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CBSE Board Exam Class 10 2019 : Mathematics (Series 3)

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SET-1 H$moS> Z . Series JMS/3 Code No. amob Z . 30/3/1 narjmWu H$moS >H$mo C ma-nwp VH$m Ho$ _wI-n >na Ad ` {bIo & Roll No. Candidates must write the Code on the title page of the answer-book. H $n`m Om M H$a b| {H$ Bg Z-n _o _w{ V n > 11 h & Z-n _| Xm{hZo hmW H$s Amoa {XE JE H$moS >Z ~a H$mo N>m C ma -nwp VH$m Ho$ _wI-n > na {bI| & H $n`m Om M H$a b| {H$ Bg Z-n _| >30 Z h & H $n`m Z H$m C ma {bIZm ew $ H$aZo go nhbo, Z H$m H $_m H$ Ad ` {bI| & Bg Z-n H$mo n T>Zo Ho$ {bE 15 {_ZQ >H$m g_` {X`m J`m h & Z-n H$m {dVaU nydm _| 10.15 ~Oo {H$`m OmEJm & 10.15 ~Oo go 10.30 ~Oo VH$ N>m Ho$db Z-n H$mo n T>|Jo Am a Bg Ad{Y Ho$ Xm amZ do C ma-nwp VH$m na H$moB C ma Zht {bI|Jo & Please check that this question paper contains 11 printed pages. Code number given on the right hand side of the question paper should be written on the title page of the answer-book by the candidate. Please check that this question paper contains 30 questions. Please write down the Serial Number of the question before attempting it. 15 minute time has been allotted to read this question paper. The question paper will be distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the question paper only and will not write any answer on the answer-book during this period. J{UV MATHEMATICS {ZYm [aV g_` : 3 K Q>o A{YH$V_ A H$ : 80 Time allowed : 3 hours 30/3/1 Maximum Marks : 80 1 P.T.O. gm_m ` {ZX}e : (i) g^r Z A{Zdm` h & (ii) Bg Z-n _| 30 Z h Omo Mma I S>m| A, ~, g Am a X _| {d^m{OV h & (iii) I S> A _| EH$-EH$ A H$ dmbo 6 Z h & I S> ~ _| 6 Z h {OZ_| go `oH$ 2 A H$ H$m h & I S> g _| 10 Z VrZ-VrZ A H$m| Ho$ h & I S> X _| 8 Z h {OZ_| go `oH$ 4 A H$ H$m h & (iv) Z-n _| H$moB g_J {dH$ n Zht h & VWm{n 1 A H$ dmbo 2 Zm| _|, 2 A H$m| dmbo 2 Zm| _|, 3 A H$m| dmbo 4 Zm| _| Am a 4 A H$m| dmbo 3 Zm| _| Am V[aH$ {dH$ n XmZ {H$E JE h & Eogo Zm| _| AmnH$mo {XE JE {dH$ nm| _| go Ho$db EH$ Z hr H$aZm h & (v) H $bHw$boQ>am| Ho$ `moJ H$s AZw_{V Zht h & General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided into four sections A, B, C and D. (iii) Section A contains 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 8 questions of 4 marks each. (iv) There is no overall choice. However, an internal choice has been provided in two questions of 1 mark each, two questions of 2 marks each, four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions. (v) Use of calculators is not permitted. I S> A SECTION A Z g `m 1 go 6 VH$ `oH$ Z 1 A H$ H$m h & Question numbers 1 to 6 carry 1 mark each. 1. { KmV g_rH$aU (x + 5)2 = 2 (5x 3) H$m {d{d $H$a (discriminant) {b{IE 2. & Write the discriminant of the quadratic equation (x + 5)2 = 2 (5x 3). 27 kmV H$s{OE {H$ g `m 23 . 5 4 . 