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ICSE Class X Prelims 2019 : Mathematics

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SET-1 H$moS> Z . Series HRK Code No. amob Z . 30/1 narjmWu H$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 _| >31 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 31 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. g H${bV narjm II SUMMATIVE ASSESSMENT II J{UV MATHEMATICS {ZYm [aV g_` : 3 K Q>o A{YH$V_ A H$ : 90 Time allowed : 3 hours 30/1 Maximum Marks : 90 1 P.T.O. gm_m ` {ZX}e : (i) g^r Z A{Zdm` h & (ii) Bg Z-n _| 31 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 4 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 _| 11 Z h {OZ_| go `oH$ 4 A H$ H$m h & (iv) H $bHw$boQ>am| Ho$ `moJ H$s AZw_{V Zht h & General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 31 questions divided into four sections A, B, C and D. (iii) Section A contains 4 questions of 1 mark each. Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each. (iv) Use of calculators is not permitted. I S> A SECTION A Z g `m 1 go 4 VH$ `oH$ Z 1 A H$ H$m h & Question numbers 1 to 4 carry 1 mark each. 1. EH$ g_m Va lo T>r, {Og_| a21 a7 = 84 h , H$m gmd A Va `m h ? What is the common difference of an A.P. in which a21 a7 = 84 ? 30/1 2 2. `{X EH$ ~m q~X P go a { `m VWm O Ho$ dmbo d m na ItMr JB Xmo ne -aoImAm| Ho$ ~rM H$m H$moU 60 hmo, Vmo OP H$s b ~mB kmV H$s{OE & If the angle between two tangents drawn from an external point P to a circle of radius a and centre O, is 60 , then find the length of OP. 3. `{X 30 _r. D $Mr EH$ _rZma, ^y{_ na H$moU `m h ? 10 3 _r. b ~r N>m`m ~ZmVr h , Vmo gy` H$m C `Z If a tower 30 m high, casts a shadow 10 3 m long on the ground, then what is the angle of elevation of the sun ? 4. 900 go~m| Ho$ EH$ T>oa _| go `m N>`m EH$ go~ MwZZo na g S>m h Am go~ {ZH$bZo H$s m{`H$Vm 0 18 h & T>oa _| g S>o h E go~m| H$s g `m `m h ? The probability of selecting a rotten apple randomly from a heap of 900 apples is 0 18. What is the number of rotten apples in the heap ? I S> ~ SECTION B Z g `m 5 go 10 VH$ `oH$ Z Ho$ 2 A H h & Question numbers 5 to 10 carry 2 marks each. 5. p H$m dh _mZ kmV H$s{OE {OgHo$ {bE { KmV g_rH$aU X gao H$m 6 JwZm h & px2 14x + 8 = 0 H$m EH$ _yb Find the value of p, for which one root of the quadratic equation px2 14x + 8 = 0 is 6 times the other. 6. lo T>r 1 1 3 20, 19 , 18 , 17 , ... H$m 4 2 4 H$m Z-gm nX W_ G$Um _H$ nX h ? 1 1 3 Which term of the progression 20, 19 , 18 , 17 , ... is the first negative 4 2 4 term ? 30/1 3 P.T.O. 7. {g H$s{OE {H$ d m H$s {H$gr Ordm Ho$ A V q~X Am| na ItMr JB ne -aoImE Ordm Ho$ gmW g_mZ H$moU ~ZmVr h & Prove that the tangents drawn at the end points of a chord of a circle make equal angles with the chord. 8. ABCD H$s AB + CD = BC + DA EH$ d m {H$gr MVw^ wO g^r Mmam| ^wOmAm| H$mo ne H$aVm h & {g H$s{OE {H$ A circle touches all the four sides of a quadrilateral ABCD. Prove that AB + CD = BC + DA 9. y-Aj VWm x-Aj H$mo H $_e: q~X Am| P VWm Q na {V N>oX (2, 5), PQ H$m _ `-q~X hmo, Vmo P VWm Q Ho$ {ZX}em H$ kmV H$s{OE & EH$ aoIm H$aVr h & `{X A line intersects the y-axis and x-axis at the points P and Q respectively. If (2, 5) is the mid-point of PQ, then find the coordinates of P and Q. 10. P(x, y) 3x = 2y. `{X H$s A(5, 1) VWm B( 1, 5) go X [a`m g_mZ hm|, Vmo {g H$s{OE {H$ If the distances of P(x, y) from A(5, 1) and B( 1, 5) are equal, then prove that 3x = 2y. I S> g SECTION C Z g `m 11 go 20 VH$ `oH$ Z Ho$ 3 A H$ h & Question numbers 11 to 20 carry 3 marks each. 11. `{X ad bc h , Vmo {g H$s{OE {H$ g_rH$aU (a2 + b2) x2 + 2 (ac + bd) x + (c2 + d2) = 0 H$m H$moB dm V{dH$ _yb Zht h & If ad bc, then prove that the equation (a2 + b2) x2 + 2 (ac + bd) x + (c2 + d2) = 0 has no real roots. 