Question Booklet Version SUBJECT : PAPER I MATHEMATICS Instruction to Candidates. This question booklet contains 50 Objective Type Questions (Single Best Response Type) in the subject of Mathematics.. The question paper and OMR (Optical Mark Reader) Answer Sheet are issued to eaminees separately at the beginning of the eamination session. 3. Choice and sequence for attempting questions will be as per the convenience of the candidate. 4. Candidate should carefully read the instructions printed on the Question Booklet and Answer Sheet and make the correct entries on the Answer Sheet. As Answer Sheets are designed to suit the OPTICAL MARK READER (OMR) SYSTEM, special care should be taken to mark appropriate entries/answers correctly. Special care should be taken to fill QUESTION BOOKLET VERSION, SERIAL No. and Roll No. accurately. The correctness of entries has to be cross-checked by the invigilators. The candidate must sign on the Answer Sheet and Question Booklet. 5. Read each question carefully. 6. Determine the correct answer from out of the four available options given for each question. 7. Fill the appropriate circle completely like this, for answering the particular question, with Black ink ball point pen only, in the OMR Answer Sheet. 8. Each answer with correct response shall be awarded two () marks. There is no Negative Marking. If the eaminee has marked two or more answers or has done scratching and overwriting in the Answer Sheet in response to any question, or has marked the circles inappropriately e.g. half circle, dot, tick mark, cross etc, mark/s shall NOT be awarded for such answer/s, as these may not be read by the scanner. Answer sheet of each candidate will be evaluated by computerized scanning method only (Optical Mark Reader) and there will not be any manual checking during evaluation or verification. 9. Use of whitener or any other material to erase/hide the circle once filled is not permitted. Avoid overwriting and/or striking of answers once marked. 0. Rough work should be done only on the blank space provided in the Question Booklet. Rough work should not be done on the Answer Sheet.. The required mathematical tables (Log etc.) are provided within the question booklet.. Immediately after the prescribed eamination time is over, the Answer Sheet is to be returned to the Invigilator. Confirm that both the Candidate and Invigilator have signed on question booklet and answer sheet. 3. No candidate is allowed to leave the eamination hall till the eamination session is over. (Pg. )
MHT-CET - 08 : Mathematics Paper and Solution ().Questions and Solutions.. If K 0, then the value of K is 8 4 (A) 3 (B) 4 (C) 3 (D) 4. (C) k 0 8 4 k 4 8 8 9 3 0 0 tan 8 3 30 3 k tan 3 tan 3k tan 0 8 0 6 tan 3k 0 6 6 tan 3k tan 3k k 4 4 3 k k. The cartesian coordinates of the point on the parabola y 6, whose parameter is, are (A) (, 4) (B) (4, ) (C) (, 4) (D) (, 4). (D) y 6 y 4a a 4 Parametric equations are at, y at i.e. 4, y 4 i.e., y 4 3. sin.cos (A) sec + log sec + tan + c (C) sec + log sec tan + c (B) sec. tan + c (D) sec + log cosec cot + c 3. (D) sin cos sin cos sin cos sin tan sec cosec cos sin sec log cosec cot c (Pg. )
