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CBSE Class X 2014 : MATHEMATICS

15 pages, 76 questions, 51 questions with responses, 137 total responses,

CBSE 10
Kendriya Vidyalaya (KV), Kamla Nehru Nagar, Ghaziabad

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H$moS> Z . Series HRS 30/1 Code No. amob Z . narjmWu H$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 > 15 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 _| >34 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 15 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 34 questions. Please write down the Serial Number of the question before attempting it. 15 minutes 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 _| 34 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 8 Z h , Omo ~h -{dH$ nr 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 _| 10 Z h {OZ_| go `oH$ 4 A H$ H$m h & (iv) H $bHw$boQ>a H$m `moJ d{O V h & General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 34 questions divided into four sections A, B, C and D. (iii) Section A contains 8 questions of 1 mark each, which are multiple choice type questions, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 10 questions of 4 marks each. (iv) Use of calculators is not permitted. I S> A SECTION A Z g `m 1 go 8 VH$ `oH$ Z 1 A H$ H$m h & Z g `m 1go 8 _| `oH$ Z Ho$ {bE Mma {dH$ n {XE JE h , {OZ_| go Ho$db EH$ ghr h & ghr {dH$ n Mw{ZE & Question numbers 1 to 8 carry 1 mark each. For each of the question numbers 1 to 8, four alternative choices have been provided, of which only one is correct. Select the correct choice. 1. `{X k, 2k 1 (A) 30/1 3 (D) EH$ g_m Va lo T>r Ho$ VrZ H $_mJV nX h , Vmo k H$m _mZ h 3 (C) 2k + 1 2 (B) VWm 5 2 If k, 2k 1 and 2k + 1 are three consecutive terms of an A.P., the value of k is (A) (B) 3 (C) 3 (D) 2. 2 5 Xmo d m na na {~ X P na ~m $n go ne H$aVo h & d mm| H$mo {~ X Am| A VWm H$aVr h B C^`{Z >> ne aoIm AB h & APB H$m _mZ h (A) 45 (C) 60 (D) na ne 30 (B) B 90 Two circles touch each other externally at P. AB is a common tangent to the circles touching them at A and B. The value of APB is (A) (B) 45 (C) 60 (D) 30/1 30 90 3 P.T.O. 3. EH$ g_H$moU { ^wO ABC _|, B g_H$moU h , BC = 12 go_r VWm Bg { ^wO Ho$ A VJ V ItMo JE d m H$s { `m (go_r _|) h (A) 3 (C) 2 (D) go_r h & 4 (B) AB = 5 1 In a right triangle ABC, right-angled at B, BC = 12 cm and AB = 5 cm. The radius of the circle inscribed in the triangle (in cm) is (A) (B) 3 (C) 2 (D) 4. 4 1 VrZ ~ m| Ho$ n[adma _|, H$_-go-H$_ EH$ b S>H$m hmoZo H$s m{`H$Vm h (A) (B) 1 8 (C) 5 8 (D) 30/1 7 8 3 4 4 In a family of 3 children, the probability of having at least one boy is (A) (B) 1 8 (C) 5 8 (D) 5. 7 8 3 4 EH$ 150 _r. D $Mo _rZma Ho$ {eIa go, g S>H$ na I S>r EH$ H$ma H$m AdZ_Z H$moU h & _rZma go H$ma H$s X ar (_r. _|) h (A) 50 3 (B) 150 3 (C) 150 2 (D) 30 75 The angle of depression of a car parked on the road from the top of a 150 m high tower is 30 . The distance of the car from the tower (in metres) is (A) (B) 150 3 (C) 150 2 (D) 30/1 50 3 75 5 P.T.O. 6. g `mAm| 1, 2, 3, ..., 15 _| go `m N>`m EH$ g `m MwZr JB & MwZr JB g `m Ho$ JwUO hmoZo H$s m{`H$Vm h (A) 2 15 (C) 1 5 (D) H$m 4 15 (B) 4 1 3 The probability that a number selected at random from the numbers 1, 2, 3, ..., 15 is a multiple of 4, is (A) (B) 2 15 (C) 1 5 (D) 7. 