CBSE QUESTION PAPER CLASS-X MATHS

Transcription

1 BSE QUESTION PPER LSS-X MTHS 'nme allowed : 3 hours RwtrrR1Fl:3 'E{TTZ Maximum Marks: 80 3:/cn: 80 General Instructions : (i) ll questions are compulsory. (ii) The question paper consists of 30 questions divided into four sections -, B, and D. Section comprises of ten questions of 1 mark each, Section B comprises of five questions of 2 marks each, Section comprises of ten questions of 3 marks each and Section D comprises of five questions of 6 marks each (iii) (iv) ll questions in Section are to be answered in one word, one sentence or as per the_ exact requirement of the question. There is no overall choice. However, an internal choice has been provided in one question of 2 marks each, three questions of 3 marks each and two questions of 6 marks each. You have to attempt only one of the alternatives in all such questions. (v) In question on construction, the drawings should be neat and exactly as per the given measurement. (vi) Use of calculators is not permitted.

2 "f1tl1p: : (i)?twjtr t, (ii) TT{ JTR-W -if 30 JT'FT t if 'cft< - 37, "ff 3#( "E:: ff fcnrfirn t I 37 ff "J(R" t "ff 'JTR I cfit!, a! ff rifr.r JTR "ff JfPU -srr t2 t, -«ff"if'fl-srrt # JTFrt3 ta?it (iii) (iv) (v) (vi) "E:: -if 'QTr.T >TFf t "ff JfPU 'JTR t6 t I 37 -if 'JTR "cfit irf< TJ:cf) TJ:cf) cwpl 37W JTR itl" 3fici 2/ctirt/riffl< f&:zrr "iift "ffcfiff r I 'JTR-W ff fcrq;f;q g I cffl'frr 2 3fri'i' cftff 'JTR #, 3 3fri'i' cftff efr- ffl" -if a?it 6 3frif?ft m ff fcrq;f;q -,m t, "ff'4t m -if amcfit ffi fcrq;f;q ft <1iRT g I fq;::rt cftff -srr -if fq;::rt rtw Rii' -,m rrrrr t 37cfi irt' I J; eyi 02fj' t ffl "Eh7 r I SETION 3l Questions number 1 to 10 carry 1 mark each. m 1-s-10rfcfi 'JTR"cfiT1 t, 1. omplete the missing entries in the following factor tree :. 2. If (x + a) is a factor of 2x 2 + 2ax + 5x + 10, find a..,.& '11 (x +a), 2 -rn,-r-,= ':rtt:m' 2x + 2ax + 5x + 10 cf)]' 2UH, Os ci, rn a cf)]' llr 'flll"l" I 3. Show that x = - 3 is a solution of x 2 + 6x + 9 = 0. fcl:i x = - 3, x 2 + 6x + 9 = 0 cf)]' I 4. The first term of an.p. is p and its common difference is q. Find its 10 th term. cf)]' '9'c!; p cf2tt mcf q t I '9'c!; I

3 5. If 5 tan =-, 12 find the value of (sin + cos ) sec. tan =. c'it 12 (sin + cos ) sec q;r llr 6. The lengths of the diagonals of a rhombus are 30 cm and 40 cm. Find the side of the rhombus. fcfcriut ctr ( WTT (i2lf 40 WTT I ctr ctr I 7. In Figure 1, PQ II B and P : PB = 1 : 2. Find ar ( PQ) ar ( B) B Figure I 1 -q, PQ II B om P: PB = 1 : 2. ar ( PQ) ar(i1b) 1 B 3Wjifrr I 8. The sur f ace area of a sphere is 616 cm 2. Find its radius. 9. die is thrown once. Find the probability of getting a number less than 3. -gm cnt WfiT I 3 "B mil" m: ltt 31 ctr >ll cfictl 10. Find the class marks of classes and ,35-55 t cf7t I

4 SETION B if Questions number 11 to 15 carry 2 marks each. w;:r ri&ff 11 "8 15 wli w;:r t 2 3T<1i!' I 11. Find all the zeros of the polynomial x 4 + x 3-34x 2-4x + 120, if two of its zeros are 2 and -2.. x 4 + x 3-34x 2-4x ' * 2 <i2it -2 t I 12. pair of dice is thrown once. Find the probability of getting the same number on each dice. 13. If sec 4 = cosee ( - 20 ), where 4 is an acute angle, find the value of. In a t.. B, right-angled at, if 1 tan =..fa, find the value of sin cos B + cos sin B. 'lfr: sec 4 = cosec ( - 20 ), 4 -'q;tot t, -al cfit llr 3l t.. B if, ;gj.tc;fi)o1 ' 'lfr: tan = ' -al sin cos B + cos sin B cfit. v3 llh I 14. Find the value of k if the points (k, 3), (6, - 2) and (- 3, 4) are collinear. k cfit lfr * <k, 3), (6, -2) (i2it <-3, 4) m 15. E is a point on the side D produced of a llgm BD and BE intersects D at F. Show that fl BE,.., 6 FB. -BD chl 71t :PfT D E t BE :PfT D cfit F Sl@-c{ I 1% t:,. BE - t:,. FB.

