Mathematics Exam Paper M-2023 PDF

M-2023 M-2023 Subject : MATHEMATICS (Booklet Number) Duration : 2 Hours...

M-2023 M-2023 Subject : MATHEMATICS (Booklet Number) Duration : 2 Hours Full Marks : 100 INSTRUCTIONS 1. All questions are of objective type having four answer options for each. 2. Category-1: Carries 1 mark each and only one option is correct. In case of incorrect answer or any combination of more than one answer, ¼ mark will be deducted. 3. Category-2: Carries 2 marks each and only one option is correct. In case of incorrect answer or any combination of more than one answer, ½ mark will be deducted. 4. Category-3: Carries 2 marks each and one or more option(s) is/are correct. If all correct answers are not marked and no incorrect answer is marked, then score = 2 × number of correct answers marked ÷ actual number of correct answers. If any wrong option is marked or if any combination including a wrong option is marked, the answer will be considered wrong, but there is no negative marking for the same and zero mark will be awarded. 5. Questions must be answered on OMR sheet by darkening the appropriate bubble marked A, B, C, or D. 6. Use only Black/Blue ink ball point pen to mark the answer by filling up of the respective bubbles completely. 7. Write Question Booklet number and your roll number carefully in the specified locations of the OMR Sheet. Also fill appropriate bubbles. 8. Write your name (in block letter), name of the examination center and put your signature (as is appeared in Admit Card) in appropriate boxes in the OMR Sheet. 9. The OMR Sheet is liable to become invalid if there is any mistake in filling the correct bubbles for Question Booklet number/roll number or if there is any discrepancy in the name/ signature of the candidate, name of the examination center. The OMR Sheet may also become invalid due to folding or putting stray marks on it or any damage to it. The consequence of such invalidation due to incorrect marking or careless handling by the candidate will be the sole responsibility of candidate. 10. Candidates are not allowed to carry any written or printed material, calculator, pen, log-table, wristwatch, any communication device like mobile phones, bluetooth device etc. inside the examination hall. Any candidate found with such prohibited items will be reported against and his/her candidature will be summarily cancelled. 11. Rough work must be done on the Question Booklet itself. Additional blank pages are given in the Question Booklet for rough work. 12. Hand over the OMR Sheet to the invigilator before leaving the Examination Hall. 13. This Booklet contains questions in both English and Bengali. Necessary care and precaution were taken while framing the Bengali version. However, if any discrepancy(ies) is /are found between the two versions, the information provided in the English version will stand and will be treated as final. 14. Candidates are allowed to take the Question Booklet after examination is over. Signature of the Candidate : ______________________________ (as in Admit Card) Signature of the Invigilator : ______________________________ A 1 P.T.O. M-2023   M-2023 SPACE FOR ROUGH WORK A 2  M-2023 MATHEMATICS Category-1 (Q. 1 to 50) (Carry 1 mark each. Only one option is correct. Negative marks: ̶ ¼)  r r – 1 1. If the matrix Mr is given by Mr =   for r = 1, 2, 3, … then  r – 1 r  det (M1) + det (M2) + … + det (M2008) =  r r – 1 jÉ¡¢VÊ„ Mr =   , r = 1, 2, 3, … ­cJu¡ B­R z ­p­r­œ r –1 r  det (M1) + det (M2) + … + det (M2008) = (A) 2007 (B) 2008 (C) (2008)2 (D) (2007)2 2. Let ,  be the roots of the equation ax2 + bx + c = 0, a, b, c real and sn = n + n and 3 1  s1 1  s 2 (a  b  c) 2 1  s1 1  s 2 1  s3 = k then k = a4 1  s 2 1  s3 1  s 4 j­e Ll, , pj£LlZ ax2 + bx + c = 0-Hl c¤¢V h£S, ­kM¡­e a, b, c h¡Ù¹h Hhw 3 1  s1 1  s 2 (a  b  c) 2 sn = n + n J 1  s1 1  s 2 1  s3 = k a¡q­m k = a4 1  s 2 1  s3 1  s 4 (A) b2 – 4ac (B) b2 + 4ac (C) b2 + 2ac (D) 4ac – b2 2 0 3    3. Let A =  4 7 11. Then 5 4 8    (A) det A is divisible by 11 (B) det A is not divisible by 11 (C) det A = 0 (D) A is orthogonal matrix 2 0 3    j­e Ll, A =  4 7 11 z ­p­r­œ 5 4 8    (A) det A, 11 à¡l¡ ¢hi¡SÉ (B) det A, 11 à¡l¡ ¢hi¡SÉ eu (C) det A = 0 (D) A HL¢V mð jÉ¡¢VÊ„ A 3 P.T.O.  