CBSE Class 1 th Mathematics Solved Paper 16 SA II
CBSE Class 1 th Mathematics Solved Paper 16 SA II Solved Questio Paper Class X Subject Mathematics All Idia: Set III Time allowed: hours Maximum Marks: 9 Geeral Istructios: (i) All questios are compulsory. (ii) The questio paper cosists of 1 questios divided ito four sectios A, B, C ad D. (iii) Sectio A cotais 4 questios of 1 mark each, sectio B cotai 6 questio of marks each, Sectio C cotais 1 questios of marks each ad Sectio D cotais 11 questios of 4 marks each. (iv) Use of calculators is ot permitted. SECTION A Questio1. A card is draw at radom from a well shuffled pack of 5 playig cards. Fid the probability of gettig either a red card or a quee. There are 6 red cards i a deck of 5 cards Also there are 4 quees i a deck of 5 cards, quees are black ad are red. Now, umbers of card which are either red or quee are = 5 (6 + ) = 4. 4 6 Probability of gettig either a red card or a quee is:. 5 1 Questio. A ladder, leaig agaist a wall, makes a agle of 6 with the horizotal. If the foot of the ladder is.5m away from the wall, fid the legth of Ladder. Let AC be a ladder, AB is the wall ad BC is the groud. The situatio give i the questio is show i the figure give below We kow that BC cos 6 AC 1.5 AC AC 5m Therefore, the legth of ladder is 5 m. Questio. I figure1, PQ is a taget at a poit C to a circle with cetre O. If AB is a diameter ad CAB =, fid PCA. Costructio: Joi OC Now i Δ AOC AO = OC (radius of the same circle) So, OAC= OCA = Also, OCP= 9 Therefore, PCA= 9 = 6. Questio4. For what value of k will k + 9, k 1 ad k + 7 are the cosecutive terms of a A.P.? If three terms x, y ad z are i A.P. the, y = x + z Sice k + 9, k 1 ad k + 7 are i A.P. k 1 k 9 k 7 4k k16 k 18.
CBSE Class 1 th Mathematics Solved Paper 16 SA II SECTION B Questio5. I figure, a quadrilateral ABCD is draw to circumscribe a circle, with cetre O, i such a way that the sides AB, BC, CD ad DA touch the circle at the poits P, Q, R ad S. Prove that AB + CD = BC + DA. Sice tagets draw from the exterior poit to a circle are equal i legth. As, DR ad DS are tagets from exterior poit D so, DR = DS (1) As, AP ad AS are tagets from exterior poit A so, AP = AS.. () As, BP ad BQ are tagets from exterior poit B so, BP = BQ... () As, CR ad CQ are tagets from exterior poit C so, CR = CQ.. (4) Addig...(1),...(),...() ad...(4), we get DR + AP + BP + CR = DS + AS + BQ + CQ (DR + CR) + (AP + BP) = (DS + AS) + (BQ + CQ) CD + AB = DA + BC AB + CD = BC + DA Hece proved. Questio6. Prove that the poits (, ), (6, 4) ad ( 1, ) are the vertices of a right agled isosceles triagle. Suppose, A (, ), B (6, 4) ad C ( 1, ) be the vertices of the triagle. From distace formula, AB 6 4 9 16 5 5 BC 1 6 4 49 1 5 5 CA 1 ( ) 16 9 5 5 As, 5 5 5 AB CA BC This implies that triagle ABC is a right agled isosceles triagle. Questio7. The 4 th term of a A.P is zero. Prove that the 5 th term of the A.P is three times its 11 th term. Let a be the first term ad d be the commo differece of a A.P. havig terms. a a 1 d. Its th term is give by, Accordig to the questio, a4 a d a d a d a a 4d 5 a d 4d a 11 5 5 11 1 d...(1) a a 1d a d 1d a11 7 d...() From equatio (1) ad () a5 a11 Hece proved. Questio8. Let P ad Q be the poits of trisectio of the lie segmet joiig the poits A (, ) ad B ( 7, 4) such that P is earer to A. Fid the coordiates of P ad Q. Sice P ad Q be the poits of trisectio of the lie segmet joiig the poits A (, ) ad B ( 7, 4) such that P is earer to A. Therefore P divides lie segmet i the rail 1: ad Q divides i :1 By sectio formula Coordiates of P are 1 7 1 4, 1 1, ( 1, ) Coordiates of Q are