(3) VIDYALANKAR : MHT-CET - 08 : Mathematics Paper and Solution 4. If log 0 (A) y 3 3 y 3 3 y then dy (B) y (C) (D) y y 4. (D) 3 3 y log0 3 3 y log ( 3 y 3 ) log ( 3 + y 3 ) dy dy 3 3y 3 3y 3 3 3 3 y y 3 3y dy 3 3y dy 3 3 3 3 3 3 3 3 y y y y 3 3 3y 3y dy 3 3 3 3 3 3 3 3 y y y y 3 3y dy 3 3 3 3 3 3 3 3 y y y y 3 3 y dy 3 3y 3 3 3 3 3 3 3 3 y y y y y dy Alternative Method 3 3 y log0 3 3 y 3 3 y 0 00 3 3 y 3 y 3 00 3 + 00 y 3 99 3 0 y 3.. () 99 (3 ) 0 3y dy dy 99 0y 3 3 dy y y y 4 5. If f : R {} R is a function defined by f(), then its range is (A) R (B) R {} (C) R {4} (D) R {, } 5. (A) 4 f Range is R (Pg. 3)
6. If f() + for 0 + for < 0 is continuous at 0 and f then + is (A) 3 6. (C) (B) 8 5 MHT-CET - 08 : Mathematics Paper and Solution (4) (C) 5 8 f, if 0 continuous at 0, if 0 f (0) 0 + + 7 f 4 4 7 4 4 7 49 50 5 4 4 6 6 8 7. If y tan d y then (D) 3 dy (A) 4 (B) (C) (D) 0 7. (B) y (tan ) dy tan dy tan y dy 4y dy dy d y dy 4 dy dy d y dy 4 4 dy d y 4 4 d y dy 8. The line 5 + y 0 coincides with one of the lines given by 5 + y k y + 0 then the value of k is (A) (B) 3 (C) (D) 3 8. (C) 5 + y 0 coincides 5 + y k y + 0 k a 5, b 0, h, g, f, c (Pg. 4)
(5) VIDYALANKAR : MHT-CET - 08 : Mathematics Paper and Solution 5 k k 0 0 k k 5 0 0 0 k k k 5 0 k 4 4 3 9. If A then (A 5A)A 4 4 3 4 3 (A) 4 (B) 4 (C) 4 4 3 (D) 4 3 4 9. (B) (A 5A) A A A 5AA A. AA 5I A 5I 3 5 0 0 0 5 0 4 0 0 5 4 3 4 0. The equation of line passing through (3,, ) and perpendicular to the lines r ˆ i ˆ j k ˆ i ˆ j ˆ k ˆ r i ˆˆ j kˆ ˆi j ˆ kˆ is and (A) 3 y z (B) 3 y z 3 3 (C) 3 y z (D) 3 y z 3 3 0. (C) Let a, b, c be d.rs of desired line. a b + c 0 a b + c 0 a b c a b c a, b 3, c 4 4 4 Hence equation of desired line is 3 y z i.e. 3 y z 3 3 (Pg. 5)
MHT-CET - 08 : Mathematics Paper and Solution (6). Letters in the word HULULULU are rearranged. The probability of all three L being together is 3 (A) (B) 3 (C) (D) 5 0 5 8 3. (C) HULULULU contains 4U, 3L, H Consider 3L together i.e. we have to arrange 6 units which contains 4U. Hence number of possible arrangements 6! 6 5 30 4! Number of ways of arranging all letters of given word Hence required probability 30 6 3 875 87 8 8! 8765 8 7 5 3!4! 3. The sum of the first 0 terms of the series 9 + 99 + 999 +., is 9 0 (A) 9 (B) 00 0 9 (C) 0 9 (D) 8 9 00 (0 0 ) 9. (B) 9 + 99 + 999 +.. 0 terms (0 ) + (00 ) + (000 ) +.. 0 terms (0 + 00 + 000 +.. 0 terms) ( + +.. 0 times) 00 0 0 0 0 0 0 0 0 90 0 9 9 0 0 00 0 0 0 9 9 9 00 0 9 3. If A, B, C are the angles of ABC then cot A. cot B + cot B. cot C + cot C. cot A (A) 0 (B) (C) (D) 3. (B) We know that tan A + tan B + tan C tan A tan B tan C tan BtanC tan AtanC tan Atan B cot B cot C + cot A cot C + cot A cot B 4. If A sin (B) + C then A + B 6 9 (A) 9 4 (B) 9 4 (C) 3 4 (D) 3 4. (D) 6 9 3 6 3 4 9 3 (Pg. 6)
(7) VIDYALANKAR : MHT-CET - 08 : Mathematics Paper and Solution 3 sin c A and B 3 4 3 4 3 A + B 3 3 3 4 5. sin e cos (A) e tan + c (B) e + tan + c (C) e tan + c (D) e tan + c 5. (A) sin sin cos e e cos cos e sec tan e tan c 6. A coin is tossed three times. If X denotes the absolute difference between the number of heads and the number of tails then P(X ) (A) (B) 3 (C) 6 (D) 3 4 6. (D) A coin is tossed 3 times. Possibilities are HHH X 3 0 3 TTT X 3 0 3 HHT X HTH X THH X HTT X TTH X THT X P (X ) 6 3 8 4 7. If sin 3 cos, then tan 6 (A) 3 (B) (C) 3 3 (D) 3 7. (D) sin cos 3 6 sin cos cos sin cos cos sin sin 3 3 6 6 sin 3 3 cos cos sin 3 sin 3 cos cos sin 3 sin cos tan 3 (Pg. 7)