M-2023 4. Let X be a nonvoid set. If 1 and2 be the transitive relations on X, then (A) 1 º 2 is transitive relation (B) 1 º 2 is not transitive relation (C) 1 º 2 is equivalence relation (D) 1 º 2 is not any relation on X (º denotes the composition of relations) j­e Ll, X HL¢V An§ZÉ ­pV z k¢c 1 J 2 X-Hl pw‘¡a pwœ²jZ pðå qu, a­h (A) 1 º 2 pwœ²jZ pðå (B) 1 º 2 pwœ²jZ pðå eu (C) 1 º 2 pja¥mÉa¡ pðå (D) 1 º 2 , X-Hl ­L¡­e¡ pðå pw‘¡a L­l e¡ (º pw­k¡SL pðå ­h¡T¡u) 1 5. Let A and B are two independent events. The probability that both A and B happen is 12 1 and probability that neither A nor B happen is. Then 2 1 j­e Ll, A J B flØfl ¢eiÑln£m eu Hje c¤¢V OVe¡ z A J B Ei­uC OV­h Hje pñ¡he¡ qm 12 1 Hhw A J B -Hl ­LEC q­h e¡ Hje pñ¡he¡ qm z ­p­r­œ 2 1 1 1 1 (A) P(A) = , P(B) = (B) P(A) = , P(B) = 3 4 2 6 1 1 2 1 (C) P(A) = , P(B) = (D) P(A) = , P(B) = 6 2 3 8 6. Let A, B, C are subsets of set X. Then consider the validity of the following set theoretic statement : j­e Ll, A, B, C ­pV X-Hl Ef­pV z ­p­r­œ ¢ej¢À m¢Ma ­pVa¡¢šL Ä ¢hhª¢a…¢ml kb¡bÑa¡ ¢eZÑu Ll : (A) A  (B \ C) = (A  B) \ (A  C) (B) (A \ B) \ C = A \ (B  C) (C) (A  B) \ A = A \ B (D) A\C=B\C A 4  M-2023 1 7. If sin , cos , tan  are in G.P, then the solution set of  is 6 1 k¢c sin , cos , tan  pj¡¿¹l fËN¢a­a b¡­L a­h -Hl pj¡d¡e l¡¢n q­h 6 π π π π (A) 2n  (B) 2n  (C) n + (–1)n (D) n  6 3 3 3 (Here/HM¡­e n  )  π 8. The equation r2 cos2  θ –  = 2 represents  3 (A) a parabola (B) a hyperbola (C) a circle (D) a pair of straight lines  π r2 cos2  θ –  = 2 p§¢Qa L­l  3 (A) HL¢V A¢dhªš (B) HL¢V fl¡hªš (C) HL¢V hªš (D) plm­lM¡ k¤Nm 9. Let S be the sample space of the random experiment of throwing simultaneously two unbiased dice and Ek = {(a, b)  S : ab = k}. If pk = P(Ek), then the correct among the following is : c¤¢V V¡mq£e R‚¡ N¢s­u ­cJu¡ qm z S OVe¡ pj§­ql ej¤e¡ ­rœ Hhw Ek = {(a, b)  S : ab = k} z k¢c pk = P(Ek) qu, a­h ¢e­jÀl pÇfLÑ…¢ml ­L¡e¢V ¢WL ? (A) p1 < p10 < p4 (B) p2 < p8 < p14 (C) p4 < p8 < p17 (D) p2 < p16 < p5 A 5 P.T.O.  M-2023 10. If 4a2 + 9b2 – c2 + 12ab = 0, then the family of straight lines ax + by + c = 0 is concurrent at (A) (2, 3) or (–2, –3) (B) (–2, 3) or (2, 3) (C) (3, 2) or (–3, 2) (D) (–3, 2) or (2, 3) k¢c 4a2 + 9b2 – c2 + 12ab = 0 qu, a­h plm­lM¡ f¢lh¡l ax + by + c = 0 ­k ¢h¾c¤­a pj¢h¾c¤ q­h a¡ q'm, (A) (2, 3) Abh¡ (–2, –3) (B) (–2, 3) Abh¡ (2, 3) (C) (3, 2) Abh¡ (–3, 2) (D) (–3, 2) Abh¡ (2, 3) 11. The straight lines x + 2y – 9 = 0, 3x + 5y – 5 = 0 and ax + by – 1 = 0 are concurrent if the straight line 35x – 22y + 1 = 0 passes through the point x + 2y – 9 = 0, 3x + 5y – 5 = 0 J ax + by – 1 = 0 plm­lM¡œu pj¢h¾c¤ q­h, k¢c 35x – 22y + 1 = 0 plm­lM¡¢V ­k ¢h¾c¤N¡j£ qu, (A) (–a, –b) (B) (a, – b) (C) (–a, b) (D) (a, b) 12. Let A be the point (0, 4) in the xy-plane and let B be the point (2t, 0). Let L be the midpoint of AB and let the perpendicular bisector of AB meet the y-axis M. Let N be the midpoint of LM. Then locus of N is (A) a circle (B) a parabola (C) a straight line (D) a hyperbola xy-a­m A(0, 4) Hhw B(2t, 0) z j­e Ll, L, AB-Hl jdÉ¢h¾c¤ Hhw j­e Ll, AB-Hl mð pj¢àMäL y-Ar­L M ¢h¾c¤­a ­Rc L­l z j­e Ll, N, LM-Hl jdÉ¢h¾c¤ z ­p­r­œ N-Hl p’¡lfb q­h (A) HL¢V hªš (B) HL¢V A¢dhªš (C) HL¢V plm­lM¡ (D) HL¢V fl¡hªš A 6  M-2023 13. Let O be the vertex, Q be any point on the parabola x2 = 8y. If the point P divides the line segment OQ internally in the ratio 1 : 3, then the locus of P is j­e Ll, A¢dhªš x2 = 8y -Hl n£oÑ¢h¾c¤ qm O Hhw Q A¢dhª­šl Ef¢lÙÛ ­L¡e HL¢V ¢h¾c¤ z k¢c P ¢h¾c¤ OQ ­lM¡wn­L 1 : 3 Ae¤f¡­a A¿¹Ñ¢hiš² L­l a­h P-Hl p’¡lfb q­h (A) x2 = y (B) y2 = x (C) y2 = 2x (D) x2 = 2y π x 2 y2 14. The tangent at point (a cos , b sin ), 0 <  < , to the ellipse 2  2 = 1 meets the 2 a b x-axis at T and y-axis at T1. Then the value of 0 min θ π (OT) (OT1) is 2 x 2 y2 π Efhªš 2  2 = 1 -Hl (a cos , b sin ), 0 <  < , ¢h¾c¤­a A¢ˆa ØfnÑL x-Ar­L T ¢h¾c¤­a a b 2 J y-Ar­L T1 ¢h¾c¤­a ­Rc L­l z ­p­r­œ 0 min θ π (OT) (OT1) Hl j¡e q­h 2 (A) ab (B) 2ab (C) 0 (D) 1 15. ABC is an isosceles triangle with an inscribed circle with centre O. Let P be the midpoint of BC. If AB = AC = 15 and BC = 10, then OP equals 5 5 (A) unit (B) unit 2 2 (C) 2 5 unit (D) 5 2 unit ABC HL¢V pj¢àh¡ý ¢œi¥S, k¡l HL¢V A¿¹xhªš haÑj¡e J I hª­šl ­L¾cÊ O z j­e Ll, P, BC Hl jdÉ¢h¾c¤ z k¢c AB = AC = 15 HLL qu J BC = 10 HLL qu, a­h OP q­h 5 5 (A) HLL (B) HLL 2 2 (C) 2 5 HLL (D) 5 2 HLL A 7 P.T.O.  