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CBSE Class X 2007 : MATHEMATICS

11 pages, 68 questions, 3 questions with responses, 3 total responses,

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

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Roll No. jksy uaCode No. Series RKM/2 dksM ua- 30/2/1 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 condidate. Please check that this question paper contains 25 questions. Please write down the serial number of the question before attempting it. i;k tk p iz'u&i=k esa i;k tk p i;k iz'u dj ysa fd bl iz'u&i=k esa eqfnzr i`"B 11 gSaA nkfgus gkFk dh vksj fn, x, dksM uEcj dks Nk=k m kj&iqfLrdk ds eq[k&i`"B ij fy[ksaA dj ysa fd bl iz'u&i=k esa 25 iz'u gSaA dk m kj fy[kuk 'kq: djus ls igys] iz'u dk ekad vo'; fy[ksaA MATHEMATICS xf.kr Time allowed : 3 hours Maximum Marks: 80 fu/kkZfjr le; % 3 ?k.Vs vf/kdre vad % 80 General Instuctions : (i) All questions are compulsory. (ii) The question paper consists of 25 questions divided into three sections A, B and C. Section A contains 7 questions of 2 marks each, Section B is of 12 questions of 3 marks each and Section C is of 6 questions of 5 marks each. (iii) There is no overall choice. However, an internal choice has been provided in two questions of two marks each, two questions of three marks each and two questions of five marks each. (iv) In question on construction, the drawing should be neat and exactly as per the given measurements. (v) Use of calculators is not permitted. However, you may ask for Mathematical tables. lekU; funsZ'k % (i) 30/2/1 lHkh iz'u vfuok;Z gSaA 1 P.T.O. (ii) (iii) (iv) (v) bu iz'u&i=k esa 25 iz'u gSa tks rhu [k.Mksa v] c vkSj l esa c Vs gq, gSaA [k.M v esa nks& nks vad okys 7 iz'u] [k.M c esa rhu&rhu vad okys 12 iz'u rFkk [k.M l esa ik p&ik p vad okys 6 iz'u 'kkfey gSaA iz'u&i=k esa dksbZ lexz O;kid fodYi ugha gSA fQj Hkh nks&nks vadksa okys nks iz'uksa] rhu&rhu vadksa okys nks iz'uksa rFkk ik p&ik p vadksa okys nks iz'uksa esa vkarfjd fodYi fn, x, gSaA jpuk okys iz'u esa vkjs[ku LoPN gks vkSj fn, x, ekiu ds loZFkk vuq:i gksA dSydqysVj ds iz;ksx dh vuqefr ugha gSA ysfdu ;fn vko';drk gks rks vki xf.krh; lkjf.k;ksa dh ek x dj ldrs gSaA SECTION A [k.M v Questions number 1 to 7 carry 2 marks each. iz'u la[;k 1 ls 7 rd izR;sd iz'u ds 2 vad gSaA 1. Find the GCD of the following polynomials : 12x4 + 324x ; 36x3 + 90x2 54x fuEu cgqinksa dk e-l- (GCD) Kkr dhft, % 12x4 + 324x ; 36x3 + 90x2 54x 2. Solve for x and y : OR Solve for x and y : 31x + 29y = 33, 29x + 31y = 27 x rFkk y ds fy, gy dhft, % vFkok vFkok x rFkk y ds fy, gy dhft, % 31x + 29y = 33, 3. 