4 15 1 3 ABCD EH$ Am`V h {OgHo$ VrZ erf EH$ {dH$U H$s b ~mB h (A) 4 (C) 3 (D) 30/1 5 (B) B(4, 0), C(4, 3) 25 6 VWm D(0, 3) h & Am`V Ho$ ABCD is a rectangle whose three vertices are B(4, 0), C(4, 3) and D(0, 3). The length of one of its diagonals is (A) (B) 4 (C) 3 (D) 8. 5 25 10 go_r { `m Ho$ d m H$s EH$ Ordm H|$ na g_H$moU A V[aV H$aVr h & Bg Ordm H$s b ~mB (go_r _|) h (A) 5 2 (B) 10 2 (C) (D) 5 2 10 3 A chord of a circle of radius 10 cm subtends a right angle at its centre. The length of the chord (in cm) is (A) 5 2 (B) 10 2 (C) (D) 30/1 5 2 10 3 7 P.T.O. I S> ~ SECTION B Z g `m 9 go 14 VH$ `oH$ Z Ho$ 2 A H$ h & Question numbers 9 to 14 carry 2 marks each. 9. p Ho$ dh _mZ kmV H$s{OE, {OZHo$ {bE, { KmV g_rH$aU g_mZ h & 4x2 + px + 3 = 0 Ho$ _yb Find the values of p for which the quadratic equation 4x2 + px + 3 = 0 has equal roots. 10. 101 VWm H$s{OE & 999 Ho$ ~rM 2 Am a 5 XmoZm| go {d^m ` mH $V g `mAm| H$s g `m kmV Find the number of natural numbers between 101 and 999 which are divisible by both 2 and 5. 11. AmH ${V 1 _|, H|$ O1 VWm O2 dmbo Xmo d mm| H$s C^`{Z >> ne aoImE E na H$mQ>Vr h & {g H$s{OE {H$ AB = CD. AB VWm CD {~ X AmH ${V 1 In Figure 1, common tangents AB and CD to the two circles with centres O1 and O2 intersect at E. Prove that AB = CD. Figure 1 30/1 8 12. EH$ g_{ ~mh { ^wO ABC, {Og_| AB = AC h , Ho$ A VJ V ItMm J`m d m, ^wOmAm| BC, CA VWm AB H$mo H $_e q~X Am| D, E VWm F na ne H$aVm h & {g H$s{OE {H$ BD = DC h & The incircle of an isosceles triangle ABC, in which AB = AC, touches the sides BC, CA and AB at D, E and F respectively. Prove that BD = DC. 13. Xmo {^ -{^ nmgm| H$mo EH$ gmW CN>mbm J`m & m{`H$Vm kmV H$s{OE {H$ (i) XmoZm| nmgm| na AmB g `mE g_ hm| & (ii) XmoZm| nmgm| na AmB g `mAm| H$m `moJ\$b 5 hmo & Two different dice are tossed together. Find the probability (i) (ii) 14. that the number on each die is even. that the sum of numbers appearing on the two dice is 5. `{X EH$ R>mog AY Jmobo H$m g nyU n >r` jo \$b 462 dJ go_r h , Vmo BgH$m Am`VZ kmV 22 br{OE ] 7 If the total surface area of a solid hemisphere is 462 cm2, find its volume. 22 [ Take = ] 7 H$s{OE & [ = I S> g SECTION C Z g `m 15 go 24 VH$ `oH$ Z Ho$ 3 A H$ h & Question numbers 15 to 24 carry 3 marks each. 15. x : 16 15 1 ; x 0, 1 x x 1 Solve for x : 16 15 1 ; x 0, 1 x x 1 16. EH$ g_m Va lo T>r Ho$ nm Md| VWm Zm d| nXm| H$m `moJ\$b 30 h & `{X BgH$m n rgdm nX BgHo$ 8d| nX H$m VrZ JwZm h , Vmo g_m Va lo T>r kmV H$s{OE & Ho$ {bE hb H$s{OE The sum of the 5th and the 9th terms of an AP is 30. If its 25th term is three times its 8th term, find the AP. 30/1 9 P.T.O. 17. EH$ { ^wO H$s aMZm H$s{OE, {OgH$s ^wOmE 5 go_r, 5.5 go_r VWm 6.5 go_r h & {\$a EH$ A ` { ^wO H$s aMZm H$s{OE, {OgH$s ^wOmE , {XE h E { ^wO H$s g JV ^wOmAm| H$s 3 JwZr hm| & 5 Construct a triangle with sides 5 cm, 5.5 cm and 6.5 cm. Now construct 3 another triangle, whose sides are times the corresponding sides of the 5 given triangle. 