M-2023 x 2 y2 16. If the lines joining the focii of the ellipse 2  2 = 1 where a > b, and an extremity of its a b minor axis is inclined at an angle 60º, then the eccentricity of the ellipse is x 2 y2 Efhªš  = 1, a > b -Hl e¡¢iàu J Efhªš¢Vl Ef¡­rl HL¢V fË¡¿¹¢h¾c¤l pw­k¡NL¡l£ plm­lM¡ a 2 b2 60º ­L¡­Z ea q­m, Efhªš¢Vl Ev­L¾cÊa¡ q­h 3 1 7 1 (A) (B) (C) (D) 2 2 3 3 17. If the distance between the plane x – 2y + z = k and the plane containing the lines x –1 y – 2 z – 3 x–2 y–3 z–4   and   is 6 , then | k | is 2 3 4 3 4 5 x –1 y – 2 z – 3 x–2 y–3 z–4 am x – 2y + z = k J   J   plm­lM¡à­ul d¡lL 2 3 4 3 4 5 a­ml j­dÉL¡l c§laÄ 6 HLL q­m | k | q­h (A) 36 (B) 12 (C) 6 (D) 2 3 π 18. Let A(2 sec , 3 tan ) and B(2 sec , 3 tan ) where  +  = be two points on the 2 x 2 y2 hyperbola – = 1. If (, ) is the point of intersection of normals to the hyperbola at 4 9 A and B, then  is equal to π x 2 y2 j­e Ll, A(2 sec , 3 tan ) J B(2 sec , 3 tan ),  +  = fl¡hªš – = 1-Hl Ef¢lÙÛ 2 4 9 c¤¢V ¢h¾c¤ z I A & B c¤C ¢h¾c¤­a A¢ˆa A¢imðà­ul ­Rc ¢h¾c¤l ÙÛ¡e¡ˆ (, ) q­m  q­h 12 13 12 13 (A) (B) (C) – (D) – 3 3 3 3 A 8  M-2023 19. Let f(x) = [x2] sin x, x > 0. Then (A) f is discontinuous everywhere. (B) f is continuous everywhere. (C) f is continuous at only those points which are perfect squares. (D) f is continuous at only those points which are not perfect squares. j­e Ll, f(x) = [x2] sin x, x > 0 ­p­r­œ, (A) f phÑœC Ap¿¹a (B) f phÑœC p¿¹a (C) ­kph ¢h¾c¤ f§ZÑhNÑ, ­Lhmj¡œ ­pCph ¢h¾c¤­aC f p¿¹a q­h (D) ­kph ¢h¾c¤ f§ZÑhNÑ eu, ­Lhmj¡œ ­pCph ¢h¾c¤­aC f p¿¹a q­h 20. If y = lognx, where logn means loge loge loge … (repeated n times), then dy x log x log2 x log3 x ….. logn – 1 x logn x is equal to dx logn hm­a loge loge loge … (n-pwMÉL fc fkÑ¿¹) ­h¡T¡u z k¢c y = logn x qu, a­h dy x log x log2 x log3 x ….. logn – 1 x logn x -Hl j¡e q­h dx (A) log x (B) x (C) 1 (D) logn x 21. The angle between a normal to the plane 2x – y + 2z – 1 = 0 and the X-axis is X-Ar Hhw 2x – y + 2z – 1 = 0 -a­ml A¢im­ðl j­dÉL¡l ­L¡Z q­h 2 1 3 1 (A) cos–1 (B) cos–1 (C) cos–1 (D) cos–1 3 5 4 3 A 9 P.T.O.  M-2023 22. lim  n  x  x – ( x – a1 )( x – a 2 )... ( x – a n ) where a1, a2, …, an are positive rational numbers. The limit a1  a 2 ... a n (A) does not exist (B) is n n n (C) is a1a 2... a n (D) is a1  a 2 ...  a n lim   x  x – ( x – a1 )( x – a 2 )... ( x – a n ) ­kM¡­e a1, a2, …, an de¡aÈL j§mc pwMÉ¡, p£j¡ qm n a1  a 2 ... a n (A) p£j¡¢Vl A¢Ù¹aÄ ­eC (B) q­h n n (C) q­h n a1a 2... a n (D) q­h a1  a 2 ...  a n  1, if x  1 23. Suppose f : be given by f(x) =  ( x –1) 10 1 , if x  1 then e  ( x – 1) 2 sin  x –1 (A) f ′(1) does not exist (B) f ′(1) exists and is zero (C) f ′(1) exist and is 9 (D) f ′(1) exists and is 10  1, if x  1 j­e Ll, f : ¢ejÀi¡­h pw‘¡a B­R f(x) =  ( x10 –1) 1 , if x  1 a¡q­m e  ( x – 1) 2 sin  x –1 (A) f ′(1)-Hl A¢Ù¹aÄ ­eC (B) f ′(1)-Hl A¢Ù¹aÄ B­R J j¡e q­h 0 (C) f ′(1) -Hl A¢Ù¹aÄ B­R J j¡e q­h 9 (D) f ′(1)-Hl A¢Ù¹aÄ B­R J j¡e q­h 10 A 10  M-2023 24. f(x) is a differentiable function and given f'(2) = 6 and f'(1) = 4, then f(2  2h  h 2 ) – f(2) L = hlim 0 f(1  h – h 2 ) – f(1) (A) does not exist (B) equal to –3 (C) equal to 3 (D) equal to 3/2 f(x) HL¢V AhLme ­k¡NÉ A­frL Hhw f'(2) = 6, f'(1) = 4 ­cJu¡ B­R z 2 f(2  2h  h ) – f(2) ­p­r­œ, L = hlim 0 f(1  h – h 2 ) – f(1) (A) Hl A¢Ù¹aÄ ­eC (B) –3 Hl p­‰ pj¡e (C) 3 Hl p­‰ pj¡e (D) 3/2 Hl p­‰ pj¡e n y  x 25. Let cos–1   = loge   , then Ay2 + By1 + Cy = 0 is possible for b n n j­e Ll, cos–1   = loge   z ­p­r­œ Ay2 + By1 + Cy = 0 pñh kMe y x b n d2y dy where/ ­kM¡­e y2 = 2 , y1 = dx dx (A) A = 2, B = x2, C = n (B) A = x2, B = x, C = n2 (C) A = x, B = 2x, C = 3n + 1 (D) A = x2, B = 3x, C = 2n 26. Let f : [1, 3] be continuous and be derivable in (1, 3) and f'(x) = [f(x)]2 + 4 x  (1, 3). Then (A) f(3) – f(1) = 5 holds (B) f(3) – f(1) = 5 does not hold (C) f(3) – f(1) = 3 holds (D) f(3) – f(1) = 4 holds j­e Ll, f : [1, 3] A­frL¢V [1, 3] A¿¹l¡­m p¿¹a J (1, 3) ­a AhLme­rœ A­frL z f ′(x) = [f(x)]2 + 4 pLm x  (1, 3)-Hl SeÉ z ­p­r­œ, (A) f(3) – f(1) = 5 kb¡bÑ (B) f(3) – f(1) = 5 kb¡bÑ eu (C) f(3) – f(1) = 3 kb¡bÑ (D) f(3) – f(1) = 4 kb¡bÑ A 11 P.T.O.  