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CBSE Class X 2014 : MATHEMATICS

15 pages, 70 questions, 14 questions with responses, 17 total responses,

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

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H$moS> Z . Series HRS/2 30/2/1 Code No. amob Z . narjmWu H$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 30/2/1 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. g_m Va lo T>r (A) 97 (D) 30/2/1 84 (C) ... H$m AJbm nX h 70 (B) 7 , 28, 63, 112 2 The next term of the A.P. (A) 84 (C) 97 (D) 2. 70 (B) 7 , 28, 63, ... is 112 AmH ${V 1 _|, d m Ho$ n[aJV EH$ MVw^w O AB, BC, CD AB = x go_r, VWm AD, BC = 7 ABCD d m H$mo H $_e Bg H$ma ItMm J`m h {H$ BgH$s ^wOmE P, Q, R go_r, CR = 3 go_r VWm VWm S AS = 5 go_r na ne H$aVr h & `{X hmo, Vmo x H$m _mZ h AmH ${V 1 (A) (B) 9 (C) 8 (D) 30/2/1 10 7 3 P.T.O. In Figure 1, a quadrilateral ABCD is drawn to circumscribe a circle such that its sides AB, BC, CD and AD touch the circle at P, Q, R and S respectively. If AB = x cm, BC = 7 cm, CR = 3 cm and AS = 5 cm, find x. Figure 1 (A) (B) 9 (C) 8 (D) 3. 10 7 Xmo g H|$ r` d mm| H$s { `mE 5 go_r VWm 3 go_r h & ~ S>o d m H$s Cg Ordm H$s b ~mB , Omo N>moQ>o d m H$mo ne H$aVr hmo, (go_r _|) h (A) 4 (B) 5 (C) 8 (D) 10 Two concentric circles are of radii 5 cm and 3 cm. Length of the chord of the larger circle, (in cm), which touches the smaller circle is (A) (B) 5 (C) 8 (D) 30/2/1 4 10 4 4. `{X {H$gr D$ dm Ya I ^o H$s D $MmB , ^y{_ na n S> ahr CgH$s N>m`m H$s b ~mB H$m h , Vmo Cg g_` gy` H$m C Vm e h (A) (B) 45 (D) JwZm 30 60 (C) 3 75 If the height of a vertical pole is 3 times the length of its shadow on the ground, then the angle of elevation of the Sun at that time is (A) 30 (B) (C) (D) 5. 60 45 75 EH$ W bo _| H$mS> h {OZ na 1 go 25 VH$ H$s g `mE A {H$V h & W bo _| go `m N>`m EH$ H$mS> {ZH$mbm J`m & Bg H$mS> na A {H$V g `m Ho$ 2 VWm 3 XmoZm| go {d^m{OV hmoZo H$s m{`H$Vm h (A) (B) (C) (D) 1 5 3 25 4 25 2 25 A bag contains cards numbered from 1 to 25. A card is drawn at random from the bag. The probability that the number on this card is divisible by both 2 and 3 is (A) (B) 3 25 (C) 4 25 (D) 30/2/1 1 5 2 25 5 P.T.O. 6. Xmo {^ -{^ {g $m| H$mo EH$ gmW CN>mbZo na, H$_-go-H$_ EH$ {M m m hmoZo H$s m{`H$Vm h (A) 1 4 (B) 1 8 (C) 3 4 (D) 7 8 Two different coins are tossed simultaneously. The probability of getting at least one head is (A) (B) 1 8 (C) 3 4 (D) 7. 1 4 7 8 erf {~ X Am| (A) 7+ (B) 5 10 (D) { ^wO H$m n[a_mn h 5 (C) (0, 4), (0, 0) VWm (3, 0) dmbo 12 The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is (A) (B) 10 (D) 5 5 (C) 30/2/1 7+ 12 6 8. go_r 22 go_r H$s EH$ Am`VmH$ma H$mJ O H$s erQ> H$mo _mo S> H$a EH$ ImoIbm ~obZ ~Zm`m J`m & ~obZ H$s { `m (go_r _|) h 40 (A) 7 (C) 80 7 (D) go_r D $MmB H$m 3 .5 (B) 40 5 A rectangular sheet of paper 40 cm 22 cm, is rolled to form a hollow cylinder of height 40 cm. The radius of the cylinder (in cm) is (A) 3 .5 (B) 7 (C) 80 7 (D) 5 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. x