18. ^y{_ Ho$ EH$ {~ X go EH$ dm`w`mZ H$m C `Z H$moU 60 h & `h C `Z H$moU 30 hmo OmVm h & `{X `h dm`w`mZ 3000 C S> ahm h , Vmo dm`w`mZ H$s J{V kmV H$s{OE & 30 3 goH$ S> H$s C S>mZ Ho$ ~mX _r. H$s AMa D $MmB na The angle of elevation of an aeroplane from a point on the ground is 60 . After a flight of 30 seconds the angle of elevation becomes 30 . If the aeroplane is flying at a constant height of 3000 3 m, find the speed of the aeroplane. 19. `{X {~ X H$s{OE & P(k 1, 2) {~ X Am| A(3, k) VWm B(k, 5) go g_X a W h , Vmo k Ho$ _mZ kmV If the point P(k 1, 2) is equidistant from the points A(3, k) and B(k, 5), find the values of k. 20. dh AZwnmV kmV H$s{OE {Og_| {~ X Am| A(3, 3) Am a B( 2, 7) H$mo {_bmZo dmbm aoImI S> x-Aj go {d^m{OV hmoVm h & Bg {d^mOZ {~ X Ho$ {ZX}em H$ ^r kmV H$s{OE & Find the ratio in which the line segment joining the points A(3, 3) and B( 2, 7) is divided by x-axis. Also find the coordinates of the point of division. 21. AmH ${V 2 _|, O H|$ dmbo Xmo g H|$ r` d m h {OZH$s { `mE 21 `{X AOB = 60 h , Vmo N>m`m {H$V ^mJ H$m jo \$b kmV H$s{OE & AmH ${V 2 30/1 10 42 go_r h & 22 [ = br{OE ] 7 go_r VWm In Figure 2, two concentric circles with centre O, have radii 21 cm and 22 42 cm. If AOB = 60 , find the area of the shaded region. [Use = ] 7 Figure 2 22. 7 go_r ^wOm dmbo bH$ S>r Ho$ EH$ R>mog KZ _| go EH$ ~ S>o -go-~ S>m Jmobm H$mQ>m J`m & eof ~Mr bH$ S>r H$m Am`VZ kmV H$s{OE & [ = 22 7 br{OE ] The largest possible sphere is carved out of a wooden solid cube of side 22 7 cm. Find the volume of the wood left. [Use = ] 7 23. _r. Mm S>r Am a 1.5 _r. Jhar EH$ Zha _| nmZr 4 {H$_r {V K Q>o H$s Mmb go ~h ahm h & 10 {_ZQ > _| `h Zha {H$VZo jo \$b H$s qgMmB H$a nmEJr O~{H$ qgMmB Ho$ {bE 8 go_r Jhao nmZr H$s Amd `H$Vm h ? 6 Water in a canal, 6 m wide and 1.5 m deep, is flowing at a speed of 4 km/h. How much area will it irrigate in 10 minutes, if 8 cm of standing water is needed for irrigation ? 24. EH$ g_b ~ h , {OgH$m jo \$b 24.5 dJ go_r h & Bg_| AD || BC, DAB = 90 , AD = 10 go_r VWm BC = 4 go_r h & `{X ABE EH$ d m AmH ${V 3 _|, ABCD H$m MVwWm e h , Vmo N>m`m {H$V ^mJ H$m jo \$b kmV H$s{OE & [ = 22 7 br{OE ] AmH ${V 3 30/1 11 P.T.O. In Figure 3, ABCD is a trapezium of area 24.5 sq. cm. In it, AD || BC, DAB = 90 , AD = 10 cm and BC = 4 cm. If ABE is a quadrant of a 22 circle, find the area of the shaded region. [ Take = ] 7 Figure 3 I S> X SECTION D Z g `m 25 go 34 VH$ `oH$ Z Ho$ 4 A H$ h & Question numbers 25 to 34 carry 4 marks each. 25. x Ho$ {bE hb H$s{OE : x 2 x 4 10 ; x 3, 5 x 3 x 5 3 Solve for x : x 2 x 4 10 ; x 3, 5 x 3 x 5 3 26. EH$ {d mb` Ho$ N>m m| Zo dm`w X fU H$_ H$aZo Ho$ {bE {d mb` Ho$ A Xa Am a ~mha no S> bJmZo H$m {ZU ` {b`m & `oH$ H$jm Ho$ `oH$ AZw^mJ mam AnZr H$jm H$s g `m Ho$ X JwZo Ho$ ~am~a no S> bJmZo H$m {ZU ` {b`m & `{X {d mb` _| 1 go 12 VH$ H$jmE h VWm `oH$ H$jm Ho$ Xmo AZw^mJ h , Vmo N>m m| mam bJmE JE Hw$b no S>m| H$s g `m kmV H$s{OE & Bg Z _| {H$g _y ` H$mo Xem `m J`m h ? 30/1 12 In a school, students decided to plant trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be double of the class in which they are studying. If there are 1 to 12 classes in the school and each class has two sections, find how many trees were planted by the students. Which value is shown in this question ? 