# These solutions designed by the ‘Basics in Maths‘ team. These notes to do help the intermediate First-year Maths students.

## Inter Maths – 1A two marks questions and solutions are very useful in IPE examinations.

**Hyperbolic Functions**

**Question 1**

Prove that for any x∈ R, sinh (3x) = 3 sinh x + 4 sinh^{3} x

**Sol:**

sinh (3x) = sinh (2x + x)

= sinh 2x cosh x + cosh 2x sinh x

= (2 sinh x cosh x) cosh x + (1 + 2 sinh^{2} x) sinh x

= 2sinh x cosh^{2} x + sinh x + 2 sinh^{3} x

= 2 sinh x (1 + sinh^{2} x) + sinh x + 2 sinh^{3} x

= 2 sinh x + 2 sinh^{3} x+ sinh x + 2 sinh^{3} x

= 3 sinh x + 4 sinh^{3} x

**Question 2**

If cosh x = , find the values of (i) cosh 2x and (ii) sinh 2x

**Sol:**

Cosh 2x = 2 cosh^{2} x – 1

Sinh^{2} 2x = cosh^{2} 2x – 1

**Question 3**

If cosh x = sec θ then prove that tanh^{2}= tan^{2}

**Sol: **

**Question 4**

If sinh x = 5, then show that x =

**Sol: **

Given, sinh x = 5

⟹ x = sinh^{-1}5

**Question 5**

**Sol: **

Given tanh^{-1}

**Question 6**

For x, y ∈ R prove that sinh (x + y) = sinh (x) cosh (y) + cosh (x) sinh (y)

**Sol: **

R.H.S = sinh (x) cosh (y) + cosh (x) sinh (y)

= sinh (x + y)

**Question 7**

For any x∈ R, prove that cosh^{4} x – sinh^{4} x = cosh 2x

**Sol:**

cosh^{4} x – sinh^{4} x = (cosh^{2} x)^{2} – (sinh^{2} x)^{2}

= (cosh^{2} x + sinh^{2} x) (cosh^{2} x – sinh^{2} x)

= 1. cosh 2x

= cosh 2x

**Question 8**

**Sol: **

**=** cosh x + sinh x

**Question 9**

If sin hx = ¾ find cosh 2x and sinh 2x.

**Sol: **

Given sin hx = ¾

We know that cosh^{2} x = 1 + sinh^{2} x

= 1 + (3/4)^{2}

= 1 + 9/16

= 25/16

cos hx = 5/4

cosh 2x = 2cosh^{2} x – 1

= 2(25/16) – 1

= 25/8 – 1

= 17/8

Sinh 2x = 2 sinh x cosh x

= 2 (3/4) (5/4)

= 15/8

**Question 10**

Prove that (cosh x – sinh x)^{ n} = cosh nx – sinh nx

**Sol: **

∴ (cosh x – sinh x)^{ n} = cosh nx – sinh nx

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