Page No 4.10:
Question 1:
Find the domain of definition of .
Answer:
For to be defined
Hence, the domain of .
Page No 4.10:
Question 2:
Find the domain of .
Answer:
For to be defined.
For to be defined.
Domain of
.
Page No 4.10:
Question 3:
Find the domain of .
Answer:
For to be defined.
Now, cosx is defined for all real values.
So, domain of cosx is R.
Domain of .
Page No 4.10:
Question 4:
Find the principal values of each of the following:
(i)
(ii)
(iii)
(iv)
Answer:
(i) Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
(ii) Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
(iii) Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
(iv) Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
Page No 4.10:
Question 5:
For the principal values, evaluate each of the following:
(i)
(ii)
(iii)
(iv)
Answer:
(ii)
Page No 4.115:
Question 1:
Evaluate the following:
(i)
(ii)
(iii)
(iv)
Answer:
(i)
(ii)
(iii)
(iv)
Page No 4.115:
Question 2:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
Answer:
Page No 4.115:
Question 3:
Answer:
Now,
Page No 4.115:
Question 4:
Prove that
(i)
(ii)
(iii)
Answer:
(i)
(ii)
(iii)
To prove:
Let us consider
Taking R.H.S.
Hence, proved.
Page No 4.115:
Question 5:
Answer:
Let:
Then,
Page No 4.115:
Question 6:
Show that 2 tan−1 x + sin−1 is constant for x ≥ 1, find that constant.
Answer:
We have
Page No 4.116:
Question 7:
Find the values of each of the following:
(i)
(ii)
Answer:
(i) Let
Then,
(ii)
We have
Page No 4.116:
Question 8:
Solve the following equations for x:
(i)
(ii)
(iii)
(iv) 2 () 1 (2 ), .
(v)
(vi)
Answer:
(i) We know
(ii)
(iii) We know
(iv) 2 () 1 (2 ),
(v)
(vi)
Page No 4.116:
Question 9:
Answer:
Page No 4.116:
Question 10:
Prove that:
where α = ax − by and β = ay + bx.
Answer:
We know
Page No 4.116:
Question 11:
For any a, b, x, y > 0, prove that:
where α = − ax + by, β = bx + ay
Answer:
Then,
Page No 4.116:
Question 1:
Write the value of
Answer:
Page No 4.116:
Question 2:
Write the difference between maximum and minimum values of sin−1 x for x ∈ [− 1, 1].
Answer:
The maximum value of in is at 1.
So, the maximum value is
Again, the minimum value is at 1.
Thus, the minimum value is
So, the difference between the maximum and the minimum value is
Page No 4.116:
Question 3:
If sin−1 x + sin−1 y + sin−1 z = , then write the value of x + y + z.
Answer:
Page No 4.117:
Question 4:
If x > 1, then write the value of sin−1 in terms of tan−1 x.
Answer:
Page No 4.117:
Question 5:
If x < 0, then write the value of cos−1 in terms of tan−1 x.
Answer:
Let
Then,
The value of x is negative.
So, let x = where a > 0.
Now,
Page No 4.117:
Question 6:
Write the value of tan−1x + tan−1 for x > 0.
Answer:
Page No 4.117:
Question 7:
Write the value of tan−1 x + tan−1 for x < 0.
Answer:
When , then both are negative.
Let x = y, y>0
Then,
Page No 4.117:
Question 8:
What is the value of cos−1?
Answer:
Page No 4.117:
Question 9:
If −1 < x < 0, then write the value of .
Answer:
Let
where
Then,
Page No 4.117:
Question 10:
Write the value of sin (cot−1 x).
Answer:
We know
Now, we have
Hence,
Page No 4.117:
Question 11:
Write the value of .
Answer:
We have
∴
Page No 4.117:
Question 12:
Write the range of tan−1 x.
Answer:
The range of is.
Page No 4.117:
Question 13:
Write the value of cos−1 (cos 1540°).
Answer:
We know that
Now,
Page No 4.117:
Question 14:
Write the value of sin−1 .
Answer:
We know that .
