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ICSE Class X Prelims 2025 : Mathematics (City Montessori School (CMS), Mahanagar, Lucknow)

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Rishi Yadav
Hoerner College, Lucknow

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THE SECOND PRE-BOARD EXAMINATION 2024-25 Class X (ICSE) MATHEMATICS fhnc: Three hours Maximum marks: 80 Instructions: Answers to this paper must be written on the answer script provid ed separa tely. You will NOT be allowe d to write during the first 15 mi11utes. This time is to be spent in readin g the questio n paper. The time given at the head of this paper is the time allowe d for writing the answers. Attemp t all questio ns from Section A and any four questio ns from Section B. All workin g, includi ng rough work, must be clearly shown and must be done on the same sheet as the rest of the answer. Omission of essenti al workin g will result in loss of marks. The intende d marks for questio ns or parts of questio ns are given in bracke ts [ ] . Mathe matica l tables and graph papers shall be provid ed to you. SECTION A (40 marks] Attemp t all questio ns from this Section. Question 1 Choose the correct answer s to the questions from the given options. (Do not copy the questio n, write the correct answers only.) "l:>+ o') 'J'l. '6 -S p .. - (. (3 +- \ ~ - lu - (. [ I 5] r) - ) : ._, - (i) (ii) SX-b ... : e, 'c ,. t;--t. A polyno mial in x is x 3 + ax2 - 5x - 6. Which of the following is a factor of the given polyno mial so that the value of a is 2? (a) (x-1) ~ (x-2) (c) (x - 3) (d) (x- 4) Prakha r deposi ted ~ 1000 per month in a recurring deposit account for 1 year. The qwdify ing sum of money for the c~lculation of interest is: (a) ~ 120 (c) ~ 12065 Jbr (d) ( iii) In the given diagram , LB = LF and BA - ....l- FD = BC FE ', ' , ~ ,,1 . !'? 1,_: "\ 11 r _., , ,g ~ r r 11 ~ 12000 -:- :f IC~ Jl 12.. ~ -c--t lloP 78000 . E - - -- - - ~ F (Figures not drawn to scale) r,,, \ 0 Which of the following options is correct? (a) MBC.~ 6DFE (b) M.BC ~ 6DEF (c) 6A C ~ 6EFD ,.,.,.,..-(d) The similarity of given triangles cannot be determ ined. This paper consists of Ten printed pages. Turn over w 1n the given figun :, PA and PB are tan~i :nts at point s A and B respe ctive ly to a circle with centr e 0 . Lf LAP B= ( i v) J~l ./ 1 -i l 1. :. ; I \', o I-= / ..J - ~ , . C. -:(:., -~1 nd jven that A a B=L ~l G . n matri ces is possible . 1,.sse rtion (A): Prod uct BA of the two ~,ve . ssib\ e if and on\y if the P,rod uct of the two mat~ ce~~ t;atr ix is equa l to th_e numb Reas on (ll): er numb er of colum ns ID 40 , then the meas ure of reflex LAO B is : . t e . . of row s i.n -the secon d matn x . (Fig ure not drawn to scale) ~ 4 0 (v) (d) R . the corre ct reaso n for A. ts ( Both A and R are trU e and d R is the incor rect reaso n for A. '""(d ) Both A and Rare tn1e . an . 320 If the angle o f depre ss ion ofan objec t from a 75 m high vertic al towe r is 60 , then the dista nce of the objec t from the towe r i.s: ~ 3 m (c) (vi) . mie R is false . (a) A ts ) A is false , R is tnJe . (b ,M2 20 (c) 280 :;f '.