STRAIGHT LINES EXERCISE – I Q1.
If the point P(x, y) be equidistant from the points A(a + b, b – a) and B(a – b, a + b) then (a) ax = by (b) bx = ay (c) ax = – by (d) bx = – ay
Q2.
The ratio in which the line y – x + 2 = 0 divides the line joining (3, – 1) and (8, 9) is (a) 2 : 3 (b) 3 : 2 (c) – 2 : 3 (d) – 3 : 2
Q3.
The extremities of the diagonal of a parallelogram are the points (3, – 4) and (– 6, 5). If the third vertex is the point (– 2, 1), then the coordinates of the fourth vertex are (a) (1, 0) (b) (– 1, 0) (c) (0, 1) (d) (0, – 1)
Q4.
The coordinates of the circucentre of the triangle having vertices (– 2, – 3), (– 1, 0) and (7, – 6) are (a) (– 3, 3) (b) (3, 3) (c) (3, – 3) (d) (–3, – 3)
Q5.
The coordinates of A, B, C are (6, 3), (– 3, 5) and (4, – 2) respectively and P is any point (x, y). The ratio of the areas of triangles PBC and ABC is (a)
x + y −2 7
(b)
| x + y −2| 7
(c)
x+ y+2 7
(d) None of these
Q6.
The distance between two parallel lines is unity. A point P lies between the lines at a distance a from one of them. The length of a side of an equilateral triangle PQR, vertex Q of which lies on one of the parallel lines and vertex R lies on the other line, is 2 2 1 1 . a 2 + a +1 a 2 − a +1 a2 + a +1 a 2 − a +1 (a) (b) (c) (d) 3 3 3 3
Q7.
The coordinates of two points A and B are (3, 4) and (5, – 2) respectively. If PA = PB and area of ∆PAB = 10, then the coordinates of the point P are (a) (7, 2) (b) (7, – 2) (c) (1, 0) (d) (13, – 4)
Q8.
x-coordinates of two points B and C are the roots of equation x 2 + 4x + 3 = 0 and their y – coordinates are the roots of equation x2 – x – 6 = 0. If x – coordinate of B is less than x – coordinate of C and y – coordinate of B is greater than the y – coordinate of C and coordinates of a third point A be (3, – 5), then the length of the bisector of the interior angle at A is 7 2 14 2 5 2 (a) (b) (c) (d) None of these 3 3 3
Q9.
The area of the triangle, formed by the straight lines 7x – 2y + 10 = 0, 7x + 2y – 10 = 0 and 9x + y + 2 = 0, is (a)
686 275
(b)
798 263
(c)
668 275
(d) None of these
Q10.
If A, B, C, D are points whose coordinates are (– 2, 3), (8, 9), (0, 4) and (3, 0) respectively, then the ratio in which AB is divided by CD, is (a) 11 : 23 (b) 11 : 47 (c) 23 : 47 (d) None of these
Q11.
If α, β, γ are the real roots of the equation x3 – 3px2 + 3qx – 1 = 0, then the centroid of the triangle having vertices α,
Q12.
1 1 β, β and γ, γ are (a) (p, q) (b) (p, – q) (c) (– p, q) (d) (– p, – q) The area of the quadrilateral whose vertices are (– 3, 2), (7, – 6), (– 5, – 4) and (5, 4) is (a) 68 (b) 70 (c) 80 (d) None of these The locus of the moving point P such that 2PA = 3PB, where A is (0, 0) and B is (4, – 3), is (a) 5x2 + 5y2 + 72x + 54y + 225 = 0 (b) 5x2 + 5y2 – 72x – 54y + 225 = 0 2 2 (c) 5x + 5y – 72x + 54y + 225 = 0 (d) None of these
Q13.
1 , α
Q14.
If (1, 4) be the CG of a triangle and the coordinate of its any two vertices be (4, – 8) and (– 9, 7), then the area of the triangle is 333 (a) 84 (b) 132 (c) 2 (d) None of these
Q15.
The coordinate of incentre of the triangle whose sides are 3x – 4y = 0, 5x + 12y = 0 and y – 15 = 0, are (a) (1, 8) (b) (1, – 8) (c) (– 1, 8) (d) (– 1, – 8)
Q16.
The coordinates of the orthocenter of the triangle, having sides 3x – 2y = 6, 3x + 4y + 12 = 0 and 3x – 8y + 12 = 0, are
(a) − Q17.
36 45 , 7 7
1 23 ,− 6 9
1 23 , 6 9
(c)
(b) −
36 45 ,− 7 7
(d) None of these
45 36 ,− 7 7
(c)
3 , 2 4 (c) − , 3
If a triangle has its orthocenter at (1, 1) and circumcentre at
5 4 ,− 6 3
(a) Q19.
(b) −
Two vertices of a triangle are (3, – 1) and (– 2, 3) and its orthocenter is origin, the coordinates of the third vertex are (a) −
Q18.
1 23 , 6 9
4 5 , 3 6
(b)
36 45 , 7 7
(d)
3 , then the coordinates of the centroid of the triangle are 4 5 5 4 (d) − , − 6 6 3
If G be the centroid and I be the incentre of the triangle with vertices A(– 36, 7), B(20, 7) and C(0, – 8) and
GI =
25 3
205 λ , then the value of λ is
(a) 25
(b)
1 25
(c)
4 25
(d) None of these
Q20.
A ladder of length ‘a’ rests against the floor and a wall of a room. If the ladder begins to slide on the floor, then the locus of its middle point is (a) x2 + y2 = a2 (b) 2(x2 + y2) = a2 (c) x2 + y2 = 2a2 (d) 4(x2 + y2) = a2
Q21.
A and B are two fixed points. The locus of a point P such that ∠APB is a right angle, is (a) x2 + y2 = a2 (b) x2 – y2 = a2 (c) 2x2 + y2 = a2 (d) None of these
Q22.
If the point A is symmetric to the point B(4, – 1) with respect to the bisector of the first quadrant, then the length of AB is (a) 5 (b) 5 2 (c) 3 2 (d) 3
Q23.
The equation of the straight line cutting off an intercept 8 from x-axis and making an angle of 60° with the positive direction of y-axis is (a) x + 3 y =8 (b) x − 3 y =8 (c) y − 3 x =8 (d) None of these
Q24.
