Rao IIT Academy/ SSC - Board Exam 2018 / Mathematics Code-A / QP + Solutions JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS SSC - BOARD

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1 Rao IIT Academy/ SSC - Board Exam 08 / Mathematcs Code-A / QP + Solutons JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS SSC - BOARD - 08 Date: MATHEMATICS - PAPER- - SOLUTIONS Q. Attempt any FIVE of the followng sub-questons : [5] () Fnd next two terms of an A.P. 4, 9, 4,... Ans. Gven sequence 4, 9, 4,... Gven sequence s an A.P. wth a = 4, d = 5 t4 a 3d t5 a 4d Topc:Arthmetc Progresson_; Sub-topc: L- SSC Board Test_Mathematcs () Ans. State whether the gven equaton s quadratc or not. Gve reson m m. Here maxmum ndex of the varable s. 5 Here a, b 0, c 7 are real numbers and a 0 4 So t s a quadratc equaton n varable m. Topc:Quadratc Equaton_; Sub-topc: L- SSC Board Test_Mathematcs () If D x = 5, D = 5 are the values of the determnants for certan smultaneous equatons n x and y, fnd x. Ans. By Cramer s Rule, D 5 x x 5 D 5 Topc:Lnear equaton n two varables_; Sub-topc: L- SSC Board Test_Mathematcs

2 Rao IIT Academy/ SSC - Board Exam 08 / Mathematcs Code-A / QP + Solutons (v) If S = {, 4, 6, 8, 0, } and A = {4, 8, }, fnd A '. Ans. S, 4, 6,8,0, and A 4,8, A', 4, 6,8,0, 4,8, A', 6,0 Topc:Probablty_; Sub-topc: L- SSC Board Test_Mathematcs (v) Wrte any one soluton of equaton x + y = 7. Ans. x y 7 Substtutng x = and y = 3 L.H.S = + (3) = 7 = R.H.S x and y 3 s the soluton of x y 7 Topc:Lnear equaton n two varables_; Sub-topc: L- SSC Board Test_Mathematcs (v) If S 5 = 5 and S 6 =, fnd t 6. Ans. Sn Sn tn t6 S6 S5 5 6 Topc:Arthmetc Pregresson_; Sub-topc: L- SSC Board Test_Mathematcs Q. Attempt any FOUR of the followng subquestons : [8] () Fnd n f the n th term of the followng A.P. s 68 : 5, 8,, 4,... Ans. Gven that a 5, d 3, t n 68 t a n d n 68 5 n 3 63 n 3 n n Topc:Arthmetc Pregresson_; Sub-topc: L- SSC Board Test_Mathematcs () If one of the roots of the quadratc equaton x x k 0 s 9, then fnd the value of k. Ans. x x k 0 Gven that One root of gven equaton s k k 0 k 8 Topc:Quadratc Equaton_; Sub-topc:Formaton of roots_l- SSC Board Test_Mathematcs

3 Rao IIT Academy/ SSC - Board Exam 08 / Mathematcs Code-A / QP + Solutons () A box contans 0 cards marked wth numbers to 0. One card s drawn at random. Event A s the number of the card whch s multple of 5. Wrte S, n(s), A and n(a). Ans. S = {,, 3, 4,...0} ns 0 A = The number on the card s multple of 5 A = {5, 0, 5, 0} n(a) = 4 Topc:Probablty_; Sub-topc: L- SSC Board Test_Mathematcs (v) Fnd the value of x y f 4x 3y 5, 3x 4y 4. Ans. 4x 3y 5...() 3x 4 y 4...() Equaton () s multply by 3 and () by 4 x 9y 75 x 6y 96 7 y y = 3 4x x = 6 x = 4 x y 4 3 Topc:Lnear equaton n two varables_; Sub-topc: L- SSC Board Test_Mathematcs (v) Form the quadratc equaton f ts roots are 3 and 4. Ans. Gven that 3, The quadratc equaton whch roots are and s x x 0 x x 0 Topc:Quadratc Equaton_; Sub-topc:Formaton of roots_l- SSC Board Test_Mathematcs (v) For a certan frequency dstrbuton, the values of mean and medan are 7 and 78 respectvely. Fnd the value of mode. Ans. Mean = 7 Medan = 78 Mean Mode = 3(Mean Medan) 7 Mode = 3(7 78) Mode = = 90 Topc:Statstcs I_; Sub-topc:Mean, Medan and Mode_L- SSC Board Test_Mathematcs 3 3

