THE ASSOCIATION OF MATHEMATICS TEACHERS OF INDIA Screeig Test - Bhaskara Cotest (NMTC at JUNIOR LEVEL I & Stadards) Saturday, 7th August 06. Note : Note : () Fill i the respose sheet with your Name, Class, the istitutio through you appear i the specified places. () Diagrams are oly visual aids; They are ot draw to scale. () You are free to do rough work o separate sheets. () Duratio of the test : p.m. to p.m- hours. PART-A Oly oe of the choices A,B,C,D is correct for each questio, shade that alphabet of your choice i the respose sheet. (If you have ay doubt i the method of aswerig seek the guidace of your supervisor). For each correct respose you ge mark; for each icorrect respose you lose / Marks.. The sum of the values, y that satisfy the equatios ( + y) y, ( + y) y simultaeously is (A) (B) (C) 5 (D) 7 ( + y) y...(i) ( + y) y...(ii) by eq. (i) & (ii) ( + y) ( + y) (y ) (y ) ( + y) ( + y) (y ) y y...(iii) Now by equatio (iii) (y ) 0 (y ) 0 (y ) or y Now by equatio (iv) & equatio (ii) ( + y) or + y so sum of all possible values of y + 5 NMTC_STAGE-I _ PAPER-06_PAGE #
a. If a a 6 a ( a ) (a ) a a a 06 the value of a is (A) 007 (B) 06 (C) 07 (D) Noe of these a a a 6 a ( a ) (a ) a a a 06 8a a 6 a (a ) (a ) (a ) / (a ) a a 06 (a ) a a a a a 06 [a + ] 06 a a + 008 a a 007. ABCD is a square iscribed i a circle of radius uit. The taget to the circle at C meets AB produced at P. The legth of PD is (A) (B) (C) (D) 0 AC, side of square so,, so three values of are possible. AC, side of square ABC PBC AB PB APD PD AP 0 AD NMTC_STAGE-I _ PAPER-06_PAGE #
. Quadrilateral ABCD is iscribed i a circle with radius uit. AC is the diameter of the circle ad BD AB. The diagoals cut at P. If PC 5 the the legth of CD is equal to (A) (B) 7 (C) 8 (D) B A q O q 90- q P q C 90- q q D O is cetre of circle, ABC 90 Let BAC BCA 90 BDC BAC (Agle i a same segmet) BDA BCA 90 (Agle i a same segmet) sice AB BD ADB BAD 90 CAD 90 90 APD (Agle sum of property of ) ACD I APD I PAD PD PC si si (i) (ii)...(i) PD AP cos cos...(ii) PC cot cot AP ta ta PC AP ta ta ta ta ta DC I ADC cos DC AC cos AC AC ta ta NMTC_STAGE-I _ PAPER-06_PAGE #
8 5. The umber of atural umbers for which the epressio is also a atural umber is (A) (B) (C) (D) 0 8 + 8 + So,,, so three value of are possible. 6. The cost price of 6 orages is equal to the sellig price of orages. The there is a (A) 0% profit (B) 0 % loss (C) S P 6CP CP + CP % profit (D) % profit CP Profit % 00 %. CP 7. The umber of positive iteger pairs (a, b) such that ab b is (A) 6 (B) 7 (C) 8 (D) 9 ab b a. b b is factor of b,,,, 6, 8,, So b ca take 8 values so umber of pairs (a, b) is 8. 8. A ( + ) ( + ) ( + )... ( 06 + ). The value of (A + ) /06 is (A) (B) 06 (C) 0 (D) Bous because here the power of are i G.P. i.e.,,,, 8, 6,... but accordig to this 06 will ot come Let istead of 06 if we take 08 A ( + ) ( + ) ( + )... ( 08 +) ( ) A ( ) ( + ) ( + )... ( 08 + ) A ( ) ( + ).. ( 08 + ) A ( 08 ) ( 08 + ). A 096 08 A 08 096. As. 9. The sum of two umbers a, b where a < b is 5 ad their H.C.F. is 8. The umber of pairs of such pairs (a, b) is (A) (B) (C) (D) HCF 8 a 8 b 8y ad y are coprime. 8 + 8 y 5 + y 5 y pairs. As. 7 8 NMTC_STAGE-I _ PAPER-06_PAGE #
0. The first Republic Day of Idia was celebrated o 6 th Jauary 950. What was the day of the week o that date? (A) Tuesday (B) Wedesday (C) Thursday (D) Friday Time period Number of odd days up to 600 0 60 to 900 st Ja 90 to st Dec 99 + 7 5 st Ja 6 Ja 950 5 total odd day S M T W Th Fr Sat odd day 0 5 6 so 6 th ja 950 day is Thursday.. The umbers a, a,..., a are i arithmetical progressio. The sum of all these umbers is 5. Let P a + a +... + a ad Q a + a +... + a. If the ratio P : Q is : 7, the commo ratio of the progressio is (A) (B) (C) (D) 5 6 [a d a d] P 6 Q [a a 0d] d 5a 7 d S 5 [a + a + d] 5 6 d 5 d 5. A Shopkeeper marks the prices of his goods at 0% higher tha the origial price. There is a icrease i demad of the goods, ad he further icreases the price by 0%. The total profit % is (A) 0 (B) 8 (C) (D) Let CP 00 00 0 SP 00 00 00 0 00 Profit %.. A circle passes through the vertices A ad D ad touches the side BC of a square. The side of the square is cm. The radius of the circle (i cm) is (A) 5 (B) 5 (C) (D) 5 OAE R ( R) + R + R R + R 5.. There are four balls - oe gree, oe red, oe blue ad oe yellow ad there are four boes oe gree, oe red, oe blue ad oe yellow. A child playig with the balls decides to put the balls i the boes, oe ball i each bo. The umber of ways i which the child ca put the balls i the boes such that o ball is i a bo of its ow color is (A) (B) 9 (C) (D) 6 Number of ways! 9.!!!! NMTC_STAGE-I _ PAPER-06_PAGE # 5