A line through the point A(2, 0), which makes an angle of 30° with the positive direction x-axis is rotated about A in clockwise direction through an angle 15°. The equation of the straight line in the new position is (a) (2 − 3 ) x − y −4 +2 3 =0 (b) ( 2 − 3 ) x + y −4 + 2 3 =0
Q25. Q26. Q27.
(c) (2 − 3 ) x − y + 4 + 2 3 =0 (d) None of these The coordinates of a point on the line x + y = 4 that lies at a unit distance from the line 4x + 3y – 10 = 0 are (a) (3, 1) (b) (– 7, 11) (c) (3, – 1) (d) (7, – 11) The equation of the internal bisector of ∠BAC of ∆ABC with vertices A(5, 2), B(2, 3) and C(6, 5) is (a) 2x + y + 12 = 0 (b) x + 2y – 12 = 0 (c) 2x + y – 12 = 0 (d) None of these The range of values of θ in the interval (0, π) such that the points (3, 5) and (sin θ, cos θ) lie on the same side of the line x + y – 1 = 0, is
(a) 0, Q28.
π 2
(b) 0,
π 4
π π , 4 2
(c)
(d) None of these
If m1 and m2 are the roots of the equation x 2 + ( 3 + 2) x + ( 3 −1) = 0 then the area of the triangle formed by the lines
y = m1x, y = m2x and y = 2 is (a) 33 − 11 (b) Q29.
33 + 11
(c)
The equation of the straight line upon which the length of perpendicular from origin is 3 2 units and this perpendicular makes an angle of 75° with the positive direction of x – axis, is (a) ( 3 −1) x +( 3 +1) y −12 =0 (b) ( 3 −1) x +( 3 +1) y +12 =0 (c) ( 3 +1) x +( 3 −1) y −12 =0
Q30.
(d) None of these
33 + 7
(d) None of these
If the straight line drawn through the point P( 8=0 at Q, then the length of PQ is (a) 4 (b) 5
3 , 2) and making an angle
π with the x – axis meets the line 6
(c) 6
3 x – 4y +
(d) None of these
Q31.
A line joining two points A(2, 0) and B(3, 1) is rotated about A in anticlockwise direction through an angle 15 °. If B goes to C in the new position, then the coordinates of C are 1 3 3 3 2, 2, − , (a) (b) (c) 2 + (d) None of these 2 2 2 2
Q32.
The acute angle between the line x + y = 3 and the line joining the points (1, 1) and (– 3, 4) is
3 7
−1 (a) tan
Q33.
1 7
−1 (c) tan
The equation of the straight line, having x – intercept equal to − (a) 2x + 5y + 4 = 0
Q34.
3 7
−1 (b) π − tan
(b) 5x – 2y + 4 = 0
1 7
−1 (d) π − tan
4 and is perpendicular to the line 2x – 5y + 8 = 0, is 5
(c) 2x – 5y + 4 = 0
(d) 5x + 2y + 4 = 0
The coordinates of the foot of the perpendicular drawn from the point (2, 3) to the line y = 3x + 4 are
1 37 , 10 10
37 1 , − 10 10
(a) −
(b)
37 1 ,− 10 10
(c) −
(d) None of these
Q35.
The image of the point (– 8, 12) with respect to the line mirror 4x + 7y + 13 = 0 is (a) (16, – 2) (b) (– 16, 2) (c) (16, 2) (d) (– 16, – 2)
Q36.
The image of the point (3, – 8) under the transformation (x, y) → (2x + y, 3x – y) is (a) (– 2, 17) (b) (2, 17) (c) (– 2, – 17) (d) (2, – 17)
Q37.
The image of the point P(3, 5) with respect to the line y = x is the point Q and the image of Q with respect to the line y = 0 is the point R(a, b), then (a, b) = (a) (5, 3) (b) (5, – 3) (c) (– 5, 3) (d) (– 5, – 3)
Q38.
The equation of the straight line passing through the point of intersection of lines 3x – 4y – 7 = 0 and 12x – 5y – 13 = 0 and perpendicular to the line 2x – 3y + 5 = 0 is (a) 33x + 22y + 13 = 0 (b) 33x + 22y – 13 = 0 (c) 33x – 22y + 13 = 0 (d) None of these
Q39.
If the family of lines x(a + 2b) + y(a + 3b) = a + b passes through the point for all values of a and b, then the coordinates of the points are (a) (2, 1) (b) (2, – 1) (c) (– 2, 1) (d) None of these If the lines ax + y + 1 = 0, x + by + 1 = 0 and x + y + c = 0 (a, b and c being distinct and different from 1) are concurrent,
Q40.
then (a) 1 Q41.
1 1 1 + + = 1−a 1−b 1−c
(b) – 1
(c) 0
(d) None of these
The sum of the abscissas of all the points on the line x + y = 4 that lie at a unit distance from the line 4x + 3y – 10 = 0, is (a) 3 (b) – 3 (c) 4 (d) – 4
Q42.
If p1 and p2 are the lengths of the perpendiculars from the origin to the straight lines x sec θ + y cosec θ = a and x cos θ – y sin θ = a cos 2θ respectively, then the value of 4 p12 + p 22 is (a) 4a2 (b) 2a2 (c) a2
(d) None of these
Q43.
For the straight lines 4x + 3y – 6 = 0 and 5x + 12y + 9 = 0, the equation of the bisector of the obtuse angle between them is (a) 9x – 7y – 41 = 0 (b) 7x + 9y – 3 = 0 (c) 7x + 9y – 3 = 0 (d) None of these
Q44.
For the straight line 4x + 3y – 6 = 0 and 5x + 12y + 9 = 0, the equation of the bisector of the angle which contains the origin is (a) 7x + 9y + 3 = 0 (b) 7x – 9y + 3 = 0 (c) 7x + 9y – 3 = 0 (d) None of these
Q45.
The value of k so that the lines 2x – 3y + k = 0, 3x – 4y – 13 = 0 and 8x – 11y – 33 = 0 are concurrent, is (a) 7 (b) – 7 (c) 5 (d) – 5
Q46.
If the equal side AB and AC (each equal to a) of a right angled isosceles triangle ABC be produced to P and Q so that BP.CQ = AB2, then the line PQ always passes through the fixed point (a) (a, 0) (b) (0, a) (c) (a, a) (d) None of these
Q47.
Without changing the direction of coordinates axes, origin is transferred to (α, β) so that the linear terms in the equation x 2 + y2 + 2x – 4y + 6 = 0 are eliminated. The point (α, β) is (a) (– 1, 2) (b) (1, – 2) (c) (1, 2) (d) (– 1, – 2)
Q48.