29x + 31y = 27 Find the sum of all three digit whole numbers which are multiples of 7. lHkh rhu vadh; iw.kZ la[;kvksa dk ;ksxQy Kkr dhft, tks 7 ds xq.kt gSaA 30/2/1 2 4. In Figure 1, PQ | | AB and PR | | AC. Prove that QR | | BC. OR In Figure 2, incircle of ABC touches its sides AB, BC and CA at D, E and F respectively. If AB = AC, prove that BE = EC. vk fr 1 esa] PQ | | AB rFkk PR | | AC fl) dhft, fd QR | | BC. vFkok vFkok vk fr 2 esa] ABC dk vUr%o` k Hkqtkvksa AB, BC rFkk CA dks e'k% fcUnqvksa D, E rFkk F ij Li'kZ djrk gSA ;fn AB = AC gS] rks fl) dhft, fd BE = EC. 30/2/1 3 P.T.O. 5. If the mean of the following frequency distribution is 49, find the missing frequency p : Class Frequency 0 - 20 2 20 - 40 6 40 - 60 p 60 - 80 5 80 - 100 2 ;fn fuEu ckjackjrk caVu dk ek/; 49 gS] rks yqIr ckjackjrk p Kkr dhft, % oxZ 0 - 20 2 20 - 40 6 40 - 60 p 60 - 80 5 80 - 100 6. ckjEckjrk 2 A wrist-watch is available for Rs. 1,000 cash or Rs. 500 as cash down payment followed by three equal monthly instalments of Rs. 180. Calculate the rate of interest charged under the instalment plan. ,d gkFk&?kM+h dk udn ewY; 1]000 #- vFkok og 500 #- udn Hkqxrku ds lkFk 180 #- dh rhu leku ekfld fdLrksa esa Hkh miyC/k gSA fdLr ;kstuk ds vUrxZr C;kt dh nj ifjdfyr dhft,A 30/2/1 4 7. An unbiased die is tossed once. Find the probability of getting (i) a multiple of 2 or 3. (ii) a prime number greater than 2. ,d vufHkur ik lk ,d ckj mNkyk x;kA fuEu ds vkus dh izkf;drk Kkr dhft, % (i) 2 vFkok 3 dk xq.ktA (ii) 2 ls cM+h vHkkT; la[;kA SECTION B [k.M c Questions number 8 to 19 carry 3 marks each. iz'u la[;k 8 ls 19 rd izR;sd iz'u ds 3 vad gSaA 8. Solve the following system of equations graphically : 2x + y = 8; x + 1 = 2y fuEu lehdj.k fudk; dks xzkQ+ dh lgk;rk ls gy dhft, % 2x + y = 8; 9. x + 1 = 2y Simplify the following rational expression in the lowest terms : fuEu ifjes; O;atd dks mlds U;wure :i esa izdV dhft, % 10. If the sum to first n terms of an A.P. is given by Sn = n (n + 1), find the 20th term of the A.P. ;fn fdlh lekUrj Js<+h ds izFke n inksa dk ;ksxQy Sn = n (n + 1) }kjk ifjHkkf"kr gS] rks lekUrj Js<+h dk 20ok in Kkr dhft,A 30/2/1 5 P.T.O. 11. In a cyclic quadrilateral ABCD, diagonal AC bisects C. Prove that the tangent to the circle at A is parallel to the diagonal BD. OR In Figure 3, O is any point in the interior of ABC. OD, OE and OF are drawn perpendiculars to the sides BC, CA and AB respectively. Prove that AF2 + BD2 + CE2 = OA2 + OB2 + OC2 OD2 OE2 OF2 ,d p h; prqHkqZt ABCD esa] fod.kZ AC dks.k C dks lef}Hkkftr djrk gSA fl) dhft, fd fcUnq A ij o` k dh Li'kZ js[kk fod.kZ BD ds lekUrj gSA vFkok vk fr 3 esa] O f=kHkqt ABC dk ,d vkUrfjd fcUnq gSA Hkqtkvksa BC, CA rFkk AB ij e'k% yEc OD, OE rFkk OF Mkys x, gSaA fl) dhft, fd AF2 + BD2 + CE2 = OA2 + OB2 + OC2 OD2 OE2 OF2 12. Construct a ABC in which base BC = 6 cm, circumcircle of ABC. ABC dh jpuk dhft, ftlesa vk/kkj BC = 6 lseh] dk ifjo` k Hkh [khafp,A 30/2/1 6 B = 45 and B = 45 rFkk C = 60 . Draw a C = 60 . ABC 13. The diameter of a solid copper sphere is 18 cm. It is melted and drawn into a wire of uniform cross-section. If the length of the wire is 108 m, find its diameter. ,d Bksl rk cs ds xksys dk O;kl 18 lseh gSA bldks fi?kykdj ,d rkj] tks ,dleku vuqizLFk&ifjPNsn (cross-section) dh gS] ds :i esa [khapk x;k gSA ;fn rkj dh yEckbZ 108 eh- gS] rks mldk O;kl Kkr dhft,A 14. The expenditure (in rupees) of a family for a month is as follows : Item Expenditure Rent Food Education Electricity and Water Others 800 3000 1200 400 1800 Represent