Now,
∴
Page No 4.117:
Question 15:
Write the value of cos .
Answer:
Let
Then,
Now, ,
∴
Page No 4.117:
Question 16:
Write the value of sin−1 (sin 1550°).
Answer:
We know that .
Now,
∴
Page No 4.117:
Question 17:
Evaluate sin .
Answer:
We know that
∴
Page No 4.117:
Question 18:
Evaluate sin .
Answer:
We know that
∴
Page No 4.117:
Question 19:
Write the value of cos−1 .
Answer:
We have
∴
Page No 4.117:
Question 20:
Write the value of cos .
Answer:
We have, cos =
Page No 4.117:
Question 21:
Write the value of cos−1 (cos 350°) − sin−1 (sin 350°)
Answer:
We have
∴
Page No 4.117:
Question 22:
Write the value of cos2 .
Answer:
Now,
∴
Page No 4.117:
Question 23:
If tan−1 x + tan−1 y = , then write the value of x + y + xy.
Answer:
We know that .
Now,
∴
Page No 4.117:
Question 24:
Write the value of cos−1 (cos 6).
Answer:
We know that .
Now,
Page No 4.117:
Question 25:
Write the value of sin−1 .
Answer:
Consider,
∴
Page No 4.117:
Question 26:
Write the value of sin .
Answer:
We have
∴
Page No 4.117:
Question 27:
Write the value of tan−1 .
Answer:
We have
∴
Page No 4.118:
Question 28:
Write the value of .
Answer:
Page No 4.118:
Question 29:
Write the value of .
Answer:
We know that .
Now,
∴
Page No 4.118:
Question 30:
Write the value of cos−1 .
Answer:
as does not lie between .
We have
Page No 4.118:
Question 31:
Show that .
Answer:
We have
Page No 4.118:
Question 32:
Evaluate: .
Answer:
We know that .
We have
∴
Page No 4.118:
Question 33:
If find x.
Answer:
We know that .
We have
∴
Page No 4.118:
Question 34:
If then find x.
Answer:
We know that .
We have
∴
Page No 4.118:
Question 35:
Write the value of .
Answer:
We know that and .
∴
Page No 4.118:
Question 36:
If 4 sin−1 x + cos−1 x = π, then what is the value of x?
Answer:
We know that
∴
Page No 4.118:
Question 37:
If x < 0, y < 0 such that xy = 1, then write the value of tan−1 x + tan−1 y.
Answer:
We know
such that
Let x = a and y = b where both a and b are positive.
Page No 4.118:
Question 38:
What is the principal value of ?
Answer:
Let
Then,
Here, is the range of the principal value branch of inverse sine function.
∴
Page No 4.118:
Question 39:
Write the principal value of
Answer:
Let
Then,
Here, is the range of the principal value branch of the inverse sine function.
∴
Page No 4.118:
Question 40:
Write the principal value of
Answer:
∴
Page No 4.118:
Question 41:
Write the value of
Answer:
Page No 4.118:
Question 42:
Write.the principal value of
Answer:
Page No 4.118:
Question 43:
Write the value of
Answer:
Page No 4.118:
Question 44:
Write the principal value of
Answer:
We know
Page No 4.118:
Question 45:
Write the principal value of
Answer:
Page No 4.118:
Question 46:
Write the value of
Answer:
Page No 4.118:
Question 47:
Write the value of .
Answer:
The value of is undefined as it is outside the range i.e., R – (–1, 1) .
Page No 4.118:
Question 48:
Write the value of
Answer:
Page No 4.118:
Question 49:
Write the value of
Answer:
We have
Now,
Page No 4.118:
Question 50:
Wnte the value of the expression
Answer:
Page No 4.119:
Question 51:
Write the principal value of
Answer:
Page No 4.119:
Question 52:
The set of values of
Answer:
The value of is undefined as it is outside the range i.e., R – (–1, 1) .
Page No 4.119:
Question 53:
Write the value of for x < 0 in terms of
Answer:
Page No 4.119:
Question 54:
Write the value of for all in terms of
Answer:
We know that
Therefore, the value of for all in terms of is .