12. EH$ g_m Va lo T>r H$m W_ nX 5, A {V_ nX 45 VWm BgHo$ g^r nXm| H$m `moJ\$b Bg g_m Va lo T>r Ho$ nXm| H$s g `m VWm gmd A Va kmV H$s{OE & 400 h & The first term of an A.P. is 5, the last term is 45 and the sum of all its terms is 400. Find the number of terms and the common difference of the A.P. 30/1 4 13. EH$ _rZma Ho$ nmX go Jw OaZo dmbr grYr aoIm na nmX go H $_e: 4 _r. VWm 16 _r. H$s X [a`m| na Xmo q~X C d D p WV h & `{X C d D go _rZma Ho$ {eIa Ho$ C `Z H$moU EH$-X gao Ho$ nyaH$ hm|, Vmo _rZma H$s D $MmB kmV H$s{OE & On a straight line passing through the foot of a tower, two points C and D are at distances of 4 m and 16 m from the foot respectively. If the angles of elevation from C and D of the top of the tower are complementary, then find the height of the tower. 14. EH$ W bo _| 15 g \o$X VWm Hw$N> H$mbr J|X| h & `{X W bo _| go EH$ H$mbr J|X {ZH$mbZo H$s m{`H$Vm EH$ g \o$X J|X {ZH$mbZo H$s m{`H$Vm H$s VrZ JwZr hmo, Vmo W bo _| H$mbr J|Xm| H$s g `m kmV H$s{OE & A bag contains 15 white and some black balls. If the probability of drawing a black ball from the bag is thrice that of drawing a white ball, find the number of black balls in the bag. 15. q~X 24 , y , 11 q~X Am| P(2, 2) AZwnmV _| {d^m{OV H$aVm h ? y H$m VWm Q(3, 7) H$mo {_bmZo dmbo aoImI S> H$mo {H$g _mZ ^r kmV H$s{OE & 24 In what ratio does the point , y divide the line segment joining the 11 points P(2, 2) and Q(3, 7) ? Also find the value of y. 16. Xr JB AmH ${V _|, `oH$ 3 go_r `mg Ho$ VrZ AY d m, 4 5 go_r `mg H$m EH$ d m VWm 4 5 go_r { `m H$m EH$ AY d m ~ZmE JE h & N>m`m {H$V ^mJ H$m jo \$b kmV H$s{OE & 30/1 5 P.T.O. Three semicircles each of diameter 3 cm, a circle of diameter 4 5 cm and a semicircle of radius 4 5 cm are drawn in the given figure. Find the area of the shaded region. 17. Xr JB AmH ${V _|, O H|$ dmbo Xmo g H|$ r` d mm| H$s { `mE 21 go_r VWm 42 go_r h & `{X AOB = 60 h , Vmo N>m`m {H$V ^mJ H$m jo \$b kmV H$s{OE & 22 [ = `moJ H$s{OE ] 7 In the given figure, two concentric circles with centre O have radii 21 cm and 42 cm. If AOB = 60 , find the area of the shaded region. 22 [ Use = ] 7 30/1 6 18. 5 4 _r. Mm S>r Am a 1 8 _r. Jhar EH$ Zha _| nmZr 25 {H$_r/K Q>m H$s J{V go ~h ahm h & Bggo 40 {_ZQ> _| {H$VZo jo \$b H$s qgMmB hmo gH$Vr h , `{X qgMmB Ho$ {bE 10 go_r Jhao nmZr H$s Amd `H$Vm h ? Water in a canal, 5 4 m wide and 1 8 m deep, is flowing with a speed of 25 km/hour. How much area can it irrigate in 40 minutes, if 10 cm of standing water is required for irrigation ? 19. EH$ e Hw$ Ho$ {N> H$ H$s {V` H $ D $MmB 4 go_r h VWm BgHo$ d mr` {gam| Ho$ n[a_mn Am a 6 go_r h & Bg {N> H$ H$m dH $ n R>r` jo \$b kmV H$s{OE & 18 go_r The slant height of a frustum of a cone is 4 cm and the perimeters of its circular ends are 18 cm and 6 cm. Find the curved surface area of the frustum. 20. EH$ R>mog bmoho Ho$ KZm^ H$s {d_mE 4 4 _r. 2 6 _r. 1 0 _r. h & Bgo {nKbmH$a 30 go_r Am V[aH$ { `m Am a 5 go_r _moQ>mB H$m EH$ ImoIbm ~obZmH$ma nmBn ~Zm`m J`m h & nmBn H$s b ~mB kmV H$s{OE & The dimensions of a solid iron cuboid are 4 4 m 2 6 m 1 0 m. It is melted and recast into a hollow cylindrical pipe of 30 cm inner radius and thickness 5 cm. Find the length of the pipe. I S> X SECTION D Z g `m 21 go 31 VH$ `oH$ Z Ho$ 4 A H$ h & Question numbers 21 to 31 carry 4 marks each. 21. x Ho$ {bE hb H$s{OE : 1 3 5 1 + = , x 1, , 4 x 1 5x 1 x 4 5 Solve for x : 1 3 5 1 + = , x 1, , 4 x 1 5x 1 x 4 5 30/1 7 P.T.O. 22. Xmo Zb EH$ gmW EH$ Q> H$ H$mo 3 1 13 K Q>o _| ^a gH$Vo h & `{X EH$ Zb Q> H$ H$mo ^aZo _| X gao Zb go 3 K Q>o A{YH$ boVm h , Vmo `oH$ Zb Q> H$ H$mo ^aZo _| {H$VZm g_` boJm ? 