MATHEMATICS J1fu1a CBSE QUESTION PAPER CLASS-X SUMMATIVE ASSESSMENT - II. '<:icbfftd Qfta:rr - II. Maximum Marks : 90 3H&Jcfirl1 3Tcli : 90

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1 CBSE QUESTION PAPER CLASS-X SUMMATIVE ASSESSMENT - II '<:icbfftd Qfta:rr - II. MATHEMATICS Jfua Time allowed : 3 hours trp:lflt"f!lzt:3 efuz Maximum Marks : 90 3H&Jcfirl 3Tcli : 90 Hit/Ir</ f.#: ( i) 8eft w.r 3lFfclTe I I ( ii) w,r-f/;f ff 34 w.r I -;;:n- - 3f,?:r J:/tn: ff I

2 (iii) (iv)? fu5 3 -q - 3Tcfi cft8" 8 W-, "!, a! -l,;<4/ W-, f I? fu5 af -q 6 w-rf # 2 3TcficfiT'CI? IU5U-q 0 w-,'fff;r-'fff;r3tc/il" f I? IU5cf-q ow-rf # 4 3TcficfiT'CI &j, &2< c/it 'Jfll7T cfmt-ci General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 34 questions divided into four sections - A, B, CandD. (iii) (iv) Section A contains 8 questions of mark each, which are multiple choice type questions, Section B contains 6 questions of 2 marks each, Section C contains 0 questions of 3 marks each and Section D contains 0 questions of 4 marks each. Use of calculators is not permitted. Section-A :w.r 'fr llt ff 8 a?li 'JTrWi :w.r 3lcFi c/it 'B" I :w.r 'fr llt ff 8 -q 'JTrWi :w.r #fr[ WR: Rl?/Jf<I W: 'ff, ff ll!fi 'lllft 'B" I 'lllft I Question numbers to 8 carry mark each. For each of these questions four alternative choices have been provided of which only one is correct. Select the correct choice. :sif4cfia t": (A) (C) (B) (D) 6 4 2

3 If two different dice are rolled together, the probability of getting an even number on both dice, is : (A) (B) (C) (D) cfilcftcffi R, 60 cfif cfiuf i I cfil 9Gfai cftcffi 2 lft l:ri, ell ( lfrw-q ) i: (A) (B) (C) 4 (D) 4 A ladder makes an angle of 60 with the ground when placed against a wall. If the foot of the ladder is 2 m away from the wall, then the length of the ladder (in meters) is : (A) 4 (B) (C) 4../3 2-J'i (D) 4 3

4 3. -q, QR cfrr oot T:R t I T:R oo QR qi) PR t Im PT= 3.8 M-t m QR cfrr (wfi-q) t (A) 3.8 (C) 5.7 (B) 7.6 (D).9 In Fig., QR is a common tangent to the given circles, touching externally at the point T. The tangent at T meets QR at P. If PT= 3.8 cm, then the length of QR (in cm) is : (A) 3.8 (C) 5.7 (B) 7.6 (D).9 4

5 4. 2 #, PQ am PR Ocr@ wnt i I LQPR=46 t if LQOR cfitllr t : p (A) 67 (C) 44 (B) 34 (D) 46 In Fig. 2, PQ and PR are two tangents to a circle with centre 0. If LQPR=46, then LQOR equals: p (A) 67 (C) 44 (B) 34 (D) 46 5

6 5. >f p:nfh, ]film: 3y -, 3y + 5 am 5y + t my cnp = rrt-: (A) -3 (B) 4 (C) 5 (D) 2 The first three terms of an AP respectively are 3y -, 3y + 5 and 5y +. Then y equals: (A) -3 (B) 4 (C) 5 (D) wfi am 4 wfi im r-pit og 6 wfi im I cfit t": (A) 3 (B) 5 (C) 4 (D) 6 The number of solid spheres, each of diameter 6 cm that can be made by melting a solid metal cylinder of height 45 cm and diameter 4 cm, is : (A) 3 (B) 5 (C) 4 (D) 6 6