Page No 4.119:
Question 55:
Wnte the value of
Answer:
Page No 4.119:
Question 56:
If , find the value of x.
Answer:
Page No 4.119:
Question 57:
Find the value of
Answer:
Page No 4.119:
Question 58:
If , find the value of x.
Answer:
Page No 4.119:
Question 59:
Find the value of
Answer:
Page No 4.119:
Question 60:
Find the value of
Answer:
Page No 4.119:
Question 61:
Find the value of .
Answer:
Page No 4.119:
Question 1:
If = α, then x2 =
(a) sin 2 α
(b) sin α
(c) cos 2 α
(d) cos α
Answer:
(a) sin 2α
Page No 4.120:
Question 2:
The value of tan is
(a)
(b)
(c)
(d)
Answer:
(d)
Page No 4.120:
Question 3:
2 tan−1 {cosec (tan−1 x) − tan (cot−1 x)} is equal to
(a) cot−1 x
(b) cot−1
(c) tan−1 x
(d) none of these
Answer:
(c) tan−1 x
Let
So,
Page No 4.120:
Question 4:
If
(a) sin2 α
(b) cos2 α
(c) tan2 α
(d) cot2 α
Answer:
(a) sin2 α
We know that .
Page No 4.120:
Question 5:
The positive integral solution of the equation
(a) x = 1, y = 2
(b) x = 2, y = 1
(c) x = 3, y = 2
(d) x = −2, y = −1.
Answer:
(a) x = 1, y = 2
Page No 4.120:
Question 6:
If sin−1 x − cos−1 x = , then x =
(a)
(b)
(c)
(d) none of these
Answer:
(b)
We know that .
Page No 4.120:
Question 7:
sin is equal to
(a) x
(b)
(c)
(d) none of these
Answer:
(a) x
Let
Then,
Page No 4.120:
Question 8:
The number of solutions of the equation
is
(a) 2
(b) 3
(c) 1
(d) none of these
Answer:
(a) 2
We know that .
Therefore, there are two solutions.
Page No 4.120:
Question 9:
If α = , then
(a) 4 α = 3 β
(b) 3 α = 4 β
(c) α − β =
(d) none of these
Answer:
(a) 4 α = 3 β
We know that .
and
∴
Page No 4.120:
Question 10:
The number of real solutions of the equation
is
(a) 0
(b) 1
(c) 2
(d) infinite
Answer:
(c) 2
Page No 4.120:
Question 11:
If x < 0, y < 0 such that xy = 1, then tan−1 x + tan−1 y equals
(a)
(b)
(c) − π
(d) none of these
Answer:
(b)
We know that .
such that
Let x = a and y = b, where a and b both are positive.
Page No 4.120:
Question 12:
(a)
(b)
(c) tan θ
(d) cot θ
Answer:
(a)
Let
Then,
Page No 4.120:
Question 13:
(a) 36
(b) 36 − 36 cos θ
(c) 18 − 18 cos θ
(d) 18 + 18 cos θ
Answer:
(c) 18 − 18 cosθ
We know
Squaring both the sides, we get
Page No 4.120:
Question 14:
If α = then α − β =
(a)
(b)
(c)
(d)
Answer:
(a)
We have
α =
Page No 4.121:
Question 15:
Let f (x) = . Then, f (8π/9) =
(a) e5π/18
(b) e13π/18
(c) e−2π/18
(d) none of these
Answer:
(b) e13π/18
Given:
Then,
Page No 4.121:
Question 16:
is equal to
(a) 0
(b) 1/2
(c) − 1
(d) none of these
Answer:
(d) none of these
We know that .
Now,
Page No 4.121:
Question 17:
If then 9x2 − 12xy cos θ + 4y2 is equal to
(a) 36
(b) −36 sin2 θ
(c) 36 sin2 θ
(d) 36 cos2 θ
Answer:
(c) 36 sin2 θ
We know
Now,
Page No 4.121:
Question 18:
If tan−1 3 + tan−1 x = tan−1 8, then x =
(a) 5
(b) 1/5
(c) 5/14
(d) 14/5
Answer:
(b)
We know that .