M-2023 dx 1  | x – 3 |3 | x  1 |  27. If  ( x  1) ( x – 2) ( x – 3) k e  ( x – 2)4  + c, then the value of k is  log   dx 1  | x – 3 |3 | x  1 |  k¢c   log e   + c qu, a­h k-Hl j¡e q­h ( x  1) ( x – 2) ( x – 3) k  ( x – 2) 4  (A) 4 (B) 6 (C) 8 (D) 12 n  [ x ] dx 0 28. The expression n , where [x] and {x} are respectively integral and fractional part of  {x} dx 0 x and n  , is equal to n  [ x ] dx 0 n -Hl j¡e q­h (HM¡­e [x] J {x}h¢m­a x-Hl kb¡œ²­j f§ZÑpwMÉ¡ j¡e J Awnj¡e h¤T¡C­h)  {x} dx 0 Hhw n  , 1 1 (A) (B) (C) n (D) n – 1 n –1 n x 2 dx 29. If I =  ( x sin x  cos x)2 = f(x) + tan x + c, then f(x) is x 2 dx k¢c I =  ( x sin x  cos x)2 = f(x) + tan x + c qu, a­h f(x) q­h sin x 1 (A) (B) x sin x  cos x ( x sin x  cos x) 2 –x 1 (C) (D) cos x ( x sin x  cos x) sin x ( x cos x  sin x) A 12  M-2023 π 2 30. If In =  cosn x cos nxdx, then I1, I2, I3 … are in 0 (A) A.P. (B) G.P. (C) H.P. (D) no such relation π 2 k¢c In =  cosn x cos nxdx, a¡q­m I1, I2, I3 … 0 (A) pj¡¿¹l fËN¢a­a b¡L­h, (B) …­Z¡šl fËN¢a­a b¡L­h, (C) ¢hfl£a fËN¢a­a b¡L­h, (D) Hje ­L¡­e¡ pÇfLÑ b¡L­h e¡ x dy y  x x 31. If y = is the solution of the differential equation      , then    is log e | cx | dx x  y y given by dy y  x x x      A¿¹lLm pj£Ll­Zl pj¡d¡e y = q­m,    q­h dx x  y log e | cx | y y2 y2 x2 x2 (A) (B) – (C) (D) – x2 x2 y2 y2 1/2 dx 32. The value of  1 – x 2n is (n  ) 0 π (A) less than or equal to (B) greater than or equal to 1 6 1 π (C) less than (D) greater than 2 6 1/2 dx  1– x 2n (n  )-Hl j¡e 0 π (A) -Hl ­Q­u ­R¡V h¡ pj¡e (B) 1-Hl ­Q­u h­s¡ h¡ pj¡e 6 1 π (C) -Hl ­Q­u ­R¡V (D) -Hl ­Q­u h­s¡ 2 6 A 13 P.T.O.  M-2023  d 2 y dy   dy  dy 33. The function y = ekx satisfies  2    – y  = y. It is valid for  dx  dx   dx  dx  (A) exactly one value of k. (B) two distinct values of k. (C) three distinct values of k. (D) infinitely many values of k.  d 2 y dy  dy A­frL y = ekx,  2    – y  = y -­L ¢pÜ L­l z H¢V °hd q­h dy  dx dx   dx  dx (A) k-Hl HL¢V j¡œ j¡­el SeÉ (B) k-Hl c¤¢V fªbL fªbL j¡­el SeÉ (C) k-Hl ¢ae¢V fªbL fªbL j¡­el SeÉ (D) k-Hl Ap£j pwMÉL j¡­el SeÉ d2y dy 34. Given 2 + cot x + 4y cosec2 x = 0. Changing the independent variable x to z by the dx dx x substitution z = log tan , the equation is changed to 2 d2y dy 2 x = 0 z ü¡d£e Qml¡¢n x-Hl hc­m z = log tan x q­m, pj£LlZ¢V + cot x + 4y cosec dx 2 dx 2 f¢lh¢aÑa q­h d2 y 3 d2y (A)  =0 (B) 2 2 + ey = 0 dz 2 y dz d2y d2y (C) – 4y = 0 (D) + 4y = 0 dz 2 dz 2 A 14  M-2023 35. A missile is fired from the ground level rises x meters vertically upwards in t sec, where 25 2 x = 100t – t. The maximum height reached is 2 i¨¢j ­b­L HL¢V ­rfZ¡Ù» Eõði¡­h Ev¢rç qu, ­k¢V t ­p­L­ä x ¢jV¡l fb A¢aœ²j L­l ­kM¡­e 25 2 x = 100t – t z ­rfZ¡Ù»¢V ­k p­hÑ¡µQ EµQa¡ f¢lœ²j L­l a¡l f¢lj¡f 2 (A) 100 m (B) 300 m (C) 200 m (D) 125 m 36. If a hyperbola passes through the point P( 2 , 3 ) and has foci at ( 2, 0), then the tangent to this hyperbola at P is P( 2 , 3 ) ¢h¾c¤N¡j£ HL¢V fl¡hª­šl e¡¢iàu ( 2, 0) q­m, P ¢h¾c¤­a A¢ˆa ØfnÑ­Ll pj£LlZ qm (A) y=x 6 – 3 (B) y=x 3– 6 (C) y=x 6 3 (D) y=x 3 6  x  1, – 1  x  0 37. Let f(x) =   – x, 0  x 1 (A) f(x) is discontinuous in [–1, 1] and so has no maximum value or minimum value in [–1, 1]. (B) f(x) is continuous in [–1, 1] and so has maximum value and minimum value. (C) f(x) is discontinuous in [–1, 1] but still has the maximum and minimum value. (D) f(x) is bounded in [–1, 1] and does not attain maximum or minimum value.  x  1, – 1  x  0 j­e Ll, f(x) =  z ­p­r­œ,  – x, 0  x 1 (A) f(x), [–1, 1]-H Ap¿¹a J ­pL¡l­Z p­hÑ¡µQ J phÑ¢ejÀ j¡e f¢lNËq L­l e¡ (B) f(x), [–1,1]-H p¿¹a J ­pL¡l­Z p­hÑ¡µQ J phÑ¢ejÀ j¡e f¢lNËq L­l (C) f(x), [–1, 1] -H Ap¿¹a ¢L¿¹¥ a¡p­šJ Ä A­frL¢V p­hÑ¡µQ J phÑ¢ejÀ j¡e f¢lNËq L­l (D) f(x), [–1,1] -H p£j¡hÜ A­frL Hhw p­hÑ¡µQ J phÑ¢ejÀ j¡e f¢lNËq L­l e¡ A 15 P.T.O.  