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CBSE Board Exam Class 10 2019 : Mathematics (Series 5)

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SET-1 H$moS> Z . Series JMS/5 Code No. amob Z . 30/5/1 narjmWu H$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 30/5/1 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$m h & Question numbers 1 to 6 carry 1 mark each. 1. Xmo g `mAm| a VWm b H$m _.g. JwUZ\$b ab kmV H$s{OE & (HCF) 5 VWm CZH$m b.g. (LCM) 200 h & The HCF of two numbers a and b is 5 and their LCM is 200. Find the product ab. 2. k H$m dh _mZ kmV H$s{OE, {OgHo$ {bE x=2 g_rH$aU kx2 + 2x 3 = 0 H$m EH$ hb 3x2 + kx + 3 = 0 Ho$ _yb h & AWdm k Ho$ do _mZ kmV H$s{OE, {OZHo$ {bE { KmV g_rH$aU dm V{dH$ VWm g_mZ hm o & 30/5/1 2 Find the value of k for which x = 2 is a solution of the equation kx2 + 2x 3 = 0. OR Find the value/s of k for which the quadratic equation 3x2 + kx + 3 = 0 has real and equal roots. 3. `{X EH$ g_m Va lo T>r _| a = 15, d = 3 VWm an = 0 h , Vmo n H$m _mZ kmV H$s{OE If in an A.P., a = 15, d = 3 and an = 0, then find the value of n. 4. `{X sin x + cos y = 1; x = 30 VWm y EH$ `yZ H$moU h , Vmo y H$m _mZ kmV H$s{OE & AWdm (cos 48 sin 42 ) H$m _mZ kmV H$s{OE & & If sin x + cos y = 1; x = 30 and y is an acute angle, find the value of y. OR Find the value of (cos 48 sin 42 ). 5. Xmo g_ $n { ^wOm| Ho$ jo \$b AZwnmV kmV H$s{OE & 25 dJ go_r VWm 121 dJ go_r h & BZH$s g JV ^wOmAm| H$m The area of two similar triangles are 25 sq. cm and 121 sq. cm. Find the ratio of their corresponding sides. 6. `{X q~X (3, a), 2x 3y = 5 mam {Z ${nV aoIm na p WV h , Vmo a H$m _mZ kmV H$s{OE & Find the value of a so that the point (3, a) lies on the line represented by 2x 3y = 5. I S> ~ SECTION B Z g `m 7 go 12 VH$ `oH$ Z Ho$ 2 A H h & Question numbers 7 to 12 carry 2 marks each. 7. `{X EH$ g_m Va lo T>r Ho$ W_ BgH$m ndm nX kmV H$s{OE & n nXm| H$m `moJ\$b Sn, Sn = 2n2 + n AWdm `{X EH$ g_m Va lo T>r H$m 17dm nX BgHo$ 10do nX go H$s{OE & 7 mam X m h , Vmo A{YH$ h , Vmo gmd A Va kmV If Sn, the sum of the first n terms of an A.P. is given by Sn = 2n2 + n, then find its nth term. OR If the 17th term of an A.P. exceeds its 10th term by 7, find the common difference. 30/5/1 3 P.T.O. 8. {~ X Am| A(2a, 4) VWm B( 2, 3b) H$mo {_bmZo dmbo aoImI S> H$m _ `-q~X h & a VWm b Ho$ _mZ kmV H$s{OE & (1, 2a + 1) The mid-point of the line segment joining A(2a, 4) and B( 2, 3b) is (1, 2a + 1). Find the values of a and b. 9. EH$ ~ o Ho$ nmg Eogm nmgm h {OgHo$ A B C A 6 A \$bH$m| na {Z Z{b{IV Aja A {H$V h : B Bg nmgo H$mo EH$ ~ma \ o$H$m OmVm h & BgH$s `m m{`H$Vm h {H$ hmo ? (i) A m hmo (ii) B m A child has a die whose 6 faces show the letters given below : A B C A A B The die is thrown once. What is the probability of getting (i) A (ii) B ? 10. A^m ` JwUZI S> {d{Y mam 612 VWm 1314 H$m _.g. (HCF) kmV AWdm Xem BE {H$ H$moB YZ {df_ nyUm H$ 6m + 1 `m 6m + 3 `m Ohm m H$moB nyUm H$ h & H$s{OE & 6m + 5 Ho$ $n _| hmoVm h , Find the HCF of 612 and 1314 using prime factorisation. 