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CBSE Class 12 Board Exam 2018 : Mathematics

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SET-1 H$moS> Z . Series SGN Code No. amob Z . 65/1 narjmWu H$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 > 12 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 _| >29 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 12 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 29 questions. Please write down the Serial Number of the question before attempting it. 15 minute 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. J{UV MATHEMATICS {ZYm [aV g_` : 3 K Q>o A{YH$V_ A H$ : 100 Time allowed : 3 hours 65/1 Maximum Marks : 100 1 P.T.O. gm_m ` {ZX}e : (i) g^r Z A{Zdm` h & (ii) Bg Z-n _| 29 Z h Omo Mma I S>m| _| {d^m{OV h : A, ~, g VWm X & I S> A _| 4 Z h {OZ_| go `oH$ EH$ A H$ H$m h & I S> ~ _| 8 Z h {OZ_| go `oH$ Xmo A H$ H$m h & I S> g _| 11 Z h {OZ_| go `oH$ Mma A H$ H$m h & I S> X _| 6 Z h {OZ_| go `oH$ N > : A H$ H$m h & (iii) I S> A _| g^r Zm| Ho$ C ma EH$ e X, EH$ dm ` AWdm Z H$s Amd `H$VmZwgma {XE Om gH$Vo h & (iv) nyU Z-n _| {dH$ n Zht h & {\$a ^r Mma A H$m| dmbo 3 Zm| _| VWm N> A H$m| dmbo 3 Zm| _| Am V[aH$ {dH$ n h & Eogo g^r Zm| _| go AmnH$mo EH$ hr {dH$ n hb H$aZm h & (v) H $bHw$boQ>a Ho$ `moJ H$s AZw_{V Zht h & `{X Amd `H$ hmo, Vmo Amn bKwJUH$s` gma{U`m _m J gH$Vo h & General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 29 questions divided into four sections A, B, C and D. Section A comprises of 4 questions of one mark each, Section B comprises of 8 questions of two marks each, Section C comprises of 11 questions of four marks each and Section D comprises of 6 questions of six marks each. (iii) All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question. (iv) There is no overall choice. However, internal choice has been provided in 3 questions of four marks each and 3 questions of six marks each. You have to attempt only one of the alternatives in all such questions. (v) 65/1 Use of calculators is not permitted. You may ask for logarithmic tables, if required. 2 I S> A SECTION A Z g `m 1 go 4 VH$ `oH$ Z 1 A H$ H$m h & Question numbers 1 to 4 carry 1 mark each. 1. tan 1 3 cot 1( 3 ) H$m _mZ kmV H$s{OE & Find the value of tan 1 3 cot 1( 3 ). 2. `{X Am `yh 0 A 2 b a 3 1 0 0 1 {df_ g_{_V h , Vmo a VWm b Ho$ _mZ kmV H$s{OE & 0 If the matrix A 2 b and b . 3. a 0 1 3 1 is skew symmetric, find the values of a 0 Xmo g{Xem| a VWm b , {OZHo$ n[a_mU g_mZ h , _| go `oH$ H$m n[a_mU kmV H$s{OE, O~{H$ CZHo$ ~rM H$m H$moU 60 h VWm CZH$m A{Xe JwUZ\$b 9 h & 2 Find the magnitude of each of the two vectors a and b , having the same magnitude such that the angle between them is 60 and their scalar 9 product is . 2 4. `{X a * b, a VWm b _| go ~ S>r g `m H$mo Xem Vm h VWm `{X a b = (a * b) + 3 h , Vmo (5) (10) H$m _mZ {b{IE, Ohm * VWm { AmYmar g {H $`mE h & If a * b denotes the larger of a and b and if a b = (a * b) + 3, then write the value of (5) (10), where * and o are binary operations. 