M-2023 x 2 y2 38. The average length of all vertical chords of the hyperbola 2 – 2 = 1, a  x  2a, is a b x 2 y2 fl¡hªš – = 1, a  x  2a z fl¡hª­šl pLm Eõð SÉ¡-Hl Ns °cOÉÑ qm a 2 b2 (A)  b 2 3  ln(2  3 )  (B)  b 3 2  ln(3  2 )  (C)  a 2 5 – ln(2  5 )  (D)  a 5 2  ln(5  2 )  39. The value of ‘a’ for which the scalar triple product formed by the vectors ^ ^ ^ ^ ^ α = ^i + a j + k , β = j + a k and γ = a ^i + k is maximum, is ^ ^ ^ ^ ^ α = ^i + a j + k , β = j + a k , γ = a ^i + k -Hl scalar triple product p­hÑ¡µQ q­m ‘a’ -Hl j¡e q­h (A) 3 (B) –3 1 1 (C) – (D) 3 3 40. A, B are fixed points with coordinates (0, a) and (0, b) (a > 0, b > 0). P is variable point (x, 0) referred to rectangular axis. If the angle APB is maximum, then A J B c¤¢V ¢ÙÛl ¢h¾c¤, a¡­cl ÙÛ¡e¡ˆ kb¡œ²­j (0, a) J (0, b) (a > 0, b > 0) z Bua¡L¡l Ar hÉhÙÛ¡u, P HL¢V N¢an£m ¢h¾c¤ (x, 0) z k¢c ­L¡Z APB p­hÑ¡µQ qu, a­h (A) x2 = ab (B) x2 = a + b \ 1 ab (C) x= (D) x= ab 2 A 16  M-2023 41. If 1, log9 (31 – x + 2), log3 (4.3x – 1) are in A.P, then x equals 1, log9 (31 – x + 2), log3 (4.3x – 1) pj¡¿¹l fËN¢a­a b¡L­m x Hl j¡e q­h (A) log3 4 (B) 1 – log3 4 (C) 1 – log4 3 (D) log4 3 42. Reflection of the line a– z + az– = 0 in the real axis is given by h¡Ù¹h A­r a– z + az– = 0 ­lM¡l fË¢agme q­h a a (A) az + az = 0 (B) a– z – az– = 0 (C) az – az = 0 (D)  =0 z z 43. If the vertices of a square are z 1, z2, z3 and z4 taken in the anti-clockwise order, then z 3 = O¢sl Ly¡V¡l ¢hfl£a ¢cL ¢e¢l­M HL¢V hNÑ­r­œl ­L±¢ZL ¢h¾c¤ Qa¥øu kb¡œ²­j z1, z2, z3 J z4 q­m z3 q­h (A) –iz1 – (1 + i) z2 (B) z1 – (1 + i) z2 (C) z1 + (1 + i) z2 (D) –iz1 + (1 + i) z2 44. If the n terms a1, a2, ……, an are in A.P. with increment r, then the difference between the mean of their squares & the square of their mean is n pwMÉL fc a1, a2, ……, an pj¡¿¹l fËN¢a­a (A.P.) B­R, ­kM¡­e p¡d¡lZ A¿¹l qm r z ­p­r­œ pwMÉ¡…¢ml hNÑ pj§­ql jdÉL J pwMÉ¡…¢ml jdÉ­Ll h­NÑl A¿¹l qm r 2 {(n – 1) 2 – 1} r2 r 2 (n 2 – 1) n2 –1 (A) (B) (C) (D) 12 12 12 12 A 17 P.T.O.  M-2023 45. The number of ways in which the letters of the word ‘VERTICAL’ can be arranged without changing the order of the vowels is ‘VERTICAL’ në¢Vl ülhZÑ…¢ml œ²j f¢lhaÑe e¡ L­l I në¢Vl Arl…¢m­L kai¡­h ¢heÉÙ¹ Ll¡ k¡u, a¡l pwMÉ¡ qm 8! (A) 6!  3! (B) 3 8! (C) 6! 3 (D) 3! 46. n objects are distributed at random among n persons. The number of ways in which this can be done so that at least one of them will not get any object is n pwMÉL hÉ¢š²l j­dÉ Ljf­r HLSe ­L¡e hÙ¹¥C f¡­he e¡ -HC naÑ¡d£­e n pwMÉL hÙ¹¥ kcªµR i¡­h pj­ha hÉ¢š²­cl j­dÉ h¾Ve Ll¡ k¡­h ­k pwMÉL fÜ¢a­a, a¡ qm (A) n! – n (B) nn – n (C) nn – n2 (D) nn – n! 47. If one root of x2 + px – q2 = 0, p and q are real, be less than 2 and other be greater than 2, then x2 + px – q2 = 0 pj£Ll­Zl (p J q h¡Ù¹h) HL¢V h£S 2-l ­Q­u ­R¡­V¡ J Afl¢V 2-l ­Q­u hs q­m (A) 4 + 2p + q2 > 0 (B) 4 + 2p + q2 < 0 (C) 4 + 2p – q2 > 0 (D) 4 + 2p – q2 < 0 A 18  M-2023 48. Let A be a set containing n elements. A subset P of A is chosen, and the set A is reconstructed by replacing the elements of P. A subset Q of A is chosen again. The number of ways of choosing P and Q such that Q contains just one element more than P is A HL¢V n pcpÉ ¢h¢nø ­pV z P, A -Hl HL¢V Ef­pV NWe Ll¡ qm z P Ef­p­Vl pcpÉ…¢m ¢c­u A ­pV¢V f¤el¡u NWe Ll¡ qm z Q, A Hl Bl HL¢V Ef­pV NWe Ll¡ qm z P J Q ka lLj i¡­h NWe Ll¡ k¡­h k¡­a Q-Hl pcpÉ pwMÉ¡ P-Hl pcpÉ pwMÉ¡l ­b­L HL¢V ­hn£ qu a¡l pwMÉ¡ q­h (A) 2nC (B) 2nC (C) 2nC (D) 22n + 1 n–1 n n+2 49. Let A and B are orthogonal matrices and det A + det B = 0. Then (A) A + B is singular (B) A + B is non-singular (C) A + B is orthogonal (D) A + B is skew symmetric j­e Ll, A J B c¤¢V mð jÉ¡¢VÊ„ Hhw det A + det B = 0 z ­p­r­œ (A) A + B ¢h¢nø jÉ¡¢VÊ„ (B) A + B A¢h¢nø jÉ¡¢VÊ„ (C) A + B HL¢V mð jÉ¡¢VÊ„ (D) A + B ¢hfË¢apj jÉ¡¢VÊ„ 50. Let P(n) = 32n + 1 + 2n + 2 where n . Then (A) P(n) is not divisible by any prime integer. (B) there exists prime integer which divides P(n). (C) P(n) is divisible by 5 for all n . (D) P(n) is divisible by 3 for all n . j­e Ll, pLm n  Hl SeÉ P(n) = 32n + 1 + 2n + 2 z ­p­r­œ (A) P(n) ­L¡e ­j±¢mL pwMÉ¡ à¡l¡ ¢hi¡SÉ eu (B) Hje ­j±¢mL pwMÉ¡l A¢Ù¹aÄ l­u­R k¡l à¡l¡ P(n) ¢hi¡SÉ q­h (C) pLm n  Hl SeÉ P(n), 5 à¡l¡ ¢hi¡SÉ q­h (D) pLm n  Hl SeÉ P(n), 3 à¡l¡ ¢hi¡SÉ q­h A 19 P.T.O.  