11. OR Show that any positive odd integer is of the form 6m + 1 or 6m + 3 or 6m + 5, where m is some integer. H$mS> {OZ na 5 go 50 VH$ H$s g `mE (EH$ H$mS> na EH$ g `m) A {H$V h H$mo EH$ ~ go _| S>mbH$a A N>r H$ma {_bm`m J`m & Bg ~ go _| go `m N>`m EH$ H$mS> {ZH$mbm J`m & m{`H$Vm kmV H$s{OE {H$ {ZH$mbo JE H$mS> na A {H$V g `m (i) 10 go H$_ H$s A^m ` g `m h , (ii) EH$ nyU dJ g `m h & Cards marked with numbers 5 to 50 (one number on one card) are placed in a box and mixed thoroughly. One card is drawn at random from the box. Find the probability that the number on the card taken out is (i) a prime number less than 10, (ii) a number which is a perfect square. 12. k Ho$ {H$g _mZ Ho$ {bE, a {IH$ g_rH$aU {ZH$m` 2x + 3y = 7 (k 1) x + (k + 2) y = 3k Ho$ An[a{_V $n go AZoH$ hb h ? For what value of k, does the system of linear equations 2x + 3y = 7 (k 1) x + (k + 2) y = 3k have an infinite number of solutions ? 30/5/1 4 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. {g H$s{OE {H$ Prove that 14. 15. 4 ~h nX x 2 VWm 5 EH$ An[a_o` g `m h & 5 is an irrational number. 3 + x 14x2 2x + 24 2 Ho$ g^r ey `H$ kmV H$s{OE O~{H$ {X`m J`m h {H$ BgHo$ Xmo ey `H$ h & Find all the zeroes of the polynomial x4 + x3 14x2 2x + 24, if two of its zeroes are 2 and 2 . q~X P, q~X Am| A(2, 1) VWm B(5, 8) H$mo {_bmZo dmbo aoImI S> H$mo Bg H$ma {d^m{OV H$aVm h {H$ AP 1 h & `{X P aoIm 2x y + k = 0 na p WV h , Vmo k H$m _mZ kmV AB 3 H$s{OE & AWdm p H$m dh _mZ kmV H$s{OE {OgHo$ {bE q~X (2, 1), (p, . 1) VWm ( 1, 3) gaoI h & Point P divides the line segment joining the points A(2, 1) and B(5, 8) AP 1 such that . If P lies on the line 2x y + k = 0, find the value of k. AB 3 OR For what value of p, are the points (2, 1), (p, 1) and ( 1, 3) collinear ? 16. {g H$s{OE {H$ : tan cot cos sin 1 tan 1 cot cos sin `{X cos + sin = AWdm 2 cos h , Vmo Xem BE {H$ cos sin = 2 sin h & Prove that : tan cot cos sin 1 tan 1 cot cos sin OR If cos + sin = 17. 2 cos , show that cos sin = 2 sin . EH$ H$m boO N>m mdmg (hostel) Ho$ _m{gH$ N>m mdmg `` H$m EH$ ^mJ {Z`V h VWm eof Bg na {Z^ a H$aVm h {H$ N>m Zo {H$VZo {XZ _og _| ^moOZ {b`m h & EH$ {d mWu A H$mo, Omo 25 {XZ ^moOZ H$aVm h , < 4,500 AXm H$aZo n S>Vo h O~{H$ EH$ {d mWu B H$mo, Omo 30 {XZ ^moOZ H$aVm h , < 5,200 AXm H$aZo n S>Vo h & _m{gH$ {Z`V `` Am a {V{XZ Ho$ ^moOZ H$m _y ` kmV H$s{OE & 30/5/1 5 P.T.O. A part of monthly hostel charges in a college hostel are fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 25 days, he has to pay < 4,500, whereas a student B who takes food for 30 days, has to pay < 5,200. Find the fixed charges per month and the cost of food per day. 18. EH$ ABC _|, B = 90 VWm q~X D ^wOm BC H$m _ `-q~X h & {g H$s{OE {H$ AC2 = AD2 + 3CD2. AWdm AmH ${V 1 _|, EH$ g_{ ~mh { ^wO ABC, {Og_| AB = AC h , H$s ~ T>mB JB ^wOm CB na EH$ q~X E p WV h & `{X AD BC VWm EF AC h , Vmo {g H$s{OE {H$ ABD ECF. AmH ${V 1 In ABC, B = 90 and D is the mid-point of BC. Prove that AC2 = AD2 + 3CD2. OR In Figure 1, E is a point on CB produced of an isosceles ABC, with side AB = AC. If AD BC and EF AC, prove that ABD ECF. Figure 1 30/5/1 6 19. {g H$s{OE {H$ {H$gr d m Ho$ n[aJV g_m Va MVw^w O g_MVw^w O hmoVm h & Prove that the parallelogram circumscribing a circle is a rhombus. 