Now,
Page No 4.121:
Question 19:
The value of is
(a)
(b)
(c)
(d)
Answer:
(b)
Page No 4.121:
Question 20:
The value of is
(a)
(b)
(c)
(d) 0
Answer:
(d) 0
We have
Page No 4.121:
Question 21:
sin is equal to
(a)
(b)
(c)
(d)
Answer:
(d)
Let
Then,
Now,
Page No 4.121:
Question 22:
If θ = sin−1 {sin (−600°)}, then one of the possible values of θ is
(a)
(b)
(c)
(d)
Answer:
(a)
We know
Now,
Page No 4.121:
Question 23:
If 3 is equal to
(a)
(b)
(c)
(d)
Answer:
(a)
Let
Then,
Page No 4.121:
Question 24:
If 4 cos−1 x + sin−1 x = π, then the value of x is
(a)
(b)
(c)
(d)
Answer:
(c)
We know that .
Page No 4.121:
Question 25:
It (−7), then the value of x is
(a) 0
(b) −2
(c) 1
(d) 2
Answer:
(d) 2
We know that .
So, we get
Page No 4.121:
Question 26:
If , then
(a)
(b)
(c)
(d) x > 0
Answer:
We know that the maximum value of cosine fuction is 1.
Hence, the correct answer is option(a).
Page No 4.121:
Question 27:
In a ∆ ABC, if C is a right angle, then
(a)
(b)
(c)
(d)
Answer:
(b)
We know
Page No 4.121:
Question 28:
The value of sin is
(a)
(b)
(c)
(d)
Answer:
(c)
Let
Then,
Now, we have
Page No 4.122:
Question 29:
(a) 7
(b) 6
(c) 5
(d) none of these
Answer:
(a) 7
Let
Then,
Page No 4.122:
Question 30:
If tan−1 (cot θ) = 2 θ, then θ =
(a)
(b)
(c)
(d) none of these
Answer:
(c)
Page No 4.122:
Question 31:
If , then, the value of x is
(a) 0
(b)
(c) a
(d)
Answer:
Hence, the correct answer is option(d).
Page No 4.122:
Question 32:
The value of is equal to
(a) 0.75
(b) 1.5
(c) 0.96
(d)
Answer:
Hence, the correct answer is option (c).
Page No 4.122:
Question 33:
If x > 1, then is equal to
(a)
(b) 0
(c)
(d)
Answer:
Hence, the correct answer is option (a)
Page No 4.122:
Question 34:
The domain of is
(a) [3, 5]
(b) [−1, 1]
(c)
(d)
Answer:
The domain of is [−1, 1]
Hence, the correct answer is option (c).
Page No 4.122:
Question 35:
The value of
(a)
(b)
(c)
(d)
Answer:
Hence, the correct answer is option (a).
Page No 4.14:
Question 1:
Find the principal values of each of the following:
(i)
(ii)
(iii)
(iv)
Answer:
(i) Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
(ii) We have
Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
(iii) Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
(iv) Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
Page No 4.14:
Question 2:
For the principal values, evaluate each of the following:
(ii)
Answer:
(ii)
Page No 4.14:
Question 3:
Evaluate each of the following:
(ii)
(iii)
Answer:
(i) Let
Then,
We know that the range of the principal value branch is .
Thus,
Now,
Let
Then,
We know that the range of the principal value branch is .
Thus,
So,
(ii)
(iii)
Page No 4.18:
Question 1:
Find the principal values of each of the following:
(i)
(ii)
(iii)
(iv)
Answer:
(i) Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
(ii) Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
(iii)
Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
(iv)
Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
Page No 4.18:
Question 2:
For the principal values, evaluate the following:
(i)
(ii)
Answer:
(i)
(ii)
Page No 4.18:
Question 3:
Find the domain of
(i)
(ii)
Answer:
(ii)
Let f(x) = g(x) − h(x), where g(x)=cotx and h(x)=cot−1x
Therefore, the domain of f(x) is given by the intersection of the domain of g(x) and h(x)
The domain of g(x) is [0, π/2) ⋃ [ π, 3π/2)
The domain of h(x) is
Therfore, the intersection of g(x) and h(x) is R − { nπ, n ⋵ Z}
Page No 4.21:
Question 1:
Find the principal values of each of the following:
(i)
(ii)
(iii)
(iv)
Answer:
(i) Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
(ii)
Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
(iii) Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
(iv)
Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
Page No 4.21:
Question 2:
Find the set of values of
Answer:
The value of is undefined as it is outside the range i.e., R – (–1, 1) .