MHT-CET - 08 : Mathematics Paper and Solution (8) 8. The area of the region bounded by 4y, y, y 4 and the yais lying in the first quadrant is square units. (A) 3 (B) 8 3 (C) 30 (D) 4 8. (B) We have 4y y 4 3 y A ydy 3 4 4 8 3 3 3 y y 8 4 Y 4y y 4 y X 9. If f() e cos, for 0 is continuous at 0, then value of f(0) is (A) 3 (B) 5 (C) (D) 3 9. (D) e cos f e cos f 0 lim 0 sin e lim 0 lim 0 sin 3 lim 0 4 4 0. The maimum value of + y subject to 3 + 5y 6 and 5 + 3y 30, 0, y 0 is (A) (B).5 (C) 0 (D) 7.33 0. (A) Y 6 0, 5 0 8 6 4 (0, 0) 9 5, X 4 6 8 0 3 + 5y 6 (6, 0) 5 + 3y 30 (Pg. 8)
(9) VIDYALANKAR : MHT-CET - 08 : Mathematics Paper and Solution Corner points Value of z + y (0, 0) z 0 (6, 0) z (6) + 0 9 5 9 5, z.5 6 0, 5 z (0) + 6 5. 5. If a, b, c are mutually perpendicular vectors having magnitudes,, 3 respectively, then [a bc ba c]? (A) 0 (B) 6 (C) (D) 8. (C) a, b, c 3 a b b ca c 0 a b cb ac a b c (b a c a b cb ca c abc bac abc a b c a b c cos0 a b c a b c sin90 () () (3). If points P(4, 5, ), Q(3, y, 4) and R(5, 8, 0) are collinear, then the value of + y is (A) 4 (B) 3 (C) 5 (D) 4. (D) PQ (, y 5, 4 ) QR (, 8 y, 4) P, Q, R are collinear y 5 4 8 y 4 8 + y y 0; 4 (4 ) y ; 4 + y 4 3. If the slope of one of the lines given by a + hy + by 0 is two times the other then (A) 8h 9ab (B) 8h 9ab (C) 8h 9ab (D) 8h 9ab 3. (A) a + hy + by 0 m m If slope of one line is k times the other then (Pg. 9)
4kh ab( + k) Here k 4() h ab( + ) 8h 9ab MHT-CET - 08 : Mathematics Paper and Solution (0) 4. The equation of the line passing through the point (3, ) and bisecting the angle between coordinate aes is (A) + y + 0 (B) + y + 0 (C) y + 4 0 (D) + y + 5 0 4. Question is Wrong. 5. The negation of the statement: Getting above 95% marks is necessary condition for Hema to get the admission is good college (A) Hema gets above 95% marks but she does not get the admission in good college (B) Hema does not get above 95% marks and she gets admission in good college (C) If Hema does not get above 95% marks then she will not get the admission in good college. (D) Hema does not get above 95% marks or she gets the admission in good college. 5. (B) p : Hema gets the admission in good college q : Hema gets 95 % marks Given statement can be written in symbolic form as p q Its negation is p ~ q 6. Cos Cos Cos3 Cos79 (A) 0 (B) (C) (D) 6. (A) cos cos cos 3.. cos 79 0 (As cos 90 0 product 0) 7. If planes cy bz 0, c y + az 0 and b + ay z 0 pass through a straight line then a + b + c (A) abc (B) abc (C) abc (D) abc 7. (C) Planes cy bz 0 c y + az 0 b + ay z 0 Planes are concurrent c b c a 0 b a ( a ) + c ( c ab) b (ac + b) 0 a c abc abc b 0 a + b + c + abc a + b + c abc 8. The point of intersection of lines represented by y + + 3y 0 is 3 (A) (, 0) (B) (0, ) (C), (D), (Pg. 0)
() VIDYALANKAR : MHT-CET - 08 : Mathematics Paper and Solution 8. (C) y + + 3y 0 3 a, h 0, b, g, f, c hf bg gh af Point of intersection, ab h ab h 3 0 0 3,, OR by partial differentiation w.r.t. + 0 by partial differentiation w.r.t. y 3 y + 3 0 y 3 Point of intersection, 9. A die is rolled. If X denotes the number of positive divisors of the outcome then the range of the random variable X is (A) {,, 3} (B) {,, 3, 4} (C) {,, 3, 4, 5, 6} (D) {, 3, 5} 9. (B) When we get, number of positive divisors are When we get, number of positive divisors are When we get 3, number of positive divisors are When we get 4, number of positive divisors are 3 When we get 5, number of positive divisors are When we get 6, number of positive divisors are 4 Hence range of r.v. X is {,, 3, 4} 30. A die is thrown four times. The probability of getting perfect square in at least one throw is (A) 6 (B) 65 (C) 3 (D) 58 8 8 8 8 30. (B) P (getting perfect square in atleast one throw) P (not getting perfect square in any throw) 4 4 4 4 6 6 6 6 4 6 65 3 8 8 3. /4 sec? 0 (A) 4 + log (B) 4 log (C) + log (D) log 3. (B) /4 sec 0 /4 /4 d sec 0 0 sec (Pg. )
MHT-CET - 08 : Mathematics Paper and Solution () /4 0 tan 0 /4 /4 log sec 0 0 /4 tan tan 0 log sec log sec0 4 4 log 4 log log 4 C 3. In ABC, with usual notations, if a, b, c are in A.P. then a cos + c A cos? (A) 3 a (B) 3 c (C) 3 b (D) 3abc 3. (C) a, b, c are in A.P. b a + c C A a cos ccos cosc cos A a c a c a cosc ccos A a c b.[ b a cosc + c cos A] b b 3b 33. If e (sin cos ), y e (sin + cos ) then dy at is 4 (A) (B) 0 (C) (D) 33. (A) e sin cos, y e sin cos e cos sin sin cose d dy e cos sin sin cose d dy dy / d e [cos ] / d e [sin ] dy cot 4 dy cot 4 (Pg. )
(3) VIDYALANKAR : MHT-CET - 08 : Mathematics Paper and Solution 34. The number of solutions of sin + sin 3 + sin 5 0 in the interval,3 is (A) (B) 3 (C) 4 (D) 5 34. (B) sin + sin 3 + sin 5 0 sin 5 + sin + sin 3 0 sin 3. cos + sin 3 0 sin3 [ cos +] 0 sin3 0 or cos + 0 sin 3 sin n or cos 3 n or cos / n cos cos/3 3 cos cos( /3) n, n 3 3 3, gives 80, 0, 40, 4, 3 3 35. If tan + tan 3 4, then cos cos 3 n 3 35. (C) (A) (B) 3 tan - + tan 3 4 3 tan 6 4 5 tan 6 4 5 6 5 6 6 + 5 0 6 + 6 0 6( + ) ( + ) 0 ( + ) (6 ) 0 (C) 6 (D), 6 When, given equation is satisfied. 6 When, we get sum of two negative angles which cannot be equal to positive angle. 6 (Pg. 3)
MHT-CET - 08 : Mathematics Paper and Solution (4) 3 36. Matri A 5 then the value of a 3 A 3 + a 3 A 3 + a 33 A 33 is 4 7 (A) (B) 3 (C) (D) 3 36. (C) 3 A 5 4 7 a 3 A 3 + a 3 A 3 + a 33 A 33 A +(7 0) (7 0) + 3(4 ) 3 + 6 + 6 37. The contrapositive of the statement: If the weather is fine then my friends will come and we go for a picnic. (A) The weather is fine but my friends will not come or we do not go for a picnic (B) If my friends do not come or we do not go for picnic then weather will not be fine (C) If the weather is not fine then my friends will not come or we do not go for a picnic (D) The weather is not fine byt my friends will come and we go for a picnic 37. (B) p The weather is fine q My friends will come and we go for a picnic. Given statement p q Contrapositive ~q ~p i.e. If my friends do not come or we do not go for picnic then weather will not be fine. 38. If f() is increasing function then the value of lies in (A) R (B) (, ) (C) (, ) (D) (, ) 38. (D) f () ( )() ()() f () 0 ( ) f () 0 ( ) Here + 0, > 0, < (, ) 39. If X {4 n 3n : n N} and Y {9 (n ) : n N}, then X Y (A) X (B) Y (C) (D) {0} 39. (A) X 4 n 3n n N Y 9(n ) n N X {0, 9, 54, 43,. } Y {0, 9, 8, 7, 36, 45, 54,.. } X Y X (Pg. 4)
