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CBSE Board Exam Class 10 2019 : Mathematics (Series 2)

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SET-1 H$moS> Z . Series JMS/2 Code No. amob Z . 30/2/1 narjmWu H$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 30/2/1 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$m h & Question numbers 1 to 6 carry 1 mark each. 1. `{X _.g. (HCF) (336, 54) = 6 h , Vmo b.g. (LCM) (336, 54) kmV H$s{OE & If HCF (336, 54) = 6, find LCM (336, 54). 2. { KmV g_rH$aU 2x2 4x + 3 = 0 Ho$ _ybm| H$s H ${V kmV H$s{OE & Find the nature of roots of the quadratic equation 2x2 4x + 3 = 0. 1 3 a 3 2a , , , ... (a 0) Ho$ {bE gmd A Va kmV H$s{OE a 3a 3a Find the common difference of the Arithmetic Progression (A.P.) 1 3 a 3 2a , , , ... (a 0) a 3a 3a 3. g_m Va lo T>r 4. _mZ kmV H$s{OE : sin2 60 + 2 tan 45 cos2 30 AWdm 30/2/1 2 & `{X sin A = 3 4 h , Vmo sec A n[aH${bV H$s{OE & Evaluate : sin2 60 + 2 tan 45 cos2 30 OR 3 , calculate sec A. 4 p WV {~ Xw P Ho$ {ZX oem H$ If sin A = 5. x-Aj na g_mZ X ar na hmo & {b{IE Omo {~ X A( 2, 0) VWm {~ X B(6, 0) go Write the coordinates of a point P on x-axis which is equidistant from the points A( 2, 0) and B(6, 0). 6. _|, ABC EH$ g_{ ~mh { ^wO h {OgH$m H$moU AC = 4 cm h & AB H$s b ~mB kmV H$s{OE & AmH ${V 1 C g_H$moU h VWm AmH ${V 1 AWdm AmH ${V 2 _|, DE BC h & ^wOm AD H$s b ~mB kmV H$s{OE O~{H$ {X`m J`m h AE = 1 8 go_r, BD = 7 2 go_r VWm CE = 5 4 go_r & AmH ${V 2 In Figure 1, ABC is an isosceles triangle right angled at C with AC = 4 cm. Find the length of AB. Figure 1 OR 30/2/1 3 P.T.O. In Figure 2, DE BC. Find the length of side AD, given that AE = 1 8 cm, BD = 7 2 cm and CE = 5 4 cm. Figure 2 I S> ~ SECTION B Z g `m 7 go 12 VH$ `oH$ Z Ho$ 2 A H h & Question numbers 7 to 12 carry 2 marks each. 7. dh g~go N>moQ>r g `m {b{IE Omo 306 VWm 657 XmoZm| go nyU V`m {d^m{OV hmo & Write the smallest number which is divisible by both 306 and 657. 8. x Am a y _| EH$ g ~ Y kmV H$s{OE Vm{H$ {~ X A(x, y), B( 4, 6) VWm C( 2, 3) . gaoIr` hm| & AWdm Cg { ^wO H$m jo \$b kmV H$s{OE {OgHo$ erf (1, 1) ( 4, 6) VWm ( 3, 5) h & Find a relation between x and y if the points A(x, y), B( 4, 6) and C( 2, 3) are collinear. OR Find the area of a triangle whose vertices are given as (1, 1) ( 4, 6) and ( 3, 5). 9. EH$ Oma _| Ho$db Zrbo, H$mbo VWm hao H $Mo h & Bg Oma _| go `m N>`m EH$ Zrbo H $Mo Ho$ {ZH$mbZo H$s m{`H$Vm 1 h VWm Cgr Oma _| go EH$ H$mbo H $Mo Ho$ `m N>`m {ZH$mbZo H$s 5 1 m{`H$Vm h & `{X Oma _| 11 hao a J Ho$ H $Mo h , Vmo Oma _| Hw$b H $Mm| H$s g `m kmV 4 H$s{OE & The probability of selecting a blue marble at random from a jar that 1 contains only blue, black and green marbles is . The probability of 5 1 selecting a black marble at random from the same jar is . If the jar 4 contains 11 green marbles, find the total number of marbles in the jar. 30/2/1 4 10. k Ho$ {H$Z _mZm| ({H$g _mZ) Ho$ {bE {Z Z g_rH$aUm| Ho$ `w _ H$m EH$ A{ Vr` hb h x + 2y = 5 Am a 3x + ky + 15 = 0 : Find the value(s) of k so that the pair of equations x + 2y = 5 and 3x + ky + 15 = 0 has a unique solution. 