Page No 4.21:
Question 3:
For the principal values, evaluate the following:
(i)
(ii)
(iii)
(iv)
Answer:
(i)
(ii)
(iii)
(iv)
Page No 4.24:
Question 1:
Find the principal values of each of the following:
(i)
(ii)
(iii)
(iv)
Answer:
(i) Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
(ii) Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
(iii) Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
(iv)
Let
Then,
We know that the range of the principal value branch is .
Thus,
Hence, the principal value of .
Page No 4.24:
Question 2:
Find the domain of
Answer:
Let f(x) = g(x) + h(x), where
Therefore, the domain of f(x) is given by the intersection of the domain of g(x) and h(x)
The domain of g(x) is R − { nπ, n ⋵ Z}
The domain of h(x) is (0, π )
Therfore, the intersection of g(x) and h(x) is R − { nπ, n ⋵ Z}
Page No 4.24:
Question 3:
Evaluate each of the following:
(i)
(ii)
(iii)
(iv)
Answer:
(i)
(ii)
(iii)
(iv)
Page No 4.42:
Question 1:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
Answer:
We know
(i) We have
(ii) We have
(iii) We have
(iv) We have
(v) We have
(vi) We have
(vii) We have
(viii)We have
(ix) We have
(x) )We have
Page No 4.42:
Question 2:
Evaluate each of the following:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
Answer:
We know
(i) We have
(ii) We have
(iii) We have
(iv) We have
(v) We have
(vi)We have
(vii) We have
(viii) We have
Page No 4.42:
Question 3:
Evaluate each of the following:
(i)
(ii)
(iii)
(iv)
(v)
(v)
(v)
(v)
Answer:
We know that
(i) We have
(ii) We have
(iii) We have
(iv) We have
(v) We have
(vi) We have
(vii) We have
(viii) We have
Page No 4.42:
Question 4:
Evaluate each of the following:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
Answer:
We know that
, [0, π/2) ⋃ ( π/2, π]
(i) We have
(ii) We have
(iii) We have
(iv)We have
(v)We have
(vi) We have
(vii)We have
(viii)We have
Page No 4.42:
Question 5:
Evaluate each of the following:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Answer:
We know that
, [−π/2, 0) ⋃ ( 0, π/2]
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Page No 4.43:
Question 6:
Evaluate each of the following:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Answer:
We know that
, ( 0, π)
(i) We have
(ii) We have
(iii) We have
(iv) We have
(v) We have
(vi) We have
Page No 4.43:
Question 7:
Write each of the following in the simplest form:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
Answer:
(i) Let
Now,
(ii) Let
Now,
(iii) Let
Now,
(iv) Let
Now,
(v) Let
Now,
(vi) Let
Now,
(vii) Let
Now,
(viii) Let
Now,
(ix) Let
Now,
(x) Let
Now,
Page No 4.54:
Question 1:
Evaluate each of the following:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
Answer:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
Page No 4.54:
Question 2:
Prove the following results
(i)
(ii)
(iii)
(iv)
Answer:
(i)
(ii)
(iii)
The question is wrong as we can't have arc sin greater than 1
(iv)
Page No 4.54:
Question 3:
Solve:
Answer:
Page No 4.54:
Question 4:
Solve:
Answer:
Given,
We know,
Therefore,
Page No 4.58:
Question 1:
Evaluate:
(i)
(ii)
(iii)
Answer:
(i)
(ii)
(iii)
Page No 4.58:
Question 2:
Evaluate:
(i)
(ii)
(iii)
Answer:
(i)
(ii)
(iii) We have
Let
Now,
Page No 4.59:
Question 3:
Evaluate:
Answer:
Page No 4.6:
Question 1:
Find the principal value of each of the following:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Answer:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Page No 4.66:
Question 1:
Evaluate:
(i)
(ii)
(iii)
(iv)
(v)
Answer:
(i)
(ii)
(iii)
(iv)
(v)
Page No 4.66:
Question 2:
If , find the value of
Answer:
Page No 4.66:
Question 3:
If and , find the values of x and y.
