Maths

61. Let T and C be sets of students taking tea and coffee.

Then, n (T) = 150, number of students taking tea

n (C) = 225, number of students taking coffee

n (T$\cap$C) = 100, number of students taking both tea and coffee.

So, Number of students taking either tea or coffee is.

n (T$\cup$C) = n (T) + n (C) n (T$\cap$C)

= 150 + 225 100

= 275

Number of students taking neither tea coffee

= Total number of students No of students taking either tea or coffee

= 600 275

= 325.

L.H.S = tan ^-1$\frac{2}{11}$ + tan ^-1 $\frac{7}{24}$

Using tan ^-1x+ tan ^-1y= tan ^-1 $\frac{x+y}{1-xy}$ , xy<1

L.H.S =tan ^-1 $\frac{\frac{2}{11}+\frac{7}{24}}{1-\frac{2}{11}*\frac{7}{24}}$ = tan ^-1 $\frac{\frac{2*24+7*2}{11*24}}{\frac{11*24-7*2}{11*24}}$

tan ^-1 $\frac{48+14}{264-14}$ = tan ^-1 $\frac{125}{250}$ = tan ^-1 $\frac{1}{2}$ = R.H.S

60. Let A = {x, y}

B = {y, z}

C = {x, z}

So, A$\cap$B = {x, y}$\cap$ {y, z} = {y}≠$?$

B$\cap$C = {y, z} $\cap$ {x, z} = {z}≠$?$

A$\cap$C = {x, y}$\cap$ {x, z} = {x}≠$?$

But A$\cap$B$\cap$C = (A$\cap$B)$\cap$ C

= {y} $\cap$ (x, z}

=$?$

We know that,

cos3θ= 4cos³θ - 3cosθ

Letx = cosθ Then θ = cos^-1x. We have,

Cos3 (cos^-1x) = 4x³-3x

3cos^-1x = cos^-1 (4x³- 3x)

Hence Proved

59. Let A, B and x be sets such that,

A$\cap$x = B$\cap$x =$?$ and A$\cup$x = B$\cup$x.

We know that,

A = A$\cap$ (A$\cup$x)

= A$\cap$ (B$\cup$x) [? A$\cup$x = $\cup$Bx]

= (A$\cap$B) (A$\cap$x) [by distributive law]

= (A$\cap$B) ∪? [? A∩x =? ]

=> A = A∩ B [? A ∪? = A]

And B = $\cap$ (B$\cup$x)

= B$\cap$ (A$\cup$x) [? B$\cup$x = A$\cup$x]

= (B$\cap$A)$\cup$ (B$\cap$x) [By distributive law]

= (B$\cap$A) ∪$?$ [? B$\cap$x =? ]

B = B$\cap$A [? A$\cup$ $?$= A]

So, A = B = A$\cap$B.

We know that.

sin 3θ =3 sin θ 4sin³θ (identity).

(E) Let x = sinθ. Then, sin ⁻¹x=θ . We have,

Sin3 (sin ⁻¹x) = 3x−4x³

3sin ⁻¹x =sin^-1 (3x−4x³)

Hence proved.

=tan⁻¹ $\left(\frac{tan\pi }{3}\right)$ − $\pi$ + sec⁻¹$\left(sec\frac{\pi }{3}\right)$

= $\frac{\pi }{3}-\pi +\frac{\pi }{3}$

= $\frac{\pi -3\pi +\pi }{3}$

= $\frac{-\pi }{3}.$

Option B is correct.

58. Let A = {a}, B = {a, b}, C = {a, c}

So, A$\cap$B = {a} $\cap$ {a, b} = {a}

A$\cap$C = {a} $\cap$ {a, c} = {a}

i.e., A$\cap$B = A$\cap$C = {a}

But B ≠C. as b$\cancel{\in }$B but $b\cancel{\ne }C$ vice-versa

Given, Sin⁻¹x=y.

(E) We know that the principal value branch of Sin⁻¹ is

$[\frac{-\pi }{2},\frac{\pi }{2}]$ Hence,  $\frac{-\pi }{2}$ ≤ y ≤ $\frac{\pi }{2}$

Option B is correct.

57. (i) We know that,

A $\subset$A

(A $\cap$B)$\subset$ A

[A$\cup$ (A $\cap$B)] $\subset$ (A)

[A $\cup$ (A $\cap$B)] $\subset$A

and also

A$\subset$ [A $\cup$ (A $\cap$B)]

So, A$\cup$ (A$\cap$ B) = A.

(ii) A$\cap$ (A$\cup$ B) = (A$\cap$ A) $\cup$ (A$\cap$ B) [By distributive law]

= A$\cup$ (A$\cap$ B)

= A as (A $\cap$B) $\subset$A