11. Xmo g nyaH$ H$moUm| _| go ~ S>o H$moU H$m _mZ N>moQ>o H$moU Ho$ _mZ go 18 A{YH$ h & XmoZm| H$moUm| Ho$ _mZ kmV H$s{OE & AWdm gw{_V H$s Am`w CgHo$ ~oQ>o H$s Am`w H$s VrZ JwZr h & nm M df Ho$ ~mX, CgH$s Am`w AnZo ~oQ>o H$s Am`w H$s T>mB JwZm hmo OmEJr & Bg g_` gw{_V H$s Am`w {H$VZo df h ? The larger of two supplementary angles exceeds the smaller by 18 . Find the angles. OR Sumit is 3 times as old as his son. Five years later, he shall be two and a half times as old as his son. How old is Sumit at present ? 12. : {Z Z{b{IV ~ma ~maVm ~ Q>Z H$m ~h bH$ kmV H$s{OE dJ A Vamb : 25 30 30 35 35 40 40 45 45 50 50 55 25 34 50 42 38 14 ~ma ~maVm : Find the mode of the following frequency distribution : Class Interval : 25 30 30 35 35 40 40 45 45 50 50 55 25 34 50 42 38 14 Frequency : I S> g SECTION C Z g `m 13 go 22 VH$ `oH$ Z Ho$ 3 A H$ h & Question numbers 13 to 22 carry 3 marks each. 13. {g H$s{OE {H$ g `m h & 2+5 3 EH$ An[a_o` g `m h , {X`m J`m h {H$ `yp bS> Eo Jmo[a _ Ho$ `moJ go 2048 AWdm VWm 960 H$m _.g. (HCF) kmV 3 H$s{OE & Prove that 2 + 5 3 is an irrational number, given that irrational number. OR Using Euclid s Algorithm, find the HCF of 2048 and 960. 30/2/1 5 EH$ An[a_o` 3 is an P.T.O. 14. EH$ hr Va\$ Xmo g_H$moU { ^wO ABC VWm DBC ~ZmE JE h & `{X BD EH$ X gao H$mo q~X P na {V N>oX H$aVo h , Vmo {g H$s{OE {H$ H$U BC na AC VWm AP PC = BP DP. AWdm EH$ g_b ~ PQRS {Og_| PQ RS h , Ho$ {dH$U na na {~ X O na {V N>oX H$aVo h & `{X PQ = 3RS hmo, Vmo { ^wOm| POQ VWm ROS Ho$ jo \$bm| H$m AZwnmV kmV H$s{OE & 15. Two right triangles ABC and DBC are drawn on the same hypotenuse BC and on the same side of BC. If AC and BD intersect at P, prove that AP PC = BP DP. OR Diagonals of a trapezium PQRS intersect each other at the point O, PQ RS and PQ = 3RS. Find the ratio of the areas of triangles POQ and ROS. AmH ${V 3 _|, PQ VWm RS, O Ho$ dmbo {H$gr d m na Xmo g_m Va ne -aoImE h Am a ne {~ X C na n m -aoIm AB, PQ H$mo A VWm RS H$mo B na {V N>oX H$aVr h & {g H$s{OE {H$ AOB = 90 h & AmH ${V 3 In Figure 3, PQ and RS are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting PQ at A and RS at B. Prove that AOB = 90 . Figure 3 16. aoIm x 3y = 0 {~ X Am| ( 2, 5) VWm (6, 3) H$mo Omo S>Zo dmbo aoImI S> H$mo {H$g AZwnmV _| {d^m{OV H$aVr h ? Bg {V N>oX {~ X Ho$ {ZX oem H$ ^r kmV H$s{OE & Find the ratio in which the line x 3y = 0 divides the line segment joining the points ( 2, 5) and (6, 3). Find the coordinates of the point of intersection. 