65/1 3 P.T.O. I S> ~ SECTION B Z g `m 5 go 12 VH$ `oH$ Z Ho$ 2 A H$ h & Question numbers 5 to 12 carry 2 marks each. 5. {g H$s{OE {H$ : 3 sin 1 x = sin 1 (3x 4x3), x Prove that : 3 sin 1 x = sin 1 (3x 4x3), x 6. {X`m J`m h {H$ 2A 1 = 9I A. 7. 2 A 4 3 7 h , Vmo 1 , 2 1 2 1 , 2 1 2 A 1 kmV H$s{OE VWm Xem BE {H$ 2 Given A 4 3 , compute A 1 and show that 2A 1 = 9I A. 7 1 cos x tan 1 sin x H$m x Ho$ gmnoj AdH$bZ H$s{OE & 1 cos x with respect to x. Differentiate tan 1 sin x 8. BH$mB`m| Ho$ C nmXZ go g ~p YV Hw$b bmJV C(x), C(x) = 0 005x 0 02x2 + 30x + 5000 go X m h & gr_m V bmJV kmV H$s{OE O~{H$ 3 BH$mB C nm{XV H$s OmVr h , Ohm gr_m V bmJV (marginal cost) go A{^ m` h C nmXZ Ho$ {H$gr Va na g nyU bmJV _| Vm H$m{bH$ n[adV Z H$s Xa & {H$gr d Vw H$s x 3 The total cost C(x) associated with the production of x units of an item is given by C(x) = 0 005x3 0 02x2 + 30x + 5000. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output. 65/1 4 9. _y `m H$Z H$s{OE : cos 2x 2 sin2 x cos 2 x dx Evaluate : 10. cos 2x 2 sin2 x cos 2 x dx dH $ Hw$b y = a ebx+5 H$mo {Z ${nV H$aZo dmbm EH$ AdH$b g_rH$aU kmV H$s{OE, Ohm a VWm b do N> AMa h & Find the differential equation representing the family of curves y = a ebx+5, where a and b are arbitrary constants. 11. `{X Xmo g{Xem| kmV H$s{OE & ^ ^ ^ i 2 j + 3k VWm 3 ^i ^ ^ 2j + k Ho$ ~rM H$m H$moU h , Vmo sin ^ ^ ^ ^ ^ ^ If is the angle between two vectors i 2 j + 3 k and 3 i 2 j + k , find sin . 12. EH$ H$mbm VWm EH$ bmb nmgm EH$ gmW CN>mbo OmVo h & nmgm| na AmZo dmbr g `mAm o H$m `moJ\$b 8 AmZo H$s g {V~ Y m{`H$Vm kmV H$s{OE, {X`m J`m h {H$ bmb nmgo na AmZo dmbr g `m 4 go H$_ h & A black and a red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4. I S> g SECTION C Z g `m 13 go 23 VH$ `oH$ Z Ho$ 4 A H$ h & Question numbers 13 to 23 carry 4 marks each. 13. 65/1 gma{UH$m| Ho$ JwUY_m o H$m `moJ H$aHo$ {g H$s{OE {H$ 1 1 1 3x 1 3y 1 1 1 1 3z 1 9 (3xyz xy yz zx) 5 P.T.O. Using properties of determinants, prove that 14. `{X 1 1 1 3x 1 3y 1 1 1 1 3z 1 (x2 + y2)2 = xy h , Vmo dy dx 9 (3xyz xy yz zx) kmV H$s{OE & AWdm `{X x = a (2 sin 2 ) = 3 VWm y = a (1 cos 2 ) h , Vmo dy dx kmV H$s{OE O~{H$ h & If (x2 + y2)2 = xy, find dy . dx OR If x = a (2 sin 2 ) and y = a (1 cos 2 ), find 15. `{X y = sin (sin x) h , If y = sin (sin x), prove that 16. dH $ 16x2 + 9y2 = 145 H$s{OE, Ohm x1 = 2 d 2y Vmo {g H$s{OE {H$ VWm Ho$ {~ X d 2y dx 2 + tan x (x1, y1) y1 > 0 h dx 2 + tan x dy + y cos2 x = 0. dx dy + y cos2 x = 0. dx na ne -aoIm VWm A{^b ~ Ho$ g_rH$aU kmV & AWdm dh A Vamb kmV H$s{OE {OZ na \$bZ x4 f(x) = x3 5x2 + 24x + 12 4 (A) {Za Va dY _mZ h , (~) {Za Va mg_mZ h & 65/1 dy when = . dx 3 6 Find the equations of the tangent and the normal, to the curve 16x2 + 9y2 = 145 at the point (x1, y1), where x1 = 2 and y1 > 0. OR Find the intervals in which the function f(x) = x4 x3 5x2 + 24x + 12 is 4 (a) strictly increasing, (b) strictly decreasing. 