~ 75 3 m (b) 75m (d) 150 m ~ The table given be\ow !:ow the value s o f x a s (dire ctly prop ortio nal) ay \ 24 \ 15? (, o (x.) ./1 ~ 9- E~>:~' \I \ ~ ~:, .h As s ume th a t the vo lumes ofa ../j ?; so lid cylin der and a solid cone of same radii are same . St.at. emen l I: The heigh t of the solid cylin der is three times the heigh t of the ~ _:I ~o lid cone . . ~1 S t.al.cm .cnl 2 : . The r.u10 betw een the heigh . . . ~2 <;-h. ts of the solid cyhn der and the sohd cone i.s 3 : I . --==.... ,2 (xi) ,, (viii) Ts n card s (id..:n tical in a.JI respe cts) are numb ered I to IO as show n . . ' l. ~ ~ ~ -!:;,' at rando m. The proba bility ,.c\cct<.-d card bca~ u co11\1> that~ <>,. it.: nun1b e.r is: ., ,, 2 (d ) - 2'6 ~ '-'_-At ;. ,.,'.s .. ,-:i t- "' 7 10 ,(. 1.'-' ;c , I c; ~IL C?0, ~ t h-:c Vi (b) a"" 32 and b = 9 (d) a=\ 8an db= \6 al r umer or '"'460 ' (b) ! ~ (x.iii) If .t 4 2 =- ~ '!,~ ~,920 . The oint of inlcr sccli on o_f th~ hncs x mx 2y = 0 . Thev a\uc. otmi :s . (a) \~ - -; ,, ... 1,, ]-J (xii) (c) (b) I 2 l.. :) + l -.::. and sells it to a , an articl A retail er purch ases ft Thee for' 800_0 fro01 _a who l~e ~te ofGS T i~ 10% cons ume r at \ 5% pro t . GSTsa\es are intra- s tate and fi the purch ase ,s: ) paid b y the cons (c) , 680 ::> - is propo rtion al -) '1.. 0 Cl. 2 -1 (a) '\20 I I ,r Q ri. The amou nt of tax. (und er 2. 5 nd y wher e . (d) S1a1..-m.:n1 I is false and St.al.c ment 2 is true. -( vii) Wlwt is th e. mean oftJ1 e rnynb ers in prog r.:ssio n 4 . 6. 8, . .... , 40? 'La.. (a} 20 ~ 21 (c ) 41 (d) 42 (a) \ (a) a=\6 and b = l8 ~ a = 9 and b = 3 2 1' "J 2 is false . CJ . b The value s of a and bare : Whic h of the follo wing optio ns is valid ? (a) BotJ1 tJie sU<tc mcots are truc _..__ {b) Buth tb.c state ment s arc fa lse . (c) Sw1c m,;n1 I is 1.rue and State ment ~ l2_l ~ ~ ~ ~ A ca1-d is s.:k,: t..:d frum th.:s.: cards . +y =S a nd x - Y (b) = o lies on the \in~ l- ~~ (d) x2 + 3k - .-.- = 0. the value of k is: is a root of the equa tio n 2 2 (b) } (d) -_i --; .c_ , 1.,= o 3 u- ' ' - l\ \ T'r'l '-\ - 'l.. ' :.J c u=O m - <-, .,,.,, ~ ,::-~ 2 Y-: ., -l-: i= 0 ~ )-.._ ... l. : 0 :..'<- cc: - b " " - 1.. (2) 'TUl' I\ o, et ,d--=-.".) f' ~ -1J ~-i j i " , fl, l ., 0 ~ - ..,.,., l j , 5," - ,.,, "' ~ (b) The rate of return for Mr Agarwal is 24%. I.,,~~," , R (xiY ) Mr Gupta invests in~ 100, 18% shares o f Company A avai lable at~ 60 each. Mr Agarwal invests in~ 50, 24% shares o f Comp any B availab le at~ 40 each . Use this information to state which of the following statements is true . r-J \'" '1' 161 D t-= 1 ~ 1 ,.... 1"i'v, r- J=-n, o ~ (a) The rate o fretum for Mr Gupta is 18%. -~ ( 111 the given figure, I is the incentre of 6ABC. Bl when produced meets the ~ ...... circle of 6ABC at D. CiJ'CLU-" J' Y '5 0 Nv :. ,so - 7l 1,.."' ,Qo ~ ..(tj Both Mr Gupta and Mr Agarwal have the same rate o fretum of 30%. (d) The given information is insufficient to compute the rate of return for Mr Gup ta and Mr Agarwal. (xv) The real number lines for two linear inequations A and B are as given below :,. . A : -t---+--9--t--+--+--..