APROGRESS GETTER L O G O N – Topic Specific MATHEMATICS Chapter Test Series –2430 PART IV ANALYTICAL GEOMETRY AND VECTOR ANALYSIS CHAPTER–3 CONICS II - PARABOLA, ELIPSE, HYPERBOLA (MCE) MAXIMUM MARKS … 155. TIME ALLOWED … 2:15 hours GENERAL INSTRUCTIONS… # This booklet is your Question Paper. # The Question Paper Code is printed on the top right- hand corner of this sheet. # The Test Series Code is mentioned above i.e. 2430. # This Question Paper contains blank spaces for your rough work. No additional sheets will be provided. # Blank papers, Clip Board, Log tables, Slide rule, Calculators, Cellular Phones, Electronic gadgets, in any form are NOT … allowed. # Write your Name, E.No. and Date in the space provided at the bottom of this sheet. # This booklet also contains the answer sheet ( i.e. a machine gradeable Response Sheet ) ORS. Handle it with care. INSTRUCTIONS FOR PAPER … # This paper contains 32 questions divided into FIVE SECTIONS viz. SECTION I ( Q.01 to Q.15 ), SECTION II ( Q.01 to Q.05), … SECTION III ( Q.01 to Q.06 ), SECTION IV (Q.01 to Q.02), SECTION V (Q.01 to Q.04). Attempt as many as possible . # Each question in SECTION II has four options out of which one or more than one is correct. Each question in SECTION I, III has four options out of which one and only one is correct. Select the correct option and shade the oval space corresponding to the select option in the OBJECTIVE RESPONSE SHEET ( ORS ). # For question in SECTION IV, V you have to write correct answer ( see separate instructions ) in the space provided in the ORS. # Each right answer in SECTION I gets 4, in SECTION II, III gets 5, in SECTION IV gets 8, in SECTION V gets 6 marks. # Each wrong answer in SECTION I, III has a penalty of 1 and in SECTION V of 2 marks. INSTRUCTIONS FOR ORS … # Please fill the information asked in ORS immediately. Make Sure the code on the ORS is same as that on this booklet. # Use a soft HB Pencil to darken the required bubble/s in ORS against the selected question number. # In case you wish to change the answer, erase the old answer completely using a good soft eraser. Name: …………………………………......................................... E.No. Date: -ALL THE BEST- DO NOT BREAK THE SEALS ON THIS BOOKLET, AWAIT INSTRUCTIONS FROM THE INVIGILATOR for ACE AXIS – S.C.O. 58, I-II Floor, Sec. 20C, Chandigarh(U.T.)–160020. Phs. (O) 0172 4670083 (M) 92165 28383, 93165 28383, 90416 28383, 90415 28383 (F) 0172 3010083 (R) 0172 4610083, 0183 2220083. www.aciax.com(in),
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[email protected] aps correspondence desk : “BEYOND”, 83A, Rani Ka Bagh, Near FCI Office, Amritsar 143001, Punjab–State (INDIA). units. that gets bisected at (8 . whose equation is … (A) 16 + 33x = 0 (B) 32x + 13 = 0 (C) 13x + 22 = 0 (D) 16x + 33 = 0 Q.01 Circle drawn with diameter being any focal chord of the parabola y2 – 4x – y – 4 = 0 will always touch a fixed line. is equal to … (A) 8 (B) 4 (C) 2 (D) none of these Q. Locus of ‘C’ is … (A) y2 = 4a(x – 4a) (B) y2 = 4a(x – 8a) (C) y2 = 2a(x – 8a) (D) y2 = 2a(x – 4a) Q.05 Slope of the normal chord of y2 = 8x. y2 = x – 4 and (x – 6)2 + y2 = 4 is … (A) y = 3 Space for Scratch Work (B) x = 4 (C) y = –3 (D) y = – 4 .03 Length of the shortest normal chord of the parabola y2 = 4ax is … (A) a 27 (B) 3a 3 (C) 2a 27 (D) none of these Q. 2) is … (A) 1 (B) –1 (C) 2 (D) –2 Q.07 A common tangent to 9x2 + 16y2 = 144. These tangents meet the y-axis at the points A 1 .SECTION–A STRAIGHT OBJECTIVE QUESTIONS (ONLY ONE CORRECT OPTION) Q. where t ∈ R. B 1 respectively. then locus of P is … (A) a2(y2 + 2x) x2 = 8 (B) a2(y2 + 4x) x2 = 16 2 2 2 (C) a (y – 4x) x = 8 (D) a2(y2 – 4x) x2 = 16 Q. If the area of triangle PA 1 B 1 is 2 sq.06 Length of the latus rectum of the parabola whose parametric equation is x = t2 + t + 1. y = t2 – t + 1.02 Two perpendicular chords OA and OB of y2 = 4ax (where ‘O’ being the origin) are drawn and … rectangle OACB is completed.04 PA and PB are the tangents drawn to y2 = 4x from point P. 15 If the tangent at the point P( h . 5) (C) c ∈(0 . locus of point ‘P’ is . 6) ( x − 2) 2 + (y − 1 ) − ( x + 2) 2 + y 2 = c will represent a hyperbola if … 2 [02] (B) c ∈ (0 .14 The vertices of a triangle ABC lie on the hyperbola xy = c2. then the value of b2 is … + 2 = 1 and the hyperbola − = 25 b 144 81 25 (A) 4 (B) 16 (C) 9 (D) 81 Q.. k ) on the hyperbola x2 a 2 − y2 b 2 = 1 cuts the circle x2 + y2 = a2 at the .11 Total number of common tangents of y2 = 4ax and xy = c2...08 The equation (A) c ∈ (0 . 17 ) (D) none of these x 2 y2 x2 y2 1 Q. Hyperbola Q.13 If sum of the slopes of normals drawn from the point ‘P’ to the hyperbola xy = c2 is equal to 4.. then smallest positive value of ‘θ’ is … π π π (A) (B) (C) (D) none of these 6 4 3 Q.12 Equation of one of the latus rectum of the hyperbola (10x – 5)2 + (10y – 2)2 =9(3x + 4y – 7)2 is . then is always equal to … y1 y2 (A) 2 h (B) 2 k (C) 1 h (D) 1 k Space for Scratch Work . 1 1 + points P 1 (x 1 . Ellipse.A –RP L : PIV C3 (MCE) / A – Parabola. then ... is equal to … (A) 1 (B) 2 (C) 3 (D) 4 Q. then triangle ABC is necessarily … (A) equilateral (B) isosceles (C) right angled (D) scalene Q..10 If the eccentricity of the hyperbola x2 – y2 sec2θ = 5 is 3 times the eccentricity of the ellipse … x2sec2 θ + y2 =25. y1 ) and P 2 (x 2 . (A) 30x + 40y – 23 = 0 (B) 40x + 30y – 23 = 0 (C) 30x + 40y + 23 = 0 (D) 40x + 30y + 23 = 0 Q.09 If the ellipse are confocal. If the tangent drawn to hyperbola at ‘A’ is right angle to the side BC of the triangle. y2 ).. (A) x2 = 2c2 (B) x2 = 4c2 (C) x2 + y2 = 2c2 (D) x2 + y2 = 4c2 Q. y = et – e–t (D) x = 2(cost + 3). –c 2 ) (C) ( c .02 For all admissible values of the parameter ‘a’ the straight line 2ax + y 1 − a 2 = 1 will touch an ellipse whose eccentricity is NOT equal to … 1 1 2 3 (A) (B) (C) (D) 2 3 3 2 Q.04 Which of the following expressions (t being the parameter) can represent a hyperbola ? a 1 b 1 tx y x ty − + t = 0. It is given. π 1 1 −1 1 −1 −1 1 + tan −1 (A) cos − (B) (C) π − tan (D) π − cos 2 5 5 5 5 Q.03 The normal to the ellipse 4x2 + 5y2 = 20 at the point ‘P’ touches the parabola y2 = 4x. that is parallel to the line y = x. whose coordinates are … (A) (c 2 ... Circles are drawn having this chord as diameter..01 A circle is drawn to pass through the extremities of the latus rectum of the parabola y2 = 8x. Hyperbola SECTION–B MULTIPLE OBJECTIVE QUESTIONS (MORE THAN ONE CORRECT OPTION) [03] Q. y = t − 2 t 2 t a b a b 2 2 2 t (C) x = et + e–t . + −1 = 0 (A) (B) x = t + . that this circle also touches the directrix of the parabola. – c ) Space for Scratch Work .05 Consider any chord of the hyperbola xy = c2. c ) (D) ( – c .A –RP L : PIV C3 (MCE) / A – Parabola. All these circles will pass through two fixed points. y = 2 2 cos − 1 2 Q. c 2 ) (B) (–c 2 . Ellipse.. The eccentric angle of the point ‘P’ is . Radius of this circle is NOT equal to … (A) 4 (B) 21 (C) 3 (D) 26 Q. . with the axes.A –RP L : PIV C3 (MCE) / A – Parabola. drawn through the points. Q.. R on the ellipse π x2 y2 + 2 = 1 be α . Also when (3) represents a pair of straight lines. Above description can be applied identically for a hyperbola and a circle.s 01 to 03 Read the following statements and answer the questions based on it...02 Suppose two lines are drawn through the common points of intersection of hyperbola x2 y2 − =1 a 2 b2 and the circle x2 + y2 + 2gx + 2fy + c = 0. Now answer the followings . 2 2 a b π + 3α 2 respectively.. β to x–axis. then … (A) α = β (B) α + β = π 2 (C) α + β = π −1 b (D) α + β = 2 tan a Q. If these lines inclined at angles α. showing that three pair of straight lines can be . In general we shall get three values of λ. they are parallel to the lines… x2 y2 + 2 + λ x 2 + y 2 = 0 . which represents a pair of lines equally inclined to the axes.03 The number of pair of straight lines through the points of intersection of rectangular hyperbola … x2 – y2 = 1 and the circle x2 + y2 – 4x = 5 is … (A) 0 Space for Scratch Work (B) 1 (C) 2 (D) 3 . then the eccentric angle of S is… (A) π − 3α (B) 3π − 3α 2 (C) π − 3α 2 (D) − Q. Ellipse. so that(3) represents a pair of straight lines. Q. R cuts the ellipse again at S. Statement : Suppose that an ellipse and a circle are respectively given by the equation… x2 y2 + =1 a2 b2 …(1) and x2 + y2 + 2gx + 2fy + c = 0 …(2) x2 y2 + λ (x2 + y2 + 2gx + 2fy + c) = 0 …(3) represents a curve which passes then the equation 1 + − a2 b2 through the common points of the ellipse (1) and the circle (2). + α and π + α .. Q.01 Let the eccentric angles of three point P. Hyperbola SECTION–C COMPREHENSION LINKED OBJECTIVE QUESTIONS (ONLY ONE CORRECT OPTION) [04] P 01–03 : Paragraph for Question No. A circle through P. We can choose λ.. 2 a b ( ) Hence two straight lines through the points of intersection of an ellipse and any circle make equal angles . . parabola x2 = 4y at A.. O being the … centre of the hyperbola.. 4 respectively.s 04 to 05 Statement : Let x = a 1 t2+ b 1 t. (iii) locus of the points of intersection of tangents … drawn at the ends of all normal chords of the … parabola y2 = 4ax is . 3.. D having co-ordinates ( x i ...A –RP L : PIV C3 (MCE) / A – Parabola.B. (ii) locus of the mid point of the chords of the … hyperbola x2 – y2 = b2 of slope ‘a’ is .. B..06 If the equation of the axis to the conic at the vertex is a 2 x – a 1 y + k = 0. 