Phase Test (Co-ordinate Geometry) Paper: I & II F/M: 160 Time : 3 hr PAPER – I SECTION – I (Total Marks: 21) Straight Objective Type This section contains 7 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct. +3 for correct, -1 for wrong 1. The point ( ) 1 , 2 is shifted through a distance of 2 3 units measured parallel to the line 1 = + y x in the direction of decreasing ordinates, to reach Q. the image of Q with respect to given line is. (a) ( ) 4 , 3 − (b) ( ) 2 , 3 − (c) ( ) 1 , 0 − (d) none of these 2. The point P on the parabola ax y 4 2 = for which RQ PR − is maximum, where ( ) 0 , a R − º , ( ) a Q , 0 º is (a) ( ) a a 2 , (b) ( ) a a 2 ,− (c) ( ) a a 4 , 4 (d) ( ) a a 4 , 4 − 3. The circles on focal radii of a parabola as diameter touch (a) the tangent at the vertex (b) the axis (c) the directrix (d) none of these 4. The curve described parametrically by 1 2 + + = t t x , 1 2 + − = t t y represents (a) a pair of straight line (b) an ellipse (c) a parabola (d) a hyperbola 5. Tangents are drawn at the point of intersection P of ellipse 50 2 2 2 = + y x and hyperbola 1 9 16 2 2 = − y x , in the first quadrant. Then the area of the circle passing through the point P and which cuts the intercepts of 2 unit length each from these tangents. (a) p (b) p 2 (c) 2 3 p (d) none of these 6. The equation of a line passing through the centre of a rectangular hyperbola is 0 1 = − − y x . If one of its asymptotes is 0 6 4 3 = − − y x , the equation of the other asymptote is (a) 0 17 3 4 = + − y x (b) 0 17 3 4 = + − − y x (c) 0 1 3 4 = + + − y x (d) 0 17 3 4 = + + y x 7. If P is any arbitrary point on the ellipse with eccentricity and area of the ellipse be 21 and 4 units respectively and 1 S and 2 S are the two foci then maximum area of 2 1S S P D is (a) p4 (b) p2 (c) p 1 (d) 4p SECTION – II (Total marks: 16) Multiple Correct Answers Type This section contains 4 multiple correct answer(s) type questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONE OR MORE is/are correct. +4 for correct, no negative marking. 8. If the conics whose equations are ( ) ( ) ( ) 0 19 16 32 cos tan 2 sin : 2 2 2 2 1 = + + + + + y x y xy h x S q q q ( ) ( ) ( ) 0 19 32 16 sin cot 2 cos : 2 2 2 2 2 = + + + + ¢ − y x y xy h x S q q q intersect in four concyclic points, where Î 2 , 0 p q , then the correct statement(s) can be (a) 0 = ¢ + h h (b) 0 = ¢ − h h (c) 4p q = (d) none of these 9. If two concentric ellipse be such that the foci of one be on the other and their major axes are equal. Let 1 e and 2 e be their eccentricities, then (a) the quadrilateral formed by joining the foci of the 2 ellipse is a parallelogram (b) the angle q between their axes is given by 2 2 2 1 2 2 2 1 1 1 1 cos e e e e − + = q (c) if 1 2 2 2 1 = + e e , then the angle between the axes of the 2 ellipse is 0 90 (d) none of the these 10. Let ( ) 1 1 , y x P and ( ) 2 2 , y x Q , 0 , 0 2 1 < < y y , be the end points of the latus rectrum of the ellipse 4 4 2 2 = + y x . The equations of parabolas with latus rectrum PQ are (a) 3 3 3 2 2 + = + y x (b) 3 3 3 2 2 + = − y x (c) 3 3 3 2 2 − = + y x (d) 3 3 3 2 2 − = − y x 11. If the circle 16 : 2 2 1 = + y x C intersects another circle 2 C of radius 5 units in such a manner that the common chord is of maximum length and has a slope equal to 43 the co-ordinates of the centre of 2 C can be (a) 5 12 , 59 (b) − 5 12 , 59 (c) − 5 12 , 59 (d) − − 5 12 , 59 SECTION – III (Total marks: 15) Linked Comprehension Type This section contains 2 paragraphs. Based upon one paragraph, 2 multiple choice questions and based on second paragraph 3 multiple choice questions have to be answered. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct. +3 for correct, -1 for wrong. Paragraph for Question Nos. 12 to 13 Consider the circle 9 2 2 = + y x and the parabola x y 8 2 = . They intersect at P and Q in the first & fourth quadrants respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents to the parabola at P and Q intersect the x-axis at S 12. The ratio of the areas of the triangles PQS and PQR is (a) 2 : 1 (b) 2 : 1 (c) 4 : 1 (d) 8 : 1 13. The radius of the circumcircle of the triangle PRS is (a) 5 (b) 3 3 (c) 2 3 (d) 3 2 Paragraph for Question Nos. 14 to 16 Consider an ellipse (E) 1 5 10 2 2 = + y x . There is a hyperbola (H) whose one asymptote is major axis of E. If the eccentricities of E and H are reciprocal of each other, both have same centre and both touch each other in first quadrant, then 14. Equation of the hyperbola is (a) 10 = xy (b) 2 5 = xy (c) 5 2 = xy (d) 2 5 = xy 15. Slope of their common tangents is (a) 21 − (b) 3 − (c) 2 1 − (d) 3 1 − 16. Perpendicular distance between their common tangents is (a) 10 (b) 35 2 (c) 35 4 (d) 25 SECTION – IV (Total marks: 28) Integer Answer Type This section contains 7 questions. The answer to each of the questions is a single digit integer, ranging from 0 to 9. +4 for correct, no negative making. 