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CBSE Board Exam Class 10 2019 : Mathematics (Series 4)

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SET-2 H$moS> Z . Series JMS/4 Code No. amob Z . 30/4/2 narjmWu H$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 30/4/2 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$m| dmbo Xmo Zm| _|, 3 A H$m| dmbo Mma Zm| _| Am a 4 A H$m| dmbo VrZ Zm| _| Am V[aH$ {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$m h & Question numbers 1 to 6 carry 1 mark each. 1. k Ho$ {H$Z _mZm| Ho$ {bE { KmV g_rH$aU Zht h ? 4x2 12x k = 0 Ho$ H$moB dm V{dH$ _yb For what values of k does the quadratic equation 4x2 12x k = 0 have no real roots ? 2. q~X Am| (a, b) VWm ( a, b) Ho$ ~rM H$s X ar kmV H$s{OE & Find the distance between the points (a, b) and ( a, b). 3. 2 7 Ho$ ~rM p WV EH$ n[a_o` g `m kmV H$s{OE & AWdm g `m 22 53 32 17 H$mo gab $n _| {bIZo na, BgHo$ A V _| {H$VZo ey ` AmE Jo, {b{IE & 30/4/2 VWm 2 Find a rational number between 2 and 7 . OR Write the number of zeroes in the end of a number whose prime factorization is 22 53 32 17. 4. _mZm ABC DEF h VWm CZHo$ jo \$b H $_e: h & `{X EF = 15 4 go_r h , Vmo BC kmV H$s{OE & 64 dJ go_r VWm 121 dJ go_r Let ABC DEF and their areas be respectively, 64 cm2 and 121 cm2. If EF = 15 4 cm, find BC. 5. _mZ kmV H$s{OE : tan 65 cot 25 AWdm (sin 67 + cos 75 ) H$mo 0 go 45 Ho$ ~rM Ho$ H$moU Ho$ { H$moU{_Vr` AZwnmVm| Ho$ nXm| _| ` $ H$s{OE & Evaluate : 6. tan 65 cot 25 OR Express (sin 67 + cos 75 ) in terms of trigonometric ratios of the angle between 0 and 45 . g_m Va lo T>r : 18, 15 1 , 13, ..., 47 Ho$ nXm| H$s g `m kmV H$s{OE & 2 1 Find the number of terms in the A.P. : 18, 15 , 13, ..., 47. 2 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. EH$ W bo _| 15 J|X| h {OZ_| go Hw$N> g\o$X VWm A ` H$mbo a J H$s h & `{X Bg W bo _| go `m N>`m EH$ H$mbo a J H$s J|X {ZH$mbZo H$s m{`H$Vm 2 3 h , Vmo W bo _| {H$VZr g\o$X J|X| ? A bag contains 15 balls, out of which some are white and the others are black. If the probability of drawing a black ball at random from the bag is 2 , then find how many white balls are there in the bag. 3 h 30/4/2 3 P.T.O. 8. 52 n mm| H$s Vme H$s J >r _| go `m N>`m EH$ n mm {ZH$mbm J`m & EH$ Eogo n mo Ho$ AmZo H$s m{`H$Vm kmV H$s{OE Omo Z Vmo h Hw$_ H$m n mm hmo Am a Z hr ~mXemh hmo & A card is drawn at random from a pack of 52 playing cards. Find the probability of drawing a card which is neither a spade nor a king. 9. {Z Z g_rH$aU `w _ H$m hb kmV H$s{OE : 3 8 1 2 1; 2, x y x y x, y 0 AWdm k Ho$ do _mZ kmV H$s{OE {OZHo$ {bE g_rH$aU `w _ kx 2y 3 3x 6y 10 H$m EH$ A{ Vr` hb h & Find the solution of the pair of equations : 3 8 1 2 1; 2, x, y 0 x y x y OR kx 2y 3 Find the value(s) of k for which the pair of equations 3x 6y 10 has a unique solution. 