3 2 Ho$ Xe_bd $n H$m Xe_bd Ho$ {H$VZo WmZm| Ho$ ~mX A V hmoJm & AWdm g `m 429 H$mo BgHo$ A^m ` JwUZI S>m| Ho$ JwUZ\$b Ho$ $n _| ` $ H$s{OE & Find after how many places of decimal the decimal form of the number 27 will terminate. 3 2 . 54 . 32 OR Express 429 as a product of its prime factors. 30/3/1 2 3. 4. 6 Ho$ W_ 10 JwUOm| H$m `moJ\$b kmV H$s{OE & Find the sum of first 10 multiples of 6. `{X {~ X A(0, 0) VWm {~ X B(x, 4) Ho$ ~rM H$s X ar 5 BH$mB h , Vmo x Ho$ _mZ kmV H$s{OE & 5. 6. Find the value(s) of x, if the distance between the points A(0, 0) and B(x, 4) is 5 units. { `mE a VWm b (a > b) Ho$ Xmo g Ho$ r d m {XE JE h & ~ S>o d m H$s Ordm, Omo N>moQ>o d m H$s ne -aoIm h , H$s b ~mB kmV H$s{OE & Two concentric circles of radii a and b (a > b) are given. Find the length of the chord of the larger circle which touches the smaller circle. AmH ${V 1 _|, PS = 3 go_r, QS = 4 go_r, PRQ = , PSQ = 90 , PQ RQ VWm RQ = 9 go_r h & tan H$m _mZ kmV H$s{OE & AmH ${V 1 AWdm 5 h , Vmo sec H$m _mZ kmV H$s{OE & 12 In Figure 1, PS = 3 cm, QS = 4 cm, PRQ = , PSQ = 90 , PQ RQ and RQ = 9 cm. Evaluate tan . `{X tan = Figure 1 OR If tan = 30/3/1 5 , find the value of sec . 12 3 P.T.O. I S> ~ SECTION B Z g `m 7 go 12 VH$ `oH$ Z Ho$ 2 A H h & Question numbers 7 to 12 carry 2 marks each. 7. q~X A(3, 1), B(5, 1), C(a, b) VWm D(4, 3) EH$ g_m Va MVw^w O ABCD Ho$ erf q~X h & a VWm b Ho$ _mZ kmV H$s{OE & AWdm q~X Am| A( 2, 0) VWm B(0, 8) H$mo Omo S>Zo dmbo aoImI S> H$mo q~X P VWm q~X Q g_{ ^m{OV H$aVo h , Ohm P q~X A Ho$ {ZH$Q> h & q~X Am| P VWm Q Ho$ {ZX oem H$ kmV H$s{OE & Points A(3, 1), B(5, 1), C(a, b) and D(4, 3) are vertices of a parallelogram ABCD. Find the values of a and b. OR Points P and Q trisect the line segment joining the points A( 2, 0) and B(0, 8) such that P is near to A. Find the coordinates of points P and Q. 8. {Z Z a {IH$ g_rH$aU `w _ H$mo hb H$s{OE : 3x 5y = 4 2y + 7 = 9x Solve the following pair of linear equations : 3x 5y = 4 2y + 7 = 9x 9. `{X 65 VWm 117 Ho$ _.g. H$m _mZ kmV H$s{OE & (HCF) H$mo 65n 117 Ho$ $n _| Xem `m Om gH$Vm h , Vmo n AWdm VrZ bmoJ gw~h H$s g a Ho$ {bE EH$ gmW ~mha {ZH$bo Am a CZHo$ H$X_ H$s b ~mB H $_e: 30 cm, 36 cm VWm 40 cm h & `oH$ H$mo `yZV_ {H$VZr X ar V` H$aZr hmoJr {H$ g^r AnZo nyU H$X_m| _| g_mZ X ar Mb| ? If HCF of 65 and 117 is expressible in the form 65n 117, then find the value of n. OR On a morning walk, three persons step out together and their steps measure 30 cm, 36 cm and 40 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps ? 30/3/1 4 10. EH$ nmgo H$mo EH$ ~ma \|$H$m OmVm h & m{`H$Vm kmV H$s{OE g `m h , (ii) m g `m EH$ A^m ` g `m h & (i) m g `m EH$ ^m ` A die is thrown once. Find the probability of getting (i) a composite number, (ii) a prime number. 11. nyU dJ ~ZmZo H$s {d{Y H$m `moJ H$aVo h E Xem BE {H$ g_rH$aU H$moB hb Zht h & Using completing the square x2 8x + 18 = 0 has no solution. 