M-2023 Category-2 (Q. 51 to 65) (Carry 2 marks each. Only one option is correct. Negative marks: ̶ ½) 51. Let  be a relation defined on set of natural numbers , as  = {(x, y)   : 2x + y = 41}. Then domain A and range B are ü¡i¡¢hL pwMÉ¡l ­pV -H pÇfLÑ  pw‘¡a B­R ­k  = {(x, y)   : 2x + y = 41} z ­p­r­œ pw‘¡l A’m A J X-H ¢hÙ¹¡l B qm (A) A  {x  : 1  x  20} and B  {y  : 1  y  39} (B) A = {x  : 1  x  15} and B = {y  : 2  y  30} (C) A ,Bℚ (D) A = ℚ, B = ℚ 52. From the focus of the parabola y2 = 12x, a ray of light is directed in a direction making an 3 angle tan–1 with x-axis. Then the equation of the line along which the reflected ray 4 leaves the parabola is 3 A¢dhªš y2 = 12x-Hl e¡¢i ­b­L HL¢V B­m¡Ll¢nÈ x-A­rl p­‰ tan–1 ­L¡­Z ea A¢ij¤­M 4 d¡¢ha qu z ­p­r­œ fË¢ag¢ma l¢nÈ ­k m¡Ce hl¡hl A¢dhªš aÉ¡N L­l a¡q¡l pj£LlZ qm (A) y=2 (B) y = 18 (C) y=9 (D) y = 36 0 0 1 0 1 0 0 1 0       53. Let A =  1 0 0  , B =  0 0 1  and P =  x 0 0  be an orthogonal matrix such that 0 0 0 0 0 0 0 0 y       B = PAP–1 holds. Then 0 0 1 0 1 0 0 1 0       j­e Ll, A =  1 0 0  , B =  0 0 1  J P =  x 0 0  HL¢V mð jÉ¡¢VÊ„ (orthogonal 0 0 0 0 0 0 0 0 y       matrix) Hl©f ­k B = PAP–1 ¢pÜ L­l z ­p­r­œ (A) x=1=y (B) x = 1, y = 0 (C) x = 0, y = 1 (D) x = –1, y = 0 A 20  M-2023 lim  1  1    1  1  ...   1  1  54. The value of n       2 · 3 22 · 3   22 · 32 23 · 32    is   2n · 3n 2n  1 · 3n  lim  1  1    1  1  ...   1  1      n   2 · 3 22 · 3   22 · 32 23 · 32    Hl j¡e   2n · 3n 2n  1 · 3n  3 3 3 3 (A) (B) (C) (D) 8 10 14 16 55. The family of curves y = ea sin x, where ‘a’ is arbitrary constant, is represented by the differential equation hœ²­lM¡ f¢lh¡l y = ea sin x, ‘a’ - kcªµR dËh¤ L, ­k AhLm pj£LlZ à¡l¡ pw‘¡a q­h ­p¢V qm dy dy (A) y log y = tan x (B) y log x = cot x dx dx dy dy (C) log y = tan x (D) log y = cot x dx dx 56. The locus of points (x, y) in the plane satisfying sin2 x + sin2 y = 1 consists of (A) a circle centered at origin (B) infinitely many circles that are all centered at the origin (C) infinitely many lines with slope  1 (D) finitely many lines with slope  1 sin2 x + sin2 y = 1 pj£LlZ­L ¢pÜ L­l, Hje (x, y) am¢ÙÛa pLm ¢h¾c¤l p’¡lfb qm (A) j§m ¢h¾c¤­a ­L¾cÊ Hje hªš (B) j§m ¢h¾c¤­a ­L¾cÊ Hje Ap£j pwMÉL hªš (C) e¢a  1 pð¢ma Ap£j pwMÉL ­lM¡l f¢lh¡l (D) e¢a  1 pð¢ma pp£j pwMÉL ­lM¡l f¢lh¡l A 21 P.T.O.  M-2023 57. If x = sin  and y = sin k, then (1 – x2)y2 – xy1 – y = 0, for α = x = sin  J y = sin k fËcš z (1 – x2)y2 – xy1 – y = 0 q­m, -Hl j¡e q­h (A) k (B) –k (C) –k2 (D) k2  1  58. In the interval (–2, 0), the function f(x) = sin  3 . x  (A) never changes sign. (B) changes sign only once. (C) changes sign more than once but finitely many times. (D) changes sign infinitely many times.  1  (–2, 0) A¿¹l¡­m A­frL qm f(x) = sin  3  z A­frL¢V x  (A) LMeC ¢Qq² f¢lhaÑe L­l e¡ (B) j¡œ HLh¡l ¢Qq² f¢lhaÑe L­l (C) HLh¡­ll ­Q­u ­h¢n ¢L¿¹¥ pp£j pwMÉL h¡l ¢Qq² f¢lhaÑe L­l (D) Ap£j pwMÉL h¡l ¢Qq² f¢lhaÑe L­l 2π  θ sin 6 59. θ cos θ dθ is equal to 0 2π  θ sin 6 θ cos θ dθ -Hl j¡e qm 0 π 3π 16π (A) (B) (C) (D) 0 16 16 3 A 22  M-2023 60. The average ordinate of y = sin x over [0, ] is [0, ] -Hl Jf®l y = sin x hœ²­lM¡l ­L¡¢V pj§­ql °c­OÑÉl Ns qm 2 3 (A) (B) π π 4 (C) (D)  π 2 2 2 61. The portion of the tangent to the curve x 3 + y 3 = a 3 , a > 0 at any point of it, intercepted between the axes (A) varies as abscissa (B) varies as ordinate (C) is constant (D) varies as the product of abscissa and ordinate 2 2 2 x 3 + y 3 = a 3 , a > 0 hœ²­lM¡l Ef¢lÙÛ ­L¡e ¢h¾c¤­a A¢ˆa ØfnÑ­Ll ­k Awn Arà­ul j­dÉ ­R¢ca qu, ­p¢V (A) i¥­Sl p­‰ plm­i­c B­R (B) ­L¡¢Vl p­‰ plm­i­c B­R (C) dˤhL (D) i¥S J ­L¡¢Vl …Zg­ml p­‰ plm­i­c B­R A 23 P.T.O.  