10. 10 Am a 205 Ho$ ~rM 4 Ho$ {H$VZo JwUO p WV h AWdm dh g_m Va lo T>r kmV H$s{OE {OgH$m Vrgam nX go 12 A{YH$ h & ? 16 h VWm {OgH$m 7dm nX BgHo$ 5d| nX How many multiples of 4 lie between 10 and 205 ? OR Determine the A.P. whose third term is 16 and 7th term exceeds the 5th term by 12. 11. `yp bS> {d^mOZ Eo Jmo[a _ Ho$ `moJ go 255 VWm 867 H$m _.g. (HCF) kmV Use Euclid s division algorithm to find the HCF of 255 and 867. 30/4/2 4 H$s{OE & 12. q~X R aoImI S> AB, Ohm A( 4, 0) B(0, 6) VWm h , H$mo Bg H$ma {d^m{OV H$aVm h 3 AB h & R Ho$ {ZX oem H$ kmV H$s{OE & 4 The point R divides the line segment AB, where A( 4, 0) and B(0, 6) 3 such that AR AB . Find the coordinates of R. 4 {H$ AR 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$s{OE {H$ : (sin + 1 + cos ) (sin 1 + cos ) . sec cosec = 2 AWdm {g H$s{OE {H$ : sec 1 sec 1 sec 1 2 cosec sec 1 Prove that : (sin + 1 + cos ) (sin 1 + cos ) . sec cosec = 2 OR Prove that : sec 1 sec 1 14. sec 1 2 cosec sec 1 dh AZwnmV kmV H$s{OE {Og_| q~X P( 4, y), q~X Am| A( 6, 10) VWm B(3, 8) H$mo {_bmZo dmbo aoImI S> H$mo {d^m{OV H$aVm h & AV: y H$m _mZ kmV H$s{OE & AWdm . p H$m dh _mZ kmV H$s{OE {OgHo$ {bE q~X ( 5, 1), (1, p) VWm (4, 2) gaoI h & In what ratio does the point P( 4, y) divide the line segment joining the points A( 6, 10) and B(3, 8) ? Hence find the value of y. OR Find the value of p for which the points ( 5, 1), (1, p) and (4, 2) are collinear. 30/4/2 5 P.T.O. 15. g_H$moU { ^wO h {Og_ o B = 90 h & `{X AB = 8 go_r VWm h , Vmo Bg { ^wO Ho$ A VJ V ItMo JE d m H$m `mg kmV H$s{OE & ABC EH$ BC = 6 go_r ABC is a right triangle in which B = 90 . If AB = 8 cm and BC = 6 cm, find the diameter of the circle inscribed in the triangle. 16. AmH ${V 1 _|, BL VWm CM, ABC, {Og_| H$s{OE {H$ 4 (BL2 + CM2) = 5 BC2. A g_H$moU h , H$s _mp `H$mE h & {g AmH ${V 1 AWdm {g H$s{OE {H$ EH$ g_MVw^w O H$s ^wOmAm| Ho$ dJm o H$m `moJ\$b BgHo$ {dH$Um o Ho$ dJm o Ho$ `moJ\$b Ho$ ~am~a hmoVm h & In Figure 1, BL and CM are medians of a ABC right-angled at A. Prove that 4 (BL2 + CM2) = 5 BC2. Figure 1 OR Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals. 30/4/2 6 17. _|, Xmo g Ho $ r d mm| H$s { `mE 21 go_r VWm 42 go_r h VWm Ho$ AOB = 60 h , Vmo N>m`m {H$V jo H$m jo \$b kmV H$s{OE & AmH ${V 2 O h & `{X AmH ${V 2 In Figure 2, two concentric circles with centre O, have radii 21 cm and 42 cm. If AOB = 60 , find the area of the shaded region. Figure 2 18. {Z Z ~ Q>Z H$m ~h bH$ dJ : ~ma ~maVm : (mode) n[aH${bV H$s{OE : 10 15 15 20 20 25 25 30 30 35 4 7 20 8 1 Calculate the mode of the following distribution : Class : Frequency : 30/4/2 10 15 15 20 20 25 25 30 30 35 4 7 20 8 1 7 P.T.O. 