JEE MAIN 2016 ONLINE EXAMINATION DATE : SUBJECT : MATHEMATICS TEST PAPER WITH SOLUTIONS & ANSWER KEY

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1 JEE MAIN 06 ONLINE EXAMINATION DATE : SUBJECT : MATHEMATICS TEST PAPER WITH SOLUTIONS & ANSWER KEY CORPORATE OFFICE : CG TOWER, A-6 & 5, IPIA, NEAR CITY MALL, JHALAWAR ROAD, KOTA (RAJ.) REG. OFFICE : J-, JAWAHAR NAGAR, MAIN ROAD, KOTA (RAJ.)-005 PH. NO.: FAX NO. : PH.NO. : , TO KNOW MORE : SMS RESO AT WEBSITE : CONTACT@RESONANCE.AC.IN CIN : U800RJ007PLC009 THIS SOLUTION WAS DOWNLOAD FROM RESONANCE JEE Main 06 SOLUTION PORTAL

2 PAPER- (B.E./B. TECH.) OF ONLINE JEE (MAIN) CBT Let C be a curve given by y() +,. If P is a point on C, such that the tangent at P has slope, then a point through which the normal at P passes, is () (, ) () (, 7) () (, ) () (, ) Ans. () y() +, > Let P(, + be the point. at which dy d ATP 9 Hence P(,) slope of normal at P(,) is equation of normal Y (X ) y y 7 clearly it is passes through (,7). Let a,b R, (a 0). If the function f defined as Ans. (), 0 a f() a, b b, () f(), (), (), (), a a b b is continuous in [0,) 0 CORPORATE OFFICE : CG TOWER, A-6 & 5, IPIA, NEAR CITY MALL, JHALAWAR ROAD, KOTA (RAJ.) REG. OFFICE : J-, JAWAHAR NAGAR, MAIN ROAD, KOTA (RAJ.)-005 PH. NO.: FAX NO. : PH.NO. : , TO KNOW MORE : SMS RESO AT WEBSITE : CONTACT@RESONANCE.AC.IN CIN : U800RJ007PLC009 THIS SOLUTION WAS DOWNLOAD FROM RESONANCE JEE Main 06 SOLUTION PORTAL PAGE #

3 PAPER- (B.E./B. TECH.) OF ONLINE JEE (MAIN) CBT continuous at and Lim f() Lim f() f() a a a () and a Lim f() b b b b a Lim f() f If a then b b 0 b ± If a then b b + 0 b is imaginary which is not possible (a, b), or,. Let a, a, a,..., a n,...be in A.P. If a + a 7 + a + a 5 7 then the sum of its first 7 terms is equal to () 5 () 06 () 6 () 0 Ans. () a, a, a... a n... are in A.P. a + a 5 a 7 + a a + a 7 6 sum of first 7 term 7 (a + a 7 ) If A > 0, B > 0 and A + B 6, then the minimum value of tana + tanb is () () Ans. () A, B > 0 and A + B 6 () () Let y tana + tanb dy da sec A sec A 6 Hence tana + tanb A, 6 CORPORATE OFFICE : CG TOWER, A-6 & 5, IPIA, NEAR CITY MALL, JHALAWAR ROAD, KOTA (RAJ.) REG. OFFICE : J-, JAWAHAR NAGAR, MAIN ROAD, KOTA (RAJ.)-005 PH. NO.: FAX NO. : PH.NO. : , TO KNOW MORE : SMS RESO AT WEBSITE : CONTACT@RESONANCE.AC.IN CIN : U800RJ007PLC009 THIS SOLUTION WAS DOWNLOAD FROM RESONANCE JEE Main 06 SOLUTION PORTAL PAGE #

4 PAPER- (B.E./B. TECH.) OF ONLINE JEE (MAIN) CBT and tana + tanb A 0, clearly tana + tanb is minimum when A B y min tan 5. The contrapositive of the following statement, " If the side of a square doubles, then its area increases four times", is () If the area of a square does not increase four times, then its side is not doubled. () If the area of a square increases four times, then its side is not doubled. () If the area of a square increases four times, then its side is doubled. () If the side of a square is not doubled, then its area does not increase four times. Ans. () p The side of a square doubles q Area of square increases four time so the contrapositive of p q is ~ q ~ P 6. Let A be a matri such that A 5A Statement : A (5 A). 7 Statement : The polynomial A A A + can be reduced to 5(A ). Then () Statement- is false, but Statement- is true. () Both the statements are false. () Both the statements are true. () Statement- is true, but Statement- is false. Ans. () A 5A A ± 0 A 5 7A A (5 A) 7 Hence statement is true Now A A A + A(A ) A A + A(5A 7) A A + A 0A + 5A 0 ((5A 7) 0A + I 5(A ) Statement also correct CORPORATE OFFICE : CG TOWER, A-6 & 5, IPIA, NEAR CITY MALL, JHALAWAR ROAD, KOTA (RAJ.) REG. OFFICE : J-, JAWAHAR NAGAR, MAIN ROAD, KOTA (RAJ.)-005 PH. NO.: FAX NO. : PH.NO. : , TO KNOW MORE : SMS RESO AT WEBSITE : CONTACT@RESONANCE.AC.IN CIN : U800RJ007PLC009 THIS SOLUTION WAS DOWNLOAD FROM RESONANCE JEE Main 06 SOLUTION PORTAL PAGE #