Answer:
Solving and , we will get
Page No 4.66:
Question 4:
If , find the values of x.
Answer:
Page No 4.66:
Question 5:
If , find x
Answer:
Page No 4.66:
Question 6:
Answer:
Page No 4.66:
Question 7:
Answer:
Page No 4.66:
Question 8:
Answer:
Page No 4.66:
Question 9:
Answer:
Page No 4.66:
Question 10:
Answer:
Page No 4.7:
Question 2:
(i)
(ii)
Answer:
(i)
(ii)
Page No 4.7:
Question 3:
Find the domain of each of the following functions:
Answer:
(i)
To the domain of which is [−1, 1]
as x2 can not be negative
Hence, the domain is [−1, 1]
(ii)
Let f(x) = g(x) + h(x), where g(x)=cotx and h(x)=cot−1x
Therefore, the domain of f(x) is given by the intersection of the domain of g(x) and h(x)
The domain of g(x) is [−1, 1]
The domain of h(x) is (−∞, ∞)
Therfore, the intersection of g(x) and h(x) is [−1, 1]
Hence, the domain is [−1, 1].
(iii)
To the domain of which is [−1, 1]
as square root can not be negative
Hence, the domain is
(iv)
Let f(x) = g(x) + h(x), where g(x)=cotx and h(x)=cot−1x
Therefore, the domain of f(x) is given by the intersection of the domain of g(x) and h(x)
The domain of g(x) is [−1, 1]
The domain of h(x) is
Therfore, the intersection of g(x) and h(x) is
Hence, the domain is
Page No 4.7:
Question 4:
If , then find the value of x2 + y2 + z2 + t2
Answer:
We know that the maximum value of
Now,
Now,
Page No 4.7:
Question 5:
If , find the value of x2 + y2 + z2
Answer:
We know that the maximum value of and minimum value of
Now,
For maximum value
and For minimum value
Now, For maximum value
and for minimum value
Page No 4.82:
Question 1:
Prove the following results:
(i)
(ii)
(iii)
Answer:
(iii)
Page No 4.82:
Question 2:
Find the value of
Answer:
We know
Now,
Page No 4.82:
Question 3:
Solve the following equations for x:
(i) tan−12x + tan−13x = nπ +
(ii) tan−1(x + 1) + tan−1(x − 1) = tan−1
(iii) tan−1(x −1) + tan−1x tan−1(x + 1) = tan−13x
(iv) tan−1tan−1x = 0, where x > 0
(v) cot−1x − cot−1(x + 2) = , x > 0
(vi) tan−1(x + 2) + tan−1(x − 2) = tan−1, x > 0
(vii)
(viii)
(ix)
(x)
Answer:
(i) We know
(ii) We know
(iii) We know
and
(iv)
(v)
(vi) We know
(vii) We know
(viii)
We know
(ix)
We know
(x)
Page No 4.82:
Question 4:
Sum the following series:
Answer:
Page No 4.89:
Question 1:
Evaluate:
Answer:
Page No 4.89:
Question 2:
(i)
(ii)
(iii)
Answer:
(i)
(ii)
(iii)
Page No 4.89:
Question 3:
Solve the following:
(i) sin−1x + sin−12x =
(ii)
Answer:
(i) We know
(ii)
Page No 4.92:
Question 1:
If then prove that 9x2 − 12xy cos α + 4y2 = 36 sin2 α.
Answer:
We know
Now,
Page No 4.92:
Question 2:
Solve the equation
Answer:
Page No 4.92:
Question 3:
Solve
Answer:
Page No 4.92:
Question 4:
Prove that:
Answer:
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