7 sif4c:fia t: (A) - 3 (B) - (C) (D) 3 30 A number is selected at random from the numbers to 30. The probability that it is a prime number is : (A) 3 (B) (C) (D) A(x, 2), B(-3, -4) om C(7, -5) m@i, ot xcfifl=!ht": (A) - 63 (B) 63 (C) 60 (D) - 60 If the points A(x, 2), B(-3, -4) and C(7, -5) are collinear, then the value of x is : (A) - 63 (B) 63 (C) 60 (D)

8 Section B Question numbers 9 to 4 carry 2 marks each. 9. Qq@ l:f\ attw P GTm PQW-TTPRW>lofiR LQPR= 20 "ITT, mm 2PQ=POt I If from an external point P of a circle with centre 0, two tangents PQ and PR are drawn such that LQPR = 20, prove that 2PQ = PO. 0. m GTlf'ITTRmoo3lf m 3TTcfil - T n lslllslo:s [ ti Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre..,(,liftcfi{oi 2x 2 + ax - a 2 = 0 cfit x I Solve the quadratic equation 2x 2 + ax - a 2 = 0 for x. 2. ufllj GT m-m IBffl cfit 3@ kll t I cfilj cfilj -qc: 3lH Sllf4cfial I Rahim tosses two different coins simultaneously. Find the probability of getting at least one tail. 8

9 3. 3 #, OPBQ qrf OABC t OA = 20 t al" 4ifeti< mlcfil ffl I (n = 3.4 ) In Fig. 3, a square OABC is inscribed in a quadrant OPBQ of a circle. If OA = 20 cm, find the area of the shaded region. (Use n = 3.4) 4. cfil s wn 65 t '9Gl"cfil<Wl" 7 30 t, al" m3lm: ffl I The first and the last terms of an AP are 8 and 65 respectively. If the sum of all its terms is 730, find its common difference. 9

10 Section C Question numbers 5 to 24 carry 3 marks each. 5. 4#, 2M" ffl ABC FTTt o t 3la: cfil" am ;..iifcha mr cfit I [t=3.4<fm J3= l.73 ] B C In Fig 4, a circle is inscribed in an equilateral triangle ABC of side 2 cm. Find the radius of inscribed circle and the area of the shaded region. [Use t = 3.4 and J3 =.73] B 0

11 6. - 3ffi t fcfi om - oo-ijt I - ftlffl 60 om 45 t I lft" t m ffi I [J3 = l.73 ] Two ships are there in the sea on either side of a light house in such a way that the ships and the light house are in the same straight line. The angles of depression of two ships as observed from the top of the light house are 60 and 45. If the height of the light house is 200 m, find the distance between the two ships. [Use J3 =.73] 7. 3ltR -ij 0 lfl" cxjm ffl 2 lfl" Hlcf>R ofll" cxjm 20 wft- :fwfw t I :fwf-ij'cfrt 4 M">!Rrefcr 'q@ wrt m -ij? A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank which is 0 m in diameter and 2 m deep. If the water flows through the pipe at the rate of 4 km per hour, in how much time will the tank be filled completely? 8. l-tl-lc:bloi ABC AB = 6 wft", BC = 8 wft" om LB = 90 t I B ACll\ BD om mb,comd cf@t I A ll\ 003TT I Draw a right triangle ABC in which AB = 6 cm, BC = 8 cm and LB = 90. Draw BD perpendicular from B on AC and draw a circle passing through the points B, C and D. Construct tangents from A to this circle.