M-2023 62. If the volume of the parallelopiped with a  b , b  c and c  a as coterminous edges is 9 cu. units, then the volume of the parallelopiped with ( a  b )  ( b  c ), ( b  c )  ( c  a ) and ( c  a )  ( a  b ) as coterminous edges is (A) 9 cu. units (B) 729 cu. units (C) 81 cu. units (D) 243 cu. units 9 Oe HLL Buae ¢h¢nø HL¢V Qa¥ÙÛm­Ll pjfË¡¢¿¹L h¡ý…¢m a  b , b  c Hhw c  a z ­k Qa¥ÙÛm­Ll pjfË¡¢¿¹L h¡ý…¢m, ( a  b )  ( b  c ), ( b  c )  ( c  a ) Hhw ( c  a )  ( a  b ) a¡l Buae q­h (A) 9 Oe HLL (B) 729 Oe HLL (C) 81 Oe HLL (D) 243 Oe HLL 63. Given f(x) = esin x + ecos x. The global maximum value of f(x) (A) does not exist. 1  π (B) exists at a point in  0,  and its value is 2e 2.  2 (C) exists at infinitely many points. (D) exists at x = 0 only. f(x) = esin x + ecos x A­fr­Ll global p­hÑ¡µQ j¡e (A) -Hl A¢Ù¹aÄ ­eC 1  π (B)  0,  -A¿¹l¡­m HL¢V ¢h¾c¤­a A¢Ù¹aÄ B­R Hhw a¡l j¡e qm 2e 2  2 (C) Ap£j pwMÉL ¢h¾c¤­a I p­hÑ¡µQ j¡e ¢hcÉj¡e (D) öd¤j¡œ x = 0 -­a A¢Ù¹aÄ B­R A 24  M-2023 64. Consider a quadratic equation ax2 + 2bx + c = 0 where a, b, c are positive real numbers. If the equation has no real root, then which of the following is true ? (A) a, b, c cannot be in A.P. or H.P. but can be in G.P. (B) a, b, c cannot be in G.P. or H.P. but can be in A.P. (C) a, b, c cannot be in A.P. or G.P. but can be in H.P. (D) a, b, c cannot be in A.P., G.P. or H.P. a, b, c- de¡aÈL h¡Ù¹h pwMÉ¡ q­m ax2 + 2bx + c = 0 ¢àO¡a pj£LlZ¢V ¢h­hQe¡ Ll z pj£LlZ¢Vl h£S…¢m h¡Ù¹h e¡ q­m, ¢ejÀ¢m¢Ma ¢hhª¢a…¢ml ­L¡e¢V paÉ : (A) a, b, c, A.P. h¡ H.P. ­a b¡L­h e¡ ¢L¿¹¥ G.P. ­a b¡L­a f¡­l (B) a, b, c, G.P. h¡ H.P. ­a b¡L­h e¡ ¢L¿¹¥ A.P. ­a b¡L­a f¡­l (C) a, b, c, A.P. h¡ G.P. ­a b¡L­h e¡ ¢L¿¹¥ H.P. ­a b¡L­a f¡­l (D) a, b, c, A.P., G.P. h¡ H.P. ­a ­eC 65. Let a1, a2, a3, …, an be positive real numbers. Then the minimum value of a1 a 2 a  ....  n is a 2 a3 a1 a1 a 2 a j­e Ll, a1, a2, a3, …, an de¡aÈL h¡Ù¹h pwMÉ¡ z ­p­r­œ  ....  n -Hl phÑ¢ejÀ j¡e q­h a 2 a3 a1 (A) 1 (B) n (C) nC (D) 2 2 A 25 P.T.O.  M-2023 Category-3 (Q. 66 to 75) (Carry 2 marks each. One or more options are correct. No negative marks) 66. Let f be a strictly decreasing function defined on such that f ( x ) > 0 ,  x . Let x2 y2  = 1 be an ellipse with major axis along the y-axis. The value of f(a 2  5a  3) f(a  15) ‘a’ can lie in the interval (s) x2 y2 f, -H kb¡bÑ œ²jqÊÊ¡pj¡e A­frL J f(x) > 0, x  z 2  = 1 Efhªš¢Vl f(a  5a  3) f(a  15) fl¡r y-Ar hl¡hl z ­p­r­œ ‘a’ ­kM¡­e b¡L­a f¡­l ­p¢V qm (A) (–, –6) (B) (–6, 2) (C) (2, ) (D) (–, ) 67. A rectangle ABCD has its side parallel to the line y = 2x and vertices A, B, D are on lines y = 1, x = 1 and x = –1 respectively. The coordinate of C can be Bua­rœ ABCD-Hl HL¢V h¡ý y = 2x-Hl pj¡¿¹l¡m Hhw n£oÑ¢h¾c¤œu A, B, D kb¡œ²­j y = 1, x = 1 J x = –1 -Hl Ef¢l¢ÙÛa z C-Hl ÙÛ¡e¡ˆ q­h (A) (3, 8) (B) (–3, 8) (C) (–3, –1) (D) (3, –1) 68. If R and R1 are equivalence relations on a set A, then so are the relations (A) R–1 (B) R  R1 (C) R  R1 (D) All of these ­pV A-­a R J R1 pja¥mÉ pðå pw‘¡a B­R z Ae¤l©f pÇfLÑ q­h (A) R–1 (B) R  R1 (C) R  R1 (D) ph L¢V A 26  M-2023 69. Let f(x) = xm, m being a non-negative integer. The value of m so that the equality f'(a + b) = f'(a) + f'(b) is valid for all a, b > 0 is f(x) = xm, m A-GZ¡aÈL f§ZÑpwMÉ¡ z f'(a + b) = f'(a) + f'(b) q­h, k¢c a, b > 0 qu (A) 0 (B) 1 (C) 2 (D) 3 70. Which of the following statements are true ? aT (A) If f(x) be continuous and periodic with periodicity T, then I =  f(x) dx depend on a ‘a’. aT (B) If f(x) be continuous and periodic with periodicity T, then I =  f(x) dx does not a depend on ‘a’.  1 , if x is rational (C) Let f(x) =   0 , if x is irrational, then f is periodic of the periodicity T only if T is rational. (D) f defined in (C) is periodic for all T. ¢ejÀ¢hhª¢a…¢ml ­L¡e¢V paÉ ? aT (A) k¢c f(x) p¿¹a Hhw T-fkÑ¡hª­šl HL¢V fkÑ¡hªš£u A­frL qu, a­h I =  f(x) dx, ‘a’-Hl a Efl ¢eiÑln£m q­h aT (B) k¢c f(x) p¿¹a Hhw T-fkÑ¡hª­šl HL¢V fkÑ¡hªš£u A­frL qu, a­h I =  f(x) dx, ‘a’-Hl a Efl ¢eiÑln£m eu  1 , x j§mc (C) j­e Ll, f(x) =  0 , x Aj§mc, f-fkÑ¡hªš T-Hl HL¢V fkÑ¡hªš£u A­frL q­h HLj¡œ k¢c T  j§mc qu (D) pLm T-Hl SeÉ (C) H h¢ZÑa f-fkÑ¡hªš A­frL q­h A 27 P.T.O.  