2 2 2 2 (iv) y1 + y2 + y 3 + y 4 is equal to … (A) 2 (B) 44 (C) 56 (D) 100 . 2. Ellipse..02 Let the circle (x – 1)2 + (y – 1)2 = 25 cuts a rectangular hyperbola with transverse axis along the line y = x at four points A.. Column I Column II (i) x 1 + x 2 + x 3 + x 4 is equal to … 2 2 2 2 (ii) x 1 + x 2 + x 3 + x 4 is equal to … 2 2 2 2 (iii) OA + OB + OC + OD is equal to . yi ).04 The cartesian equation of the given curve should be a conic … (A) ( a 2 x + a 1 y )2 = ( a 1 b 2 – a 2 b 1 )( b 2 x + b 1 y ) 2 (B) ( a 2 x – a 1 y ) = ( a 1 b 2 – a 2 b 1 )( b 2 x – b 1 y ) (C) ( a 2 x – a 1 y )2 = – ( a 1 b 2 – a 2 b 1 )( b 2 x – b 1 y ) (D) none of these Q. Hyperbola P 04–05 : Paragraph for Question No. (iv) locus of the foot of the perpendicular from the … centre of the hyperbola xy = a2 on a variable tangent is … 2 (A) (B) (C) y2(x + 2a) + 4a3 = 0 x = ay (a + 2)x2 = 4y (D) (x2 + y2)2 = 4a2xy Q. y = a 2 t2+ b 2 t be a parametric curve where t is a parameter and … a1b2 – a2 b1 ≠ 0 Q.. The locus of midpoint of AB is . C. Now match the following columns . i = 1.. then k must be … a1b 2 − a 2b1 a 1b 2 − a 2 b1 3 (A) (B) a 12 + a 2 2 a 12 + a 2 2 2 (a 1b 2 − a 2 b1 )(a 1b1 + a 2 b 2 ) (C) − (D) none of these 2 2 2 a1 + a 2 ( ) ( ) SECTION–D MATRIX MATCH TYPE QUESTIONS Q.01 Match the following statements in Column I with their answer(s) in Column II … Column I Column II (i) a tangent to the parabola x + 4ay = 0 cuts the .05 The latus rectum of the conic in the above case is … (A) [05] (a1b2 − a 2b1 )2 a 12 + a 2 2 (B) (a1b 2 − a 2b1 )2 (a 1 2 + a2 3 2 2 ) (C) (a1b 2 − a 2b1 )2 a 12 + a 2 2 (D) none of these Q. The appropriate bubbles below the respective question numbers in the OMR have to be darkened. 3 rectangle formed by joining the vertices of the latus rectum of the ellipse is equal to . Hyperbola SECTION–E INTEGER ANSWER TYPE QUESTIONS This section Contains 4 questions. 2aT). Ellipse.01 If the fourth term in the expansion of . then [(TC)2 + (TD)2]2 is … Q. 5t4 + 1286 is … Q.. n Space for Scratch Work .. 2at ) meets the parabola again at (aT2.02 If the normal chord at the point ‘t’ of the parabola y2 = 4ax subtends a right angle at the vertex.. A square ABCD is constructed on this chord lying inside the parabola. [06] x 1 5 2 Q.04 If the length of the semi–major axis of an ellipse is 72 and the eccentricity is 1 . n ∈ N is 2 and the normal to the parabola y = 4ax at p + x (at2. then minimum value of T2 is … Q.. then ... parabola y2 = 4x intersects at the point T..A –RP L : PIV C3 (MCE) / A – Parabola. then the area of the .03 Tangents to the parabola at the extremities of a common chord AB of the circle x2 + y2 = 5 and the .. 04 Q.03 Q.14 Q.Answers ANSWERS ROGRESS GETTER CHAPTER TEST SERIES : CONICS II – 2 4 3 0 SECTION A Q.01 Q.15 D D C D C D B C B B A A B C B Q.07 Q.02 Q.06 SECTION C C C C B B C SECTION D Q.01 Q.02 Q.01 Q.04 Q.02 (I) (II) (III) (IV) A B C D SECTION E Q.03 Q.04 0008 1306 6400 6936 .02 Q.03 Q.04 Q.01 Q.13 Q.08 Q.09 Q.05 SECTION B BCD BCD AD ABC CD Q.05 Q.06 Q.03 Q.12 Q.02 Q.10 Q.01 (I) (II) (III) (IV) A B C D Q.11 Q.05 Q.