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QUESTION: 1

Solution:

QUESTION: 2

If cot^{-1} x + cot^{-1} y + cot^{-1} then x + y + z is equal to

Solution:

QUESTION: 3

Solution:

QUESTION: 4

If a, b, c be positive real numbers and the value of then tanθ is equal to -

Solution:

=> θ = tan^{-1} 0

=> tan θ = 0

QUESTION: 5

Solution:

QUESTION: 6

The value of tan^{–1}(1) + cos^{–1}(–1/2) + sin^{–1}(–1/2) is equal to -

Solution:

Since, here we are considering only principle solutions

tan-1(1) = π/4

cos-1(-1/2) = 2π/3

sin-1(-1/2) = -π/6

Sol : π/4 +2π/3 -π/6

: 3π/4

QUESTION: 7

Solution:

L.H.S. and R.H.S. of the given equation are defined if

QUESTION: 8

Solution:

QUESTION: 9

Value of

Solution:

QUESTION: 10

The Greatest value among tan1, tan^{-1}1, sin1, sin^{-1}1, cos1

Solution:

Greatest value sin^{-1}1 = π/2

QUESTION: 11

Where a and b are in their lowest form, then (a + b) equal to

Solution:

On solving, we get

QUESTION: 12

If domain of function f:x→x² + 1 is {0,1}, then its range is

Solution:

QUESTION: 13

The sum of the series cot^{–1}2 + cot^{–1}8 + cot^{–1}18 + cot^{–1}32 + ….. is

Solution:

QUESTION: 14

Solution:

From the given equation

[we take only the +ve sign before the square root since

QUESTION: 15

If tan (x + y) = 33 and x = tan^{–1} 3, then y will be

Solution:

tan (X + Y) = 33 --- eqn1

X = tan^{–1} 3

So,

tan X = 3

eqn1 ⇒ 33 = (tanX + tanY)/(i-tanX*tanY)

33 = ( 3 + tanY )/( 1 – 3*tanY)

33 – 99*tanY = 3 + tanY

30 = 100 * tanY

tanY = 0.3

So, Y = tan^{-1}(0.3)

QUESTION: 16

Solution:

QUESTION: 17

The number of solutions of the equation belonging to the interval ( 0,1) is

Solution:

⇒ x = 1(not possible)

QUESTION: 18

The value of

Solution:

QUESTION: 19

If cos^{-1} x - cos^{-1} y/2 = α, then 4x^{2} − 4xy cos α + y^{2} is equal to

Solution:

QUESTION: 20

If [sin^{-1} cos^{-1} sin^{-1} tan^{-1}x] = 1, ëû where [.] denotes the greatest integer function, then x belongs to the interval.

Solution:

[x] = K, K ∈ Z ⇒ K __<__ x < K + 1, where [x] represents integer part of x.

QUESTION: 21

The value of sin^{-1} [cos(cos^{-1}(cosx) + sin^{-1} (sinx))], where

Solution:

sin^{-1} [cos(cos^{-1}(cosx) + sin^{-1} (sinx))], where x€(π/2,π)

=> sin^{-1}[cos(x + π-x)]

=> sin^{-1}[cos(π)]

=> sin^{-1}[-1]

=> -π/2

QUESTION: 22

The principal value of

Solution:

QUESTION: 23

If x_{1}, x_{2} , x_{3} , x_{4} are roots of the equation x^{4} - x^{3} sin 2β + x^{2} cos2β - x cosβ - sinβ Tan^{-1} (x_{1}) + Tan^{-1} (x_{2}) + Tan^{-1} (x_{3}) + Tan^{-1}(x_{4}) can be equal to

Solution:

QUESTION: 24

The number of solutions of

Solution:

**Correct Answer :- D**

**Explanation : **sin(2cos^{-1}(cot(2tan^{-1}x)))=0

2cos^{-1}(cot(2tan^{-1}x))=sin^{-1}0=0

cos^{-1}(cot(2tan^{-1}x))=0

cot(2tan^{-1}x)=cos0

cot(2tan^{-1}x)=1

cot(tan^{-1}(2x/(1−x^{2})))=1

(1−x^{2})/2x=1

1−x^{2}=2x

x^{2}+2x−1=0

x=[−2±√(4+4)]/2

x=−1±√2, +-1

QUESTION: 25

If xy + yz + zx = 1, then, tan^{–1}x + tan^{–1}y + tan^{–1}z =

Solution:

If xy + yz + zx = 1,

Then, 1 – xy – yz – zx = 0 ----- eqn1

tan^{-1}x + tan^{-1}y = tan^{-1}((x+y)/(1-xy))

tan^{-1}x + tan^{-1}y + tan^{-1}z = tan-1((x+y)/(1-xy)) + tan^{-1}z

= tan^{-1}( ( ((x+y)/(1-xy)) + z ) / (1 – ((x+y)*z)/(1+xy)))

= tan^{-1}((x+y+z-xyz)/(1-xy-yz-zx))

= tan^{-1}((x+y+z-xyz)/0) [From eqn1]

= π/2

QUESTION: 26

then the value of tan^{-1} (sin A) + tan^{-1}(sin^{3} A) + tan^{-1} (cot A cos A)

Solution:

QUESTION: 27

The equation

Solution:

Cancelling x both sides (One root x=0)

25 = 3*(25-16x^{2})^{1/2} + 4*(25-9x^{2})^{1/2}

25 - 3*(25-16x^{2})^{1/2} = 4*(25-9x^{2})^{1/2}

Squaring both sides,

625 + 9*(25-16x^{2}) – 2*25*3*(25-16x^{2})^{1/2} = 16*(25-9x^{2})

625 + 225 – 144x^{2} – 2*25*3*(25-16x^{2})1/2 = 400 – 144x^{2}

450 = 2*25*3*(25-16x^{2})^{1/2}

3 = (25-16x^{2})^{1/2}

Squaring both sides

9 = 25 – 16x^{2}

x^{2} = 1

x = ±1

So, roots of the equation are

x = -1, 0, 1

Sum of the roots is zero.

QUESTION: 28

The number of real solution of

Solution:

From function it is clear that

(1) x(x + 1) __>__ 0 ∴ Domain of square root function.

(2) x^{2} + x + 1__>__0 ∴ Domain of square root function.

Domain of sin^{-1} function. From (2) and (3)

QUESTION: 29

If 0 < x < 1, the numberof solutions of the equation tan^{-1}(x-1) + tan^{-1} x + tan^{-1} (x+1) = tan^{-1} 3x is

Solution:

The given equation can be written as

tan^{-1}(x-1) + tan^{-1} (x +1) = tan^{-1} 3x - tan^{-1} x

QUESTION: 30

If x^{2} + y^{2} + z^{2} = r^{2}, then

Solution:

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