30/2/1 6 17. _mZ kmV H$s{OE : 2 3 sin 43 cos 37 cosec 53 tan 5 tan 25 tan 45 tan 65 tan 85 cos 47 Evaluate : 2 3 sin 43 cos 37 cosec 53 tan 5 tan 25 tan 45 tan 65 tan 85 cos 47 18. 4 _|, EH$ OA = 15 go_r h , AmH ${V d m Ho$ MVwWm e OPBQ Ho$ A VJ V EH$ dJ OABC ~Zm h Am h & `{X Vmo N>m`m {H$V jo H$m> jo \$b kmV H$s{OE & ( = 3 14 `moJ H$s{OE) AmH ${V 4 AWdm AmH ${V 5 _|, 2 2 go_r ^wOm dmbm dJ ABCD EH$ d m Ho$ A VJ V ~Zm h Am h & N>m`m {H$V jo H$m jo \$b kmV H$s{OE & ( = 3 14 `moJ H$s{OE) AmH ${V 5 In Figure 4, a square OABC is inscribed in a quadrant OPBQ. If OA = 15 cm, find the area of the shaded region. (Use = 3 14) Figure 4 OR 30/2/1 7 P.T.O. In Figure 5, ABCD is a square with side 2 2 cm and inscribed in a circle. Find the area of the shaded region. (Use = 3 14) Figure 5 19. EH$ R>mog ~obZ Ho$ AmH$ma H$m h {OgHo$ XmoZm| {gao AY JmobmH$ma h & R>mog H$s Hw$b b ~mB 20 go_r h VWm ~obZ H$m `mg 7 go_r h & R>mog H$m Hw$b Am`VZ kmV H$s{OE & 22 `moJ H$s{OE) 7 A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 20 cm and the diameter of the cylinder is 7 cm. Find 22 the total volume of the solid. (Use = ) 7 ( = 20. ZrMo {X`m h Am ~ Q>Z 100 {d m{W `m| mam EH$ narjm _| m V A H$m o H$mo Xem ahm h : m Vm H$ : 30 35 35 40 40 45 45 50 50 55 55 60 {d m{W `m| H$s 14 16 28 23 18 8 g `m : {d m{W `m| Ho$ _m ` A H$ kmV H$s{OE & 60 65 3 The marks obtained by 100 students in an examination are given below : Marks : Number of Students : 30 35 35 40 14 16 40 45 45 50 50 55 55 60 60 65 28 23 18 8 3 Find the mean marks of the students. 21. k Ho$ {H$g _mZ Ho$ {bE, ~h nX f(x) = 3x4 9x3 + x2 + 15x + k, 3x2 5 go nyU V`m {d^m{OV hmoVm h ? AWdm 11 { KmV ~h nX 7y2 y 2 Ho$ ey `H$ kmV H$s{OE Am a ey `H$m| VWm JwUm H$m| Ho$ ~rM 3 3 Ho$ g ~ Y H$s g `Vm H$s Om M H$s{OE &$ 30/2/1 8 For what value of k, is the polynomial f(x) = 3x4 9x3 + x2 + 15x + k completely divisible by 3x2 5 ? OR 11 2 and verify y 3 3 the relationship between the zeroes and the coefficients. Find the zeroes of the quadratic polynomial 7y2 22. p Ho$ Eogo g^r _mZ {b{IE {H$ { KmV g_rH$aU hm| & m g_rH$aU Ho$ _yb kmV H$s{OE & x2 + px + 16 = 0 Ho$ ~am~a _yb Write all the values of p for which the quadratic equation x2 + px + 16 = 0 has equal roots. Find the roots of the equation so obtained. 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. `{X {H$gr { ^wO H$s EH$ ^wOm Ho$ g_m Va A ` Xmo ^wOmAm| H$mo {^ -{^ {~ X Am| na {V N>oX H$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 & If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio. 24. A{_V Omo {H$ EH$ g_Vb O_rZ na I S>m h , AnZo go 200 _r. X a C S>Vo h E njr H$m C `Z H$moU 30 nmVm h & XrnH$ Omo {H$ 50 _r. D $Mo ^dZ H$s N>V na I S>m h , Cgr njr H$m C `Z H$moU 45 nmVm h & A{_V Am a XrnH$ njr Ho$ {dnarV {Xem _| h & XrnH$ go njr H$s X ar kmV H$s{OE & Amit, standing on a horizontal plane, finds a bird flying at a distance of 200 m from him at an elevation of 30 . Deepak standing on the roof of a 50 m high building, finds the angle of elevation of the same bird to be 45 . Amit and Deepak are on opposite sides of the bird. Find the distance of the bird from Deepak. 