12 9. 5 'ij, PSR, RTQ am PAQ i, = 0 M-, 3 M- am 7 M- -w ifct:;a mr cfit Jfu:ntr I [t=3.4 7wft s R T In Fig.5, PSR, RTQ and PAQ are three semicircles of diameters 0 cm, 3 cm and 7 cm respectively. Find the perimeter of the shaded region. [Use t = 3.4] R T 20. cfit mwri :G.!. % am -;ffcfl :JG.!. t it 63 cft :G 9 7 I If the seventh term of an AP is.!. and its ninth term is.!., find its 63 rd term M" 60 fttfch)o cf@ ir:r at -dl4 cfit - wr mttt 'ij cfitcj TT(JT % wr % I mlf cfil _!_ M" cf@ citt: 'ij t 2 it cm: A solid metallic right circular cone 20 cm high and whose vertical angle is 60, is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter _!_ cm, find the 2 length of the wire. 2

13 22. A(-, -4), B(b, c) am C(5, -) t am 2b + c = 4 i, it b am c l=l"h I If the points A(-, -4), B(b, c) and C(5, -) are collinear and 2b + c = 4, find the values of band c. 23. P(2, 2), * A(-2, k) am B(-2k, -3) "ij Wi til i, it k cfit l=fh IAPc f,:ft- I If the point P(2, 2) is equidistant from the points A(-2, k) and B(-2k, -3), find k. Also find the length of AP x+3 x+l '84cfr(OI --=-;x:;t-3,-, cfil x I Solve the equation!_± = - 5 -; x :;t -3, -, for x. x+3 x+l SectionD Question numbers 25 to 34 carry 4 marks each. 25. wmi- wrr cfil ffli wftam "mtc t 2.5 wfti I ffi-qr-fl WT 3lli I ffi-q 0.5 wft olfmc f JLJ lcfir JLJR-!.!.!i ffi -qr-flcfif mr 3lT I 5 fofi ffl-qfona-ft JLJR-!.!.!i I WITT!" mmfofiffl -qr-ft m c t l R i on) zjfq I WITT!" cfit M? 3

14 Sushant has a vessel, of the form of an inverted cone, open at the top, of height cm and radius of top as 2.5 cm and is full of water. Metallic spherical balls each of diameter 0.5 cm are put in the vessel due to which th of the water in 5 the vessel flows out. Find how many balls were put in the vessel. Sushant made the arrangement so that the water that flows out irrigates the flower beds. What value has been shown by Sushant? 26. 6olfl"½cfi RlwiJ amtig am cfitut : 30 am 60 I R!wiJ am Ji=q 'i cfit 3TTR:am ffi I The angles of elevation and depression of the top and the bottom of a tower from the top of a building, 60 m high, are 30 and 60 respectively. Find the difference between the heights of the building and the tower and the distance between them. Prove that a parallelogram circumscribing a circle is a rhombus. 28. P(x, 2), * A(2, 5) am B(4,- 3) cfil c!tr{@l@os cfil" -q t I xcfilllr I Find the ratio in which the point P(x, 2) divides the line segment joining the points A(2, 5) and B(4,- 3). Also find the value of x. 4

15 29. 5ow.Wft owc:fil<:rtt 20am 5Wc:fil<WT 2565 t I.WJfi I In an AP of 50 terms, the sum of first 0 terms is 20 and the sum of its last 5 terms is Find the A.P. 3o. 2.s wft 3fu: 4.2 wft om 3fu: q@t icfcllcfir efitc IBm t I om c:fit I [ t = 22 ] 7 From a solid cylinder of height 2.8 cm and diameter 4.2 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid. [Take t = 22 ] 7 3. p l=l"r,ai::flcfi(oi (2p+l).x2- (p + 2)x + (p- 3) = 0 Wf"B="Rtl Wf I Find the value of p for which the quadratic equation (2p+ l).x2- (p + 2)x + (p - 3) = 0 has equal roots. Also find these roots ocfi 3TT ofili t I ofili 4 -i04 R FM t, ot :sllflcfidl f.:roti@ ofili "CR = (i) t (ii) qrf t (iii) 5 i (iv) 20 cfilf t Cards numbered from to 60 are kept in a box. If a card is drawn at random from the box, find the probability that the number on the drawn card is (i) an odd number (ii) a perfect square number (iii) divisible by 5 (iv) a prime number less than 20 5

16 33. fcfi R 00, - cfrvft R mdlt" I Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. The difference of two natural numbers is 5 and the difference of their reciprocals is 2-. Find the numbers. 4 6