2 b2 d2
a3 b3 c3 d3 b3 c3 a3 d3 c3 a3 b3 d3
Dx Dy Dz
∴x= ,y = and z =
D D D

❖ Solve using Determinants: x + 2y−z = 5, 3x−y + 3z = 7, 2x + 3y + z = 11

[BUTEX’10-11]
 Solution to Equations Using Determinants (Cramer’s Rule)

❖ Say, λ ∈ ℝ and x + y + 7 = 6, 4x + λy − λz = λ − 2 and 3x + 2y − 4z = −5 equation

system has infinite amounts of solutions, λ is the root of which of the following
equation? [2019 Main, 10 April II]

(a) λ − 3λ − 4 = 0
2

(b) λ + 3λ − 4 = 0
2

(c) λ − λ − 6 = 0
2

(d) λ + λ − 6 = 0
2
 Solution to Equations Using Determinants (Cramer’s Rule)

❖ If the roots (x, y, z) of the system of equations x + ky + 3z = 0, 3x + ky − 2z = 0,

xz
2x + 4y − 3z = 0 are non zero, then what is the value of ? [2018 Main]
y2

(a) −10

(b) 10

(c) −30

(d) 30
Characteristic Equation of Matrix
 Characteristic Equation of Matrix

0 1 0
❖ Show that the matrix A = 0 0 1 , satisfies the equation λI − A = 0. Hence
1 0 0
find A .
12
[BUET’21-22]
 Characteristic Equation of Matrix

1 2
❖ If A = is a matrix, then determine the value of A − 4A + 8A − 12A +
6 5 4 3
−1 3
2
14A .