27. ^y{_ na p WV {~ X A go 120 _r. H$s X ar na p WV EH$ _rZma Ho$ {eIa H$m C `Z H$moU 45 h & `{X _rZma Ho$ {eIa na bJo EH$ dOX S> Ho$ D$nar {gao H$m {~ X A na C `Z H$moU 60 h , Vmo dOX S> H$s D $MmB kmV H$s{OE & [ 3 = 1.73 br{OE ] The angle of elevation of the top of a tower at a distance of 120 m from a point A on the ground is 45 . If the angle of elevation of the top of a flagstaff fixed at the top of the tower, at A is 60 , then find the height of the flagstaff. [ Use 3 = 1.73 ] 28. 52 n mm| H$s Vme H$s EH$ J >r _| go bmb a J H$s ~oJ_| VWm H$mbo a J Ho$ Jwbm_ {ZH$mb {XE JE & eof n mm| H$mo A N>r H$ma \|$Q>Zo Ho$ ~mX CZ_| go `m N>`m EH$ n mm {ZH$mbm J`m & m{`H$Vm kmV H$s{OE {H$ {ZH$mbm J`m n mm (i) EH$ ~mXemh h (ii) bmb a J H$m hmo (iii) EH$ V dra dmbm n mm hmo (iv) EH$ ~oJ_ hmo Red queens and black jacks are removed from a pack of 52 playing cards. A card is drawn at random from the remaining cards, after reshuffling them. Find the probability that the drawn card is (i) a king (ii) of red colour (iii) a face card (iv) a queen 29. `{X A( 3, 5), B( 2, 7), C(1, 8) VWm BgH$m jo \$b kmV H$s{OE & D(6, 3) EH$ MVw^w O ABCD Ho$ erf h , Vmo If A( 3, 5), B( 2, 7), C(1, 8) and D(6, 3) are the vertices of a quadrilateral ABCD, find its area. 30/1 13 P.T.O. 30. EH$ _moQ>a-~moQ>, {OgH$s p Wa Ob _| Mmb 18 {H$_r {V K Q>m h , 24 {H$_r Ymam Ho$ {VHy$b OmZo _|, dhr X ar Ymam Ho$ AZwHy$b OmZo H$s Anojm 1 K Q>m A{YH$ boVr h & Ymam H$s Mmb kmV H$s{OE & A motorboat whose speed in still water is 18 km/h, takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream. 31. AmH ${V 4 _|, 10 go_r { `m Ho$ EH$ d m H$s 16 go_r b ~r EH$ Ordm PQ h & P Am a Q na ne aoImE na na EH$ {~ X T na {V N>oX H$aVr h & TP H$s b ~mB kmV H$s{OE & AmH ${V 4 In Figure 4, PQ is a chord of length 16 cm, of a circle of radius 10 cm. The tangents at P and Q intersect at a point T. Find the length of TP. Figure 4 32. {g H$s{OE {H$ d m Ho$ {H$gr {~ X na ne aoIm ne {~ X go OmZo dmbr { `m na b ~ hmoVr h & Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. 30/1 14 33. Ho$ EH$ ~obZmH$ma ~V Z, {Og_| Hw$N> nmZr h , _| 1.4 go_r `mg dmbr Jmo{b`m S>mbr JB Omo {H$ nmZr _| nyU V`m Sy>~ JB & kmV H$s{OE {H$ ~V Z _| nmZr Ho$ Va _| {H$VZr d { h B & 7 go_r `mg 150 JmobmH$ma 150 spherical marbles, each of diameter 1.4 cm, are dropped in a cylindrical vessel of diameter 7 cm containing some water, which are completely immersed in water. Find the rise in the level of water in the vessel. 34. D$na go Iwbm EH$ ~V Z e Hw$ Ho$ {N> H$ Ho$ AmH$ma H$m h , {OgH$s D $MmB 24 go_r h VWm {ZMbo VWm D$nar d mr` {gam| H$s { `mE H $_e 8 go_r VWm 20 go_r h & < 21 {V brQ>a H$s Xa go Bg ~V Z H$mo nyam ^a gH$Zo dmbo X Y H$m _y ` kmV H$s{OE & [ = 22 7 br{OE ] A container open at the top, is in the form of a frustum of a cone of height 24 cm with radii of its lower and upper circular ends as 8 cm and 20 cm respectively. Find the cost of milk which can completely fill the container 22 at the rate of < 21 per litre. [Use = ] 7 30/1 15 P.T.O.

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