20. AmH ${V 2 _|, 7 go_r { `m dmbo d m Ho$ VrZ { `I S> Omo H|$ na 60 , 80 VWm 40 Ho$ H$moU ~ZmVo h , H$mo N>m`m {H$V {H$`m J`m h & N>m`m {H$V ^mJ H$m jo \$b kmV H$s{OE & AmH ${V 2 In Figure 2, three sectors of a circle of radius 7 cm, making angles of 60 , 80 and 40 at the centre are shaded. Find the the area of the shaded region. Figure 2 21. {Z Z Vm{bH$m EH$ `moJm H $ n _| ^mJ boZo dmbm| H$s g `m H$mo Xem Vr h Am`w (dfm o _|) : ^mJ boZo dmbm| H$s g `m : : 20 30 30 40 40 50 50 60 60 70 8 40 58 90 83 ^mJ boZo dmbm| H$s ~h bH$ Am`w kmV H$s{OE & The following table gives the number of participants in a yoga camp : Age (in years) : No. of Participants : 20 30 30 40 40 50 50 60 60 70 8 40 58 90 83 Find the modal age of the participants. 30/5/1 7 P.T.O. 22. EH$ Oyg ~oMZo dmbm AnZo J mhH$m| H$mo AmH ${V 3 _ o Xem E JE {Jbmgm| _| Oyg XoVm h & ~obZmH$ma {Jbmg H$m Am V[aH$ `mg 5 go_r Wm, na Vw {Jbmg Ho$ {ZMbo AmYma _| EH$ C^am h Am AY Jmobm Wm, {Oggo {Jbmg H$s Ym[aVm H$_ hmo OmVr Wr & `{X EH$ {Jbmg H$s D $MmB 10 go_r Wr, Vmo {Jbmg H$s Am^mgr Ym[aVm VWm CgH$s dm V{dH$ Ym[aVm kmV H$s{OE & ( = 3 14 `moJ H$s{OE) AmH ${V 3 AWdm EH$ b S>H$s aoV go ^ar EH$ ~obZmH$ma ~m Q>r H$mo, {OgH$s AmYma { `m 18 go_r VWm D $MmB 32 go_r h , \$e na Bg H$ma Imbr H$aVr h {H$ aoV H$m e dmH$ma T>oa ~ZVm h & `{X Bg e dmH$ma T>oa H$s D $MmB 24 go_r h , Vmo {V` H$ D $MmB (Xe_bd Ho$ 1 WmZ VH$ R>rH$) kmV H$s{OE & A juice seller was serving his customers using glasses as shown in Figure 3. The inner diameter of the cylindrical glass was 5 cm but bottom of the glass had a hemispherical raised portion which reduced the capacity of the glass. If the height of a glass was 10 cm, find the apparent and actual capacity of the glass. (Use = 3 14) Figure 3 OR A girl empties a cylindrical bucket full of sand, of base radius 18 cm and height 32 cm on the floor to form a conical heap of sand. If the height of this conical heap is 24 cm, then find its slant height correct to one place of decimal. 30/5/1 8 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. EH$ aobJm S>r 360 {H$_r H$s X ar EH$g_mZ Mmb go V` H$aVr h & `{X `h Mmb 5 {H$_r/K Q>m A{YH$ hmoVr, Vmo Cgr `m m _| 1 K Q>m H$_ g_` boVr & aobJm S>r H$s Mmb kmV H$s{OE & AWdm x Ho$ {bE hb H$s{OE : 1 1 1 1 ; a b 0, x 0, x (a + b) a b x a b x A train travels 360 km at a uniform speed. If the speed had been 5 km/hr more, it would have taken 1 hr less for the same journey. Find the speed of the train. OR Solve for x : 1 1 1 1 ; a b 0, x 0, x (a + b) a b x a b x 24. `{X EH$ g_m Va lo T>r Ho$ W_ p nXm| H$m `moJ\$b q h VWm W_ q nXm| H$m `moJ\$b Vmo Xem BE {H$ BgHo$ W_ (p + q) nXm| H$m `moJ\$b { (p + q)} hmoJm & p h ; If the sum of the first p terms of an A.P. is q and the sum of the first q terms is p; then show that the sum of the first (p + q) terms is { (p + q)}. 25. `{X {H$gr { ^wO _|, EH$ ^wOm H$m dJ , A ` Xmo ^wOmAm| Ho$ dJm o Ho$ `moJ\$b Ho$ ~am~a h , Vmo {g H$s{OE {H$ nhbr ^wOm H$m g _wI H$moU g_H$moU hmoJm & In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then prove that the angle opposite to the first side is a right angle. 