P(3, 1), Q(6, 5) and R(x, y) are three points such that the angle RPQ is a right angle and the area of ∆RPQ = 7, then the number of such point R is (a) 0 (b) 1 (c) 2 (d) 4
Q49.
A line L passes through the point (1, 1) and (2, 0). Another line M which is perpendicular to L passes through the point
1 , 0 . Then the area of the triangle formed by the lines L, M and y – axis is 2 25 25 25 25 (a) (b) (c) (d) 8 16 4 32 Q50.
P is a point on either of the two lines y − 3 | x |=2 at a distance of 5 units from their point of intersection. The coordinates of the foot of the perpendicular from P on the bisector of the angle between them are
1 1 (4 + 5 3 ) or 0, (4 − 5 3 ) depending on which line the point P is taken 2 2 5 5 3 1 1 (b) 0, ( 4 + 5 3 ) (c) 0, (4 − 5 3 ) (d) , 2 2 2 2 (a) 0,
Q51.
Let P be the image of the point (– 3, 2) with respect to x-axis. Keeping the origin as same, the coordinate axes are rotated through an angle 60° in the clockwise sense. The coordinates of point P with respect to the new axes are 2 3 −3 (3 3 + 2) , − (a)
2
2
2 3 −3 3 3 + 2 , (b)
( 2 3 − 3) (3 3 + 2) − , (c)
2
2
2
2
(d) None of these
Q52.
A square is constructed on the portion of the line x + y = 5 which is intercepted between the axes, on the side of the line away from origin. The equations to the diagonals of the square are (a) x = 5, y = – 5 (b) x = 5, y = 5 (c) x = – 5, y = 5 (d) x – y = 5, x – y = – 5
Q53.
If one of the diagonals of a square is along the line x = 2y and one of its vertices is (3, 0), then its sides through this vertex are given by the equations (a) y – 3x + 9 = 0, 3y + x – 3 = 0 (b) y + 3x + 9 = 0, 3y + x – 3 = 0 (c) y – 3x + 9 = 0, 3y – x + 3 = 0 (d) y – 3x + 3 = 0, 3y + x + 9 = 0
Q54.
Q55.
The area of the region enclosed by 4 | x | + 5 | y | ≤ 20 is (a) 10 (b) 20 (c) 40 If a, b, c are in HP then the straight line (a) (– 1, – 2)
(b) (– 1, 2)
(d) None of these
x y 1 + + = 0 always passes through a fixed point, that point is a b c 1 (c) (1, – 2) (d) 1, − 2
x y + = 1 , the condition a–2 + b–2 = c–2 (c is a constant) is satisfied, then the locus of foot of the a b perpendicular drawn from origin to this is c2 (a) x2 + y2 = (b) x2 + y2 = 2c2 (c) x2 + y2 = c2 (d) x2 – y2 = c2 2
Q56.
If for a variable line
Q57.
The point (3, 2) is reflected in the y – axis and then moved a distance 5 units towards the negative side of y – axis. The coordinates of the point thus obtained are (a) (3, – 3) (b) (– 3, 3) (c) (3, 3) (d) (– 3, – 3)
Q58.
The vertices of a ∆OBC are O(0, 0), B(– 3, – 1) and C(– 1, – 3). The equation of a line parallel to BC and intersecting sides OB and OC whose distance from the origin is (a) x + y +
1 2
=0
(b) x + y −
1 2
1 , is 2
=0
(c) x + y −
1 =0 2
(d) x + y +
1 =0 2
Q59.
The point (– 4, 5) is the vertex of a square and one of its diagonals is 7x – y + 8 = 0. The equation of the other diagonal is (a) 7x – y + 23 = 0 (b) x + 7y = 31 (c) x – 7y = 31 (d) None of these
Q60.
The condition to be imposed on β so that (0, β) lies on or inside the triangle having sides y + 3x + 2 = 0, 3y – 2x – 5 = 0 and 4y + x – 14 = 0 is (a) 0 < β <
5 3
(b) 0 < β <
7 2
(c)
5 7 ≤β≤ 3 2
(d) None of these
Q61.
The new coordinates of a point (4, 5) when the origin is shifted to the point (1, – 2) are (a) (5, 3) (b) (3, 5) (c) (3, 7) (d) None of these
Q62.
A trapezium has vertices (– 1, 8), (– 2, 4), (2, 4) and (1, 8) taken in order. The equation of the line passing through (1, 8) and dividing the trapezium in two equal areas is (a) 2x – y + 6 = 0 (b) 2x + y + 6 = 0 (c) x – 2y + 6 = 0 (d) x + 2y + 6 = 0
Q63.
The limiting position of the point of intersection of the lines 3x + 4y = 1 and (1 + c)x + 3c2y = 2 as c tends to 1 is (a) (5, – 4) (b) (– 5, – 4) (c) (5, 4) (d) (– 5, 4) Through the point (1, 1), a straight line is drawn so as to form with coordinate axes a triangle of area S. The intercepts made by the line on the coordinate axes are the roots of the equation (a) x2 – | S |x + 2| S | = 0 (b) x2 + | S |x + 2| S | = 0 (c) x2 – 2| S |x + 2| S | = 0 (d) None of these t If the parametric equation of a line is given by x = 4 + and y =−1 + 2 t , where t is the parameter then 2
Q64.
Q65.
1 2
−1 (b) slope of the line is tan
(a) slope of the line is tan–1 2 (c) intercept made by the line on the x-axis = Q66.
9 2
(d) intercept made by the line on the y-axis = – 9
Let P(2, 0) and Q(0, 2) be two points and O be the origin. If A(x, y) is a point such that xy > 0 and x + y < 2, then (a) A cannot be inside the ∆OPQ (b) A lies outside the ∆OPQ (c) A lies either inside ∆OPQ or in the 3rd quadrant (d) None of these
Q67.
If the point (2 cos θ, 2 sin θ) does not fall in that angle between the line y = | x – 2 | in which the origin lies then θ belongs to
π 3π , 2 2
(b) −
(a) Q68.
π π , 2 2
(c) (0, π)
(d) None of these
Let S1, S2, … be squares such that for each n ≥ 1, the length of a side of S n equals the length of a diagonal of S n + 1. If the length of a side of S1 is 10 cm, then for which of the following values of n is the area of Sn less than 1 sq cm? (a) 7 (b) 8 (c) 9 (d) 10 EXERCISE – II
Q1.
