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 co-ordinates of mid-point 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 co-ordinates 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 co-ordinates 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 co-ordinates of a variable point P w.r.t. an orthonormal co-ordinate 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 co-ordinates with slope / and / (C) a line pair through the origin of co-ordinates with slope / and (D) a line pair through the origin of co-ordinates 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 co-ordinate 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 co-ordinates are (, ) and (5, 4) respectivel. The co-ordinates 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 Column-I Column-II (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

PREPARED BY: ER. VINEET LOOMBA (B.TECH. IIT ROORKEE) 60 Best JEE Main and Advanced Level Problems (IIT-JEE). Prepared by IITians.

www. Class XI TARGET : JEE Main/Adv PREPARED BY: ER. VINEET LOOMBA (B.TECH. IIT ROORKEE) ALP ADVANCED LEVEL PROBLEMS Straight Lines 60 Best JEE Main and Advanced Level Problems (IIT-JEE). Prepared b IITians.