17. EH$ dJm H$ma AmYma d D$ dm Ya Xrdmam| dmbr D$na go Iwbr EH$ Q> H$s H$mo YmVw H$s MmXa go ~Zm`m OmZm h Vm{H$ dh EH$ {XE JE nmZr H$s _m m H$mo O_m aI gHo$ & Xem BE {H$ Q> H$s H$mo ~ZmZo H$m `` `yZV_ hmoJm O~{H$ Q> H$s H$s JhamB CgH$s Mm S>mB H$s AmYr hmo & `{X Bg nmZr H$mo nmg _| ahZo dmbo H$_ Am` dmbo bmoJm| Ho$ n[admam| H$mo Cnb Y H$amZm hmo VWm CgHo$ ~ZmZo H$m `` B ht n[admam| H$mo XoZm hmo, Vmo Bg Z _| `m _y ` Xem `m J`m h ? An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. If the cost is to be borne by nearby settled lower income families, for whom water will be provided, what kind of value is hidden in this question ? 18. kmV H$s{OE : 2 cos x (1 sin x) (1 sin2 x) dx Find : 19. 2 cos x (1 sin x) (1 sin2 x) dx ex tan y dx + (2 ex) sec2 y dy = 0 y= O~ x = 0 h & 4 AdH$b g_rH$aU {X`m J`m h {H$ H$m {d{e Q> hb kmV H$s{OE, AWdm dy + 2y tan x = sin x dx y = 0 O~ x = h & 3 AdH$b g_rH$aU {H$ 65/1 7 H$m {d{e Q> hb kmV H$s{OE, {X`m J`m h P.T.O. Find the particular solution of the differential equation ex tan y dx + (2 ex) sec2 y dy = 0, given that y = when x = 0. 4 OR Find the particular solution of the differential dy + 2y tan x = sin x, given that y = 0 when x = . dx 3 20. ^ ^ ^ ^ ^ ^ a = 4 i + 5 j k , b = i 4 j + 5 k VWm c = g{Xe d kmV H$s{OE Omo c VWm b XmoZm| na b ~ h VWm d . _mZm equation ^ ^ ^ 3i + j k a = 21 h & h & EH$ ^ ^ ^ ^ ^ ^ ^ ^ ^ Let a = 4 i + 5 j k , b = i 4 j + 5 k and c = 3 i + j k . Find a vector d which is perpendicular to both c and b and d . a = 21. 21. aoImAm| ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ r = (4 i j ) + ( i + 2 j 3 k ) VWm r = ( i j + 2 k ) + (2 i + 4 j 5 k ) Ho$ ~rM `yZV_ X ar kmV H$s{OE & Find the shortest distance between the lines ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ r = (4 i j ) + ( i + 2 j 3 k ) and r = ( i j + 2 k ) + (2 i + 4 j 5 k ). 22. _mZ br{OE H$moB b S>H$s EH$ nmgm CN>mbVr h & `{X Cgo 1 `m 2 m V hmo, Vmo dh EH$ {g Ho$ H$mo 3 ~ma CN>mbVr h Am a nQ>m| H$s g `m ZmoQ> H$aVr h & `{X Cgo 3, 4, 5 AWdm 6 m V hmo, Vmo dh EH$ {g Ho$ H$mo EH$ ~ma CN>mbVr h Am a ZmoQ> H$aVr h {H$ Cgo {MV `m nQ > m V h Am & `{X Cgo R>rH$ EH$ nQ > m V hmo, Vmo CgHo$ mam CN>mbo JE nmgo na 3, 4, 5 AWdm 6 m V H$aZo H$s m{`H$Vm `m h ? Suppose a girl throws a die. If she gets 1 or 2, she tosses a coin three times and notes the number of tails. If she gets 3, 4, 5 or 6, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one tail , what is the probability that she threw 3, 4, 5 or 6 with the die ? 65/1 8 23. W_ nm M YZ nyUm H$m| _| go Xmo g `mE `m N>`m ({~Zm {V WmnZ Ho$) MwZr JB & _mZ br{OE X m V XmoZm| g `mAm| _| go ~ S>r g `m H$mo ` $ H$aVm h & X H$m _m ` VWm gaU kmV H$s{OE & Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X. I S> X SECTION D Z g `m 24 go 29 VH$ `oH$ Z Ho$ 6 A H$ h & Question numbers 24 to 29 carry 6 marks each. 24. _mZm A = {x Z : 0 x 12}. Xem BE {H$ R = {(a, b) : a, b A, |a b|, 4 go ^m ` h } EH$ Vw `Vm g ~ Y h & g^r Ad`dm| H$m g_w ` kmV H$s{OE & Vw `Vm dJ AWdm Xem BE {H$ \$bZ f: Omo g^r x Ho$ {bE Z Vmo EH $H$s h Am a Z hr Am N>mXH$ h & `{X n[a^m{fV h , Vmo fog(x) [2] ^r g: 1 go g ~ {YV {b{IE & f(x) = x mam n[a^m{fV h , x2 1 , g(x) = 2x 1 mam ^r kmV H$s{OE & Let A = {x Z : 0 x 12}. Show that R = {(a, b) : a, b A, |a b| is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2]. OR Show that the function f : neither one-one nor onto. Also, if g : x , x is x 1 is defined as g(x) = 2x 1, defined by f(x) = 2 find fog(x). 