4 Rao IIT Academy/ SSC - Board Exam 08 / Mathematcs Code-A / QP + Solutons Q.3 Attempt any THREE of the followng subquestons : [9] () For an A.P., fnd S 7 f a = 5 and d = 4. Ans. a 5, d 4 n Sn a n d S S7 9 Topc:Arthmetc Progresson_; Sub-topc: L- SSC Board Test_Mathematcs () Solve the followng quadratc equaton by usng formula method : x 3x Ans. Gven quadratc equaton x 3x x 3x 0 a =, b = 3, c = By Formula method, x a b b 4ac x or 4 x =, 3 5 x Topc:Quadratc Equaton_; Sub-topc:Soluton of QE_L- SSC Board Test_Mathematcs 4 4

5 Rao IIT Academy/ SSC - Board Exam 08 / Mathematcs Code-A / QP + Solutons () Solve the followng smultaneous equatons usng Cramer s rule : 3x y 3; x y 6 Ans. 3x y 3...() x y 6...() 3 D = 3() ( ) = 7 D x 3 = 3() 6( ) = = Dy = 3(6) 3() = 48 6 = 4 6 Now, D x x D D y y D 35 4 x y 7 7 x = 5 y = 6 Topc:Lnear equaton n two varables_; Sub-topc:Cramer s Rule_L- SSC Board Test_Mathematcs (v) A de s thrown, fnd the probablty of the event of gettng a number less than 3. Ans. Sample space when a de s thrown S = {,, 3, 4, 5, 6} n(s) = 6 Let A = Gettng a number less than 3 A n A, n P(gettng a number less than 3) A P( A) ns 6 3 Topc:Probablty_; Sub-topc: L- SSC Board Test_Mathematcs 5 5

6 Rao IIT Academy/ SSC - Board Exam 08 / Mathematcs Code-A / QP + Solutons (v) The marks obtaned by a student n an examnaton out of 00 are gven below. The total marks obtaned n varous subjects are as follows : Subject Marks Marath 85 Englsh 85 Scence 90 Mathematcs 00 Total 360 Represent the above data usng pe dagram. Ans. Frst of all, we compute the central angle for each subject as shown n followng table. Sr.No. Subject Marks Measure of central angle Marath = Englsh = Scence = Mathematcs = Total Englsh Scence 00 Marath 85 Mathematcs Topc:Statstcs II_; Sub-topc:Pe Dagram_L- SSC Board Test_Mathematcs Q.4 Attempt any TWO of the followng subquestons : [8] () 3 3 If 5 and 35, fnd the quadratc equaton whose roots are and. Ans. Here and are the roots of the quadratc equaton, so requred equatons s x x 0...() 3 3 We have 5 and So from () requred quadratc equaton s x 5x 6 0 Topc:Quadratc Equaton_; Sub-topc:Formaton of QE_L-3 SSC Board Test_Mathematcs 6 6

7 Rao IIT Academy/ SSC - Board Exam 08 / Mathematcs Code-A / QP + Solutons () Two dce are thrown. Fnd the probablty of gettng : (a) The sum of the numbers on ther upper faces s at least 9. (b) The sum of the numbers on ther upper faces s 5. (c) The number on the upper face of the second de s greater than the number on the upper face of the frst de. Ans. S {(,) (,) (,3) (,4) (,5) (,6) (,) (, ) (,3) (, 4)(,5)(,6) (3,)(3, ) (3,3) (3, 4)(3,5) (3,6) (4,) (4, ) (4,3) (4, 4)(4,5)(4,6) (5,)(5, )(5,3)(5, 4)(5,5) (5,6) (6,) (6, )(6,3) (6, 4) (6,5) (6,6)} n( S) 36 Let A sum of the numbers n ther upper faces s at least 9. A{(3,6) (4,5)(4,6)(5, 4) (5,5) (5,6) (6,3) (6, 4) (6,5)(6,6)} n( A) 0 n( A) 0 5 P( A) n( S) 36 8 Let B sum of the number on ther upper faces s 5. B (Null set) n( B) 0 n( B) 0 P( B) 0 n( S) 36 Let C number on the upper face of second de s greater than the number on the upper face of frst de. C {(, ) (, 3) (, 4) (, 5) (, 6) (, 3) (, 4) (, 5) (, 6) (3, 4) (3, 5) (3, 6) (4,5) (4, 6) (5, 6)} n( C) 6 n( C) 5 5 P( C) n( S) 36 Topc:Probablty_; Sub-topc:Probablty_L- SSC Board Test_Mathematcs 7 7