The straight line y = x – 2 rotates about a point where it cuts the x – axis and becomes perpendicular to the straight line ax + by + c = 0. Then its equation is (a) ax + by + 2a = 0 (b) ax – by – 2a = 0 (c) bx + ay – 2b = 0 (d) ay – bx + 2b = 0
Q2.
If the line y = origin ) is (a) a – 2b + c
3 x cuts the curve x 4 + ax2y + bxy + cx + dy + 6 = 0 at A, B, C and D, then OA.BO.OC.OD (where O is the
(b) 2c2d
(c) 96
(d) 6
Q3.
A curve with an equation of the form y = ax4 + bx3 + cx + d has zero gradient at the point (0, 1) and also touches the x – axis at the point (–1, 0). Then the values of x for which the curve has negative gradients are (a) x > – 1 (b) x < 1 (c) x < – 1 (d) – 1 ≤ x ≤ 1
Q4.
If the straight lines ax + by + p = 0and x cos α + y sin α = p are inclined at an angle x sin α – y cos α = 0, then the value of a2 + b2 is (a) 0 (b) 1
Q5.
(c) 2
π and concurrent with the straight line 4
(d) none of these
The area of the square formed by the lines | y | = (1 – x ) and | y | = x + 1 equals to (a)
1 sq. unit. 6
(b)
1 sq. unit. 12
(c) 2 sq. unit.
(d)
1 sq. unit.
Q6.
If a2 + b2 – c2 – 2ab = 0, then the family of straight lines ax + by + c = 0 is concurrent at the points (a) (–1, 1) (b) (1, –1) (c) (1, 1) (d) (–1, –1)
Q7.
The number of points on the line 3x + 4y = 5, which are at a distance of sec2 θ + 2 cosec2 θ, θ ∈ R, from the point (1, 3) is (a) 1 (b) 2 (c) 3 (d) infinite
Q8.
Two sides of a rhombus OABC (lying entirely in first quadrant or fourth quadrant) of area equal to 2 sq. units, are y = y = 3 x. Then possible coordinates of B is / are (‘O’ being the origin ) (a) (1 + 3 , 1 + 3 ) (b) (– 1 – 3 , – 1 – 3 ) (c) ( 3 – 1 , 3 – 1)
If the lines x = a + m, y = – 2 and y = mx are concurrent, the least value of | a | is (a) 0 (b) 2 (c) 2 2 (d) none of these
Q10.
Through the point P(α, β), where αβ > 0, the straight line
Q11.
3
,
(d) none of these
Q9.
of area s. If ab > 0, then least value of s is (a) 2αβ (b) 1/2 αβ
x
x y + = 1 is drawn so as to form with coordinate axes a triangle a b
(c) αβ
(d) none of these
Let ax + by + c = 0 be a variable straight line, where a, b and c are 1 st, , 3rd and 7th terms of some increasing A.P. Then the variable straight line always passes through a fixed point which lies on.
(a) x2 + y2 = 13
(b) x2 +y2 = 5
(c) y2 =
9 x 2
(d) 3x + 4y = 9
Q12.
The point (4, 1) undergoes the following three transformations successively (a) Reflection about the line y = x (b) Transformation through a distance 2 units along the positive direction of the x – axis. (c) Rotation through an angle π/4 about the origin in the anti-clockwise direction. The final position of the point is given by the co-ordinates. 4 1 1 3 1 7 7 4 , , , , (a) (b) (c) (d) − 2 2 2 2 2 2 2 2
Q13.
If the quadratic equation ax2 + bx + c = 0 has -2 as one of its roots then ax + by + c = 0 represents (a) a family of concurrent lines (b) a family of parallel lines (c) a single line (d) a line perpendicular to x – axis
Q14.
If A(cos α, sin α) B(sin α, - cos α) , C(2, 1) are the vertices of a ∆ ABC, then as α varies the locus of its centroid is (a) x2 + y2 – 2x – 4y + 1 = 0 (b) 3(x2 +y2) - 2x – 4y + 1 = 0 2 2 (c) x + y – 2x – 4y + 3 = 0 (d) none of these
Q15.
It is desired to construct a right angled triangle ABC (∠C = π/2) is xy plane so that it’s sides are parallel to coordinates axis and the medians through A and B lie on the lines y = 3x +11 and y = mx + 2 respectively. The values of ‘m’ for which such a triangle is possible is / are (a) 12 (b) 3/4 (c) 4/3 (d) 1/12
Q16.
If 3a + 2b + 6c = 0, the family of lines ax + by + c = 0 passes through a fixed point whose coordinates are given by
1 1 , 2 3
(a)
(b) (2, 3)
(c) (3, 2)
1 1 3 2
(d) ,
Q17.
Area of the parallelogram whose sides are x cos α + y sin α = p , x cos α + y sin α = q, x cos β + y sin β = r ,x cos β+ y sin β = s is (a) pq + rs (b) | pq tan α + rs tan β | (c) | (p - q) (r - s) cosec (α - β) | (d) | (p - q) (r - s) tan (α + β) |
Q18.
Let 2x – 3y = 0 be a given line and P (sin θ, 0) and Q(0, cos θ) be the two point. Then P and Q lie on the same side of the given line, if θ lies in the (a) 1st quadrant (b) 2nd quadrant (c) 3rd quadrant (d) 4th quadrant
Q19.
Points on the line x + y = 4 that lie at a unit distance from the line 4x + 3y – 10 = 0 are (a) (3, 1) and (–7, 11) (b) (–3, 7) and (2, 2) (c) (–3, 7) and (–7, 11) (d) None of these
Q20.
The algebraic sum of perpendicular distances from A(x1, y1), B(x2 , y2) , C(x3, y3) to a variable line is zero. Then the line passes through. (a) the orthocentre of ∆ ABC (b) centroid of ∆ ABC (c) incenter of ∆ ABC (d) circumcentre of ∆ ABC
Q21.
If the two pairs of lines x2 – 2mxy – y2 = 0 and x2 – 2nxy – y2 = 0 are such that one of them represents the bisector of the angles between the other, then (a) mn + 1 = 0
Q22.
(c)
1 1 + =0 m nb
(d)
1 1 − =0 m n
The lines (p + 2q) x + (p – 3q)y = p – q, for different values pf p and q, passes through the fixed point
3 5 , 2 2
(a) Q23.