19. _m S>b ~ZmZo dmbr {_ >r go ~Zo EH$ e Hw$ H$s D $MmB 24 go_r VWm AmYma H$s { `m 6 go_r h & EH$ ~ m BgH$m AmH$ma ~Xb H$a Bgo EH$ Jmobo _| ~Xb XoVm h & Bg Jmobo H$s { `m kmV H$s{OE, AV: Bg Jmobo H$m n R>r` jo \$b kmV H$s{OE & AWdm EH$ {H$gmZ AnZo IoV _| ~Zr 10 _r. `mg dmbr VWm 2 _r. Jhar EH$ ~obZmH$ma Q> H$s H$mo Am V[aH$ `mg 20 go_r dmbo EH$ nmBn mam EH$ Zha go Omo S>Vm h & `{X nmBn _| nmZr 3 {H$_r/K Q>m H$s Mmb go ~h ahm h , Vmo {H$VZo g_` ~mX Q> H$s nyar ^a OmEJr ? A cone of height 24 cm and radius of base 6 cm is made up of modelling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere and hence find the surface area of this sphere. OR A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in his field which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 km/hr, in how much time will the tank be filled ? 20. {g H$s{OE {H$ 2 An[a_o` g `m h & + 3 3 EH$ An[a_o` g `m h O~{H$ {X`m J`m h {H$ Prove that 2 + 3 3 is an irrational number when it is given that an irrational number. 21. Xmo dJm o Ho$ jo \$bm| H$m `moJ\$b 157 dJ _r. h & `{X CZHo$ n[a_mnm| H$m `moJ\$b hmo, Vmo XmoZm| dJm o H$s ^wOmE kmV H$s{OE & 3 EH$ 3 is 68 _r. Sum of the areas of two squares is 157 m2. If the sum of their perimeters is 68 m, find the sides of the two squares. 22. EH$ { KmV ~h nX kmV H$s{OE {OgHo$ ey `H$m| H$m `moJ\$b VWm JwUZ\$b H $_e: 20 hmo & Bg ~h nX Ho$ ey `H$ ^r kmV H$s{OE & 1 VWm Find the quadratic polynomial, sum and product of whose zeroes are 1 and 20 respectively. Also find the zeroes of the polynomial so obtained. 30/4/2 8 I S> X SECTION D Z g `m 23 go 30 VH$ `oH$ Z Ho$ 4 A H$ h & Question numbers 23 to 30 carry 4 marks each. 23. EH$ {d_mZ AnZo {ZYm [aV g_` go 30 {_ZQ> H$s Xoar go MbVm h & 1500 {H$_r H$s X ar na g_` na nh MZo Ho$ {bE Cgo AnZr Mmb {ZYm [aV Mmb go 250 {H$_r/K Q>m ~ T>mZr n S>Vr h & {d_mZ H$s gm_m ` Mmb kmV H$s{OE & AWdm EH$ Am`VmH$ma nmH $ H$s {d_mE kmV H$s{OE {OgH$m n[a_mn 60 _rQ>a VWm jo \$b 200 _r.2 h & A plane left 30 minutes later than the scheduled time and in order to reach its destination 1500 km away on time, it has to increase its speed by 250 km/hr from its usual speed. Find the usual speed of the plane. OR Find the dimensions of a rectangular park whose perimeter is 60 m and area 200 m2. 24. x H$m _mZ kmV H$s{OE O~{H$ {Z Z 2 + 6 + 10 + ... + x = 1800 h & g_m Va lo T>r Find the value of x, when in the A.P. given below 2 + 6 + 10 + ... + x = 1800. 25. `{X sec + tan = m h , If sec + tan = m, show that 26. AmH ${V 3 _|, ABC _| m2 1 Vmo Xem BE {H$ AD BC h m2 1 m2 1 m2 1 = sin . = sin . & {g H$s{OE {H$ AC2 = AB2 + BC2 2BC BD AmH ${V 3 30/4/2 9 P.T.O. In ABC (Figure 3), AD BC. Prove that AC2 = AB2 + BC2 2BC BD Figure 3 27. 