### Eamcet - Maths - Geometry

If N denotes the set of all positive integers and if f:N->N is defined by f(n) = thesum of positive divisors of n then, f (2k, 3), where k is a positive integers is

A.  2k+1 -1

B.  2(2k+1 -1 )

C.  3(2k+1 -1 )

D.  4(2k+1 -1 )

The area (in square units) of the triangle formed by the lines x = 0,y = 0 and 3x + 4y = 12, is

A.  3

B.  4

C.  6

D.  12

If a point P moves such that its distances from the point A (1, 1) and the line x+ y + 2 = 0 are equal, then the locus of  P is

A.  a straight line

B.  a pair of straight lines

C.  a parabola

D.  an ellipse

For a binomial variate X with n = 6, if P(X = 2) = 9P(X = 4), then its variance is

A.  8 / 9

B.  1 / 4

C.  9 / 8

D.  4

Box A contains 2 black and 3 red balls, while Box B contains 3 black and 4 red balls. Out of these two boxes one is selected at random; and the probability of choosing Box A is double that of Box B. If a red ball is drawn from the selected box, then the probability that it has come from Box B, is

A.  21/41

B.  10/31

C.  12/31

D.  13/41

A coin and six faced die, both unbiased, are thrown simultaneously. The probability of getting a head on the coin and an odd number on the die, is

A.  1/2

B.  3/4

C.  1/4

D.  2/3

In a ∆ABC, a(cos2 B + cos2 C) + cos A (c cos C + b cos B) is equal to

A.  a

B.  b

C.  c

D.  a + b + c

2 tan h -1 1/2  is equal to

A.  0

B.  log 2

C.  log 3

D.  log 4

sin -1(4/5) + 2 tan -1(1/3)  is equal to

A.  0

B.  π/3

C.  π/4

D.  π/2

If A+C =2B then(cos C - cos A) / (sin A -sin C) is equal to

A.  cot B

B.  cot 2B

C.  tan 2B

D.  tan B

The extreme values of  4 cos(x2) cos(π/3  + x2 ) cos(π/3 - x2) over IR are

A.  -1, 1

B.  -2, 2

C.  -3, 3

D.  -4, 4

The roots of the equation x3 - 3x - 2 = 0 are

A.  -1,-1,2

B.  -1 ,1,-2

C.  -1,2,-3

D.  -1,-1,-2

The function f : c → c defined by  f(x) = (ax+b) / (cx+d) for x ε c where bd ≠ 0 reduces to a constant function, if

A.  a = c

B.  b = d

D.  ab = cd

If f(x) = 10 cos x +(13+2x) sin x then f"(x) + f(x) is equal to

A.  cos x

B.  4cos x

C.  sin x

D.  4 sin x

A stone thrown upward has its equation of motion s =490t - 4.9t2 then the maximum height reached by it is

A.  24500

B.  12500

C.  12250

D.  25400

The radius of a circle plate is increasing at the rate of  0.01 cm/s when the radius is at 12 cm . then the rate at which the area increases is

A.  0.24 π sq.cm/sec

B.  60 π sq.cm/sec

C.  24 π sq.cm/sec

D.  12 π sq.cm/sec

Observe the following statements A: f'(x) = 2x3 - 9x2  + 12x - 3 is increasing outside the interval (1, 2)R: f'(x) < 0 for x belongs to (1,2).Then which of the following is true

A.  Both A and R are true, and R is not the correct reason for A

B.  Both A and R are true, and R is the correct reason for A

C.  A is true but R is false

D.  A is false but R is true

For any integer n ≥ 1, the number of positive divisors of n is denoted by d(n). Then for   a prime P, d(d(d(P7}}) =

A.  1

B.  2

C.  3

D.  P

The line among the following which touches the parabola y2=4ax, is

A.  x + my + am3 = 0

B.  x-my +am2 = 0

C.  x + my - am2 = 0

D.  y + mx + am2 = 0

Which of the following equations gives a circle

A.  r = 2 sinθ

B.  r2 cos2θ = 1

C.  r(4 cosθ + 5 sinθ) = 3

D.  5 = r (1 + √2  cosθ)

The Cartesian form of the polar equation θ = tan -1 2 is

A.  x=2y

B.  y=2x`

C.  x=4y

D.  y=4x

The area (in square units) bounded by the curves y2 = 4x and x2 = 4y in the plane is :

A.  8/3

B.  16/3

C.  32/3

D.  64/3

If dx + dy =(x + y) ( dx- dy ) then log ( x  +  y ) is equal to

A.  x + y + c

B.  x +2 y + c

C.  x - y + c

D.  2x + y + c

If PM is the perpendicular from P (2, 3) onto the line x + y = 3, then the co-ordinates of M are

A.  (2, 1)

B.  (-1,4)

C.  (I, 2)

D.  (4,-1)

The equation of the straight line perpendicular to 5x - 2y = 7 and passing through the point of intersection of the lines 2x + 3y = 1 and 3x + 4y = 6, is

A.  2x + 5y + 17 = 0

B.  2x + 5y - 17 = 0

C.  2x - 5y + 17 = 0

D.  2x - 5y = 17

The area of the triangle formed by the pair of straight lines (ax + by)2 - 3(bx - ay)2 = 0 and ax + by + c = 10, is

A.  c2 / (a2 + b2)

B.  c2 / 2(a2 + b2)

C.  c2 / √2(a2 + b2)

D.  csup>2 / √3(a2 + b2)

The product of the perpendicular distances from the origin on the pair of straight lines12x2 + 25xy + 12y2 + 10x + 11y + 2 = 0, is

A.  1/25

B.  2/25

C.  3/25

D.  4/25

The direction cosines of the line passing through P (2, 3, - 1) and the origin are

A.  2/√14 , 3/ √14, 1/ √14

B.  2/√14 , -3/ √14, 1/ √14

C.  -2/√14 , -3/ √14, 1/ √14

D.  2/√14 ,- 3/ √14, -1/ √14

The point collinear with (1,-2,-3) and (2,0, 0) among the following is

A.   (0,4, 6)

B.  (0,-4,-5)

C.  (0,-4,-6)

D.   (0,-4, 6)

The equation of the circle whose diameter is the common chord of the circlesx2 + y2 + 2x + 3y + 2 = 0 and x2 + y2 + 2x-4y-4 = 0 is

A.  x2 + y2 + 2x + 2y + 2 = 0

B.  x2 + y2 + 2x + 2y-l =0

C.  x2 + y2 + 2x + 2y + l =0

D.  x2 + y2 + 2x + 2y+ 3= 0