CBSE Class 1 th Mathematics Solved Paper 16 SA II 7 1 4 1, 1 1 1 6, ( 4,) Questio9. I figure, from a exteral poit P, two tagets PT ad PS are draw to a circle with cetre O ad radius r. If OP = r, show that OTS = OST = I OTP, OTP 9 (taget to a circle perpedicular to the radius through the poit of cotact) Perpedicular OT siopt Hypoteus OP r siopt r 1 siopt OPT Also, OTP + OPT + TOP = 18 o (sum of agles of a triagle) 9 o + o + TOP = 18 o TOP = 18 o 1 o = 6 o I OTP ad OSP OT OS ( radius) PT PS (tagets from same exteral poit) OP OP (commo) So, OTP OSP Therefore, TPO = SPO OP is bisector of TPS. Now, ΔTQP ad ΔSQP will also be cogruet by SAS cogruecy. Therefore, OQT = OQS = 9 o. I Δ OTQ, OQT + OTQ + TOQ = 18 o (agles of a triagle) 9 o + OTQ + 6 o = 18 o OTQ = o OTS = o. Similarly, OST = o. Questio1. Solve for x: x x We have 6x 7 x 7 6x 7 x 7 Squarig both sides x 6x 7 7 6x 7 4x 49 8x 4x 4x 4 x 17x 1 Usig quadratic formula, x x b b 4ac a 17 17 4 1 17 89 168 x 4 17 11 x 4 17 11 x 4 8 6 x, 4 4 x 7,1.5 6 7 7 SECTION C Questio11. A Coical vessel with base of radius 5cm ad height 4 cm, is full of water. This water is emptied ito cylidrical vessel of base 1cm. Fid the height to which water will rise i the cylidrical vessel. Let r, h, l be the radius, height ad slat height of the coe. Let R ad H be the radius ad height of the cylider. Sice volume will remai same Volume of water i coical vessel = Volume i cylidrical vessel Therefore
CBSE Class 1 th Mathematics Solved Paper 16 SA II 1 r h R H 1 5 (4) 1 H H cm. Questio1. I figure 4, O is the cetre of a circle such that diameter AB = 1cm ad AC = 1 cm. BC is joied. Fid the area of shaded regio. I Δ ABC C 9 (agle i a semicircle) therefore, BC AC AB BC 1 1 BC BC 169 144 5 BC 5 Area of shaded regio Area of semicircle Area of triagle 1 1 Area of shaded regio 1 1 1 1 Area of shaded regio 51 7 11 1 1 Area of shaded regio 7 Area of shaded regio 6.9 cm radius base height Questio1. If the poit P(x, y) is equidistat from the poits A (a + b, b a) ad B (a b, a + b). Prove that b x = a y. Sice the poit P (x, y) is equidistat from the poits A (a + b, b a) ad B (a b, a + b) PA PB From distace formula x a b y b a x a b y a b x a b y b a x a b y a b x a b x a b y b a y b a x a b x a b y x a b y b a x a b y b a ax bx by ay ax bx by ay 4bx 4ay bx ay a b y b a Questio14. I figure 5, a tet is i the shape of a cylider surmouted by a coical top of same diameter. If the height ad diameter of cylidrical part are.1 m ad m respectively ad the slat height of coical part is.8m, fid the cost of cavas eeded to make the tet if the cavas is available at the rate of 5/Sq meters. Cavas eeded to make the tet = Curved surface area of the coical part + curved surface area of cylidrical part. Radius of the coical part = radius of cylidrical part = r =/ m Slat height of the coical part is = l =.8 m. Height of cylidrical part, h =.1m Curved surface area of coical part = rl.8 7 Curved surface area of cylidrical part = rh.1 7 Total surface area =