--+ 2 I O -I -2 -J -4 -5 _g_ ' g~ :. )Pf '/ 1 - - 7 .lG")... .;::. l.. B: -t--.--+--+---+---- ---t---t -5 -J -4 -2 -1 1 '. J (Figure not drawn to scale) R :.- "'' =-.L -e,.,_~ t<1_j I Given that LBAC = 600 and LACB = 700, find: 2 - !" <'.'.'.,( {:. I) 0 A r. B can be represented by : (a) 5 -4 -3 -2 -I 0 (b) fa -5 -4 -3 -2 -I 0 -5 -4 .J -2 -1 0 -5 -4 -3 -2 -I 0 LDCA ,~.-/ cis . (b) LDAC ~ /" S (c) LOCI ::- 1'-'-'f{; . (d) LAIC. mun>' (4) Question 3 (i) (d) Question 2 (a) 2 (ii) 'l. -1 , 3 ~~factorised the polynomia l 2x + 5x2 - I Ix - 14 and found the result as (x + I) ( x .- ]) (2x + 3 ). Using remainder and factor theorem, verify whether her result 1s correct or noL If not. write the correct factorisation o f the polynomial. showing essential working. / A:ili Ria has a rectangular piece of paper of dimensions 33 c~ by 24 cm. Find the vo lume of the largest cylinder that Ria can form by ~o ll mg the paper a long its longe r side as shown in the given figure . ~"' 2 4 t ') ~ ..,.., I (Taken= (4) th The 4 h tenn ofan A.P. is equal to 3 times its first term . The 7 term of the exceeds tw ice its 3rd tenn by I . Fmd the first term, the common d1ference an ~ =2 [ "-t3 ) s~ of its' first IO terms . ., ~- :c 1'- " e (4) [4) 33..) 7 ~ In the given figure. M is the mid point of the line segment joining the points A(0, 4) and 8(6. 0). M also divides the line segment OP in the ratio J : 3. "'- , 1.. y Find : (a) the coordinates ofM .,__-(S , t ) (b) the coordinates of P. i...----( 1'2 ,'i) (c) the equation of the line passing through Mand perpendicu lar to OP. JJ 24 cm IO. J ) A [4] 3:1. .. l-.:, l~:O --L....-------< ........................... 33 cm-, (Figure 1101 c _] drawn to scale) ( Fi g un: nvt drJ~il tu .),4,.J.k J (4) (S) T urn o, er (iii) Study the gi\'CO graph and answer the questions that follow: :~~ms r~l~_: J~:: :irtB< u; . :i-;..:-:- ~ u = ' -';+/ . (iii) ......... : '<.rt.;..- ! ~--=1~t:.: I :=-;2:: :~ -' suresh invested t _ qQQO int 100 shares. payin~)0 dividend quo ted at t gQ_. ~-J After a year, when the price of the shares rose to ~lQ_0 per share, Suresh sold all the shares and invested the er9ceeds in 12% ~ 400 shares at t 500. Calculate: 1 }A the sale proceeds. "1,1/ ~ ~ ~ ," "' .(f\) the m1;mber o_f~ ~00 shares he ~ought. {c) .::=:: ~ ~ r1'.. " 2.ai> the change in his annual income from dividend .~ 4 1,(b ~stion5 riVThe scale of map is 1 : 50000. In the map. a triangular plot PQR of land h as the t Y' tf 1 dimensions as shown 111 the followmg figure. p H t3.( \ \<.CO ;;. ',O, nfl - 1 1 3cm ~L-..L..--------"' R _!=l:Ltt Q<- : --------- 4 cm : - > (Figure not Jrawn to sl:ale) s l tttl =llJ jj.i:li-t. (a) ~ Calculate: the actual length of side QR. in km , oftbe land. Wn!c the coordinates of points p and S. (b) Given th at point Q is the image of oint p . . . u,n er reflect,on in a lme, write the -,: ~ c4uat1on of the line ofr fl " t. c ,:c; ,o n. -~~ d Write . , tJ1c coordin .. . a te so ftJ ie image ofS when reflected in the line found in (b). I Q . R and S 111 ordcr. f-<~el'j !Yl'\ - [5 1 (a) the numb e r or monthly instalments deposited by Dhceraj. (b) the total amount of money deposited by him in the accatiut.