30/5/1 9 P.T.O. 26. EH$ g_{ ~mh { ^wO H$s aMZm H$s{OE, {OgH$m AmYma 8 go_r VWm D $MmB 4 go_r h & A~ EH$ A ` { ^wO H$s aMZm H$s{OE {OgH$s ^wOmE Bg g_{ ~mh { ^wO H$s g JV ^wOmAm| H$s 3 JwZr hmo & 4 Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and 3 then another triangle whose sides are times the corresponding sides of 4 the isosceles triangle. 27. g_Vb na I S>m EH$ b S>H$m AnZo go 100 _r. H$s X ar na p WV EH$ C S>Vo h E njr H$m C `Z H$moU 30 nmVm h & EH$ b S>H$s, Omo EH$ 20 _r. D $Mo ^dZ Ho$ {eIa na I S>r h , Bgr njr H$m C `Z H$moU 45 nmVr h & b S>H$m VWm b S>H$s njr H$s {dnarV {XemAm| _| h & njr H$s b S>H$s go X ar kmV H$s{OE & ({X`m J`m h 2 = 1 414) AWdm ^y{_ na Ho$ EH$ q~X A go EH$ C S>Vo h E {d_mZ H$m C `Z H$moU 60 h & 30 goH$ S H$s C S>mZ Ho$ n MmV >, C `Z H$moU 30 hmo OmVm h & `{X `h {d_mZ EH$ AMa D $MmB 3600 3 _rQ>a na C S> ahm h , Vmo {d_mZ H$s Mmb kmV H$s{OE & A boy standing on a horizontal plane finds a bird flying at a distance of 100 m from him at an elevation of 30 . A girl standing on the roof of a 20 m high building, finds the elevation of the same bird to be 45 . The boy and the girl are on the opposite sides of the bird. Find the distance of the bird from the girl. (Given 2 = 1 414) OR The angle of elevation of an aeroplane from a point A on the ground is 60 . After a flight of 30 seconds, the angle of elevation changes to 30 . If the plane is flying at a constant height of 3600 3 metres, find the speed of the aeroplane. 28. {Z Z ~ma ~maVm ~ Q>Z gmaUr _| ~ma ~maVmE x VWm y Ho$ _mZ kmV H$s{OE O~{H$ VWm _m `H$ = 32 h & A H$ : {d m{W `m| H$s g `m : 0 10 10 20 20 30 30 40 40 50 50 60 10 x 25 AWdm 30/5/1 N = 100 10 30 y 10 `moJ 100 {Z Z ~ma ~maVm ~ Q>Z H$m EH$ go A{YH$ H$ma H$m g M`r ~ma ~maVm dH $ (VmoaU) It{ME, AV: Bg ~ Q>Z H$m _m `H$ _mZ kmV H$s{OE & dJ : 0 10 10 20 20 30 30 40 40 50 50 60 5 ~ma ~maVm : 15 20 23 17 60 70 11 9 Find the values of frequencies x and y in the following frequency distribution table, if N = 100 and median is 32. 0 10 10 20 20 30 30 40 40 50 50 60 Total Marks : 10 No. of Students : x 25 30 y 10 100 OR For the following frequency distribution, draw a cumulative frequency curve (ogive) of more than type and hence obtain the median value. Class : 0 10 10 20 20 30 30 40 40 50 50 60 Frequency : 29. 5 15 20 17 11 9 {g H$s{OE {H$ : (1 cot tan ) (sin cos ) 3 3 (sec cos ec ) Prove that : (1 cot tan ) (sin cos ) 3 3 (sec cos ec ) 30. 23 60 70 sin2 cos 2 sin2 cos 2 EH$ YmVw H$s D$na go Iwbr ~m Q>r, e Hw$ Ho$ {N> H$ Ho$ AmH$ma H$s h & `{X BgHo$ D$nar VWm {ZMbo d mr` {gam| Ho$ `mg H $_e: 45 go_r VWm 25 go_r h VWm ~m Q>r H$s grYr (D$ dm Ya) D $MmB 24 go_r h , Vmo Bg ~m Q>r H$mo ~ZmZo _| bJr YmVw H$s MmXa H$m jo \$b kmV H$s{OE & `h ^r kmV H$s{OE {H$ Bg_| {H$VZm nmZr Am gH$Vm h & ( 22 `moJ 7 H$s{OE) An open metallic bucket is in the shape of a frustum of a cone. If the diameters of the two circular ends of the bucket are 45 cm and 25 cm and the vertical height of the bucket is 24 cm, find the area of the metallic sheet used to make the bucket. Also find the volume of the water it can 22 hold. (Use ) 7 30/5/1 11 P.T.O.

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