CBSE Class 1 th Mathematics Solved Paper 16 SA II.8m.1 7 7.8.1 7 7 7 7 m Therefore cost of cavas used = 5 = Rs.165. Questio15. A sphere of diameter 1 cm is dropped i a right circular cylidrical vessel, partly filled with water. If the sphere is completely submerged i water, the water level i 5 the cylidrical vessel rises by.fid the diameter of the cylidrical vessel. 9 Icrease i the height of water level i the cylidrical vessel due to sphere (h) = 5 =/9 9 cm. Radius of sphere (R) = 6cm Let radius of cylider be r Rise i volume of water i cylider = Volume of sphere 4 r h R 4 r 6 9 r 81 r 9 cm Hece diameter of the cylidrical vessel is 9 =18 cm. Questio16. A ma stadig o the deck of a ship, which is 1 m above water level, observes the agle of elevatio of the top of a hill as 6 ad the agle of depressio of the base of hill as. Fid the distace of the hill from the ship ad height of the hill. Suppose the ma is stadig o the deck of the ship at poit A. Let CE be the hill with base at C. It is give that the agle of elevatio of poit E from A is 6 o ad the agle of depressio of poit C from A is So, DAE=6 CAD= CAD= ACB= (alterate agles) AB =1 m Suppose ED = h m ad BC = x m I ΔEAD, we have ED ta 6 AD h (BC = AD x) x h x...(1) I ABC, we have AB ta BC 1 1 x x 1...() Therefore distace of the hill from the ship is 1 m Ad height of the hill is h x 1 h 1 1 h m. Questio17. I figure 6, fid the area of the shaded regio, eclosed betwee two cocetric circles of radii 7cm ad 14cm where AOC=4
CBSE Class 1 th Mathematics Solved Paper 16 SA II We have Area of shaded regio = Area of circular rig Area of regio ABDC 4 14 7 14 7 6 1 14 7 1 9 8 14 714 7 7 9 696 cm 9 1 cm 41.67 cm. Questio18. There are hudred cards i a bag o which umbers from 1 to 1 are writte. A card is take out from the bag at radom. Fid the probability that the umber o the selected card (i) is divisible by 9 ad is a perfect square. (ii) is a prime umber greater tha 8. (i) Total cards =1 The umbers those are divisible by 9 ad are perfect squares are 9, 6, 81. Therefore probability that the umber o the selected card is divisible by 9 ad is a perfect square is 1 (ii) Total cards =1 The prime umbers greater tha 8 are, 8, 89, 97 Therefore probability that the umber o the selected card is a prime umber greater tha 1 is. 1 Questio19. Three cosecutive atural umbers are such that the square of the middle umber exceeds the differece of the squares of other two by 6.Fid the umbers. Let x, x + 1 ad x + be the three cosecutive umbers, The accordig to the questio, x 1 x x 6 x x x x x 1 4 4 6 x x 6 x x x x x x x x x 7 x 9 9 7 6 9 7 6 9 7 9 x 7,9. The root 7 will be rejected, as atural umber caot be egative. Therefore, the umbers are 9, 1, 11. Questio. The sum of first terms of three arithmetic progressios are S 1, S ad S respectively. The first term of each A.P is 1 ad their commo differeces are 1, ad. Prove that S1 S S. If a is the first term ad d is the commo differece of a A.P. havig terms the,
CBSE Class 1 th Mathematics Solved Paper 16 SA II the sum of its terms is give by, S a 1 d Therefore S1 1 11...(1) S 1 1 S S...() S 1 1...() Now from...(1) ad...(), 1 1 S1 S 1 11 1 1 S1 S 1 S1 S 1 1 S1 S 1 1 S1 S 4 S S S S S from...() SECTION D Questio1. Due to heavy floods i a state, thousads were redered homeless. 