(5) VIDYALANKAR : MHT-CET - 08 : Mathematics Paper and Solution 40. The statement pattern p (~ p q) is (A) a tautology (C) equivalent to p q (B) a contradiction (D) equivalent to p q 40. (B) p ( p q) (p p) q.. (Associative law) F q.. (Compliment law) F.. (Identity law) 4. If the line y 4 5 touches to the curve y a 3 + b at the point (, 3) then 7a + b (A) 0 (B) (C) (D) 4. (A) line y 4 5 slope of line m 4 curve y a 3 + b Differentiating w.r.t. y dy 3a dy 3a slope of tangent y (,3) (i) dy 3a 4 a (ii) 3 from (i) and (ii) 4 a a y a 3 + b at (, 3) 9 8 + b b 9 6 7 b 7 7a + b 7 + (7) 0 4. The sides of a rectangle are given by a and y b. The equation of the circle passing through the vertices of the rectangle is (A) + y a (B) + y a + b (C) + y a b (D) ( a) + (y b) a + b 4. (B) a Y a O b a y b X y b Centre (0, 0) r equation of circle + y a + b a b (Pg. 5)
MHT-CET - 08 : Mathematics Paper and Solution (6) 43. The minimum value of the function f() log is (A) (B) e (C) (D) e e e 43. (A) f () log f () + log f () 0 + log 0 log e min value f e.log e e (log log e) e (0 ) e e 44. If X ~ B (n, p) with n 0, p 0.4 the E (X )? (A) 4 (B).4 (C) 3.6 (D) 8.4 44. (D) n 0, p 0.4, q 0.6 E() np 4 V() npq 0(0.4) (0.6).4 V() E( ) [E()].4 E( ) (4) E( ) 8.4 45. The general solution of differential equation dy cos ( + y) is y (A) tan y + c (B) tan y + c y (C) cot y + c (D) cot y + c 45. (A) cos ( + y) dy dy cos ( y) Put + y V Differentiating w.r.t. + dy dv dy dv dv cosv (Pg. 6)
(7) VIDYALANKAR : MHT-CET - 08 : Mathematics Paper and Solution dv cosv dv cos V cosv cosv dv ( cosv) ( cosv) dv cos V V dv cos V tan V + C y + y tan + C y tan y + C.. [C C ] 46. If planes r pi ˆ ˆj kˆ + 3 0 and r i ˆ pj ˆ kˆ 46. (D) 5 0 include angle 3 then the value of p is (A), 3 (B), 3 (C) 3 (D) 3 We have ˆ ˆ ˆ ˆ ˆ ˆ cos r. pi j k 3 0 and r. i pj k 5 0 include angle 3 n.n n. n pi ˆ ˆ j k ˆ. i ˆ pj ˆ k ˆ cos 3 p p p p 3p p 5 p 5 p 5 p + 5 6p 4 p 6p + 9 0 (p 3) 0 p 3 47. The order of the differential equation of all parabolas, whose latus rectum is 4a and ais parallel to the ais, is (A) one (B) four (C) three (D) two 47. (D) Equation of parabola whose ais is parallel to X ais and latus rectum is 4a (y k) 4a( h) h & k are arbitrary constants order. 48. If lines y z and 3 y k z intersect then the value of k is 3 4 (A) 9 (B) (C) 5 (D) 7 (Pg. 7)
MHT-CET - 08 : Mathematics Paper and Solution (8) 48. (A) Points on the line are (,, ) and (3, k, 0) and direction ratios of lines are, 3, 4 and,, Since lines intersect, then lines are coplanar y y z z a b c a b c k 3 4 0 0 (5) (k + ) () () 0 + k + 0 k 9 49. If a line makes angles 0 and 60 with the positive directions of X and Z aes respectively then the angle made by the line with positive Yais is (A) 50 (B) 60 (C) 35 (D) 0 49. (C) cos + cos + cos (cos 0) + cos + (cos 60) cos cos cos 35 50. L and M are two points with position vectors aband ab respectively. The position vector of the point N which divides the line segment LM in the ratio : eternally is (A) 3b (B) 4b (C) 5b (D) 3a 4 b 50. (C) We have L (, ) and M (, ) and is divided by N in ratio : eternally. N, i.e. 5 N 0, i.e. N 0,5i.e. 5b (Pg. 8)
(9) VIDYALANKAR : MHT-CET - 08 : Mathematics Paper and Solution (Pg. 9)
MHT-CET - 08 : Mathematics Paper and Solution (0) (Pg. 0)
() VIDYALANKAR : MHT-CET - 08 : Mathematics Paper and Solution (Pg. )
MHT-CET - 08 : Mathematics Paper and Solution () (Pg. )