(b) mn – 1 = 0
2 2 , 5 5
(b)
3 3 5 5
(c) ,
The difference of the tangents of the angles which the lines x2(sec 2θ - sin 2θ) – 2xy tan θ + y2 sin 2θ = 0 with the x – axis is
2 3 , 5 5
(d)
(a) 2 tan θ Q24.
If the line y = (a)
(c) 2 cot θ
(b) 2
(d) sin 2 θ
3 x cuts the curve x3 + y3 + 3xy + 5x2 + 3y2 + 4x + 5y -1 = 0 at the points A, B, C then OA . OB. OC is
4 (3 3 −1) 13
(b) 3
3 +1
(c)
2 3
+7
(d) none of these
Q25
If a line is perpendicular to the line 5x – y = 0 and forms a triangle, with the coordinate axes of area 5 sq. units, then its equation is (a) x + 5y ± 5 2 = 0 (b) x – 5y ± 5 2 = 0 (c) 5x + y ± 5 2 = 0 (d) 5x – y ± 5 2 = 0
Q26
The area of the rhombus enclosed by the lines ax ± by ± c = 0 is (a) 2c2 / ab (b) 2ab /c2 (c) 2c /ab
Q27.
(d) none of these
If 3a – 2b + 5c = 0, family of straight lines ax + by + c = 0 are always concurrent at a point whose coordinate is
3 2 , 5 5
(b) −
(a)
3 2 , 5 5
2 3 , − 5 5
(d) −
(c)
3 2 ,− 5 5
Q28.
Consider the straight line ax + by = c, where a, b, c ∈ R+. This line meets the coordinate axes at ‘P’ and ‘Q’ respectively. If the area of triangle OPQ. ‘O’ being origin, does not depend upon a, b and c, then (a) a, b, c are in GP (b) a, c, b are in GP (c) a, b, c are in AP (d) a, c, b are in AP
Q29.
The area enclosed by | x – 1 | + | y – 3 | = 1 is equal to (a) 4 sq units (b) 6 sq units (c) 1 sq units
(d) 2 sq units
Q30.
The extremities of the base of an isosceles triangle are (2, 0) and (0, 2). If the equation of one of the equal sides is x = 2, then equation of other equal side is (a) x – y = 2 (b) x – y + 2 = 0 (c) y = 2 (d) 2x + y = 2
Q31.
A variable line is drawn through the intersection point of the lines 3x + 4y – 12 = 0 and x + 2y – 5 = 0, meeting the coordinate axes at the points A and B. Locus of midpoint of segment AB is (a) 4x + 3y = 4xy (b) 3x + 4y = 3xy (c) 3x + 4y = 4xy (d) None of these
Q32.
The straight line 3x + 4y – 12 = 0 meets the coordinate axes at A and B. An equilateral triangle ABC is constructed. One possible co-ordinate of vertex ‘C’ is 3 3 3 , 2 1 − 4 1− 1− 3 (a) 2 (b) − 2 1 + 3 , 4 3 2 3
(
(
) 32 (1 + 3 )
(c) 2 1 + 3 ,
) (
)
(d) None of these
Q33.
If the vertices of a triangle are rational points, which of the following points of the triangle is not necessarily a rational point (a) Centroid (b) Circumcentre (c) Orthocenter (d) Incentre
Q34.
P(x, y) is called a good point if x, y ∈ N. Total number of good points lying inside the quadrilateral formed by the line 2x + y = 2, x = 0, y = 0 and x + y = 5, is equal to (a) 4 (b) 2 (c) 10 (d) 6
Q35.
If the point P(a2, a) lies in the region corresponding to the acute angle between the lines 2y = x and 4y = x, then (a) a ∈ (2, 6) (b) a ∈ (4, 6) (c) a ∈ (2, 4) (d) None of these If the algebraic sum of distances of points (2, 1), (3, 2) and (– 4, 7) from the line y = mx + c is zero, then this line will always pass through a fixed point whose co-ordinate is
Q36
(a) (1, 3) Q37.
(b) (1, 10)
(c) (1, 6)
1 10 , 3 3
(d)
If x – 2y + 4 = 0 and 2x + y – 5 = 0 are the sides of an isosceles triangle having area 10 sq units, then equation of third side is (a) x + 3y = 19 (b) 3x – y + 11 = 0 (c) x – 3y = 10 (d) 3x – y + 15 = 0
Q38.
A variable line drawn through the point (1, 3) meets the x-axis at A and y-axis at B. If the rectangle OAPB is completed, where ‘O’ is the origin, then locus of ‘P’ is 1 3 1 3 + =1 + =1 (a) (b) x + 3y = 1 (c) (d) 3x + y = 1 y x x y
Q39.
A line is drawn perpendicular to the line y = 5x, meeting the coordinate axes at A and B. If the area of triangle OAB is 10 sq unit where ‘O’ is the origin, then equation of drawn line is (a) x + 5y = 10 (b) x – 5y = 10 (c) x + 4y = 10 (d) x – 4y = 10
Q40.
If the straight lines a1x + b1y + c1 = 0, a1x + b1y + c2 = 0, a2x + b2y + d1 = 0 a2x + b2y + d2 = 0 are the sides of rhombus, then (a) (a 22 + b 22 )(c1 − c 2 ) 2 = (a 12 + b12 )(d1 − d 2 ) 2 (b) (a 12 + b12 ) | d 1 − d 2 |= (a 22 + b 22 ) | c1 − c 2 | (c) (a 22 + b 22 )(d1 − d 2 ) = (a 12 + b12 )(c1 − c 2 )
Q41.
θ1 and θ2 are the inclination of lines L1 and L2 with x-axis. If L1 and L2 pass through P(x1, y1), then equation of one of the angle bisector of these lines is
x − x1 y − y1 = (a) θ − θ2 θ − θ2 cos 1 sin 1 2 2 x − x1 y − y1 = (c) θ + θ2 θ + θ2 sin 1 cos 1 2 2 Q42.
x − x1 y − y1 = (b) θ − θ2 θ − θ2 − sin 1 cos 1 2 2 x − x1 y − y1 = (d) θ + θ2 θ + θ2 − sin 1 cos 1 2 2
A light ray coming along the line 3x + 4y = 5, gets reflected from the line ax + by = 1 and goes along the line 5x – 12y = 10 then (a) a =
Q43.