5 schools collectively offered to the state govermet to provide place ad the cavas for 15 tets to be fixed by the govermet ad decided to share the whole expediture equally. The lower part of each tet is cylidrical of base radius.8m ad height.5m, with coical upper part of same base radius but of height.1 m. If the cavas used to make the tets costs Rs 1 per sq m, fid the amout shared by each school to set up the tets. What value is geerated by the above problem. The figure from the give iformatio is as follows: Let the radius of base of both cylider ad coe be r Let the height of coe be H ad that of cylider be h. Let the slat height of coe be l, The, Area of tet = CSA of cylidrical base + CSA of coe rh rl rh r H r rh r.1.8 rh r 1.5 rh r.5.8.5.5 7.8 1.5 7 9.4 m Cost of cavas used i makig oe tet = Rs. 9.4 1 = Rs.1188 Cost of cavas used i makig 15 tets = Rs. 1188 15= Rs.166 Share of oe school = Rs.166/5 = Rs. 64 The give problem shows kidess of school authorities. Questio. The houses i a row are umbered cosecutively from 1 to 49. Show that there exists a value of X such that the sum of umbers of houses precedig the house umbered X is equal to sum of the umber of houses followig X. The umber of houses precedig X is X 1 Sum of the umber of houses precedig the house umbered, X = 1 + + +... X 1. Here, a = 1, d =1 Accordig to questio
CBSE Class 1 th Mathematics Solved Paper 16 SA II S S S X1 49 X 1 49 X a X 11 d a 48d a X 1 d X 1 49 X 1 X 1 1 481 1 X 11 X 1 X 49 5 X 1 X X 1 1 X X 49 5 X 1 1 X 1 49 1 481 X 1 X 1 1 X 49 5 X 49 5 X 75 X 5 X Questio. I figure 7, the vertices of triagle are A (4, 6), B (1, 5) ad C (7, ). A lie segmet DE is draw to itersect the sides AB ad AC at D ad E respectively such that AD AE 1.Calculate the area of Δ ADE ad compare it with area of Δ ABC. AB AC We have Area of triagle ABC is: 1 x y y x y y x y y 1 4 5 1 6 7 6 5 15 sq.uits Now i ADE ad ABC AD AE 1 (give) AB AC A A (commo) 1 1 1 ADE ABC Therefore, ar ADE AD 1 ar ABC AB ar ABC ar ADE 9 15 ar ADE 9 5 ar ADE square uits. 6 Questio4. I figure 8, two equal circles, with cetre O ad O, touch each other at X.OO produced meets the circle with cetre O at A.AC is taget to the circle with cetre O, at the poit C.O D is perpedicular to AC. Fid the value of DO /CO. As, circles are equal, so their radius are also equal. That meas AO O X XO r I ADOad ACO ADO ACO A A (commo) ADO ACO Therefore
CBSE Class 1 th Mathematics Solved Paper 16 SA II AO DO AO CO r DO r r r CO r DO r CO DO 1 CO Questio5. A motor boat whose speed is 4 km/hr i still water takes 1 hr more to go km upstream tha to retur dowstream to the same spot. Fid the speed of the stream. Let the speed of stream be x. The, Speed of boat i upstream is 4 x I dowstream, speed of boat is 4 + x Accordig to questio, Time take i the upstream jourey Time take i the dowstream jourey = 1 hour 1 4 x 4 x 4 x 4 x 1 4 x x 1 576 x x x 64 576 x x x x x x x x x 8 x 7 7 8 576 7 8 576 7 8 7 x 8, 7 Sice speed caot be egative, So, speed of stream is 8km/hr. Questio6. I figure 9, is show a sector OAP of a circle with cetre O, cotaiig θ AB is perpedicular to the radius OA ad meets OP produced at B. Prove that the perimeter of shaded regio is r[ta sec 1]. 18 I ΔAOB OA cos OB OA OB cos OB r sec PB OP OB rsec r Also, AB ta OA AB OA ta AB r ta Legth of arc AP r 6 18 Perimeter of shaded regio PB AB Legth of arc AP r Perimeter of shaded regio r sec r r ta 18 Perimeter of shaded regio r sec ta 1 18 Questio7. Prove that the legth of taget draw from a exteral poit to a circle are equal. Let us draw a circle with cetre O ad two tagets PQ ad PR are draw from a exteral poit P to the circle as show i the figure give below,
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