If two vertices of an equilateral triangle have integral co-ordinates then the third vertex

STRAIGHT LINE

26. If two vertices of an equilateral triangle have integral co-ordinates then the third vertex

will have

(a) integral co-ordinates

(b) co-ordinates which are rational (c) at least one co-ordinate irrational (d) co-ordinates which are irrational

27. If a , , a2, a3, P1; P2, p3 are the values'of n for

(a) an isosceles triangle

(b) a right angled isosceles triangle (c) a right angled triangle

(d) an equilateral triangle

28. The points (a, p), (y, 5), (a, 8) and (y, P) where a , p, y, 8 are different real numbers are (a) collinear

(b) vertices of a square (c) vertices of a rhombus (d) concyclic

29. The equations of the three sides of a triangle are x = 2 , y + l = 0 and x + 2 y = 4. The co-ordinates of the circumcentre of the triangle are

(a) (4,0) (b) (2, - 1 ) (c) (0, 4) (d) None of these 30. If P (1 + a/ 2 + a / V 2 ) be any point on a

line then the range of values of t for which the point P lies between the parallel lines x + 2y = 1 and 2x + 4y = 15 is

31. If the point (a, a) fall between the lines I x + y I = 2 then

(a) I a I = 2 (b) I a I = 1

32. If a ray travelling the line x = 1 gets reflected the line x + y = 1 then the equation of the line along which the reflected ray trevels is (a) y = 0 ( b ) x - y = l

34. The diagonals of the parallelogram whose sides are Ix + my + n = 0, lx + my + n = 0, 35. The area of the triangle having vertices (-2,

1), (2, 1) and f Lim Lim cos "' (n ! 7tx); x is m —> oo n —> oo

rational, Lim Lim cos

m —> 0 0 n —» oo 2 m (n ! kx); where x is irrational) is

(a) 2 (b)3 (c) 4 (d) None of these 36. Consider the straight line ax + by = c where

a,b,ce RH this line meets the co-ordinate axes at A and B respectively. If the area of the AOAB, O being origin, does not depend upon a, b and c then

(a) a, b, c are in A.P.

(b) a, b, c are in G.P.

(c) a, b, c are in H.P.

(d) None of these

37. If A and B are two points having co-ordinates (3, 4) and (5, -2) respectively and P is a point such that PA = PB and area of triangle PAB= 10 square units, then the co-ordinates of P are

(a) (7, 4) or (13, 2) (b) (7, 2) or (1, 0) (c) (2, 7) or (4, 13) (d) None of these

38. The position of a moving point in the xy-plane at time is given by (u cos a.t, u sin at — — gt), where u, a, g are I 2 constants. The locus of the moving point is (a) a circle (b) a parabola (c) an ellipse (d) None of these 39. If the lines x + ay + a = 0,bx + y + b = 0 and

cx + cy + 1 = 0 (a, b, c being distinct * 1) are concurrent, then the value of a-I b-1 c - t is

(a) - 1 (b) 0

vertices A (0, b), B (0, 0 ) and C (a, 0) are their point of intersection. The co-ordinates of the foot of perpendicular from P on the bisector of the angle between them are (a) ^ 0 , ^ ( 4 + 5 ^ 3 ) j o r j ^ 0 , | ( 4 - 5 V 3 ) j , (depending on which line the point P is taken) ( b ) ^ ( 4 + 5V3)

(a) parabola (b) ellipse (c) hyperbola (c) circle

43. Let O be the origin, and let A (1, 0), B (0, 1)

Straight Line 171

If the point P (a, a) lies in the region corresponding to the acute angle between the lines 2y = x and 4y = x, then

(a) a e (2, 4) (b) a e (2, 6) (c) a e (4, 6) (d) a e (4, 8)

A ray of light coming along the line 3x + 4y - 5 = 0 gets reflected from the line in A and B respectively, then equation of a line parallel to L, and L2 and passes through a point P such that AP:PB = 2:1 (internally) is (P is on the line 2x + 7y - 1 = 0)

(a) 9x + 12)1 + 3 = 0 (b)9x + 1 2 y - 3 = 0 (c) 9x + 12y - 2 = 0 (d) None of these 47. If the point (1 + cos 9, sin 0) lies between the

region corresponding to the acute angle between the lines x - 3y = 0 and x-6y = 0

In a AABC, side AB has the equation 2x + 3y = 29 and the side AC has the equation x + 2y- 16. If the mid point of BC is (5, 6) then the equation of BC is

(a) 2x + y = 7 (b) x + y = 1 ( c ) 2 x - y = 1 7 (d) None of these 49. The locus of a point which moves such that

the square of its distance from the base of an isosceles triangle is equal to the rectangle under its distances from the other two sides is (a) a circle (b) a parabola

50. The equation of a line through the point (1, 2) whose distance from the point (3, 1) has the greatest possible value is

(a) y = x (b) y = 2x (c) y = - 2x (d) y = - x

51. If the point (cos 0, sin 0) does not fall in that angle between the lines y = I x - 1 I in which the origin lies then 0 belongs to

371

52. ABC is an equilateral triangle such that the vertices B and C lie on two parallel lines at a distance 6. If A lies between the parallel lines at a distance 4 from one of them then the length of a side of the equilateral triangle is (a) 8

, , 4 V 7

(b)

(d) None of these 53. The four sides of a quadrilateral are given by

the equation xy (x - 2) (y - 3) = 0. The equation of the line parallel to x - 4y = 0 that divides the quadrilateral in two equal areas is (a) x - 4y - 5 = 0 (b) x - 4y + 5 = 0 (c) x - 4y - 1 = 0 (d) x - 4y + 1 = 0 54. Two points A and B move on the x-axis and

the y-axis respectively such that the distance between the two points is always the same.