12. method, show x2 8x + 18 = 0 that the H$m equation H$mS> {OZ na 7 go 40 VH$ H$s g `mE {bIr h , EH$ noQ>r _| aIo h E h & nyZ_ CZ_| go EH$ H$mS> `m N>`m {ZH$mbVr h & m{`H$Vm kmV H$s{OE {H$ nyZ_ mam {ZH$mbo JE H$mS> na A {H$V g `m 7 H$m EH$ JwUO h & Cards numbered 7 to 40 were put in a box. Poonam selects a card at random. What is the probability that Poonam selects a card which is a multiple of 7 ? I S> g SECTION C Z g `m 13 go 22 VH$ `oH$ Z Ho$ 3 A H$ h & Question numbers 13 to 22 carry 3 marks each. 13. {H$gr { ^wO ABC Ho$ erf A go ^wOm BC na S>mbm J`m b ~ BC H$mo q~X D na Bg H$ma {_bVm h {H$ DB = 3CD h & {g H$s{OE {H$ 2AB2 = 2AC2 + BC2. AWdm AD Am a PM { ^wOm| ABC Am a PQR H$s H $_e: _mp `H$mE h O~{H$ ABC PQR AD AB = h & PM PQ The perpendicular from A on side BC of a ABC meets BC at D such that h & {g H$s{OE {H$ DB = 3CD. Prove that 2AB2 = 2AC2 + BC2. OR 14. AD and PM are medians of triangles ABC and PQR respectively where AB AD ABC PQR. Prove that = . PQ PM ~h nX p(x) H$mo ~h nX g(x) go ^mJ H$aHo$ Om M H$s{OE {H$ `m g(x) ~h nX p(x) H$m EH$ JwUZI S> h & {X`m J`m h {H$ p(x) = x5 4x3 + x2 + 3x + 1, g(x) = x3 3x + 1 Check whether g(x) is a factor of p(x) by dividing polynomial p(x) by polynomial g(x), where p(x) = x5 4x3 + x2 + 3x + 1, g(x) = x3 3x + 1 30/3/1 5 P.T.O. 15. 16. erfm o A(0, 1), B(2, 1) Am a C(0, 3) dmbo { ^wO ~ZZo dmbo { ^wO H$m jo \$b kmV H$s{OE & ABC H$s ^wOmAm| Ho$ _ `-{~ X Am| go Find the area of the triangle formed by joining the mid-points of the sides of the triangle ABC, whose vertices are A(0, 1), B(2, 1) and C(0, 3). g_rH$aUm| x y + 1 = 0 Am a 3x + 2y 12 = 0 H$m J m\$ It{ME & J m\$ mam, x Am a y Ho$ XmoZm| g_rH$aUm| H$mo g Vw Q> H$aZo dmbo _mZ kmV H$s{OE & Draw the graph of the equations x y + 1 = 0 and 3x + 2y 12 = 0. Using this graph, find the values of x and y which satisfy both the equations. 17. {g H$s{OE {H$ EH$ An[a_o` g `m h & AWdm dh ~ S>r-go-~ S>r g `m kmV H$s{OE {Oggo g `mAm| H$aZo na H $_e: 1, 2 VWm 3 eof\$b AmVm h & 3 1251, 9377 VWm 15628 H$mo ^mJ 3 is an irrational number. OR Find the largest number which on dividing 1251, 9377 and 15628 leaves remainders 1, 2 and 3 respectively. A, B Am a C { ^wO ABC Ho$ A V: H$moU h & {XImBE {H$ Prove that 18. (i) A B C sin = cos 2 2 (ii) `{X A = 90 h , Vmo B C tan H$m 2 _mZ kmV H$s{OE & AWdm `{X tan (A + B) = 1 VWm tan (A B) = 1 3 h , Ohm 0 < A + B < 90 , A > B h , A VWm B Ho$ _mZ kmV H$s{OE & A, B and C are interior angles of a triangle ABC. Show that Vmo (i) A B C sin = cos 2 2 B C If A = 90 , then find the value of tan . 2 OR 1 If tan (A + B) = 1 and tan (A B) = , 0 < A + B < 90 , A > B, then find 3 the values of A and B. (ii) 30/3/1 6 19. AmH ${V 2 _|, 5 go_r { `m Ho$ EH$ d m H$s 8 go_r b ~r EH$ Ordm PQ h & P Am a ne -aoImE na na EH$ q~X T na {V N>oX H$aVr h & TP H$s b ~mB kmV H$s{OE & Q na AmH ${V 2 AWdm {g H$s{OE {H$ d m Ho$ n[aJV ~Zr MVw^w O H$s Am_Zo-gm_Zo H$s ^wOmE , d m Ho$ Ho$ na g nyaH$ H$moU A V[aV H$aVr h & In Figure 2, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP. Figure 2 OR Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. 