the above data by a pie-chart. fdlh ifjokj ds ,d ekg ds O;; #- esa dk C;kSjk fuEu gS % en fdjk;k Hkkstu f'k{kk fctyh rFkk ikuh vU; O;; 800 3000 1200 400 1800 mi;qZ vk dM+ksa dks ikbZ&pkVZ }kjk iznf'kZr dhft,A 15. From a pack of 52 cards, red face cards are removed. After that a card is drawn at random from the pack. Find the probability that the card drawn is (i) a queen. (ii) a red card. (iii) a spade card. 52 i kksa dh rk'k dh x h esa ls] yky fp=k okys i ks (face cards) fudky fy, x,A blds ckn x h esa ls ;kn`PN;k ,d i kk fudkyk x;kA izkf;drk Kkr dhft, fd fudkyk x;k i kk (i) ,d csxe gSA (ii) ,d yky i kk gSA (iii) ,d gqdqe (spad) dk i kk gSA 16. Prove that : OR If A, B and C are the interior angles of a triangle ABC, show that 30/2/1 7 P.T.O. fl) dhft, fd vFkok v Fkok ;fn A, B rFkk C, ABC ds vkUrfjd dks.k gSa] rks fn[kkb, fd % 17. The coordinates of the mid-points of the sides of a triangle are (4, 3), (6, 0) and (7, 2). Find the coordinates of the centroid of the triangle. fdlh f=kHkqt dh Hkqtkvksa ds e/;&fcUnqvksa ds funsZ'kkad (4, 3), (6, 0) rFkk (7, 2) gSaA f=kHkqt ds dsUnzd ds funsZ'kkad Kkr dhft,A 18. If the distance of P (x, y) from two points with coordinates (5, 1) and ( 1, 5) is equal, prove that 3x = 2y. ;fn fcUnq P (x, y) nks fcUnqvksa (5, 1) rFkk ( 1, 5) ls leku nwjh ij gS] rks fl) dhft, fd 3x = 2y 19. A loan of Rs. 24,600 is to be paid back in two equal semi-annual instalments. If the interest is charged at 10% per annum, compounded semi-annually, find the instalment. 24,600 #- dk _.k nks leku v/kZokf"kZd fdLrksa esa ykSVk;k tkuk gSA ;fn C;kt dh okf"kZd nj 10% gS rFkk C;kt izfr N%ekgh la;ksftr gksrk gS] rks fdLr dh jkf'k Kkr dhft,A SECTION C [k.M l Questions number 20 to 25 carry 5 marks each. iz'u la[;k 20 ls 25 rd izR;sd iz'u ds 5 vad gSaA 20. Prove that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. Using the above, prove the following : In Figure 4, O is the centre of the circle. If of AOC. 30/2/1 BAO = 30 and 8 BCO = 40 , find the value fl) dhft, fd fdlh pki }kjk dsUnz ij vUrfjd dks.k ml pki }kjk o` k ds 'ks"k Hkkx ij fLFkr fdlh fcUnq ij vUrfjd dks.k dk nqxquk gksrk gSA mi;qZ dk iz;ksx dj fuEu fl) dhft, % vk fr 4 esa] O o` k dk dsUnz gSA ;fn Kkr dhft,A BAO = 30 rFkk BCO = 40 rks AOC dk eku 21. State and prove Pythagoras theorem. Use the above to prove the following : ABC is an isosceles right triangle, right angled at C. Prove that AB2 = 2AC2. ikbFkSxksjl izes;ds dFku dks fyf[k, rFkk mls fl) dhft,A mi;qZ dk iz;ksx dj fuEu fl) dhft, % ABC ,d lef}ckgq ledks.k f=kHkqt gS ftlesa C ij ledks.k gSA fl) dhft, fd AB2 = 2AC2. 22. The side of a square exceeds the side of another square by 4 cm and the sum of areas of two squares is 400 sq. cm. Find the dimension of the squares. OR A fast train takes 3 hours less than a slow train for a journey of 600 km. If the speed of the slow train is 10 km/hour less than that of the fast train, find the speeds of the two trains. ,d oxZ dh Hkqtk nwljs oxZ dh Hkqtk ls 4 lseh cM+h gS vkSj nksuksa oxks ds {ks=kQyksa dk ;ksxQy 400 oxZ lseh gSA oxks dh Hkqtk dh yEckb;k Kkr