17. The eccentricity of the conic 2x2 + 3y2 – 4x + 5y + 4 = 0 is k/3 then find k=? 18. Tangents are drawn from any point on the line x – y – 4 = 0 to the curve x2 + 16y2 = 16, then all the chord of contact pass through a fixed point (a, b). Find -4ab=? 19. Find the no of points on hyperbola -= 1 from where mutually perpendicular tangents can be drawn to the circle x2 + y2 = a2. 20. A variable chord PQ of the parabola y = 4x2 subtends a right angle at the vertex. If the locus of the point of intersection of normal at P and Q is y = λx2 + µ, then find the value of (λ3 + 64µ2)/16. 21. The circle x2 + y2 – 4x – 4y + 4 = 0 is inscribed in a triangle whose two of its side are along the coordinate axis. The locus of the circumcentre of the triangle is x + y – xy + k(x2 + y2)1/2 = 0, then find the value of k. 22. The area of the quadrilateral formed by a pair of tangents from the point (4, 5) to the circle (x – 2)2 + (y – 1)2 = 16 with a pair of radii where tangents touch the circle is k square unit then find k. 23. If from the point P (4, 4) perpendiculars to the straight lines 3x + 4y + 5 = 0 and y = mx + 7 meet at Q and R and the area triangle PQR is maximum then 3m is equal to ? PAPER: II SECTION – I (Total marks: 24) Straight Objective Type This section contains 8 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct. +3 for correct, -1 for wrong marking. 1. Five circles C1, C2, C3, C4, C5 with radii r1, r2, r3, r4, r5 respectively (r1 < r2 < r3 < r4 < r5) be such that Ci and C i+1 touches each other externally for all i = 1,2,3,4. If all the 5 circles touches the two lines L1 and L2 and r1 = 2 and r5 = 32 find r3? (a) 8 (b) 17 (c) 64/17 (d) none of these. 2. A tangent and a normal is drawn at the point ( ) 16 , 16 P of the parabola x y 16 2 = which cut the axis of the parabola at the points A and B respectively. If the centre of the circle through P, A and B is C then the angle between PC and the x-axis is (a) 21 tan 1 − (b) 2 tan 1 − (c) 43 tan 1 − (d) 34 tan 1 − 3. Let A(x1, y1) and B(x2, y2) are two points on the parabola y2 = 4ax. If the circle with the chord AB as diameter touches the parabola then find − =? (a) 4a (b) 8a (c) 6a (d) none of these. 4. Equation of the ellipse whose axes are the axes of coordinates and which passes through the point (−3, 1) and has eccentricity is (a) 5x2 + 3y2 = 48 (b) 3x2 + 5y2 = 15 (c) 5x2 + 3y2 = 32 (d) 3x2 + 5y2 = 32 5. From a point ' 't on the parabola ax y 4 2 = , a focal chord and a tangent is drawn. Two circles are drawn in which one circle is drawn taking focal chord as diameter and other is drawn by taking intercept of tangent between point ' 't and directrix as diameter. Then the locus of midpoint of common chord of the circle is (a) a x y + = 3 (b) 0 6 2 9 3 2 2 2 2 = + + − − a x a xy ay ax (c) 0 6 2 9 3 2 2 2 2 = − − + + a x a xy ay ax (d) none of these 6. The two circles x2 + y2 = ax and x2 + y2 = c2 (c > 0) will touch each other if (a) = c (b) a = 2c (c) = 2c (d) 2= c 7. If a line intersects the two branches of a hyperbola at A and B and intersect the asymptotes at C & D find the value of BD AC − (a) 1 (b) 0 (c) 2 1 (d) can’t be determined 8. Consider three points ( ) ( ) ( ) ( ) ( ) ( ) ( ) q b q a b b a b b a b − + − = − = − − − = sin , cos sin , cos , cos , sin R and Q P where 4 , , 0 p q b a < < . Then (a) P lies on the line segment RQ. (b) Q lies on the line segment PR. (c) R lies on the line segment QP. (d) P, Q, R are non-collinear. SECTION – II (Total marks: 16) Multiple Correct Answers Type This section contains 4 multiple correct answer(s) type questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONE OR MORE is/are correct. +4 for correct, no negative marking. 