30/2/1 9 P.T.O. 25. bmoho Ho$ EH$ R>mog I ^o _| 220 go_r D $MmB Ho$ EH$ ~obZ {OgHo$ AmYma H$m `mg 24 go_r h , Ho$ D$na 60 go_r D $MmB H$m EH$ A ` ~obZ A `mamo{nV h {OgH$s { `m 8 go_r h & Bg I ^o H$m ^ma kmV H$s{OE, O~{H$ {X`m J`m h {H$ 1 KZ go_r bmoho H$m bJ^J ^ma 8 J m_ h & ( = 3 14 `moJ H$s{OE) A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm3 of iron has approximately 8 gm mass. (Use = 3 14) 26. 5 go_r ^wOm dmbo g_~mh { ^wO ABC H$s{OE {OgH$s ^wOmE {XE h E { ^wO H$s aMZm H$s{OE & {\$a EH$ A ` { ^wO H$s aMZm ABC H$s g JV ^wOmAm| H$s 2 3 JwZr hm| & AWdm 2 go_r { `m Ho$ d m na 5 go_r { `m H$m EH$ g Ho$ r d m It{ME & ~m d m na {bE JE EH$ {~ X P go N>moQ>o d m na Xmo ne -aoImAm| PA VWm PB H$s aMZm H$s{OE & PA H$s b ~mB _m{nE & Construct an equilateral ABC with each side 5 cm. Then construct 2 another triangle whose sides are times the corresponding sides of 3 ABC. OR Draw two concentric circles of radii 2 cm and 5 cm. Take a point P on the outer circle and construct a pair of tangents PA and PB to the smaller circle. Measure PA. 27. {Z Z{b{IV ~ Q>Z H$mo "go H$_ H$ma' Ho$ ~ Q>Z _| ~X{bE Am a {\$a CgH$m VmoaU It{ME : dJ AV amb : ~ma ~maVm : 30 40 40 50 50 60 7 5 8 60 70 70 80 10 6 80 90 90 100 6 8 Change the following data into less than type distribution and draw its ogive : Class Interval : Frequency : 30/2/1 30 40 40 50 50 60 7 5 8 10 60 70 70 80 10 6 80 90 90 100 6 8 28. {g H$s{OE {H$ : tan cot 1 sec cosec 1 cot 1 tan AWdm {g H$s{OE {H$ : sin sin 2 cot cosec cot cosec Prove that : tan cot 1 sec cosec 1 cot 1 tan OR Prove that : sin sin 2 cot cosec cot cosec 29. g_m Va lo T>r 7, 12, 17, 22, ... H$m H$m Z-gm nX 82 hmoJm ? `m 100 Bg g_m Va lo T>r H$m H$moB nX hmoJm ? gH$maU C ma ~VmBE & AWdm g_m Va lo T>r 45, 39, 33, ... Ho$ {H$VZo nXm| H$m `moJ\$b 180 hmoJm ? Xmohao C ma H$s `m `m H$s{OE & Which term of the Arithmetic Progression 7, 12, 17, 22, ... will be 82 ? Is 100 any term of the A.P. ? Give reason for your answer. OR How many terms of the Arithmetic Progression 45, 39, 33, ... must be taken so that their sum is 180 ? Explain the double answer. 30. {h Xr VWm A J o Or H$s H$jm narjm _| A U Ho$ m A H$m| H$m `moJ\$b 30 h & `{X CgHo$ {h Xr _| 2 A H$ A{YH$ hmoVo Am a A J o Or _| 3 A H$ H$_ hmoVo, Vmo m A H$m| H$m JwUZ\$b 210 hmoVm & XmoZm| {df`m| _| CgHo$ mam m A H$m| H$mo kmV H$s{OE & In a class test, the sum of Arun s marks in Hindi and English is 30. Had he got 2 marks more in Hindi and 3 marks less in English, the product of the marks would have been 210. Find his marks in the two subjects. 30/2/1 11 P.T.O.

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