(d) (a 12 + b12 ) | c1 − c 2 |= (a 22 + b 22 ) | d 1 − d 2 |
64 112 ,b= 115 5
(b) a =
14 8 64 8 ,b=− (c) a = ,b=− 15 115 115 115
(d) None of these
If the point (1 + cos θ, sin θ) lies between the region corresponding to the acute angle between the lines 3y = x and 6y = x, then (a) θ ∈ R
(b) θ ∈ R ~ nπ, n ∈ I
(c) θ ∈ R ~ (2n + 1)
π ,n∈I 2
(d) None of these
Q44.
A rectangle ABCE, A ≡ (0, 0), B ≡ (4, 0), C ≡ (4, 2), D ≡ (0, 2) undergoes the following transformations successively (i) f1 (x, y) → (y, x) (ii) f2 (x, y) → (x + 3y, y) x−y x+y , (iii) f3 (x, y) → 2 2 The final figure will be (a) A square (b) A rhombus (c) A rectangle (d) A parallelogram
Q45.
In the given figure combined equation of the incident and the refracted rays is 2 2 (a) ( x − 2) − y + 2 2 (c) ( x − 2) + y +
4 3 4 3
( x − 2) y = 0
2 2 (b) ( x − 2) + y −
( x − 2) y = 0
(d) None of these
4 3
( x − 2) y = 0
Q46.
Consider the points A1 (x1, y1), A2 (x2, y2) and A3 (x3, y3). If x1, x2, x3 and y1, y2, y3 are in GP with same common ratio, then the points A1, A2, A3 are (a) Vertices of an equilateral triangle (b) Vertices of a right angled triangle (c) Vertices of an isosceles triangle (d) None of these
Q47.
If the point P(a, a2) lies completely inside the triangle formed by the lines x = 0, y = 0 and x + y = 2, then exhaustive range of ‘a’ is
(a) a ∈ (0, 1)
(b) a ∈ (1,
(c) a ∈ (
2 )
2 – 1,
2 )
(d) a ∈ (
2 – 1, 1)
Q48.
A light ray emerging from the point source placed at P(2, 3) is reflected at a point ‘Q’ on the y-axis and then passes through the point R(5, 10). Coordinate of ‘Q’ is (a) (0, 3) (b) (0, 2) (c) (0, 5) (d) None of these
Q49.
Straight lines y = mx + c1 and y = mx + c2, where m ∈ R+, meet the x–axis at A1 and A2 respectively and y–axis at B1 and B2 respectively. It is given that points A1, A2, B1 and B2 are concyclic. Locus of intersection of lines A1B2 and A2B1 is (a) y = x (b) y + x = 0 (c) xy = 1 (d) xy + 1 = 0
Q50.
A variable line ‘L’ is drawn through O(0, 0) to meet the lines L 1 : y – x – 10 = 0 and L 2 : y – x – 20 = 0 at the points A and B respectively. A point P is taken on ‘L’ such that (a) 3x + 3y = 40
Q51.
13 13 , 7 7
12 17 , 5 5 14 3 ,− 5 20
Q55. Q56.
(d) 3y – 3x = 40
23 23 , 7 7
31 31 , 7 7
(b)
33 33 , 7 7
(c)
(d)
(b) −
84 13 , 5 5
(c) −
6 17 , 5 5
(d) None of these
In the above question, coordinates of the point ‘P’ such that | PA – PB | is minimum, is (a)
Q54.
(c) 3x – 3y = 40
Consider the points A ≡ (0, 1) and B ≡ (2, 0). ‘P’ be a point on the line 4x + 3y + 9 = 0. Coordinate of the point ‘P’ such that | PA – PB | is maximum, is (a) −
Q53.
(b) 3x + 3y + 40 = 0
Consider the point A ≡ (3, 4), B ≡ (7, 13). If ‘P’ be a point of the line y = x such that PA + PB is minimum, then coordinates of ‘P’ is (a)
Q52.
2 1 1 = + . Locus of ‘P’ is OP OA OB
3 14 , 20 5
(c) −
(b)
3 14 , 20 5
9 12 , − 20 5
All chords of the curve 3x2 – y2 – 2x + 4y = 0 that subtends a right angle at the origin, pass through a fixed point whose coordinate is (a) (1, – 2) (b) (1, 2) (c) (– 1, 2) (d) (– 1, – 2) Vertices of a variable triangle are (3, 4), (5 cos θ, 5 sin θ) and (5 sin θ, – 5 cos θ), where θ ∈ R. Locus of its orthocenter is (a) (x + y – 1)2 + (x – y – 7)2 = 100 (b) (x + y – 7)2 + (x – y – 1)2 = 100 2 2 (c) (x + y – 7) + (x – y – 1) = 100 (d) (x + y – 7)2 + (x – y + 1)2 = 100 2 2 Equation ax + 2hxy + by = 0 represents a pair of lines. Combined equation of lines that can be obtained by reflecting these lines about the x – axis is (a) ax2 – 2hxy + by2 = 0 (b) bx2 + 2hxy + ay2 = 0 (c) bx2 – 2hxy + ay2 = 0 (d) None of these
Q57.
If the line y = x is one of the angle bisector of the pair of lines ax2 + 2hxy + by2 = 0, then (a) a + b = 0 (b) h = 0 (c) a – b = 0 (d) None of these
Q58.
In the adjacent figure triangle ABC is right angled at B, if AB = 4 and BC = 3 and side AC slides along the coordinate axes in such a way that ‘B’ always remains in the first quadrant, then ‘B’ always lie on the straight line (a) y = x
Q59.
(d) −
(b) 3y = 4x
(c) 4y = 3x
(d) x + y = 0
Adjacent figure represents a equilateral triangle ABC of side length 2 units. Locus of vertex ‘C’ as the side AB slides along the coordinates axes is (a) x2 + y2 – xy + 1 = 0 (c) x2 + y2 = 1 + xy 3
(b) x2 + y2 – xy (d) x2 + y2 – xy
3 =1 3 +1=0
EXERCISE – III
Q1.
The line parallel to the x – axis and passing through the intersection of the lines ax + 2by + 3b = 0 and bx – 2ay – 3a = 0 where (a, b) ≠ (0, 0) is:
2 from it 3 2 (c) above the x - axis at a distance of from it 3 (a) below the x - axis at a distance of
Q2.
If non-zero numbers a, b, c are in HP, then the straight line
(a) (– 1, – 2) Q3.