20. 6 _r. Mm S>r Am a 1 5 30 {_ZQ> _|, `h Zha _r. Jhar EH$ Zha _| nmZr 10 {H$_r/K . H$s Mmb go ~h ahm h & {H$VZo jo \$b H$s qgMmB H$a nmEJr O~{H$ qgMmB Ho$ {bE 8 go_r Jhao R>hao h E nmZr H$s Amd `H$Vm hmoVr h ? Water in a canal, 6 m wide and 1 5 m deep, is flowing with a speed of 10 km/h. How much area will it irrigate in 30 minutes if 8 cm of standing water is needed ? 30/3/1 7 P.T.O. 21. {H$gr H$jm A `m{nH$m Zo nyao g Ho$ {bE AnZr H$jm Ho$ 40 {d m{W `m| H$s AZwnp W{V {Z Z{b{IV $n _| [aH$m S> H$s & EH$ {d mWu {OVZo {XZ AZwnp WV ahm CZH$m _m ` kmV H$s{OE & {XZm| H$s g `m : {d m{W `m| H$s g `m : 0 6 6 12 10 11 12 18 18 24 24 30 30 36 36 42 7 4 4 3 1 A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent. 22. Number of days : 0 6 6 12 Number of students : 10 11 12 18 18 24 24 30 30 36 36 42 7 4 4 3 1 {H$gr H$ma Ho$ Xmo dmBna (wipers) h , Omo na na H$^r Am N>m{XV Zht hmoVo h & `oH$ dmBna H$s n mr H$s b ~mB 21 go_r h Am a 120 Ho$ H$moU VH$ Ky_H$a g\$mB H$a gH$Vm h & n{ m`m| H$s `oH$ ~whma Ho$ gmW {OVZm jo \$b gm\$ hmo OmVm h , dh kmV H$s{OE & ( 22 br{OE) 7 A car has two wipers which do not overlap. Each wiper has a blade of length 21 cm sweeping through an angle 120 . Find the total area cleaned 22 at each sweep of the blades. (Take ) 7 I S> X SECTION D Z g `m 23 go 30 VH$ `oH$ Z Ho$ 4 A H$ h & Question numbers 23 to 30 carry 4 marks each. 23. 13 _rQ>a `mg dmbo EH$ d mmH$ma nmH $ H$s n[agr_m Ho$ EH$ q~Xw na EH$ I ^m Bg H$ma Jm S>Zm h {H$ Bg nmH $ Ho$ EH$ `mg Ho$ XmoZm| A V q~X Am| na ~Zo \$mQ>H$m| A Am a B go I ^o H$s X [a`m| H$m A Va 7 _rQ>a hmo & `m Eogm H$aZm g ^d h ? `{X h , Vmo XmoZm| \$mQ>H$m| go {H$VZr X [a`m| na I ^m Jm S>Zm h ? A pole has to be erected at a point on the boundary of a circular park of diameter 13 m in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 m. Is it possible to do so ? If yes, at what distances from the two gates should the pole be erected ? 30/3/1 8 24. `{X {H$gr g_m Va lo T>r Ho$ md| nX H$m m JwUm, BgHo$ nd| nX Ho$ n JwUm Ho$ ~am~a hmo (m n), Vmo Xem BE {H$ g_m Va lo T>r H$m (m + n)dm nX ey ` hmoJm & AWdm {H$gr g_m Va lo T>r H$s W_ VrZ g `mAm| H$m `moJ\$b 18 h & `{X nhbo Am a Vrgao nX H$m JwUZ\$b gmd A Va H$m 5 JwUm hmo, Vmo VrZm| g `mAm| H$mo kmV H$s{OE & If m times the mth term of an Arithmetic Progression is equal to n times its nth term and m n, show that the (m + n)th term of the A.P. is zero. OR The sum of the first three numbers in an Arithmetic Progression is 18. If the product of the first and the third term is 5 times the common difference, find the three numbers. 