150 _r. D $Mr EH$ nhm S> H$s MmoQ>r go, BgHo$ AmYma go X a OmVr h B EH$ Zmd H$mo XoIm J`m & BgH$m AdZ_Z H$moU 2 {_ZQ> _ o 60 go 45 hmo OmVm h & Zmd H$s _r./{_ZQ> _| Mmb kmV H$s{OE & AWdm EH$ ZXr Ho$ `oH$ {H$Zmao na EH$-X gao Ho$ g _wI Xmo I ^o I S>o h & EH$ I ^o H$s D $MmB 60 _r. h VWm Bg I ^o Ho$ {eIa go X gao I ^o Ho$ {eIa VWm nmX Ho$ AdZ_Z H$moU H $_e: 30 VWm 60 h & ZXr H$s Mm S>mB VWm X gao I ^o H$s D $MmB kmV H$s{OE & A moving boat is observed from the top of a 150 m high cliff moving away from the cliff. The angle of depression of the boat changes from 60 to 45 in 2 minutes. Find the speed of the boat in m/min. OR There are two poles, one each on either bank of a river just opposite to each other. One pole is 60 m high. From the top of this pole, the angle of depression of the top and foot of the other pole are 30 and 60 respectively. Find the width of the river and height of the other pole. 28. EH$ { ^wO H$s aMZm H$s{OE {OgH$s ^wOmE 5 go_r, 6 go_r VWm 7 go_r h Am a A~ EH$ A ` { ^wO H$s aMZm H$s{OE {OgH$s ^wOmE nhbo ~Zr { ^wO H$s g JV ^wOmAm| H$s 3 JwZr h & 5 Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another 3 triangle whose sides are of the corresponding sides of the first 5 triangle. 30/4/2 10 29. {Z Z ~ma ~maVm ~ Q>Z H$m _m ` n[aH${bV H$s{OE : 10 30 30 50 50 70 70 90 90 110 110 130 dJ : 5 8 12 20 3 2 ~ma ~maVm : AWdm {Z Z{b{IV gmaUr {H$gr Jm d Ho$ 100 \$m_m o _| h Am {H$J m {V h Q>o`a _| Joh H$m C nmXZ Xem Vr h : C nmXZ ({H$J m/ h Q>o`a) : 40 45 45 50 50 55 55 60 60 65 65 70 4 6 16 20 30 24 \$m_m o H$s g `m : Bg ~ Q>Z H$mo go A{YH$ H$ma Ho$ ~ Q>Z _| ~X{bE Am a {\$a CgH$m VmoaU It{ME & Calculate the mean of the following frequency distribution : Class : 10 30 30 50 50 70 5 8 12 Frequency : 70 90 90 110 110 130 20 3 2 OR The following table gives production yield in kg per hectare of wheat of 100 farms of a village : Production yield (kg/hectare) : Number of farms : 40 45 45 50 50 55 55 60 60 65 65 70 4 6 16 20 30 24 Change the distribution to a more than type distribution, and draw its ogive. 30. YmVw H$s MmXa go ~Zm, D$na go Iwbm EH$ ~V Z e Hw$ Ho$ {N> H$ Ho$ AmH$ma H$m h {OgH$s D $MmB 16 go_r h VWm {ZMbo VWm D$nar {gam| H$s { `mE H $_e: 8 go_r VWm 20 go_r h & Bg_| nyar Vah go ^ao Om gH$Zo dmbo < 50 {V {bQ>a dmbo X Y H$m _y ` kmV H$s{OE & Bg ~V Z H$mo ~ZmZo _| bJr YmVw H$s MmXa H$m _y ` kmV H$s{OE O~{H$ BgH$s Xa < 10 {V 100 dJ go_r h & ( = 3 14 br{OE) A container opened at the top and made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of milk which can completely fill the container, at the rate of < 50 per litre. Also find the cost of metal sheet used to make the container, if it costs < 10 per 100 cm2. (Take = 3 14) 30/4/2 11 P.T.O.

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