Ho$ {bE hb H$s{OE : 3 x2 2 2 x 2 3 = 0 Solve for x : 3 x2 2 2 x 2 3 = 0 10. EH$ g_m Va lo T>r Ho$ W_ kmV H$s{OE & n nXm| H$m `moJ\$b 3n2 + 6n h & Bg g_m Va lo T>r H$m ndm nX The sum of the first n terms of an A.P. is 3n2 + 6n. Find the nth term of this A.P. 30/2/1 7 P.T.O. 11. AmH ${V 2 _|, XP VWm XQ, H|$ O dmbo d m na EH$ ~m {~ X X go ItMr JB Xmo ne aoImE h & ARB d m H$s EH$ A ` ne aoIm h Omo d m H$mo R na ne H$aVr h & {g H$s{OE {H$ XA + AR = XB + BR. AmH ${V 2 In Figure 2, XP and XQ are two tangents to the circle with centre O, drawn from an external point X. ARB is another tangent, touching the circle at R. Prove that XA + AR = XB + BR. Figure 2 12. {g H$s{OE {H$ d m Ho$ {H$gr `mg Ho$ {gam| na ItMr JB ne aoImE g_m Va hmoVr h & Prove that the tangents drawn at the ends of any diameter of a circle are parallel. 13. Xmo {^ -{^ nmgm| H$mo EH$ gmW CN>mbm J`m & XmoZm| nmgm| na AmE A H$m| H$m `moJ\$b hmoZo H$s m{`H$Vm kmV H$s{OE & 10 Two different dice are rolled simultaneously. Find the probability that the sum of numbers appearing on the two dice is 10. 30/2/1 8 AmH ${V 3 OD = 4 14. go_r h , Vmo N>m`m {H$V jo H$m jo \$b kmV H$s{OE & _|, OABC 7 go_r h 22 [ = br{OE] 7 EH$ d m H$m MVwWm e h , {OgH$s { `m & `{X AmH ${V 3 In Figure 3, OABC is a quadrant of a circle of radius 7 cm. If OD = 4 cm, 22 find the area of the shaded region. [Use = ] 7 Figure 3 30/2/1 9 P.T.O. I S> g SECTION C Z g `m 15 go 24 VH$ `oH$ Z Ho$ 3 A H$ h & Question numbers 15 to 24 carry 3 marks each. 15. `{X { KmV g_rH$aU 2x2 + px 15 = 0 H$m EH$ _yb 5 h VWm { KmV g_rH$aU p(x2 + x) + k = 0 Ho$ _yb g_mZ h , Vmo k H$m _mZ kmV H$s{OE & If 5 is a root of the quadratic equation 2x2 + px 15 = 0 and the quadratic equation p(x2 + x) + k = 0 has equal roots, find the value of k. 16. EH$ g_m Va lo T>r Ho$ W_ 7 nXm| H$m `moJ\$b 63 h VWm AJbo h & Bg g_m Va lo T>r H$m 28dm nX kmV H$s{OE & 7 nXm| H$m `moJ\$b 161 The sum of the first 7 terms of an A.P. is 63 and the sum of its next 7 terms is 161. Find the 28th term of this A.P. 17. EH$ { ^wO ABC H$s aMZm H$s{OE, {Og_| AB = 5 go_r, BC = 6 go_r VWm AC = 7 go_r hm| & {\$a EH$ A ` { ^wO H$s aMZm H$s{OE {OgH$s ^wOmE , ABC H$s 3 JwZr hm| & 5 Construct a triangle ABC, in which AB = 5 cm, BC = 6 cm and AC = 7 cm. 3 Then construct another triangle whose sides are times the 5 corresponding sides of ABC. g JV ^wOmAm| H$s 18. Xmo g_w r Ohm O EH$ H$me- V ^ H$s Amoa {dnarV {XemAm| go Am aho h & H$me- V ^ Ho$ {eIa go BZ Ohm Om| Ho$ AdZ_Z H$moU 30 VWm 45 h & `{X Ohm Om| Ho$ ~rM H$s X ar 100 _r hmo, Vmo H$me- V ^ H$s D $MmB kmV H$s{OE & [ 3 = 1.732 br{OE] Two ships are approaching a light-house from opposite directions. The angles of depression of the two ships from the top of the light-house are 30 and 45 . If the distance between the two ships is 100 m, find the height of the light-house. [Use 3 = 1.732] 19. k Ho$ dh _mZ kmV H$s{OE (k 1, k 2) g aoIr h & {OZHo$ {bE {~ X (3k 1, k 2), (k, k 7) VWm Find the value(s) of k for which the points (3k 1, k 2), (k, k 7) and (k 1, k 2) are collinear. 