25. EH$ { ^wO ABC H$s aMZm H$s{OE {Og_| ^wOm BC = 6 go_r, AB = 5 go_r Am a ABC = 60 hmo & {\$a EH$ A ` { ^wO H$s aMZm H$s{OE {OgH$s ^wOmE ABC H$s g JV ^wOmAm| H$s 3 JwZr hm| & 4 Construct a triangle ABC with side BC = 6 cm, AB = 5 cm and 3 ABC = 60 . Then construct another triangle whose sides are of the 4 corresponding sides of the triangle ABC. 26. AmH ${V 3 _|, gOmdQ> Ho$ {bE ~Zm EH$ bm H$ Xem `m J`m h Omo Xmo R>mogm o EH$ KZ VWm EH$ AY Jmobo go ~Zm h & bm H$ H$m AmYma EH$ 6 go_r ^wOm H$m KZ h VWm CgHo$ D$na EH$ AY Jmobm h {OgH$m `mg 4 2 go_r h & kmV H$s{OE (a) bm H$ H$m Hw$b n R>r` jo \$b & (b) ~Zo h E bm H$ H$m Am`VZ & ( 22 br{OE) 7 AmH ${V 3 AWdm 30/3/1 9 P.T.O. D$na go Iwbr EH$ ~m Q>r e Hw$ Ho$ {N> H$ Ho$ AmH$ma H$s h {OgH$s Ym[aVm 12308 8 go_r3 h & CgHo$ D$nar VWm {ZMbo d mmH$ma {gam| H$s { `mE H $_e: 20 go_r VWm 12 go_r h & ~m Q>r H$s D $MmB kmV H$s{OE VWm ~m Q>r H$mo ~ZmZo _| bJr YmVw H$s MmXa H$m jo \$b kmV H$s{OE & ( = 3 14 H$m `moJ H$s{OE) In Figure 3, a decorative block is shown which is made of two solids, a cube and a hemisphere. The base of the block is a cube with edge 6 cm and the hemisphere fixed on the top has a diameter of 4 2 cm. Find (a) the total surface area of the block. (b) the volume of the block formed. (Take 22 ) 7 Figure 3 OR A bucket open at the top is in the form of a frustum of a cone with a capacity of 12308 8 cm3. The radii of the top and bottom circular ends are 20 cm and 12 cm respectively. Find the height of the bucket and the area of metal sheet used in making the bucket. (Use = 3 14) 27. `{X {H$gr { ^wO H$s EH$ ^wOm Ho$ g_m Va A ` Xmo ^wOmAm| H$mo {^ -{^ q~X Am| na {V N>oX H$aZo Ho$ {bE EH$ aoIm ItMr OmE, Vmo {g H$s{OE {H$ `o A ` Xmo ^wOmE EH$ hr AZwnmV _| {d^m{OV hmo OmVr h & AWdm {g H$s{OE {H$ EH$ g_H$moU { ^wO _| H$U H$m dJ eof Xmo ^wOmAm| Ho$ dJm o Ho$ `moJ\$b Ho$ ~am~a hmoVm h & 30/3/1 10 If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio. 28. 29. OR Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. `{X 1 + sin2 = 3 sin cos h , Vmo {g H$s{OE {H$ tan = 1 AWdm tan = 1 . 2 1 If 1 + sin2 = 3 sin cos , then prove that tan = 1 or tan = . 2 ZrMo {XE JE ~ Q>Z H$mo go A{YH$ H$ma Ho$ ~ Q>Z _| ~X{bE Am a {\$a Cg ~ Q>Z H$m go A{YH$ H$ma H$m VmoaU It{ME & dJ A Vamb : ~ma ~maVm : 20 30 30 40 40 50 50 60 60 70 70 80 80 90 10 8 12 24 6 25 15 Change the following distribution to a more than type distribution. Hence draw the more than type ogive for this distribution. Class interval : Frequency : 30. 20 30 30 40 40 50 50 60 60 70 70 80 80 90 10 8 12 24 6 25 15 EH$ g_Vb O_rZ na I S>r _rZma H$s N>m`m Cg p W{V _| 40 _r. A{YH$ b ~r hmo OmVr h O~{H$ gy` H$m C Vm e (altitude) 60 go KQ>H$a 30 hmo OmVm h & _rZma H$s D $MmB kmV H$s{OE & ({X`m J`m h 3 1 732 ) The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun s altitude is 30 than when it was 60 . Find the height of the tower. (Given 3 1 732 ) 30/3/1 11 P.T.O.

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