5 SETION lf Questions number 16 to 25 carry 3 marks each. JfR "ffi!h 16 "R" 25 ffcfi JfR i!t 3 3fcfi / I 16. Use Euclid's Division Lemma to show that the square of any positive integer is either of the form 3m or (3m + 1 ) for some integer m. 51 f4cfil q;r m fen fcflm q;r cr,f fcfim m t 3m (3m + 1) t if QTdT t I 17. Represent the following pair of equations graphically and write the coordinates of points where the lines intersect y-axis X + 3y = 6 2x - 3y = 12 f.r-i:;: wi'lcfi<o, "T1l mr f.i{<zftrn <f21t 6-1 an t f--1 1icfi y-3'.re;t 51 fa-.i c:: X + 3y = 6 2x - 3y = For what value of n are the n th terms of two.p.'s 63, 65, 67,... and 3, 10, 17,... equal? If m times the m th term of an.p. is equal to n times its n th term, find the ( m + n lh term of the.p. D t BR" t 'c!,t fll-jl.-rj{ ) (> 63, 65, 67,... cf217" 3, 10, 17,... t nit 1lc!, m 31' M Bl-11.-d{ t mil 'le!. cfit m :ftt az!t nit 'le!. cfit n :ftt m t ill cfit (m + n)9t 'le!. I 19. In an.p., the first term is 8, n th term is 33 and sum to first n terms is 123. Find n and d, the common difference. "[cn cfit >f2tlr 'le!. 8, n9t 'le!. 33 cf21t >f2tlr n cfit lfl'llfici 123 I d('bfcf ) I n a21t

6 20. Prove that : (1 + cot + tan ) (sin - cos ) = sin tan - cot cos. Without using trigonometric tables, evaluate the following 2 ( cos 58 ) _ ls ( cos 38 cosec 52 sin 32 tan 15 tan 60 t n 75 ) fcti (1 + cot + tan ) (sin - cos ) = sin tan - cot cos. 2 ( cos 58 ) - ls ( cos 38 cosec 52 ) sin 32 tan 15 tan 60 tan 75. ' 21. If P divides the join of (-2, -2) and B(2, -4) such that = f, find the coordinates of P. * 3TT (-2, -2) B(2, -4) cit 9@ {@ O:S cit P i fcti P =, or p f.i 1icfi -=furr '-flll"i B The mid-points of the sides of a triangle are (3, 4), (4, 6) and (5, 7). Find the coordinates of the vertices of the triangle. ct>l 3lf t -i:rv:r- ;m t R licfi (3, 4), (4, 6) <12TT cs, 7) R licfi I % ffl 23. Draw a right triangle in which the sides containing the right angle are 5 cm and 4 cm. onstruct a similar triangle whose sides are. times the sides of the above triangle.. 3 f!chloi q)l' f!chloi qfffi :Pfl"3TT q)l' 1.1'41$-41 5 WTT 4-M)- I - :PfT cfif i 3

7 24: Prove that a parallelogram circumscribing a circle is a rhombus. In Figure 2, D.l B. Prove that B 2 + D 2 = BD B ' , Figure 2 ft -cmr@<if.ti m<tt t 2 "G", D.l B. fcfi B 2 + D 2 = BD B' , 25. In Figure 3, B is a quadrant of a circle of radius 14 cm and a semi-circle is drawn with B' as diameter. Find the area of the shaded region. Figure 3

8 3 #, B Tf1 lltttt f-jl*lc:til 14 "@TT t, cfit 'tj ;qf:.tl * WTT B chl" olfffi ithc:fi< 01-'-tifctia 'ITI1f ci)t ctlr!q, I 3 SETION D G Questions number 26 to 30 carry 6 marks each. #<!!r 26 "# 30 rrq; "JfR t 6 afrf I I 26. peacock is sitting on the top of a pillar, which is 9 m hig. From a point 27 m away from the bottom of the pillar, a snake is coming to its hole at the base of the pillar. Seeing the sn'ake the peacock pounces on it. If their speeds are equal, at what distance from the hole is. the snake caught? 4 The difference of two numbers is 4. If the difference of their reciprocals is -, find the 21 two numbers. 9 lft &i mm en: -im: t 1 21 lft cn1 u mer 3m fa@", &i 1lfc!: # W«=r, cfi1 am * I mq chl" ch< litt 'R "J:Tm I 7ITTl<TT m, m c:tilf¼t; f&i u en: mer GT 3TT cfit 4 t I* ffl &i41 cfit _±_ ' "f1t zy;rr fi@i wa- c:tilf-jlq, The angle of elevation of an aeroplane from a point on the ground is 60. fter a flight of 30 seconds, the angle of elevation changes to 30. If the plane is flying at a constant height of 3600 F3 m, find the speed, in km/hour, of the plane. ql llh cfit - U "c:fi1ut 60 t I 30 c:fiu:s cfi1. &i q ill<(, irur 30 cfit "ITT I c11 llh 3600.J3 ltt ch1 3fiR 'R WT "ITT, m c111llh ch1 71fu (f&ilft;ek:r #) chlmq, I

9 28. If a Jine is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio. Using the above, prove the following : In Figure 4, B II DE and B.II EF. Prove that II DF. Figure 4 Prove that the lengths of tangents drawn from an external point to a circle are equal. Using the above, prove the following B is an isosceles triangle in which B =, circumscribed about a circle, as shown in Figure 5. Prove that the base is bisected by the point of contact. B p Figure 5

10 <l'r qi! :PTT ifi 3Ff :fittan qi) m-m an 1n: sira w, wr. m fcn c) an qif m amm wt, qifffl R8 : "' "' 4, B II DE W-lT B II EF. Ffi II DF. rn;s: fcn ifi f<:fim Tf 1n: wran cf>1 ak11 24, i, qifffl R8cfil : 5 B B4f,<ills, B::: t, ifi lffi7to <irt<tt 1"fcIT t I Fii 3ff'EITT m ID"U ffl "ITTfil t I B p 3WfFrr 5

11 29. If the radii of the circular ends of a conical bucket, which is 16 cm high, are 20 cm and 8 cm, find the capacity and total surface area of the bucket. [Use n = fchm i tlcfir, 16 WTT t fuu cfi1 20 WTT cf2tt 8WTT ten ch1' cf2l1 J - h= 1 22 ] Find mean, median and mode of the following data : lasses Frequency T ail is!hi: