Q.2 A, B and C are points in the xy plane such that A(1, 2) ; B (5, 6) and AC = 3BC. Then. (C) 1 1 or


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1 STRAIGHT LINE [STRAIGHT OBJECTIVE TYPE] Q. A variable rectangle PQRS has its sides parallel to fied directions. Q and S lie respectivel on the lines = a, = a and P lies on the ais. Then the locus of R is (A) a straight line (B) a circle (C) a parabola (D) pair of straight lines Q. A, B and C are points in the plane such that A(, ) ; B (5, 6) and AC = BC. Then (A) ABC is a unique triangle (B) There can be onl two such triangles. (C) No such triangle is possible (D) There can be infinite number of such triangles. Q. If A (, p ) ; B (0, ) and C (p, 0) are the coordinates of three points then the value of p for which the area of the triangle ABC is minimum, is (A) (B) (C) or (D) none Q.4 Each member of the famil of parabolas = a + + has a maimum or a minimum point depending upon the value of a. The equation to the locus of the maima or minima for all possible values of 'a' is (A) a straight line with slope and intercept. (B) a straight line with slope and intercept. (C) a straight line with slope and intercept. (D) a circle Q.5 m, n are integer with 0 < n < m. A is the point (m, n) on the cartesian plane. B is the reflection of A in the line =. C is the reflection of B in the ais, D is the reflection of C in the ais and E is the reflection of D in the ais. The area of the pentagon ABCDE is (A) m(m + n) (B) m(m + n) (C) m(m + n) (D) m(m + n) Q.6 The area enclosed b the graphs of + = and = is (A) (B) 4 (C) 6 (D) 8 Q.7 If P = (, 0) ; Q = (, 0) and R = (, 0) are three given points, then the locus of the points S satisfing the relation, SQ + SR = SP is : (A) a straight line parallel to ais (B) a circle passing through the origin (C) a circle with the centre at the origin (D) a straight line parallel to ais. Q.8 Two points A(, ) and B(, ) are chosen on the graph of f () = ln with 0 < <. The points C and D trisect line segment AB with AC < CB. Through C a horizontal line is drawn to cut the curve at E(, ). If = and = 000 then the value of equals (A) 0 (B) 0 (C) (0) / (D) (0) / Q.9 What is the intercept of the line that is parallel to =, and which bisects the area of a rectangle with corners at (0, 0), (4, 0), (4, ) and (0, )? (A) (0, 7) (B) (0, 6) (C) (0, 5) (D) (0, 4) Q.0 Given A (, ) and AB is an line through it cutting the ais in B. If AC is perpendicular to AB and meets the ais in C, then the equation of locus of mid point P of BC is (A) + = (B) + = (C) + = (D) + =
2 Q. AB is the diameter of a semicircle k, C is an arbitrar point on the semicircle (other than A or B) and S is the centre of the circle inscribed into triangle ABC, then measure of (A) angle ASB changes as C moves on k. (B) angle ASB is the same for all positions of C but it cannot be determined without knowing the radius. (C) angle ASB = 5 for all C. (D) angle ASB = 50 for all C. Q. Given = and a + b = are two variable lines, 'a' and 'b' being the parameters connected b a b the relation a + b = ab. The locus of the point of intersection has the equation (A) + + = 0 (B) + + = 0 (C) = 0 (D) + = 0 Q. If the lines + + = 0 ; = 0 and + + = 0, where + =, are concurrent then (A) =, = (B) =, = ± (C) =, = ± (D) = ±, = Q.4 Let (, ) ; (, ) and (, ) are the vertices of a triangle ABC respectivel. D is a point on BC such that BC = BD. The equation of the line through A and D, is (A) + = 0 (B) + = 0 (C) + = 0 (D) + = 0 Q.5 If the straight lines, a + am + = 0, b + (m + ) b + = 0 and c + (m + )c + = 0, m 0 are concurrent then a, b, c are in : (A) A.P. onl for m = (B) A.P. for all m (C) G.P. for all m (D) H.P. for all m. Q.6 If in triangle ABC, A (, 0), circumcentre, and orthocentre 4 coordinates of midpoint of side opposite to A is : (A) (, /) (B) (, 5) (C) (, ) (D) (, 6), then the 4 Q.7 A is a point on either of two lines + = at a distance of units from their point of intersection. The coordinates of the foot of perpendicular from A on the bisector of the angle between them are (A), (B) (0, 0) (C), (D) (0, 4) Q.8 Point 'P' lies on the line l { (, ) + 5 = 5}. If 'P' is also equidistant from the coordinate aes, then P can be located in which of the four quadrants. (A) I onl (B) II onl (C) I or II onl (D) IV onl
3 Q.9 An equilateral triangle has each of its sides of length 6 cm. If (, ) ; (, ) and (, ) are its vertices then the value of the determinant, is equal to : (A) 9 (B) 4 (C) 486 (D) 97 Q.0 A graph is defined in polar coordinates as r() = cos +. The smallest coordinates of an point on the graph is (A) 6 (B) 8 (C) 4 (D) Q. Consider a parallelogram whose sides are represented b the lines + = 0; + 5 = 0; 4 = 0 and 4 =. The equation of the diagonal not passing through the origin, is (A) + 5 = 0 (B) = 0 (C) 9 5 = 0 (D) 5 = 0 Q. Triangle formed b the lines + = 0, = 0 and l + m =. If l and m var subject to the condition l + m = then the locus of its circumcentre is (A) ( ) = + (B) ( + ) = ( ) (C) ( + ) = 4 (D) ( ) = ( + ) Q. The distance between the two parallel lines is unit. A point 'A' is chosen to lie between the lines at a distance 'd' from one of them. Triangle ABC is equilateral with B on one line and C on the other parallel line. The length of the side of the equilateral triangle is (A) d d (B) d d (C) d d (D) d d Q.4 Given the famil of lines, a ( ) + b ( + + ) = 0. The line of the famil situated at the greatest distance from the point P (, ) has equation : (A) = 0 (B) = 0 (C) = 0 (D) none Q.5 A rectangular billiard table has vertices at P(0, 0), Q(0, 7), R(0, 7) and S (0, 0). A small billiard ball starts at M(, 4) and moves in a straight line to the top of the table, bounces to the right side of the table, then comes to rest at N(7, ). The coordinate of the point where it hits the right side, is (A).7 (B).8 (C).9 (D) 4 Q.6 A ra of light passing through the point A (, ) is reflected at a point B on the ais and then passes through (5, ). Then the equation of AB is : (A) = (B) 5 4 = (C) = 4 (D) 4 5 = 6 Q.7 If L is the line whose equation is a + b = c. Let M be the reflection of L through the ais, and let N be the reflection of L through the ais. Which of the following must be true about M and N for all choices of a, b and c? (A) The intercepts of M and N are equal. (B) The intercepts of M and N are equal. (C) The slopes of M and N are equal. (D) The slopes of M and N are reciprocal.
4 Q.8 In a triangle ABC, if A (, ) and = 0 and + 5 = 0 are equations of an altitude and an angle bisector respectivel drawn from B, then equation of BC is (A) + + = 0 (B) = 0 (C) = 0 (D) 5 7 = 0 Q.9 Given A(0, 0) and B(, ) with (0, ) and > 0. Let the slope of the line AB equals m. Point C lies on the line = such that the slope of BC equals m where 0 < m < m. If the area of the triangle ABC can be epressed as (m m ) f (), then the largest possible value of f () is (A) (B) / (C) /4 (D) /8 Q.0 The graph of ( ) against ( + ) is as shown. Which one of the following shows the graph of against? (A) (B) (C) (D) Q. P is a point inside the triangle ABC. Lines are drawn through P, parallel to the sides of the triangle. The three resulting triangles with the verte at P have areas 4, 9 and 49 sq. units. The area of the triangle ABC is (A) (B) (C) 4 (D) 44 Q. Through a point A on the ais a straight line is drawn parallel to ais so as to meet the pair of straight lines a + h + b = 0 in B and C. If AB = BC then (A) h = 4ab (B) 8h = 9ab (C) 9h = 8ab (D) 4h = ab Q. The equation of the pair of bisectors of the angles between two straight lines is, 7 = 0. If the equation of one line is = 0 then the equation of the other line is : (A) 4 8 = 0 (B) + = 0 (C) = 0 (D) = 0 Q.4 Consider a quadratic equation in Z with parameters and as Z Z + ( ) = 0 The parameters and are the coordinates of a variable point P w.r.t. an orthonormal coordinate sstem in a plane. If the quadratic equation has equal roots then the locus of P is (A) a circle (B) a line pair through the origin of coordinates with slope / and / (C) a line pair through the origin of coordinates with slope / and (D) a line pair through the origin of coordinates with slope / and / Q.5 The image of the pair of lines represented b a + h + b = 0 b the line mirror = 0 is (A) a h b = 0 (B) b h + a = 0 (C) b + h + a = 0 (D) a h + b = 0 Q.6 Area of the triangle formed b the line + = and the angle bisectors of the line pair = 0 is (A) / (B) (C) / (D)
5 Q.7 The distance of the point P(, ) from each of the two straight lines through the origin is d. The equation of the two straight lines is (A) ( ) = d ( + ) (B) d ( ) = + (C) d ( + ) = + (D) ( + ) = d ( + ) Q.8 Let PQR be a right angled isosceles triangle, right angled at P (, ). If the equation of the line QR is + =, then the equation representing the pair of lines PQ and PR is (A) = 0 (B) = 0 (C) = 0 (D) = 0 Q.9 The greatest slope along the graph represented b the equation 4 + = 0, is (A) (B) (C) (D) Q.40 If the straight lines joining the origin and the points of intersection of the curve = 0 and + k = 0 are equall inclined to the coordinate aes then the value of k : (A) is equal to (B) is equal to (C) is equal to (D) does not eist in the set of real numbers. Q.4 Vertices of a parallelogram ABCD are A(, ), B(, 6), C(, ) and D(, 6). If a line passing through the origin divides the parallelogram into two congruent parts then the slope of the line is (A) (B) 8 (C) 8 5 (D) 8 [REASONING TYPE] Q.4 Consider the lines, L : ; L = ; L : and L 4 : Statement: The quadrilateral formed b these four lines is a rhombus. Statement: If diagonals of a quadrilateral formed b an four lines are unequal and intersect at right angle then it is a rhombus. (A) Statement is true, statement is true and statement is correct eplanation for statement. (B) Statement is true, statement is true and statement is NOT the correct eplanation for statement. (C) Statement is true, statement is false. (D) Statement is false, statement is true. Q.4 Given a ABC whose vertices are A(, ) ; B(, ) ; C(, ). Let there eists a point P(a, b) such that 6a = + + ; 6b = + + Statement: Area of triangle PBC must be less than the area of ABC Statement: P lies inside the triangle ABC (A) Statement is true, statement is true and statement is correct eplanation for statement. (B) Statement is true, statement is true and statement is NOT the correct eplanation for statement. (C) Statement is true, statement is false. (D) Statement is false, statement is true.
6 Q.44 Let points A, B, C are represented b (a cos i, a sin i ) i =,, and cos ( ) + cos ( ) + cos ( ) =. Statement : Orthocentre of ABC is at origin Statement: ABC is equilateral triangle. (A) Statement is true, statement is true and statement is correct eplanation for statement. (B) Statement is true, statement is true and statement is NOT the correct eplanation for statement. (C) Statement is true, statement is false. (D) Statement is false, statement is true. Q.45 Given the lines + = and + = 5 cut the aes at A, B and C, D respectivel. Statement : ABDC forms quadrilateral and point (, ) lies inside the quadrilateral Statement : Point lies on same side of the lines. (A) Statement is True, Statement is True ; Statement is a correct eplanation for Statement (B) Statement is True, Statement is True ; Statement is NOT a correct eplanation for Statement (C) Statement is True, Statement is False (D) Statement is False, Statement is True Q.46 Consider a triangle whose vertices are A(, ), B(, ) and C(, ) where is a real number. Statement : The area of the triangle ABC is independent of Statement : The verte C of the triangle ABC alwas moves on a line parallel to the base AB. (A) Statement is true, statement is true and statement is correct eplanation for statement. (B) Statement is true, statement is true and statement is NOT the correct eplanation for statement. (C) Statement is true, statement is false. (D) Statement is false, statement is true. Q.47 Statement: Centroid of the triangle whose vertices are A(, ); B( 9, 8) and C(5, ) lies on the internal angle bisector of the verte A. Statement: Triangle ABC is isosceles with B and C as base angles. (A) Statement is true, statement is true and statement is correct eplanation for statement. (B) Statement is true, statement is true and statement is NOT the correct eplanation for statement. (C) Statement is true, statement is false. (D) Statement is false, statement is true. Q.48 Consider the line L: = = 0 and the points A( 5, 6) and B(, ) Statement: There is eactl one point on the line L which is equidistant from the point A and B. Statement: The point A and B are on the different sides of the line. (A) Statement is true, statement is true and statement is correct eplanation for statement. (B) Statement is true, statement is true and statement is NOT the correct eplanation for statement. (C) Statement is true, statement is false. (D) Statement is false, statement is true.
7 Q.49 Consider the following statements Statement: The equation = 0 represents two real lines on the cartesian plane. Statement: A general equation of degree two a + h + b + g + f + c = 0 denotes a line pair if abc + fgh af bg ch = 0 (A) Statement is true, statement is true and statement is correct eplanation for statement. (B) Statement is true, statement is true and statement is NOT the correct eplanation for statement. (C) Statement is true, statement is false. (D) Statement is false, statement is true. [COMPREHENSION TYPE] Paragraph for Question Nos. 50 to 5 Consider a famil of lines (4a + ) (a + ) (a + ) = 0 where a R Q.50 The locus of the foot of the perpendicular from the origin on each member of this famil, is (A) ( ) + 4( + ) = 5 (B) ( ) + ( + ) = 5 (C) ( + ) + 4( ) = 5 (D) ( ) + 4( ) = 5 Q.5 A member of this famil with positive gradient making an angle of 4 with the line 4 =, is (A) 7 5 = 0 (B) 4 + = 0 (C) + 7 = 5 (D) 5 4 = 0 Q.5 Minimum area of the triangle which a member of this famil with negative gradient can make with the positive semi aes, is (A) 8 (B) 6 (C) 4 (D) Paragraph for Question Nos. 5 to 55 Consider a general equation of degree, as = 0 Q.5 The value of '' for which the line pair represents a pair of straight lines is (A) (B) (C) / (D) Q.54 For the value of obtained in above question, if L = 0 and L = 0 are the lines denoted b the given line pair then the product of the abscissa and ordinate of their point of intersection is (A) 8 (B) 8 (C) 5 (D) 5 Q.55 If is the acute angle between L = 0 and L = 0 then lies in the interval (A) (45, 60 ) (B) (0, 45 ) (C) (5, 0 ) (D) (0, 5 ) [MULTIPLE OBJECTIVE TYPE] Q.56 If = is a line through the intersection of = and = and the lengths of the c d a b b a perpendiculars drawn from the origin to these lines are equal in lengths then : (A) = a b c d (C) a b = c d (B) (D) none = a b c d
8 Q.57 A and B are two fied points whose coordinates are (, ) and (5, 4) respectivel. The coordinates of a point P if ABP is an equilateral triangle, is/are : (A) 4, (B) 4, (C), 4 (D), 4 Q.58 Straight lines + = 5 and = intersect at the point A. Points B and C are chosen on these two lines such that AB = AC. Then the equation of a line BC passing through the point (, ) is (A) = 0 (B) + = 0 (C) + 9 = 0 (D) + 7 = 0 Q.59 The straight lines + = 0, + 4 = 0 and + 4 = 0 form a triangle which is (A) isosceles (B) right angled (C) obtuse angled (D) equilateral [MATCH THE COLUMN] Q.60 ColumnI ColumnII (A) Let 'P' be a point inside the triangle ABC and is equidistant (P) centroid from its sides. DEF is a triangle obtained b the intersection of the eternal angle bisectors of the angles of the ABC. With respect to the triangle DEF point P is its (B) Let 'Q' be a point inside the triangle ABC (Q) orthocentre If (AQ)sin A = (BQ)sin B = (CQ)sin C then with respect to the triangle ABC, Q is its (C) Let 'S' be a point in the plane of the triangle ABC. If the point is (R) incentre such that infinite normals can be drawn from it on the circle passing through A, B and C then with respect to the triangle ABC, S is its (D) Let ABC be a triangle. D is some point on the side BC such that (S) circumcentre the line segments parallel to BC with their etremities on AB and AC get bisected b AD. Point E and F are similarl obtained on CA and AB. If segments AD, BE and CF are concurrent at a point R then with respect to the triangle ABC, R is its
9 ANSWER KEY STRAIGHT LINE Q. A Q. D Q. D Q.4 A Q.5 B Q.6 D Q.7 D Q.8 A Q.9 C Q.0 A Q. C Q. A Q. D Q.4 D Q.5 D Q.6 A Q.7 B Q.8 C Q.9 D Q.0 A Q. D Q. A Q. B Q.4 A Q.5 A Q.6 A Q.7 C Q.8 B Q.9 D Q.0 C Q. D Q. B Q. A Q.4 D Q.5 D Q.6 A Q.7 A Q.8 B Q.9 C Q.40 B Q.4 B Q.4 C Q.4 A Q.44 A Q.45 D Q.46 A Q.47 A Q.48 B Q.49 D Q.50 D Q.5 A Q.5 C Q.5 B Q.54 C Q.55 D Q.56 A,C Q.57 A,B Q.58 A,B Q.59 A,C Q.60 (A) Q; (B) R; (C) S; (D) P
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