65/1 9 P.T.O. 25. `{X 2 A 3 1 3 2 1 5 4 2 h , Vmo A 1 kmV H$s{OE & BgH$m `moJ H$aHo$ g_rH$aU {ZH$m` 2x 3y + 5z = 11 3x + 2y 4z = 5 x + y 2z = 3 H$mo hb H$s{OE & AWdm ma {^H$ n { $ $nm VaUm| mam Am `yh 2 If A 3 1 3 2 1 1 A 2 2 2 5 4 3 7 5 H$m `w H $_ kmV H$s{OE & 5 4 , find A 1. Use it to solve the system of equations 2 2x 3y + 5z = 11 3x + 2y 4z = 5 x + y 2z = 3. OR Using elementary row transformations, find the inverse of the matrix 1 A 2 2 65/1 2 5 4 3 7 . 5 10 26. W_ MVwWm e _|, x-Aj, aoIm y = x VWm d m x2 + y2 = 32 mam {Kao jo H$m jo \$b, g_mH$bZm| Ho$ `moJ go kmV H$s{OE & Using integration, find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x2 + y2 = 32. 27. _y `m H$Z H$s{OE /4 : sin x cos x dx 16 9 sin 2x 0 AWdm `moJm| H$s gr_m Ho$ $n _| 3 (x 2 3x e x ) dx 1 H$m _mZ kmV H$s{OE & Evaluate : /4 sin x cos x dx 16 9 sin 2x 0 OR Evaluate 3 (x 2 3x e x ) dx, 1 as the limit of the sum. 65/1 11 P.T.O. 28. {~ X ( 1, 5, 10) ^ ^ ^ r . (i j + k ) = 5 go aoIm ^ ^ ^ ^ ^ ^ r = 2 i j + 2 k + (3 i + 4 j + 2 k ) Am a g_Vb Ho$ {V N>oXZ {~ X Ho$ _ ` H$s X ar kmV H$s{OE & Find the distance of the point ( 1, 5, 10) from the point of ^ ^ ^ ^ ^ ^ intersection of the line r = 2 i j + 2 k + (3 i + 4 j + 2 k ) and the plane ^ ^ ^ r . ( i j + k ) = 5. 29. EH$ H$maImZo _| Xmo H$ma Ho$ n oM A Am a B ~ZVo h & `oH$ Ho$ {Z_m U _| Xmo _erZm| Ho$ `moJ H$s Amd `H$Vm h , {Og_| EH$ dMm{bV h Am a X gar h VMm{bV h & EH$ n Ho$Q> n|M A Ho$ {Z_m U _| 4 {_ZQ> dMm{bV Am a 6 {_ZQ> h VMm{bV _erZ, VWm EH$ n Ho$Q> n oM B Ho$ {Z_m U _| 6 {_ZQ> dMm{bV Am a 3 {_ZQ> h VMm{bV _erZ H$m H$m` hmoVm h & `oH$ _erZ {H$gr ^r {XZ Ho$ {bE A{YH$V_ 4 K Q>o H$m_ Ho$ {bE Cnb Y h & {Z_m Vm n|M A Ho$ `oH$ n Ho$Q> na 70 n go Am a n oM B Ho$ `oH$ n Ho$Q> na < 1 H$m bm^ H$_mVm h & `h _mZVo h E {H$ H$maImZo _| {Z{_ V g^r n|Mm| Ho$ n Ho$Q> {~H$ OmVo h , kmV H$s{OE {H$ {V{XZ H$maImZo Ho$ _m{bH$ mam {H$VZo n Ho$Q> {d{^ n|Mm| Ho$ ~ZmE OmE {Oggo bm^ A{YH$V_ hmo & Cn`w $ a {IH$ moJm _Z g_ `m H$mo gy ~ H$s{OE VWm Bgo J m\$s` {d{Y go hb H$s{OE VWm A{YH$V_ bm^ ^r kmV H$s{OE & A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 minutes on the hand-operated machines to manufacture a packet of screws A while it takes 6 minutes on the automatic and 3 minutes on the hand-operated machine to manufacture a packet of screws B . Each machine is available for at most 4 hours on any day. The manufacturer can sell a packet of screws A at a profit of 70 paise and screws B at a profit of < 1. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maximize his profit ? Formulate the above LPP and solve it graphically and find the maximum profit. 65/1 12

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