1.

For each t ∈R, let [t] be the greatest integer less than or equal to t. Then

$\underset{x\to 0^{+}}{lim} x\left(\left[\frac{1}{x}\right]+\left[\frac{2}{x}\right]+......+\left[\frac{15}{x}\right]\right)$

• does not exist (in R)

• is equal to 0

• is equal to 15

• is equal to 120

If the tangent at (1, 7) to the curve x² = y – 6 touches the circle x² + y² + 16x + 12y + c = 0 then the value of c is

• 95

• 195

• 185

• 85

If α, β ∈ C are the distinct roots, of the equation x² -x + 1 = 0, then α¹⁰¹ + β¹⁰⁷ is equal to

• 2

• -1

• 0

• 1

PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR. If the angles of elevation of the top of the tower at P, Q and R are respectively 45^o, 30^o and 30^o, then the height of the tower (in m) is

• 50√2

• 100

• 50

• 100√3

The sum of the coefficients of all odd degree terms in the expansion of ${\left(x + \sqrt{x^3 -1}\right)}^5 + {\left(x - \sqrt{x^3-1}\right)}^5, (x>1) is$

• 2

• -1

• 0

• 1

Tangents are drawn to the hyperbola 4x² – y² = 36 at the points P and Q. If these tangents intersect at the point T(0, 3) then the area (in sq. units) of △PTQ is

• 36√5

• 45√5

• 54√3

• 60√3

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is

• At least 750 but less than 1000

• At least 1000

• Less then 500

• At least 500 but less than 750

Two sets A and B are as under:

A = {(a-b)∈ RxR:|a-5|<1 and |b-5|<1}

B = {(a,b)∈ Rx R: 4(a-6)² + 9 (b-5)² ≤ 36},then

• Neither A ⊂ B nor B ⊂ A

• B ⊂ A

• A ⊂ B

• A ∩ B = ∅

Tangent and normal are drawn at P(16, 16) on the parabola y² = 16x, which intersect the axis of the
parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and,∠CPB = θ then a value of tan θ is

• 4/3

• 1/2

• 2

• 3

The Boolean expression ~ (p v q )v (-p ∧q ) is equivalent to

• ~q

• ~ p

• p

• q