M-2023 71. A balloon starting from rest is ascending from ground with uniform acceleration of 4 ft/sec2. At the end of 5 sec, a stone is dropped from it. If T be the time to reach the stone to the ground and H be the height of the balloon when the stone reaches the ground, then ¢ÙÛa¡hÙÛ¡ ­b­L k¡œ¡ L­l HL¢V ­hm¤e 4 ft/sec2 aÄl­Z EÜÑN¡j£ z 5 ­p­Lä f­l ­hm¤e¢V ­b­L HL¢V f¡bl ­R­s ­cJu¡ qm z k¢c f¡bl¢V T ­p­Lä f­l i¨¢j ØfnÑ L­l Hhw kMe f¡bl¢V i¨¢j ØfnÑ L­l aMe ­hm¤e¢Vl EµQa¡ H ft.qu, a­h (A) T = 6 sec (B) H = 112.5 ft (C) T = 5/2 sec (D) 225 ft 3 72. If f(x) = 3 x 2 – x2, then (A) f has no extrema. (B) f is maximum at two points x = 1 and x = –1. (C) f is minimum at x = 0. (D) f has maximum at x = 1 only. j­e Ll, f(x) = 3 3 x 2 – x2 ­p­r­œ (A) f -Hl Qlj j¡e ­eC (B) x = 1, x = –1 ¢h¾c¤­a f -Hl p­hÑ¡µQ j¡e B­R (C) x = 0 ¢h¾c¤­a f -Hl phÑ¢ejÀ j¡e ¢hcÉj¡e (D) öd¤j¡œ x = 1 ¢h¾c¤­a f -Hl p­hÑ¡µQ j¡e B­R x 0  π 73. Let f be a non-negative function defined on 0, . If  2  ( f ′(t) – sin 2t) dt =  f(t) tan tdt, 0 x π 2 f(0) = 1, then  f(x) dx is 0 j­e Ll, 0,  -­a pw‘¡a f A-GZ¡aÈL A­frL z π  2 π x 0 2  (f ′(t) – sin 2t)dt =  f(t) tan tdt, f(0) = 1 ­p­r­œ  f(x) dx q­h 0 x 0 π π π (A) 3 (B) 3– (C) 3+ (D) 2 2 2 A 28  M-2023 z1  z 2 z 74. If z1 and z2 are two complex numbers satisfying the equation = 1, then 1 may z1 – z 2 z2 be (A) real positive (B) real negative (C) zero (D) purely imaginary z1  z 2 z z1 J z2 c¤¢V S¢Vml¡¢n Hje ­k = 1, ­p­r­œ 1 q­h z1 – z 2 z2 (A) de¡aÈL h¡Ù¹h (B) GZ¡aÈL h¡Ù¹h (C) n§ZÉ (D) f¤­l¡f¤¢l L¡Òf¢eL 75. A letter lock consists of three rings with 15 different letters. If N denotes the number of ways in which it is possible to make unsuccessful attempts to open the lock, then (A) 482 divides N (B) N is the product of two distinct prime numbers. (C) N is the product of three distinct prime numbers. (D) 16 divides N. HL¢V Arl a¡m¡u ¢ae¢V hmu B­R z fË¢a¢V hm­u 15 ¢V L­l ¢h¢iæ Arl B­R z N k¢c a¡m¡ ­M¡m¡l ­Qø¡u Ap¡g­mÉl pwMÉ¡ qu a­h (A) N, 482 à¡l¡ ¢hi¡SÉ (B) N, c¤¢V Bm¡c¡ ­j±¢mL pwMÉ¡l …Zgm (C) N, ¢ae¢V Bm¡c¡ ­j±¢mL pwMÉ¡l …Zgm (D) N, 16 à¡l¡ ¢hi¡SÉ ____________ A 29 P.T.O.  M-2023 SPACE FOR ROUGH WORK A 30  M-2023 SPACE FOR ROUGH WORK A 31 P.T.O.  M-2023 M-2023 Subject : MATHEMATICS pju: 2 O¾V¡ f§ZÑj¡e : 100 ¢e­cÑn¡hm£ 1. HC fËnÀf­œ pjÙ¹ fËnC À Ah­S¢ƒi fËnÀ Hhw fË¢a¢V fË­nÀl Q¡l¢V pñ¡hÉ Ešl ­cJu¡ B­R z 2. Category-1 : HL¢V Ešl p¢WL z p¢WL Ešl ¢c­m 1 eðl f¡­h z i¥m Ešl ¢c­m Abh¡ ­k ­L¡e HL¡¢dL Ešl ¢c­m ¼ eðl L¡V¡ k¡­h z 3. Category-2 : HL¢V Ešl p¢WL z p¢WL Ešl ¢c­m 2 eðl f¡­h z i¥m Ešl ¢c­m Abh¡ ­k ­L¡e HL¡¢dL Ešl ¢c­m ½ eðl L¡V¡ k¡­h z 4. Category-3: HL h¡ HL¡¢dL Ešl p¢WL z ph L¢V p¢WL Ešl ¢c­m 2 eðl f¡­h z k¢c ­L¡e i¥m Ešl e¡ b¡­L Hhw p¢WL EšlJ ph L¢V e¡ b¡­L a¡q­m f¡­h 2  ­k L¢V p¢WL Ešl ­cJu¡ q­u­R a¡l pwMÉ¡  Bp­m ­k L¢V Ešl p¢WL a¡l pwMÉ¡ z k¢c ­L¡­e¡ i¥m Ešl ­cJu¡ qu h¡ HL¡¢dL Eš­ll j­dÉ HL¢VJ i¥m b¡­L a¡q­m Ešl¢V i¥m d­l ­eJu¡ q­h z ¢L¿¹¥ ­p­r­œ ­L¡­e¡ eðl L¡V¡ k¡­h e¡, AbÑ¡v n§eÉ eðl f¡­h z 5. OMR f­œ A, B, C, D ¢Q¢q²a p¢WL Ol¢V il¡V L­l Ešl ¢c­a q­h z 6. OMR f­œ Ešl ¢c­a öd¤j¡œ L¡­m¡ h¡ e£m L¡¢ml hm f­u¾V ­fe hÉhq¡l Ll­h z 7. OMR f­œ ¢e¢cÑø ÙÛ¡e R¡s¡ AeÉ ­L¡b¡J ­L¡e c¡N ­c­h e¡ z 8. OMR f­œ ¢e¢cÑø ÙÛ¡­e fËnf À ­œl eðl Hhw ¢e­Sl ­l¡m eðl A¢a p¡hd¡ea¡l p¡­b ¢mM­a q­h Hhw fË­u¡Se£u Ol…¢m f§lZ Ll­a q­h z 9. OMR f­œ ¢e¢cÑø ÙÛ¡­e ¢e­Sl e¡j J fl£r¡ ­L­¾cÊl e¡j ¢mM­a q­h Hhw ¢e­Sl (Admit Card-H E­õ¢Ma) pÄ¡rl Ll­a q­h z 10. fËnÀf­œl eðl h¡ ­l¡m eðl i¥m ¢mM­m Abh¡ i¥m Ol il¡V Ll­m, fl£r¡bÑ£l e¡j, fl£r¡ ­L­¾cÊl e¡j h¡ pÄ¡r­l ­L¡e i¥m b¡L­m Ešl fœ h¡¢am q­u ­k­a f¡­l z OMR fœ¢V i¡yS q­m h¡ a¡­a Ae¡hnÉL c¡N fs­mJ h¡¢am q­u ­k­a f¡­l z fl£r¡bÑ£l HC dl­el i¥m h¡ ApaÑLa¡l SeÉ Ešlfœ h¡¢am q­m HLj¡œ fl£r¡bÑ£ ¢e­SC a¡l SeÉ c¡u£ b¡L­h z 11. ­j¡h¡Cm­g¡e, LÉ¡mL¥­mVl, pÔ¡CXl¦m, mN­Vhm, q¡aO¢s, ­lM¡¢Qœ, NË¡g h¡ ­L¡e dl­Zl a¡¢mL¡ fl£r¡ L­r Be¡ k¡­h e¡ z Be­m ­p¢V h¡­Su¡ç q­h Hhw fl£r¡bÑ£l JC fl£r¡ h¡¢am Ll¡ q­h z 12. fËnÀf­œ l¡g L¡S Ll¡l SeÉ gy¡L¡ S¡uN¡ ­cJu¡ B­R z AeÉ ­L¡e L¡NS HC L¡­S hÉhq¡l Ll¡ k¡­h e¡ z 13. fl£r¡ Lr R¡s¡l B­N OMR fœ AhnÉ C f¢lcnÑL­L ¢c­u k¡­h z 14. HC fËnÀf­œ Cwl¡S£ J h¡wm¡ Eiu i¡o¡­aC fËnÀ ­cJu¡ B­R z h¡wm¡ j¡dÉ­j fËnÀ °al£l pju fË­u¡Se£u p¡hd¡ea¡ J paLÑa¡ Ahmðe Ll¡ q­u­R z a¡ p­šÄJ k¢c ­L¡e Ap‰¢a mrÉ Ll¡ k¡u, ­p­r­œ Cwl¡S£ j¡dÉ­j ­cJu¡ fËnÀ ¢WL J Q¨s¡¿¹ h­m ¢h­h¢Qa q­h z 15. fl£r¡ ®n­o fl£r¡b£Ñl¡ fËnÀfœ¢V ¢e­u k¡­h z A 32 