30/2/1 10 20. {~ X P, Q, R VWm S, {~ X Am| A(1, 2) VWm B(6, 7) H$mo {_bmZo dmbo aoImI S> H$mo nm M g_mZ ^mJm| _| {d^m{OV H$aVo h & {~ X Am| P, Q VWm R Ho$ {ZX}em H$ kmV H$s{OE & Points P, Q, R and S divide the line segment joining the points A(1, 2) and B(6, 7) in 5 equal parts. Find the coordinates of the points P, Q and R. 21. AmH ${V 4 _|, EH$ Am`VmH$ma jo ABCD, {Og_| AB = 20 go_r h , Ho$ EH$ {gao go EH$ g_H$moU { ^wO AED H$mQ>m J`m h , Ohm AE = 9 go_r VWm DE = 12 go_r h >& X gao {gao BC H$mo `mg boH$a ~mha H$s Amoa EH$ AY d m Omo S>m J`m h & N>m`m {H$V jo H$m jo \$b kmV H$s{OE & [ = 3.14 br{OE] AmH ${V 4 In Figure 4, from a rectangular region ABCD with AB = 20 cm, a right triangle AED with AE = 9 cm and DE = 12 cm, is cut off. On the other end, taking BC as diameter, a semicircle is added on outside the region. Find the area of the shaded region. [Use = 3.14] Figure 4 30/2/1 11 P.T.O. 22. AmH ${V 5 _|, ABDC, 28 go_r { `m dmbo EH$ d m H$m MVwWm e h VWm BC H$mo `mg _mZH$a EH$ AY d m BEC ItMm J`m h & N>m`m {H$V ^mJ H$m jo \$b kmV H$s{OE & [ = 22 7 br{OE] AmH ${V 5 In Figure 5, ABDC is a quadrant of a circle of radius 28 cm and a semi circle BEC is drawn with BC as diameter. Find the area of the shaded 22 region. [Use = ] 7 Figure 5 23. EH$ e dmH$ma Q> Q>, {OgHo$ AmYma H$m `mg 14 _r VWm D $MmB 24 _r h , Ho$ ~ZmZo _| 5 _r Mm S>mB H$m H$n S>m bJm`m J`m & < 25 {V _rQ>a H$s Xa go, H$n S>m bJmZo H$m `` kmV 22 br{OE] 7 A 5 m wide cloth is used to make a conical tent of base diameter 14 m and height 24 m. Find the cost of cloth used at the rate of < 25 per metre. 22 [Use = ] 7 EH$ b S>H$s aoV go ^ar, 18 go_r AmYma { `m VWm 32 go_r D $MmB dmbr, EH$ ~obZmH$ma H$s{OE & 24. [ = ~m Q>r H$mo \$e na Imbr H$aHo$ EH$ e dmH$ma T>oar ~ZmVr h & `{X Bg e dmH$ma T>a o r H$s D $MmB 24 go_r h Vmo BgH$s {VaN>r D $MmB Xe_bd Ho$ EH$ WmZ VH$ kmV H$s{OE & A girl empties a cylindrical bucket, full of sand, of base radius 18 cm and height 32 cm, on the floor to form a conical heap of sand. If the height of this conical heap is 24 cm, then find its slant height correct upto one place of decimal. 30/2/1 12 I S> X SECTION D Z g `m 25 go 34 VH$ `oH$ Z Ho$ 4 A H$ h & Question numbers 25 to 34 carry 4 marks each. 394 h & g `mE kmV H$s{OE & The sum of the squares of two consecutive odd numbers is 394. Find the numbers. 25. Xmo H $_mJV {df_ g `mAm| Ho$ dJm] H$m `moJ\$b 26. `{X {H$gr g_m Va lo T>r Ho$ W_ n nXm| H$m `moJ\$b Sn hmo, Vmo {g H$s{OE {H$ S30 = 3 (S20 S10). If Sn denotes the sum of the first n terms of an A.P., prove that S30 = 3 (S20 S10). 27. {g H$s{OE {H$ {H$gr ~m {~ X go d m na ItMr JB ne aoImAm| H$s b ~mB`m ~am~a hmoVr h & Prove that the lengths of the tangents drawn from an external point to a circle are equal. 