5 PAPER- (B.E./B. TECH.) OF ONLINE JEE (MAIN) CBT Equation of the tangent to the circle, at the point (, ), whose centre is the point of intersection of the Ans. () straight lines y and + y is () y 0 () + y + 0 () y 0 () + y 0 Centre of circle is, equation of circle is y y y y 8 y 0 + y 8 y 0 Equation of tangent at (, ) is y ( + ) (y ) 0 y 0 + y The sum (r )(r!) is equal to r () 0 (!) () 0 (0!) () (!) () (!) Ans. () 0 (r ).r! r 0 r (r ) r r! 0 0 (r )(r )! r.r! r r 0 0 {(r )(r )! r(r!)} r.r! r r (.! ) 0 r ((r )! r!) (.! (!!) 0.! CORPORATE OFFICE : CG TOWER, A-6 & 5, IPIA, NEAR CITY MALL, JHALAWAR ROAD, KOTA (RAJ.) REG. OFFICE : J-, JAWAHAR NAGAR, MAIN ROAD, KOTA (RAJ.)-005 PH. NO.: FAX NO. : PH.NO. : , TO KNOW MORE : SMS RESO AT WEBSITE : CONTACT@RESONANCE.AC.IN CIN : U800RJ007PLC009 THIS SOLUTION WAS DOWNLOAD FROM RESONANCE JEE Main 06 SOLUTION PORTAL PAGE # 5

6 PAPER- (B.E./B. TECH.) OF ONLINE JEE (MAIN) CBT Let ABC be a triangle whose circumcentre is at P. If the position vectors of A,B,C and P are a, b, c and a b c respectively, then the position vector of the orthocentre of this triangle, is () 0 () a b c a b c () a b c () Ans. () Position vector of the centroid of ABC is a b c Now we known that centroid divides the line joining orthocentre to circum centre divided by centriod divided by centroid in the ratio in : orthocentre a b c a b c a b c 0. Let f() sin + cos. Then f is an increasing function in the interval 5 (), (), 8 () 0, Ans. () f() sin + cos f'() sin cos cos sin sincos(sin cos ) sin. cos sin > 0 sin < 0 < < < < () 5, 8. Let z + ai be a comple number, a > 0 such that z is a real number. Then the sum + z + z z is equal to () 50 i () 50 i () 65 i () 65 i Ans. () z + ai, a > 0 z a + (a a )i is a real number a a 0 a a, a > 0 z + i cos isin z Now + z + z z z CORPORATE OFFICE : CG TOWER, A-6 & 5, IPIA, NEAR CITY MALL, JHALAWAR ROAD, KOTA (RAJ.) REG. OFFICE : J-, JAWAHAR NAGAR, MAIN ROAD, KOTA (RAJ.)-005 PH. NO.: FAX NO. : PH.NO. : , TO KNOW MORE : SMS RESO AT WEBSITE : CONTACT@RESONANCE.AC.IN CIN : U800RJ007PLC009 THIS SOLUTION WAS DOWNLOAD FROM RESONANCE JEE Main 06 SOLUTION PORTAL PAGE # 6 cos isin i i 095 i 65 i

7 PAPER- (B.E./B. TECH.) OF ONLINE JEE (MAIN) CBT Let P { : sin cos cos} and Q { : sin+ cos sin} be two sets. Then () Q P () P Q () P Q and Q P () P Q Ans. () For Let P sin cos( + ) ( ) sin cos... (i) For Let Q cos ( ) sin... (ii) (i) & (ii) are same P Q. The mean of 5 observations is 5 and their variance is. If three of the observations are, and 6, then the mean deviation from the mean of the data is ().5 ().8 ().6 (). Ans. (Bonus) This question is wrong i i 5..(i) Also 5 by (i) Now (... ) (ii) + 6.(iii) (iv) Ans 5 CORPORATE OFFICE : CG TOWER, A-6 & 5, IPIA, NEAR CITY MALL, JHALAWAR ROAD, KOTA (RAJ.) REG. OFFICE : J-, JAWAHAR NAGAR, MAIN ROAD, KOTA (RAJ.)-005 PH. NO.: FAX NO. : PH.NO. : , TO KNOW MORE : SMS RESO AT WEBSITE : CONTACT@RESONANCE.AC.IN CIN : U800RJ007PLC009 THIS SOLUTION WAS DOWNLOAD FROM RESONANCE JEE Main 06 SOLUTION PORTAL PAGE # 7

8 PAPER- (B.E./B. TECH.) OF ONLINE JEE (MAIN) CBT The number of distinct real values of for which the lines y z and are coplanar is () () () () Ans. () y z 0 0 (0) + ( ) 0 [ ] 0 0 ± 5. The angle of elevation of the top of a vertical tower from a point A, due east of it is 5º. The angle of elevation of the top of the same tower from a point B, due south of A is 0º. If the distance between A and B is 5 m, then the height of the tower (in metres), is () 5 () 08 () 5 () 6 Ans. () Let height of tower is h. h O 5º 0º A E B OA G OB h Also OB OA + AB h h + (5 ) h 5 6. ( cos) lim tan tan 0 is Ans. () () () () () CORPORATE OFFICE : CG TOWER, A-6 & 5, IPIA, NEAR CITY MALL, JHALAWAR ROAD, KOTA (RAJ.) REG. OFFICE : J-, JAWAHAR NAGAR, MAIN ROAD, KOTA (RAJ.)-005 PH. NO.: FAX NO. : PH.NO. : , TO KNOW MORE : SMS RESO AT WEBSITE : CONTACT@RESONANCE.AC.IN CIN : U800RJ007PLC009 THIS SOLUTION WAS DOWNLOAD FROM RESONANCE JEE Main 06 SOLUTION PORTAL PAGE # 8

9 PAPER- (B.E./B. TECH.) OF ONLINE JEE (MAIN) CBT (sin ) sin 8 dy y tan 7. The solution of the differential equation sec, where 0 < and y(0), is given by d y Ans. () () y () y + sec tan sec tan y dy d + y sec tan put y t y dy d dt d dt d + t sec tan sec d n(sec tan ) I.F. e e sec + tan t.(sec + tan) (sec tan ) tan d sec tan d y (sec + tan) sec + tan + c y(0) c 0 y sec tan tan d () y + () y sec tan sec tan 8. P and Q are two distinct points on the parabola, y, with parameters t and t respectively. If the normal at p passes through Q, then the minimum value of t is () () 6 () 8 () Ans. () CORPORATE OFFICE : CG TOWER, A-6 & 5, IPIA, NEAR CITY MALL, JHALAWAR ROAD, KOTA (RAJ.) REG. OFFICE : J-, JAWAHAR NAGAR, MAIN ROAD, KOTA (RAJ.)-005 PH. NO.: FAX NO. : PH.NO. : , TO KNOW MORE : SMS RESO AT WEBSITE : CONTACT@RESONANCE.AC.IN CIN : U800RJ007PLC009 THIS SOLUTION WAS DOWNLOAD FROM RESONANCE JEE Main 06 SOLUTION PORTAL PAGE # 9

10 PAPER- (B.E./B. TECH.) OF ONLINE JEE (MAIN) CBT P(t, t) Q(t, t ) t t t t t + t + min of t 8 9. A hyperbola whose transverse ais is along the major ais of the conic, y and has vertices at Ans. () the foci of this conic. If the eccentricity of the hyperbola is, then which of the following points does NOT lie on it? () 5, () 5, () (0, ) () 0, (0, ) (0, ) y ellipse 6 foci (0, ± be) e e 6 for hyperbola (0, ) h H (0, ) CORPORATE OFFICE : CG TOWER, A-6 & 5, IPIA, NEAR CITY MALL, JHALAWAR ROAD, KOTA (RAJ.) REG. OFFICE : J-, JAWAHAR NAGAR, MAIN ROAD, KOTA (RAJ.)-005 PH. NO.: FAX NO. : PH.NO. : , TO KNOW MORE : SMS RESO AT WEBSITE : CONTACT@RESONANCE.AC.IN CIN : U800RJ007PLC009 THIS SOLUTION WAS DOWNLOAD FROM RESONANCE JEE Main 06 SOLUTION PORTAL PAGE # 0

11 PAPER- (B.E./B. TECH.) OF ONLINE JEE (MAIN) CBT e H equation a y b e H a b a 5 b a 5 y 5 Clearly point (5, 9 a b ) does not lies on it. 0. For R, 0, if y() is a differentiable function such that y(t) dt( ) ty(t) dt, then y() equals (where C is a constant) () Ans. () C e () C e () C e () C e y(t)dt ( + ) ty(t)dt differentiate equation ()...(i) y() + y(t)dt ( + ) y() + ty(t)dt y(t)dt y() + ty(t)dt again differentiate y() y() + y() + y() y y + dy d y( ) dy d d dy y CORPORATE OFFICE : CG TOWER, A-6 & 5, IPIA, NEAR CITY MALL, JHALAWAR ROAD, KOTA (RAJ.) REG. OFFICE : J-, JAWAHAR NAGAR, MAIN ROAD, KOTA (RAJ.)-005 PH. NO.: FAX NO. : PH.NO. : , TO KNOW MORE : SMS RESO AT WEBSITE : CONTACT@RESONANCE.AC.IN CIN : U800RJ007PLC009 THIS SOLUTION WAS DOWNLOAD FROM RESONANCE JEE Main 06 SOLUTION PORTAL PAGE #

12 solve differential equation PAPER- (B.E./B. TECH.) OF ONLINE JEE (MAIN) CBT n ny + nc n y + nc yc e / c y e /. ABC is a triangle in a plane with vertices A(,,5), B(,,) and C(, 5, µ). If the median through A is equally inclined to the coordinate aes, then the value of ( + µ + 5) is () 676 () 0 () 8 () 077 Ans. () A(,, 5) B(,, ) D D,, direction cosine of AD C(, 5, ),, AD i + j + ˆk,, ˆk AD.i AD. ĵ AD 7, CORPORATE OFFICE : CG TOWER, A-6 & 5, IPIA, NEAR CITY MALL, JHALAWAR ROAD, KOTA (RAJ.) REG. OFFICE : J-, JAWAHAR NAGAR, MAIN ROAD, KOTA (RAJ.)-005 PH. NO.: FAX NO. : PH.NO. : , TO KNOW MORE : SMS RESO AT WEBSITE : CONTACT@RESONANCE.AC.IN CIN : U800RJ007PLC009 THIS SOLUTION WAS DOWNLOAD FROM RESONANCE JEE Main 06 SOLUTION PORTAL PAGE #

13 PAPER- (B.E./B. TECH.) OF ONLINE JEE (MAIN) CBT A ray of light is incident along a line which meets another line, 7 y + 0, at the point (0, ). The ray is then reflected from this point along the line, y +. Then the equation of the line of incidence of the ray of light is () + 8y 8 0 () 8y () + 5y 5 0 () 5y Ans. () Incidene line L + L 0 (7 y +) + (y + ) 0 0,, y + /, 0,0 7 Let a point (, ) on y And image of (, ) lie on incidence line in 7 y + 0 y (7 ) 8, y (7 y + ) (y + ) y + 0 8y A straight line through origin O meets the line y 0 and 8 + 6y at points A and B respectively. Then O divides the segment AB in the ratio () : () : () : () : Ans. () Let equation of line thourgh 0(0, 0) is y r If this line meets y 0 at A then cos r, sin 0 r cos r (sin + cos) 0.(i) Again the line meets 8 + 6y +5 0 at B sin 8r cos + 6r sin r (sin + cos) 5..(ii) by r 0 r r 5 r CORPORATE OFFICE : CG TOWER, A-6 & 5, IPIA, NEAR CITY MALL, JHALAWAR ROAD, KOTA (RAJ.) REG. OFFICE : J-, JAWAHAR NAGAR, MAIN ROAD, KOTA (RAJ.)-005 PH. NO.: FAX NO. : PH.NO. : , TO KNOW MORE : SMS RESO AT WEBSITE : CONTACT@RESONANCE.AC.IN CIN : U800RJ007PLC009 THIS SOLUTION WAS DOWNLOAD FROM RESONANCE JEE Main 06 SOLUTION PORTAL PAGE #

14 PAPER- (B.E./B. TECH.) OF ONLINE JEE (MAIN) CBT The value of the integral to, is 0 [ ]d, where [] denotes the greatest integer less than or equal [ 8 96] [ ] Ans. () () () 7 () 6 () 0 [ ]d.(i) [ 8 96] [ ] Use property b a f(a b )d f()d b a 0 0 [ 8 96]d.(ii) [ ] [ 8 96] by (i) and (ii) d 0 6 I 5. If n n C P 6, then n satisfies the equation () n + n 0 0 () n + 5n 8 0 () n + n 08 0 () n + n 80 0 Ans. () n C6 n P (n )!(n )!. 6!(n )!(n )! (n+)!. 6! (n )! (n+) (n+) n(n ).6! (n+) (n+) n (n ) n + n 9 Which satifies the n + n 08 0 CORPORATE OFFICE : CG TOWER, A-6 & 5, IPIA, NEAR CITY MALL, JHALAWAR ROAD, KOTA (RAJ.) REG. OFFICE : J-, JAWAHAR NAGAR, MAIN ROAD, KOTA (RAJ.)-005 PH. NO.: FAX NO. : PH.NO. : , TO KNOW MORE : SMS RESO AT WEBSITE : CONTACT@RESONANCE.AC.IN CIN : U800RJ007PLC009 THIS SOLUTION WAS DOWNLOAD FROM RESONANCE JEE Main 06 SOLUTION PORTAL PAGE #

15 PAPER- (B.E./B. TECH.) OF ONLINE JEE (MAIN) CBT If the coefficients of and in the epansion of m n is equal to 8, ( > 0), are m and n respectively, then () 5 () 5 () 7 () 8 Ans. () T r+ 8 / 8r C() r / r 8r 8 C r For coefficient of, For coefficient of, 8 C() 6 8 C 8 r 8 r 8 r r 8 C m r 5 n 8 C If A, then the determinant of the matri (A 06 A 05 A 0 ) is () 0 () 06 () 75 () 5 Ans. () A A 9 A A I 9 And A A 06 A 05 A 0 A 0 A A I 0 5 ( ) CORPORATE OFFICE : CG TOWER, A-6 & 5, IPIA, NEAR CITY MALL, JHALAWAR ROAD, KOTA (RAJ.) REG. OFFICE : J-, JAWAHAR NAGAR, MAIN ROAD, KOTA (RAJ.)-005 PH. NO.: FAX NO. : PH.NO. : , TO KNOW MORE : SMS RESO AT WEBSITE : CONTACT@RESONANCE.AC.IN CIN : U800RJ007PLC009 THIS SOLUTION WAS DOWNLOAD FROM RESONANCE JEE Main 06 SOLUTION PORTAL PAGE # 5

16 PAPER- (B.E./B. TECH.) OF ONLINE JEE (MAIN) CBT If is a solution of the equation,,, then is equal to () () () () Ans. () + Squaring on both sides /8 Now at 5/8 5 6 / 9. An eperiment succeeds twice as often as it fails. The probability of at least 5 successes in the si trials of Ans. () this eperiment is () 9 79 () () 0 79 Given p q & we know that p+q P /, q / The problem of at least 5 successes 6 C 5 P 5 q + 6 C 6 P 6 () The integral d is equal to (where C is a constant of integration) () C () C () C () C Ans. () CORPORATE OFFICE : CG TOWER, A-6 & 5, IPIA, NEAR CITY MALL, JHALAWAR ROAD, KOTA (RAJ.) REG. OFFICE : J-, JAWAHAR NAGAR, MAIN ROAD, KOTA (RAJ.)-005 PH. NO.: FAX NO. : PH.NO. : , TO KNOW MORE : SMS RESO AT WEBSITE : CONTACT@RESONANCE.AC.IN CIN : U800RJ007PLC009 THIS SOLUTION WAS DOWNLOAD FROM RESONANCE JEE Main 06 SOLUTION PORTAL PAGE # 6

17 I d put cos d cos sind I PAPER- (B.E./B. TECH.) OF ONLINE JEE (MAIN) CBT sincos d ( cos) cos sin sec d cos d cos / tan / + C tan / tan / c tan CORPORATE OFFICE : CG TOWER, A-6 & 5, IPIA, NEAR CITY MALL, JHALAWAR ROAD, KOTA (RAJ.) REG. OFFICE : J-, JAWAHAR NAGAR, MAIN ROAD, KOTA (RAJ.)-005 PH. NO.: FAX NO. : PH.NO. : , TO KNOW MORE : SMS RESO AT WEBSITE : CONTACT@RESONANCE.AC.IN CIN : U800RJ007PLC009 THIS SOLUTION WAS DOWNLOAD FROM RESONANCE JEE Main 06 SOLUTION PORTAL PAGE # 7

Memory Based Questions & Solutions

PAPER- (B.E./B. TECH.) JEE (Main) 09 COMPUTER BASED TEST (CBT) Memory Based Questions & Solutions Date: 09 April, 09 (SHIFT-) TIME : (9.0 a.m. to.0 p.m) Duration: Hours Ma. Marks: 60 SUBJECT : MATHEMATICS