dhft,A vFkok 600 fdeh dh ,d ;k=kk esa ,d rst+ jsyxkM+h nwljh /kheh jsyxkM+h ls 3 ?kaVs de le; ysrh gSA ;fn /kheh jsyxkM+h dh xfr rst+ jsyxkM+h dh xfr ls 10 fdeh@?kaVk de gS] rks nksuksa jsyxkfM+;ksa dh xfr Kkr dhft,A 30/2/1 9 P.T.O. 23. A hollow copper sphere of external and internal diameter 8 cm and 4 cm respectively is melted into a solid cone of base diameter 8 cm. Find the height of the cone. OR If the radii of the circular ends of a bucket 45 cm high, are 28 cm and 7 cm, find the capacity and surface area of the bucket. (Use ) ,d [kks[kys rk cs ds xksys ds ck rFkk vkUrfjd O;kl e'k% 8 lseh rFkk 4 lseh gSaA bldks fi?kykdj 8 lseh vk/kkj O;kl dk ,d Bksl 'kadq cuk;k x;k gSA 'kadq dh pkbZ Kkr dhft,A vFkok ;fn 45 lseh ph ,d ckYVh ds o` kkdkj fljksa dh f=kT;k, 28 lseh rFkk 7 lseh gSa] rks ckYVh dh /kkfjrk rFkk i`"Bh; {ks=kQy Kkr dhft,A iz;ksx dhft, 24. An observer in a lighthouse observes two ships on the same side of the lighthouse, and in the same straight line with the base of the lighthouse. The angles of depression of the ships approaching it are 30 and 60 . If the height of the lighthouse is 150 m, find the distance between the ships. izdk'k&LrEHk esa cSBk ,d izs{kd viuh vksj vkrs gq, nks ty;kuksa] tks izdk'k&LrEHk ds ,d gh vksj rFkk mlds vk/kkj ls ,d gh js[kk esa fLFkr gSa] ns[k jgk gSA ty;kuksa] tks izdk'k&LrEHk dh vksj vk jgs gSa] ds voueu dks.k 30 rFkk 60 gSaA ;fn izdk'k&LrEHk dh pkbZ 150 eh- gS] rks ty;kuksa ds chp dh nwjh Kkr dhft,A 25. Satish (aged 67 years) has monthly income of Rs. 30,000 (excluding HRA). He donates Rs. 80,000 to a charitable orphanage (50% exemption). He contributes Rs. 30,000 towards Public Provident Fund and purchases NSCs worth Rs. 20,000. He pays Rs. 1,500 as income tax per month for 11 months. Calculate the income tax to be paid by him in the 12th month of the year. Use the following to calculate income tax : (a) Savings 100% exemption for permissible savings upto Rs. 1,00,000 (b) Rates of Income tax for Senior Citizens (over 65 years) Slab (i) No tax From Rs. 1,85,001 to Rs. 2,50,000 20% of the taxable income exceeding Rs. 1,85,000 (iii) 30/2/1 Upto Rs. 1,85,000 (ii) (c) Income tax From Rs. 2,50,001 and above Rs. 13,000 + 30% of the taxamble exceeding Rs. 2,50,000 Education Cess 2% of Income tax payable 10 lrh'k vk;q 67 o"kZ dh ekfld vk; 30]000 #- edku fdjk;k Hk kk NksM+dj gSA og ,d vukFkky; dks 80]000 #- nku nsrk gS NwV 50% A og 30]000 #- lkoZtfud Hkfo"; fuf/k [kkrs esa tek djkrk gS rFkk 20]000 #- ds jk"V h; cpr i=k [kjhnrk gSA og 1]500 #- ekfld 11 ekg rd vk; dj nsrk gSA crkb, o"kZ ds 12osa ekg esa mls fdruk vk; dj nsuk gksxkA vk; dj dh x.kuk gsrq fuEu dk iz;ksx djsa % v cpr vf/kdre 1]00]000 #- dh vuqer cprksa ij 100% NwV c 65 o"kZ rFkk vf/kd ds O;f ;ksa ds fy, vk; dj dh njsa LySc vk; dj (i) 1]85]000 #- rd dksbZ vk; dj ugha (ii) 1]85]001 #- ls 2]50]000 #- rd 1]85]000 #- ls vf/kd dj&;ksX; vk; dk 20% (iv) 2]50]001 #- vkSj vf/kd 13]000 #- + 2]50]000 #- ls vf/kd dj&;ksX; jkf'k dk 30% ns; vk; dj dk 2% l f'k{kk midj 30/2/1 11 P.T.O.

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