9. Two perpendicular tangents are drawn from a point P to the circle C1. Another circle C2 is drawn touching C1 and also the perpendicular tangents from P. If r1 and r2 are the radii of the two circles C1 and C2 respectively then r1/r2 is equal to (a) 2 + 1 (b) 2 -1 (c) 3 + 22 (d) 3 -22 10. If a variable tangent of circle x2 + y2 = 1 intersects the ellipse x2 + 2y2 = 4 at points P and Q, then the locus of the point of intersection of tangents at P and Q is (a) a parabola with latus rectum = 4 (b) a parabola with focus at (2, 3) (c) an ellipse with eccentricity (d) an ellipse of eccentricity greater than ½. 11. If the chord through the points whose eccentric angles are α and β on the ellipse, + = 1 passes through a focus, then the value of tan(α/2).tan (β/2) is (a) (b) (c) (d) 12. If (5, 12) and (24, 7) are the foci of a conic passing through the origin, then the eccentricity of the conic is (a) (b) (c) (d) SECTION – III (Total marks: 16) Integer Answer Type This section contains 4 questions. The answer to each of the questions is a single digit integer, ranging from 0 to 9. +4 for correct, no negative making. 13. The value of k for which the equation 12x2 – 10xy + 2y2 + 11x – 5y + k = 0 will represent a pair of straight line? 14. In a triangle ABC, A (a, b), B (1, 2) and C(2, 3) and point A lies on the line y = 2x + 3. If area of the triangle is Δ with [Δ] = 2, [.] being the Greatest integer function, find the no of possible integral coordinate of A. 15. A quadrilateral is constructed with the tangents drawn at the end point of the latus rectum of the ellipse x2/9 + y2/5 = 1. If Δ be the area of the quadrilateral then find Δ/3. 16. If a circle cuts the rectangular hyperbola xy = c2 in four points A, B, C, D and the parameters of the four points be t1, t2, t3, t4 respectively find 4t1t2t3t4. SECTION – IV (Total marks: 24) Matrix Match Type This section contains 3 questions. Each question contains statements given in two columns, which have to be matched. Statements in Column I are labelled as A, B, C and D whereas statements in Column II are labelled as p, q, r and s. The answers to these questions have to be appropriately bubbled as illustrated in the following example. If the correct matches are A–q, A–r, B–p, B–s, C–r, C–s and D–q, then the correctly bubbled matrix will look like the following : (You will be awarded +2 for each correct row) p q r s p q r s p q r s p q r s p q r s ABCD 17. Column I Column II (A) The point (1/4, 1/2) is reflected first about the line y = x and then with respect to y-axis. What is the coordinate of the final position of the point? (p) ( ) 3 , 0 (B) Three points A,B,C are taken on the ellipse ( ) ( ) 1 1441 4 641 2 2 2 = − + − y x . If area of ABC D is maximum then possible co-ordinate of the centroid of triangle ABC. (q) − 35 , 32 (C) 0 = + + c by ax be a variable straight line, where a,b,c are 1st, 4th, 6th term of an increasing A.P. Then the variable straight line always passes through a fixed point. (r) 41 , 21 (D) A ray of light emerging from the point ( ) 2 , 1 is reflected at a point Q on the yaxxis and then passes through the point ( ) 9 , 6 co-ordinate of Q is. (s) − 41 , 21 18. Column I Column II (A) 0 P is the parabola x y 4 2 = , with vertex ( ) 0 , 0 k . A and B are points on 0 P where tangents are at right angle, Let C be the centroid of triangle ABK. The locus of C is another parabola 1 P . Now this process is repeated with 1 P , then 3 2 , P P etc. The length of latus rectrum of 4 P is (p) 0 (B) If two distinct chords of a parabola ax y 4 2 = passing through the point ( ) a a 2 , are bisected by the line 1 = + y x , then the length of the latus rectrum can be (q) 2 (C) The parabola 4 5 2 + − = x x y cuts the xaxxi at P&Q. A circle is drawn through P&Q so that the origin lies outside it. The length of a tangent to the circle from the origin is equal to (r) 3 (D) If the point ( ) 1 ,− h is exteriors to both the parabolas x y = 2 , then the integral part of h can be equal to. (s) 81 4 19. Consider the lines given by 0 5 3 : 1 = − + y x L 0 1 3 : 2 = − − ky x L 0 12 2 5 : 3 = − + y x L Column I Column II (A) 3 2 1 , , L L L are concurrent if (p) 9 − = k (B) One of 3 2 1 , , L L L is parallel to at least one of the other two, if (q) 56 − = k (C) 3 2 1 , , L L L from a triangle if (r) 65 = k (D) 3 2 1 , , L L L do not from a triangle if (s) 5 = k