__.,.....- Age (years) ~n4 (ii) Provc th.: following trigo nometric id.:ntit y: (sec A - tan A)i (l + sin A) ~ I - si n A 6- 8- 8 - 10 10 - 12 12 - 14 20 l7 No . of students ckc1mnl place . + rn :;;. - 2.. q, ~ [3] [3 l 14 - 16 so ~ - 1~") f'.ind the median class of the distribution by ~~l~ng C\1muh1tivc lrcquene,cs . "I 'LC -::c ""-i.si.ew.. 1~ )_ ,,\' ',~~-~, '~-- 1 _J"-~ 1, ~ -~ \ 'I, ~ l , ( 6) , ,.,.., ~ 71 Calculah: the mean by short-cut method . ExprS:'-~Y:pur answer concct tl, one (3 J "\ ~ ~ ) The following table shows the ages of th e students ofa secondary school noted for otricial record : an_rfour q11csrio11sji o111 this Scc1io 11 . Solve lh.: following incqu ati o n and rep rcs7 n t th solut' . t on the number line : 1011 se _3_ < x - 3 x I <' c...' L::\-- - - - - < - - . whl're x E I , !"' r.,J - - J... 3 2 3 3 /-'' I 11 as interest at the time of maturity, find : . . . . J axis. obtained on JOming the points ti) 0 2 , }vY'J'\ The bank pays interest at the rate of 12% per annum . If Dhecraj gets t I 026 SECTION B 140 marks! ;/ /f('lnpl r~ --(b) the actual area of the plot in sq. km . , .1-_..,,_,',i,.,..., rw u;a><> . . ~Dheeraj deposited t 600 per month in a recurring deposit account in a bank . (d) Wntc th e coordin ates of a point which . ,s . invariant under reflection in the ,_ . (c) 5,1ate the geo,n.:trical n\lme oft.he closed figure 1- .,(a) ~ (i) Turn o a+ Question 6 Y , / Find the equation ofn str.1ight lin e AB p<"rpcndicular to the line PQ whose /4 uation is .t - 2_1 - 6 = 0. The line AB cuts nn intercept of 2 units from the pos itiv..- _1~e1.x is. Henc.: . find the point of in tersection of the two lin es AB and PQ. (ii) [3) T o celebrate N..:w Year P:uty in a schoo l. the Principal asked the Students Counci l to bu y nece ssary itc:ms for decoration from thc mnrket and to mark the bill of ra~mcnt 10 the AccounL, Section. Th..: fo llow in g bill shows the GST mte and the mark ed pncc of item s along with the di scount "-../ Prakas h Departmental Store Marked Price per packet 5% 18% hem I. Artifici a l Flowers ( I packet) 2. Balloo ns ( I packet) noo 10% 5%, 3. Buntings ( I packet) ~ no 5% ~ 360 400 ~~ l Fi nd th c 10 1.11 bill amoum including GST. (iii) Discount Rntc of GST S . No . L CAB (ii) (bj L l>AO -') (J' (c) 13 ,\ ~o-t 5 \" '- = /ao L BOD . -'1 )1,o - <\ bl-~ (r:1 !! 1111 11.,11..lr:l\ v! 1111 (iii) 1111! d niw 11 111 ,-.:ak- ) (b) a card with a prime number. (c) 11 [ 51 blue or an omng.: cnrcl with n numbcr whic h is n multiple o f 5 . 13] Using rukr and compasses only. construct a triangle PQR when: PQ = 3 c m. QR = 4 c m nncl L PQR = 90". Hence'. constnict a c ircumcirclc circumscribing th.: triangk PQR. Measure und write clown the 111clius of the circumcirc lc . I 31 If Rx - 13y = 5x + 3J . use prop.:rtics ofprupo11ion to !ind the va luc of 9x + Sy 9x - Sy. ). 0 l 'co- 111 IJ (F igu11.: a ye ll ow card . (11) [3) chords o f a circ le in1crscc1ing al the ce ntre 0 . (a) t (,'-~""' LUll.ia._ _ _l _S_Cl_11_1_....LJ11) cords arc numbcrcd I to 12. The cards arc wcll-shunlcd und 1hen _a ca rd 1s drnwn ol random from the hag. Find the probability that th.: ca rd drawn 1s : t In the gi ven figure. AB and CD ore two C Question fl A bug contai ns 12 yc ll ow cards. 