5. The 5 5 array of dots represets trees i a orchard. If you were stadig at the cetral spot marked C, you would ot be able to see 8 of the trees (show as ). If you were stadig at the ceter of a 9 9 array of trees, how may of the 80 trees would be hidde? (A) 0 (B) (C) 6 (D) Note : The tree which came i straight lie oly first tree is visible ad rest are ivisible. So total tree are hidde PART-B Write the correct aswer i the space provided i the respose sheet. For each correct respose you ge mark ; for each icorrect respose you lose / Marks. 6. a ad b are positive itegers such that a + b b + a + 5. The value of b is. a + b b + a + 5 a a b b + 5 (a ) (b ) 5 (a + b ) (a b) 5 a + b 5 a b a 8 a, b. As. 7. After full simplificatio, the value of product is.. As. NMTC_STAGE-I _ PAPER-06_PAGE # 6
8. ABCD is a rectagle with AD ad AB. DFEB is also a rectagle. The area of DFEB is. ar Rectagle ABCD sq. uits. ar DCB ar Rectagle ABCD sq. uits. Rectagle DFEB ad DCB have same base ad betwee same parallel ar Rectagle DFEB ar DBC () sq. uits. 9. The two digit umber whose uits digit eceeds the tes digit by ad such that the product of the umber ad the sum of its digits is is. te s digit uit digit + ATQ (0 + + ) ( + ) + 70 0 ( + 5) ( ) 0 required umber. 0. If q p where p, q are itegers havig o commo divisors other tha, satisfies the is. + + 5 p. 6 q NMTC_STAGE-I _ PAPER-06_PAGE # 7
. AE ad BF are medias draw to the legs of a right agled triagle ABC. The umerical value of AE BF AB is. AE a b BF a + b AE + BF 5 (a + b ) AE + BF 5 AB AE BF 5. AB. AB is a chord of a circle with ceter O. AB is produced to C such that BC OA. CO is produced to E. AOE The value of is. ACE y + y y y + y y + y AOE ACE y. As.. The umber of two digit umbers that are less tha the sum of the squares of their digits by ad eceed twice the product of their digits by 5 is. Uit digit b te s digit a ATQ ab + 5 0a + b a + b a + b ab 6 ad 0a + b ab 5 (a b) 6 (a ) (5 b) 0 (a b) ± a or b 5. a b ± ad b 5 a 9, Number 95, 5. So umber eist. NMTC_STAGE-I _ PAPER-06_PAGE # 8
. AB is a diameter of circle ad CD is a parallel chord. P is ay poit i AB. The umerical value of PC PA PD PB is. PA + PB (r ) + (r + ) (r + )...(i) PC + PD y + (d ) + y + (d + ) y + (d + ) (y + d ) + r + (r + )...(ii) (i) (ii) PA PC PB PD 5. I the sequeces,,,,,,, 8, 8, 8, 8, 8, 8, 8, 8,... the 06th term is. The. We observed that term o from to value is term o from to 7 value is term o from to 5 value is 8... term o from 0 to 07 value is 0 0 so at 06 th term value is 0 0 0 6. Each root of the equatio a + b + c 0 is decreased by. The quadratic equatio with these roots is + + 0. The umerical value of b + c is. a + b + c 0 b + a a c + 0 ( ) ( ) ( + ) + a c + a b 0 b + c 0 NMTC_STAGE-I _ PAPER-06_PAGE # 9
7. The umber of itegers such that > ( + ) > + ( ) ( + ) < 0 > is. so itegral value (0,, ). 8. P ad P are two regular polygos. The umber of sides of P ad P respectively are i the ratio : ad the respective iterior agles are i the ratio 0 : 9. The the sum of the umber of sides of P ad P is. Iterior agleof polygo P Iterior agleof polygo P ( )80 ( )80 0 9 0 9 8 + + 8 0. As. 9. I triagle ABC, F ad E are the mid poits of AB ad AC respectively. P is ay poit o the side BC. The ratio Area of ABC Area of FPE is By MPT FE BC AFE ~ ABC FE h BC h h h h h ar ABC ar FPE BC h FE (h h) h h. NMTC_STAGE-I _ PAPER-06_PAGE # 0
0., y, z are distict real umbers such that + y y + z z + y + z +. The value of y y z is. z solvig i pair we get y y z yz...() y z z z z y y () () ()...()...() ( y)(y z)(z ) ( y) y z) (z ) y z y z. NMTC_STAGE-I _ PAPER-06_PAGE #