The locus of the middle point of AB is (a) a straight line (b) a circle 56. If sum of the distances of a point from two

perpendicular lines in a plane is 1, then its locus is

(a) a square (b) a circle

59. If the area of the triangle whose vertices are (b, c), (c, a) and (a, b) is A , then the area of perpendicular, then sin 2a + sin 2b is equal to (a) sin (a-b)-2 sin (a + b)

Each question in this part, has one or more than one correct answer(s). For each question, write the letters a, b, c, d corresponding to the correct answer(s).

(a) A (b) (a + b + c) A (c) aA + bA (d) None of these 60. On the portion of the straight line x + y = 2

which is intercepted between the axes, a square is constructed away from the origin, with this portion as one of its side. If p denote the perpendicular distance of a side of this square from the origin, then the maximum value of p is

(a) V2 (b) 2 <2 (c) 3 V2 (d) 4 <2

61. The points (2, 3), (0, 2), (4, 5) and (0, t) are concyclic if the value of t is

(a) 1 (b) 1

• + f = 1 and y + ^ = 1 lies on will represent the same line if

(a) b = c (b) c = a (c) a-b (d)a + b + c = 0

64. The area of a triangle is 5. Two of its vertices are (2, 1) and (3, -2). The third vertex lies on parallelogram then remaining vertex can not be

68. Equation of a straight line passing through the point of intersection o f x - y + l = 0 and 3x + y - 5 - 0 are perpendicular to one of them is

(a) x + y + 3 = 0 (b)x + y - 3 = 0 (c) x - 3y - 5 = 0 (d) x — 3y + 5 = 0 69. If one vertex of an equilateral triangle of side

a lies at the origin and the other lies on the

Straight Line 173 line x = 0, the co-ordinates of the third

vertex are

71. If the co-ordinates of the vertices of a triangle are rational numbers then which of the following points of the triangle will always have rational co-ordinates

(a) centroid (b) incentre (c) circumcentre (d) orthocentre

72. Let 5], S2, ... be squares such that four each

73. A line passing through the point (2, 2) and the axes enclose an area X. The intercepts on the axes made by the line are given by the two roots of sign, be a line such that the area enclosed by

1 2

the line and the axes of reference is — unit 8

(a) bisector of the angle including origin (b) bisector of acute angle

76. Two roads are represented by the equations y —x = 6 and x + y = 8. An inspection bunglow has to be so constructed that it is at a distance of 100 from each of the roads.

Possible location of the bunglow is given by (a)(10(W2 + l , 7 ) (b) (1 - 100^2", 7) (c) (1, 7 + IOOV2) (d) ( 1 , 7 - 100V2) Angles made with the x-axis by two lines drawn through the point (1,2) cutting the line x + y = 4 at a distance V 6 / 3 from the point

its ends are always on two fixed perpendicular lines. The locus of the point which divides this line into portions of lengths a and b is its third vetex is

(a) (1,6) (b) ( - 1 , 6 ) (c) ( 1 , - 6 ) (d) None of these 81. Length of the median from B on AC where

A ( - 1, 3), 5 (1, - 1), C(5, 1) is (a) VTsT (b) VTo (c) 2^3" (d) 4

82. The point P ( l , 1) is translated parallel to 2x = y in the first quadrant through a unit distance. The co-ordinates of the new position of P are

(a) 1 ± ; 2 1 ± ( b ) ( l ± ^ , l ±

J _ V5 _2_

83. The mid points of the sides of a triangle are (5, 0), (5, 12) and (0, 12). The orthocentre of this triangle is

(a) (0,0) (b) (10, 0) (c) (0, 24) (d) J 1 g

3 '8

84. The algebraic sum of the perpendicular distances from A (a\ , b{)\ B (a2\ b2) and C ( a3, to a variable line is zero, then the line passes through

(a) the orthocentre of AABC (b) the centroid of AABC (c) the circumcentre of AABC (d) None of these

85. One vertex of the equilateral triangle with centroid at the origin and one side as x + y - 2 = 0 i s

(a) ( - 1 , - 1 ) ( b ) ( 2 , 2 ) (c) ( - 2, - 2) (d) none of these 86. The point (4, 1) undergoes the following two

successive transformations : (a) reflection about the line y-x

87.

(b) rotation through a distance 2 units along the positive x-axis.

Then the final co-ordinates of the point are (a) (4, 3) (b) (3, 4)

( c ) l , 4 ) (d) ( 7 / 2 , 7 / 2 )

The incentre of the triangle formed by the lines x = 0, y = 0 and 3x + 4y = 12 is at

(c) 1, 1

( b ) ( l , l )

<d) f

^{2 '}¹

88. If (a, b) be an end of a diagonal of a square

In document Arihant AIEEE Mathematics (Page 174-179)