12 blue curds and 12 ornngc cards. Ench sci o f (i) C Giwn llwt L A BC = 35". find: ln the given figure . the horizontal di stance between two vertica l to wers AB and CD on the same levcl ground is I 50 m. The angle of elevation of the top und the nngk of depression o f the bottom o f the 10,ve r AB as observed from the top of the tower CD nre 30" and 24 rcspcctively. Find the height of t.hc two towers. Give you r answers co rrcct to J significunt figures . (Use M111hcnmticnl Tubles !or thi s qu estion .) , 1,: ;1k) 1-11 Jl "- ',/.: Show your st <'rs and givc va lid reasons . f4] Question 9 11 ,so ~stion7 (i) - t (i) respectively . If the common ruti o is positivc , lind the lirsl term . common mtio Thc weight s of 50 workers arc given below : Weight (in kg) No. of workers so-60 The 4 1" . 6 1" and the In st 1c1111s of a gco1m:tric progrcssion arc 6. 24 1111d 384 l , oio,."' sonl- R 7o'L.~ 6160i--=f1on 77i 90;.i-~~~r:-=-=--~-100-110 110- 120 100 90- II 14 (ii) 6 ( H) . II is mc ltcd and rccus t (a) the radius of the srh cn: . l:l I (b) !he number ofcnncs rccast. / . (b) If weighing 95 k , . d I b g an a Juve is considen:d ovcr.vcigh 1 find th ' n k f ~ y um er . o wor er,; who arc overweight . \\ ~ (-, &>,t ( The surface arcu o fn solid mctnllic sphcrc is 1386 cm 1 into solid right circu lar cones ofrndius J.5 c m and hcighl 3 cm . C akulutc : - IO k Draw an ogive ofl11c giwn distribution using a graph sheet Take 2 g 011 cm h . . .. . one axis and 2 c:m = 5 work r . c > o11 11 ie ot er axis . Use the gr:ph lo estima te the fullowini; : \.. ,.. IV. ....--r'-\V '(> (a) lhc upper and low.:r quartiks. ~ 131 and the number of tcnns in the progn.:ss ion. 2' ) (Tukc n = -= 7 f5] ( 9) Turn ov~r -- - (iii) The histogram give n below represents the heights of the pupils of a school. Study the graph and answer the following questions: [4] (::i) Write the frequency for each c lass interval. (b) State the modal c lass . (c) Estimate the modal height. ,.,, a rn ::, 0. <.0 - 12 E ::, ;z 8 QJ ~ 4 0 Question 10 (i) Solve the following equation for x and give your ans\ver cotTect to two decimal places : [3) 3x2 + 5x - I = 0 (ii) Given A - [ cos 450 - 2 cos 0 30"] sin sin 0 and B = [tan 45'' sin O'' 90''] . cos cot 45 " (a) Evaluate AB by show ing the steps of multiplication of two mat1ices. (b) What special name can be assigned to matrix B ? ~ ~ -f\.,\o.,hi~ (c) What inference can be drawn from the result obtained in part (a)? (iii) [3] Use ruler and compasses only for this question. (a) Construct ~ABC. where AB = 3.5 cm, BC = 6 cm and L ABC = 60 . (b) Construct the locus of points inside the triangle which are equidistant from BA and BC. (c) Construct the locus of points inside the triangle which are equidistant from Band C. (cl) Mark the point P which is equidistant from AB, BC and also equidistant from Band C. Measure and record the length of PB. ( 10 ) (4)

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