8 Rao IIT Academy/ SSC - Board Exam 08 / Mathematcs Code-A / QP + Solutons () Frequency dstrbuton of daly commsson receved by 00 salemen s gven below : Daly Commsson (n Rs.) No. of Salesmen Fnd mean daly commsson receved by salemen, by assumed mean method. Ans. Daly commsson Classmark d x A d x 50 No. of salemen A f d 360 d 3.60 f 00 x A d f f 00 f x 360 Topc:Statstcs I_; Sub-topc: Mean L- SSC Board Test_Mathematcs f d Q.5 Attempt any TWO of the followng subquestons : [0] () A boat takes 0 hours to travel 30 km upsteam and 44 km downstream, but t takes 3 hours to travel 40 km upstream and 55 km downstream. Fnd the speed of the boat n stll water and the speed of the stream. Ans. Let the speed of the boat n stll water be x km/hr and the speed of the stream by y km/hr. Therefore, the speed of the boat downstream x / x / Now, y km hr dstance tme= speed y km hr and the speed of the boat upstream Therefore, tme taken by the boat to cover 30 km upstream = 30 hours x y and the tme taken by the boat to cover 4km down stream 44 hours x y But the total tme taken by the boat to cover 30 km upstream and 44 km downstream s 0 hours. 8 8

9 Rao IIT Academy/ SSC - Board Exam 08 / Mathematcs Code-A / QP + Solutons x y x y smlarly by second condton, x y x y 3... substtutng x y a equaton () and () and 30a 44b a 55b 3...v x y b n Equaton () x (v) and eqaton (v) x (), we get 0a 76b 40...v 0a 65b 39...v equaton (v) equaton (v), we get b b substtutng 0a a a 4 4 a 0 a 5 b n equaton (v), we get Now, x y 5 and x y x y 5 and x y x y... v x y 5... v 9 9

10 Rao IIT Academy/ SSC - Board Exam 08 / Mathematcs Code-A / QP + Solutons Addng equaton (v) and equaton (v), we get x 6 x 8 Subsdty x 8 n equaton (v) we get y 3 speed of the boat n stll water s 8 km/hr and speed of the stream s 3 km/hr Topc:Lnear equaton n two varables_; Sub-topc: L-3 SSC Board Test_Mathematcs () If the 9th term of an A.P. s zero, then prove that 9th term s double of 9th term. Ans. t a n d n 9 th term.e. n 9 t9 a 9 d a 8d It s gven that t 9 0 a 8d th term.e. t 9 where n 9 t9 a 9 d t9 a 8 d... a 8d 0d 0 0d...by eq () t9 0 d... t9 a 9 d t a d 9 8 a 8d 0d 0 0d t 9 0 d...() by equaton () & () t t 9 9 Topc:Arthmetc Progresson_; Sub-topc: L-3 SSC Board Test_Mathematcs 0 0

11 Rao IIT Academy/ SSC - Board Exam 08 / Mathematcs Code-A / QP + Solutons () Draw hstrogram and frequency polygon on the same graph paper for the followng frequency dstrbuton : Ans. Class Frequency Class Frequency Classmark Scale - on x axs : cm = 5 unts and y axs : cm = 5 unts Hstogram y x

12 Rao IIT Academy/ SSC - Board Exam 08 / Mathematcs Code-A / QP + Solutons Frequency Polygon curve y x Topc:Statstcs II_; Sub-topc: L- SSC Board Test_Mathematcs