3 from it 2 3 (d) above the x - axis at a distance of from it 2 (b) below the x - axis at a distance of
x y 1 + + = 0 always passes through a fixed point. That point is a b c
(c) 1, −
(b) (– 1, 2)
1 2
(d) (1, – 2)
If a vertex of a triangle is (1, 1) and the midpoints of two sides through this vertex are (– 1, 2) and (3, 2), then the centroid of the triangle is
−1 7 , 3 3
(a)
(b) −1,
7 3
1 7 3 3
(c) ,
(d) 1,
7 3
Q4.
Let A(2, – 3) and B(– 2, 1) be vertices of a triangle ABC. If the centroid of this triangle moves on the line 2x + 3y = 1, then the locus of the vertex C is the line (a) 2x + 3y = 9 (b) 2x – 3y = 7 (c) 3x + 2y = 5 (d) 3x – 2y = 3
Q5.
The equation of the straight line passing through the point (4, 3) and making intercepts on the co-ordinate axes whose sum is – 1 is:
x y y x + = −1 + = −1 and 2 3 −2 1 x y y x + = 1 and (c) + =1 2 3 −2 1 (a)
Q6.
If the sum of the slopes of the lines given by x2 – 2cxy – 7y2 = 0 is four times their product, then c has the value: (a) 1
Q7.
x y x y − = −1 and + = −1 2 3 −2 1 x y y x − = 1 and (d) + =1 2 3 −2 1 (b)
(b) – 1
(c) 2
(d) – 2
If one of the lines given by 6x2 – xy + 4cy2 = 0 is 3x + 4y = 0, then c equals: (a) 1 (b) – 1 (c) 3
(d) – 3
Q8.
If x1, x2, x3 and y1, y2, y3 are both in GP with the same common ratio, then the points (x1, y1), (x2, y2) and (x3, y3): (a) lie on a straight line (b) lie on an ellipse (c) lie on a circle (d) are vertices of a triangle
Q9.
If the equation of the locus of a point equidistant from the point (a 1, b1) and (a2, b2) is (a1 – a2)x + (b1 – b2)y + c = 0, then the value of ‘c’ is
1 2 (a 2 + b 22 − a 12 − b12 ) 2 1` 2 2 2 2 (c) (a 1 + a 2 + b1 + b 2 ) 2 (a)
(b) a 12 − a 22 + b12 − b 22 (d)
a 12 + b12 − a 22 − b 22
Q10.
Locus of the centroid of the triangle whose vertices are (a cos t, a sin t), (b sin t, – b cos t) and (1, 0), where t is a parameter is: (a) (3x – 1)2 + (3y)2 = a2 – b2 (c) (3x – 1)2 + (3y)2 = a2 + b2 2 2 2 2 (c) (3x + 1) + (3y) = a + b (d) (3x + 1)2 + (3y)2 = a2 – b2
Q11.
If the pair of straight lines x2 – 2pxy – y2 = 0 and x2 – 2qxy – y2 = 0 be such that each pair bisects the angle between the other pair, then (a) p = q (b) p = – q (c) pq = 1 (d) pq = – 1
Q12.
A square of side a lies above the x-axis and has one vertex at the origin. The side passing through the origin makes an angle
π α 0 < α < with the positive direction of x-axis. The equation of its diagonal not passing though the origin is: 4
Q13.
Q14.
Q15.
(a) y(cos α – sin α) – x(sin α – cos α) = a (c) y(cos α + sin α) + x(sin α + cos α) = a
(b) y(cos α + sin α) + x(sin α – cos α) = a (d) y(cos α + sin α) + x(cos α – sin α) = a
A triangle with vertices (4, 0), (– 1, – 1), (3, 5) is: (a) isosceles and right angled (c) right angled but not isosceles
(b) isosceles but not right angled (d) neither right angled nor isosceles
The incentre of the triangle with vertices (1, 2 1 3 1, (a) (b) 3 , 2 3
3 ), (0, 0) and (2, 0) is:
2 , (c) 3
3 2
Three straight lines 2x + 11y – 5 = 0, 24x + 7y – 20 = 0 and 4x – 3y – 2 = 0: (a) form an triangle (c) are concurrent with one line bisecting the angle between the other two
1 (d) 1, 3
(b) are only concurrent (d) none of these
Q16.
A straight line through the point (2, 2) intersects the line 3 x + y = 0 and 3 x – y = 0 at the points A and B. The equation to the line AB so that the triangle OAB is equilateral is: (a) x – 2 = 0 (b) y – 2 = 0 (c) x + y – 4 = 0 (d) none of these
Q17.
The straight lines x + y = 0, 3x + y – 4 = 0, x + 3y – 4 = 0 form a triangle which is: (a) isosceles (b) equilateral (c) right angled (d) none of these
Q18.
If P = (1, 0), Q = (– 1, 0) and R = (2, 0) are three given points then locus of the point S satisfying the relation SQ 2 + SR2 = 2SP2, is: (a) a straight line parallel to x – axis (b) a circle passing through the origin (c) a circle with the centre at the origin (d) a straight line parallel to y – axis
Q19.
Line L has intercepts a and b on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line L has intercepts p and q, then: 1 1 1 1 1 1 1 1 (a) a2 + b2 = p2 + q2 (b) 2 + 2 = 2 + 2 (c) a2 + p2 = b2 + q2 (d) 2 + 2 = 2 + 2 a b p q a p b q
Q20.
If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is: (a) square (b) circle (c) straight line (d) two intersecting line
Q21.
The orthocenter of the triangle formed by the lines xy = 0 and x + y = 1 is
1 1 , 2 2
(a)
1 1 , 3 3
(b)
(c) (0, 0)
1 1 , 4 4
(d)
Q22.
Let PQR be a right angled isosceles triangle, right angled at P(2, 1). If the equation of the line QR is 2x + y = 3, then the equation representing the pair of lines PQ and QR is: (a) 3x2 – 3y2 + 8xy + 20x + 10y + 25 = 0 (b) 3x2 – 3y2 + 8xy – 20x – 10y + 25 = 0 2 2 (c) 3x – 3y + 8xy + 10x + 15y + 20 = 0 (d) 3x2 – 3y2 – 8xy – 10x – 15y – 20 = 0
Q23.
Let PS be the median of the triangle with vertices P(2, 2), Q(6, – 1) and R(7, 3). The equation of the line passing through (1, – 1) and parallel to PS is (a) 2x – 9y – 7 = 0 (b) 2x – 9y – 11 = 0 (c) 2x + 9y – 11 = 0 (d) 2x + 9y + 7 = 0
Q24.
