If x = cos α + cos β – cos( α + β ) and y = , then (x – y) equals

1) 0

2) 1

3) – 1

4) – 2

Correct Answer :2

**Solution**

Let there exist a unique point P inside a** **D** **ABC such that ** **ÐPAB=** **ÐPBC=** **ÐPCA= α** . **If PA = x, PB = y, PC = z, D**= **area of D** **ABC and a, b, c, are** **the sides opposite to the angle A,B,C respectively, then tan α is equal to

Correct Answer :4

**Solution**

Let there exist a unique point P ...........We have

x^{2} = z^{2} + b^{2}–2bz cos α

[By cosine rule]

y^{2} = x^{2} + c^{2}–2cx cos α

z^{2} = y^{2 }+a2 – 2ay cos α

On adding, we get 2( cx + ay + bz) cos α = a^{2} + b^{2}+ c^{2} ...........(1)

Also area of DABC =area (DPAB) + area (DPBC) + area (D PAC)

=>D= 1/2( cx + bz + ay)sin α ....(2)

From (1) and (2) ,

We get

**Let f (x) = and g (x) be the inverse of f(x), then which one of the following holds good?**

1) 2g'' = g2

2) 2g'' = 3g^{2}

3) 3g'' = 2g^{2}

4) 3g'' = g^{2}

Correct Answer :2

**Solution**

Let O be centre, S, S' be foci of hyperbola. If tangent at any point P on hyperbola cuts ...

[Hyperbola]

Let O be centre, S, S' be foci of hyperbola. If tangent at any point P on hyperbola cuts asymptotes at M and N then OM + ON =

1) |SP - S'P|

2) SP + S'P

3) SS'

4) distance between vertices

Correct Answer :2

**Solution**

The solution set of values of x satisfying equation

1) all real numbers

2) (-∞ , 0]

3) [0,∞)

4) (-∞ , -1) È (1, ∞)

Correct Answer :3

**Solution**

If g(x) = 2f (2x^{3} -3x^{2}) + f (6x^{2} -4x^{3 }-3 ), " x Î R and f " (x) > 0, " x Î R, then g(x) is increasing on the interval

1) (-∞ , -1/2) È (0,1)

2) (-1/2 , 0) È (1,∞ )

3) (0,∞)

4) (-∞ , 1)

Correct Answer :1

**Solution**

**If f(x) = , then f(x) is
1) Continuous on [-1, 1] and differentiable on (-1, 1)
2) Continuous on [-1, 1] and differentiable on (-1, 0) ∪ (0, 1)
3) Continuous and differentiable on [-1, 1]
4) None**

Correct Answer :2

**Solution**

A line makes the same angle θ with each of the X and Z - axes. If it makes the angle ...

[Trigonometric Equations]

A line makes the same angle θ with each of the X and Z - axes. If it makes the angle β with Y - axis such that sin^{2} β = 3sin^{2}θ, then cos^{2}θ equals

1) 3/5

2) 1/5

3) 2/5

4) 2/3

Correct Answer :1

**Solution**

cos^{2}θ + cos^{2} θ + cos^{2} β = 1

= 2(1—sin^{2}θ)+ 1 —sìn^{2}β = 1

= 2(1—sin^{2}θ)+1—3sin^{2}θ = 1

= 3— 5sin^{2}θ =1

= sin^{2}θ = 2/5

So, cos^{2} θ =1 —2/5 =3/5

**Let and B = A ^{2 }. If (l – m)^{2} + (p – q)^{2} = 9, (m – n)^{2} + (q – r)^{2} = 16, (n – l)^{2} + (r – p)^{2} = 25, then the value of det. B equals..**

1) 100

2) 125

3) 144

4) 169

Correct Answer :3

**Solution**

**det. A is twice the area of the triangle with vertices (l , p), (m, q), (n, r) with sides 3, 4, 5.**

D^{2} = s(s – a)(s – b)(s – c)

D^{2 } = 6(6 – 3)(6 – 4)(6 – 5)

D** ^{2 } = 36 =>**D

Now det A =

=>det. B = (det A)

= 144