28. EH$ {M_Zr Ho$ {eIa H$m EH$ _rZma Ho$ nmX go C `Z H$moU 60 h VWm _rZma Ho$ erf go {M_Zr Ho$ nmX H$m AdZ_Z H$moU 30 h & `{X _rZma H$s D $MmB 40 _r h , Vmo {M_Zr H$s D $MmB kmV H$s{OE & X fU _mnX S>m| Ho$ AZwgma YwAm N>mo S>Zo dmbr {M_Zr H$s D $MmB H$_-go-H$_ 100 _r hmoZr Mm{hE & ~VmBE {H$ Cnamo $ {M_Zr H$s D $MmB X fU _mnX S>m| H$mo nyam H$aVr h `m Zht & Bg Z _| {H$g _y ` H$mo Xem `m J`m h ? The angle of elevation of the top of a chimney from the foot of a tower is 60 and the angle of depression of the foot of the chimney from the top of the tower is 30 . If the height of the tower is 40 m, find the height of the chimney. According to pollution control norms, the minimum height of a smoke emitting chimney should be 100 m. State if the height of the above mentioned chimney meets the pollution norms. What value is discussed in this question ? 30/2/1 13 P.T.O. 29. EH$ W bo _| H$mS> h {OZ na 1 go 30 VH$ H$s g `mE A {H$V h & Bg W bo _| 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$V g `m (i) 3 go ^m ` Zht h & (ii) 7 go ~ S>r A^m ` g `m h & (iii) nyU dJ g `m Zht h & Cards numbered 1 to 30 are put in a bag. A card is drawn at random from this bag. Find the probability that the number on the drawn card is (i) not divisible by 3. (ii) a prime number greater than 7. (iii) not a perfect square number. 30. A( 10, 4) VWm B( 2, 0) H$mo C( 9, 4) VWm D( 4, y) H$mo {_bmZo H$s{OE {Og_| {~ X P aoImI S> CD H$mo {~ X Am| {_bmZo dmbo aoImI S> H$m _ `-{~ X P, {~ X Am| dmbo aoImI S> na p WV h & dh AZwnmV kmV ~m Q>Vm h & y H$m _mZ ^r kmV H$s{OE & The mid-point P of the line segment joining the points A( 10, 4) and B( 2, 0) lies on the line segment joining the points C( 9, 4) and D( 4, y). Find the ratio in which P divides CD. Also find the value of y. 31. x Ho$ {bE hb H$s{OE : 2x 1 x 3 1 2 x 3 3 2x 1 = 5; x 3, 2 . Solve for x : 2x 1 x 3 1 2 x 3 3 2x 1 = 5; x 3, 2 . 32. EH$ d m Ho$ n[aJV EH$ MVw^w O ItMm J`m h & {g H$s{OE {H$ MVw^w O H$s g _wI ^wOmAm| Ho$ `moJ\$b g_mZ h & A quadrilateral is drawn to circumscribe a circle. Prove that the sums of opposite sides are equal. 30/2/1 14 33. ^wOm 7 go_r dmbo EH$ KZmH$ma bm H$ Ho$ EH$ \$bH$ H$mo A Xa H$s Amoa go H$mQ> H$a EH$ AY JmobmH$ma JS >T>m Bg H$ma ~Zm`m J`m h {H$ AY Jmobo H$m `mg KZ Ho$ EH$ {H$Zmao Ho$ 22 br{OE] 7 A hemispherical depression is cut out from one face of a cubical block of side 7 cm, such that the diameter of the hemisphere is equal to the edge 22 of the cube. Find the surface area of the remaining solid. [Use = ] 7 ~am~a h & eof ~Mo R>mog H$m n >r` jo \$b kmV H$s{OE & 34. [ = 24 go_r D $MmB H$s D$na go Iwbr YmVw H$s EH$ ~m Q>r, EH$ e Hw$ Ho$ {N> H$ Ho$ AmH$ma H$s h {OgHo$ {ZMbo VWm D$nar d mr` {gam| H$s { `mE H $_e 7 go_r VWm 14 go_r h & kmV H$s{OE : (i) ~m Q>r H$mo nyam ^aZo dmbo nmZr H$m Am`VZ & (ii) ~m Q>r H$mo ~ZmZo _| bJr YmVw H$s MmXa H$m jo \$b & [ = 22 7 br{OE] A metallic bucket, open at the top, of height 24 cm is in the form of the frustum of a cone, the radii of whose lower and upper circular ends are 7 cm and 14 cm respectively. Find : (i) the volume of water which can completely fill the bucket. (ii) the area of the metal sheet used to make the bucket. 22 [Use = ] 7 30/2/1 15 P.T.O.

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