1 hours. If one tap takes 13 3 hours more than the other to fill the tank, then how much time will Two taps running together can fill a tank in 3 each tap take to fill the tank ? 23. `{X Xmo g_m Va lo{ T>`m| Ho$ W_ n nXm| Ho$ `moJ\$bm| H$m AZwnmV Vmo CZHo$ 9d| nXm| H$m AZwnmV kmV H$s{OE & (7n + 1) : (4n + 27) h , If the ratio of the sum of the first n terms of two A.Ps is (7n + 1) : (4n + 27), then find the ratio of their 9th terms. 24. {g H$s{OE {H$ d m Ho$ {H$gr ~m q~X go d m na ItMr JB Xmo ne -aoImAm| H$s b ~mB`m g_mZ hmoVr h & Prove that the lengths of two tangents drawn from an external point to a circle are equal. 25. Xr JB AmH ${V _|, XY VWm X Y , O H|$ dmbo d m H$s Xmo g_m Va ne -aoImE h VWm EH$ A ` ne -aoIm AB, {OgH$m ne q~X C h , XY H$mo A VWm X Y H$mo B na {V N>oX H$aVr h & {g H$s{OE {H$ AOB = 90 . 30/1 8 In the given figure, XY and X Y are two parallel tangents to a circle with centre O and another tangent AB with point of contact C, is intersecting XY at A and X Y at B. Prove that AOB = 90 . 26. EH$ { ^wO ABC H$s aMZm H$s{OE {Og_| ^wOm BC = 7 go_r, B = 45 , A = 105 hmo & V~ EH$ A ` { ^wO H$s aMZm H$s{OE {OgH$s ^wOmE ABC H$s g JV ^wOmAm| H$s 3 4 JwZr hm| & Construct a triangle ABC with side BC = 7 cm, B = 45 , A = 105 . 3 Then construct another triangle whose sides are times the 4 corresponding sides of the ABC. 27. EH$ hdmB Ohm O ^yVb go D$na 300 _r. H$s D $MmB na C S> ahm h & Bg D $MmB na C S>Vo h E hdmB Ohm O go EH$ ZXr Ho$ XmoZm| {H$Zmam| na na na {dnarV {XemAm| _| p WV Xmo q~X Am| Ho$ AdZ_Z H$moU H $_e: 45 VWm 60 h & ZXr H$s Mm S>mB kmV H$s{OE & [ 3 = 1 732 `moJ H$s{OE ] An aeroplane is flying at a height of 300 m above the ground. Flying at this height, the angles of depression from the aeroplane of two points on both banks of a river in opposite directions are 45 and 60 respectively. Find the width of the river. [Use 3 = 1 732] 28. `{X q~X A(k + 1, 2k), kmV H$s{OE & B(3k, 2k + 3) VWm C(5k 1, 5k) maoI hm|, Vmo k H$m _mZ If the points A(k + 1, 2k), B(3k, 2k + 3) and C(5k 1, 5k) are collinear, then find the value of k. 30/1 9 P.T.O. 29. Xmo {d{^ nmgm| H$mo EH$ gmW \|$H$m J`m & m{`H$Vm kmV H$s{OE {H$ m V g `mAm| H$m (i) `moJ\$b g_ hmoJm, Am a (ii) JwUZ\$b g_ hmoJm & Two different dice are thrown together. Find the probability that the numbers obtained have 30. (i) even sum, and (ii) even product. Xr JB AmH ${V _|, ABCD EH$ Am`V h {OgH$s {d_mE 21 go_r 14 go_r h & BC H$mo `mg _mZ H$a EH$ AY d m ItMm J`m h & AmH ${V _| N>m`m {H$V ^mJ H$m jo \$b VWm n[a_mn kmV H$s{OE & In the given figure, ABCD is a rectangle of dimensions 21 cm 14 cm. A semicircle is drawn with BC as diameter. Find the area and the perimeter of the shaded region in the figure. 30/1 10 31. {H$gr dfm -Ob g J hU V _|, 22 _r. 20 _r. H$s N>V go dfm -Ob ~hH$a 2 _r. AmYma Ho$ `mg VWm 3 5 _r. D $MmB Ho$ EH$ ~obZmH$ma Q> H$ _| AmVm h & `{X Q> H$ ^a J`m hmo, Vmo kmV H$s{OE {H$ go_r _| {H$VZr dfm h B & Ob g ajU na AnZo {dMma ` $ H$s{OE & In a rain-water harvesting system, the rain-water from a roof of 22 m 20 m drains into a cylindrical tank having diameter of base 2 m and height 3 5 m. If the tank is full, find the rainfall in cm. Write your views on water conservation. 30/1 11 P.T.O.

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