The number of integer values of m, for which the x-coordinate of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1 is also an integer, is: (a) 2 (b) 0 (c) 4 (d) 1
Q25.
Area of the parallelogram formed by the lines y = mx, y = mx + 1, y = nx, y = nx + 1 equals: (a) |m + n| / (m – n)2 (b) 2 / |m + n| (c) 1 / (|m + n|) (d) 1 / (|m – n|)
π be a fixed angle. If P = (cos θ, sin θ) and Q = (cos (α – θ), sin (α – θ)), then Q is obtained from P by: 2
Q26.
Let 0 < α <
Q27.
(a) clockwise rotation around origin through an angle α (b) anticlockwise rotation around origin through an angle α (c) reflection in the line through origin with slope tan α (d) reflection in the line through origin with slope tan (α/2) Let P = (– 1, 0), Q = (0, 0) and R = (3, 3 3 ) be three points. Then the equation of the bisector of the angle PQR is:
3 3 x+y=0 (b) x + 3 y = 0 (c) 3 x + y = 0 (d) x + y=0 2 2 A straight line through the origin O meets the parallel line 4x + 2y = 9 and 2x + y + 6 = 0 at points P and Q respectively. Then the point O divides the segment PQ in the ratio: (a) 1 : 2 (b) 3 : 4 (c) 2 : 1 (d) 4 : 3 The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0, 0), (0, 21) and (21, 0) is: (a) 133 (b) 190 (c) 233 (d) 105 Orthocenter of triangle with vertices (0, 0), (3, 4) and (4, 0) is: (a)
Q28. Q29. Q30.
(a) 3,
5 4
(c) 3,
(b) (3, 12)
3 4
(d) (3, 9)
Q32.
Area of the triangle formed by the line x + y = 3 and angle bisectors of the pair of straight lines x2 – y2 + 2y = 1 is: (a) 2 sq units (b) 4 sq units (c) 6 sq units (d) 8 sq units Three lines px + qy + r = 0, qx + ry + p = 0 and rx + py + q = 0 are concurrent if: (a) p + q + r = 0 (b) p2 + q2 + r2 = pr + rp + pq (c) p3 + q3 + r3 = 3pqr (d) none of these
Q33.
The points 0,
Q31.
Q34.
Q35.
8 , (1, 3) and (82, 30) are vertices of: 3
(a) an obtuse angled triangle (b) an acute angled triangle (c) a right angled triangle (d) none of these All points lying inside the triangle formed by the points (1, 3), (5, 0) and (– 1, 2) satisfy: (a) 3x + 2y ≥ 0 (b) 2x + y – 13 ≥ 0 (c) 2x – 3y – 12 ≤ 0 (d) – 2x + y ≥ 0
(e) none of these
A vector a has 2p and 1 with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to the new system a has components p + 1 and 1, then: (b) p = 1 or p = −
(a) p = 0
1 3
(c) p = – 1 or p =
1 3
(d) p = 1 or p = – 1
(e) none of these
Q36.
If P(1, 2), Q(4, 6), R(5, 7) and S(a, b) are the vertices of a parallelogram PQRS, then: (a) a = 2, b = 4 (b) a = 3, b = 4 (c) a = 2, b = 3 (d) a = 3, b = 5
Q37.
The diagonals of a parallelogram PQRS are along the lines x + 3y = 4 and 6x – 2y = 7. Then PQRS must be a: (a) rectangle (b) square (c) cyclic quadrilateral (d) rhombus
Q38.
If the vertices P, Q, R of a triangle PQR are rational points, which of the following points of the triangle PQR is (are) always rational point(s)? (a) centroid (b) incentre (c) circumcentre (d) orthocenter (A rational point is a point both of whose coordinates are rational numbers)
Q39.
Let L1 be a straight line passing through the origin and L2 be the straight line x + y = 1. If the intercepts made by the circle x2 + y2 – x + 3y = 0 on L1 and L2 are equal, then which of the following equations can represents L1? (a) x + y = 0 (b) x – y = 0 (c) x + 7y = 0 (d) x – 7y = 0
ANSWERS EXERCISE – I Q1.
b
Q2.
a
Q3.
b
Q4.
c
Q5.
b
Q6.
b
Q7.
a
Q8. Q15. Q22. Q29. Q36. Q43. Q50. Q57. Q64.
b c b a a a b d c
Q9. Q16. Q23. Q30. Q37. Q44. Q51. Q58. Q65.
a b b c b c a a a,c,d
Q10. Q17. Q24. Q31. Q38. Q45. Q52. Q59. Q66.
b b a c a b b b c
Q11. Q18. Q25. Q32. Q39. Q46. Q53. Q60. Q67.
a b a,b c b c a c b
Q12. Q19. Q26. Q33. Q40. Q47. Q54. Q61. Q68.
c b c d a a c c b, c,d
Q13. Q20. Q27. Q34. Q41. Q48. Q55. Q62.
c d a a d c c a
Q14. Q21. Q28. Q35. Q42. Q49. Q56. Q63.
c a b d c b c d
EXERCISE – II Q1. Q8.
d a, b
Q2. Q9.
c c
Q3. Q10.
c a
Q4. Q11.
c a, c
Q5. Q12.
c b
Q6. Q13.
a, b a
Q7. Q14.
b d
Q15. Q22. Q29. Q36 Q43. Q50. Q57.
a, b d d d d d c
Q16. Q23. Q30. Q37. Q44. Q51. Q58.
a b c a d c b
Q17. Q24. Q31. Q38. Q45. Q52. Q59.
c a c c c b c
Q18. Q25 Q32. Q39. Q46. Q53.
b, d a a a d d
Q19. Q26 Q33. Q40. Q47. Q54.
a a d a a a
Q20. Q27. Q34. Q41. Q48. Q55.
b c d d c d
Q21. Q28. Q35. Q42. Q49. Q56.
a b c c b a
Q5. Q12. Q19. Q26. Q33.
d d b d d
Q6. Q13. Q20. Q27. Q34.
c a a c a, c
Q7. Q14. Q21. Q28. Q35.
d d c b b
EXERCISE – III Q1. Q8. Q15. Q22. Q29. Q36.
b a c b b c
Q2. Q9. Q16. Q23. Q30. Q37.
d a b d c d
Q3. Q10. Q17. Q24. Q31. Q38.
d c a a a a
Q4. Q11. Q18. Q25. Q32. Q39.
a d d d a, b, c c
