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CBSE Class 12 Board Exam 2019 : Mathematics (Series 2)

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SET-1 H$moS> Z . Series BVM/2 Code No. amob Z . 65/2/1 narjmWu H$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 > 11 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 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 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 65/2/1 Maximum Marks : 100 1 P.T.O. gm_m ` {ZX}e : 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 I S> A Ho$ 1 Z _|, I S> ~ Ho$ 3 Zm| _|, I S> g Ho$ 3 Zm| _| VWm I S> X Ho$ 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 & (i) 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 1 question of Section A, 3 questions of Section B, 3 questions of Section C and 3 questions of Section D. You have to attempt only one of the alternatives in all such questions. (v) Use of calculators is not permitted. You may ask for logarithmic tables, if required. I S> A SECTION A Z g `m 1 go 4 VH$ `oH$ Z 1 A H$ H$m h & Question numbers 1 to 4 carry 1 mark each. 1. {X A EH$ dJ Am yh h {Og | 2. A A = I h , Vmo |A| H$m mZ {b{IE & If A is a square matrix satisfying A A = I, write the value of |A|. {X y = x|x| h , Vmo x < 0 Ho$ {bE, dy kmV H$s{OE & dx dy If y = x|x|, find for x < 0. dx 3. {Z Z AdH$b g rH$aU H$s H$mo{Q> d KmV ( {X n[a^m{fV h ) kmV H$s{OE 2 : d 2y dy + x = 2x2 log 2 2 dx dx dx Find the order and degree (if defined) of the differential equation 2 2 d 2y dy 2 log d y + x = 2x dx 2 dx dx 2 d 2y 65/2/1 2 4. Cg aoIm Ho$ {XH $-H$mogmBZ kmV H$s{OE Omo {ZX}em H$ Ajm| go g mZ H$moU ~ZmVr h & AWdm EH$ aoIm {H$gr EH$ {~ X , {OgH$m p W{V g{Xe 2 ^i ^j + 4 k^ h , go Jw OaVr h Am a g{Xe ^i + ^j 2 k^ H$s {Xem | h & Bg aoIm H$m H$mVu g rH$aU kmV H$s{OE & Find the direction cosines of a line which makes equal angles with the coordinate axes. OR ^ ^ ^ A line passes through the point with position vector 2 i j + 4 k and is ^ ^ ^ in the direction of the vector i + j 2 k . Find the equation of the line in cartesian form. 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^r dm V{dH$ g mAm| Ho$ g w na n[a^m{fV g {H $ m * : a * b = a 2 b 2 m { AmYmar h , BgH$s Om M H$s{OE & {X h { AmYmar h , Vmo kmV H$s{OE {H$ m `h gmhM h m Zht & Examine whether the operation * defined on , the set of all real 2 2 numbers, by a * b = a b is a binary operation or not, and if it is a binary operation, find whether it is associative or not. 6. {X A = 4 1 4 If A = 1 7. kmV H$s{OE Find : 65/2/1 2 1 h , Vmo Xem BE {H$ (A 2I) (A 3I) = 0. 2 , show that (A 2I) (A 3I) = 0. 1 : 3 2x x 2 dx 3 2x x 2 dx 3 P.T.O. 8. kmV H$s{OE : sin3 x cos 3 x sin kmV H$s{OE 2 x cos 2 x dx AWdm : (x 1) x 3 3 e x dx Find : sin3 x cos 3 x sin 2 x cos 2 x dx OR Find : (x 1) x 3 9. 10. 3 e x dx dH $m| Ho$ Hw$b y = Ae2x + Be 2x, Ohm AdH$b g rH$aU kmV H$s{OE & A, B do N> AMa h , H$mo {Z ${nV H$aZo dmbm Find the differential equation of the family of curves y = Ae2x + Be 2x, where A and B are arbitrary constants. {X | a | = 2, | b | = 7 VWm a b = 3 ^i + 2 ^j + 6 k^ h , Vmo a Am a b Ho$ ~rM H$m H$moU kmV H$s{OE & AWdm Cg KZm^ H$m Am VZ kmV H$s{OE {OgHo$ {H$Zmao 3 ^i VWm 7 ^i 5 ^j 3 k^ mam {XE JE h & ^ ^ ^ ^ ^ + 7 j + 5k , 5i + 7 j 3k ^ ^ ^ | a | = 2, | b | = 7 and a b = 3 i + 2 j + 6 k , find the angle between a and b . OR ^ ^ ^ Find the volume of a cuboid whose edges are given by 3 i + 7 j + 5 k , ^ ^ ^ ^ ^ ^ 5 i + 7 j 3 k and 7 i 5 j 3 k . If 11. {X P(A Zht) = 0 7, P(B) = 0 7 VWm P(B/A) = 0 5 h , Vmo P(A/B) kmV If P(not A) = 0 7, P(B) = 0 7 and P(B/A) = 0 5, then find P(A/B). 65/2/1 4 H$s{OE & 12. EH$ {g H$m 5 ~ma CN>mbm J m & (i) 3 ~ma {MV AmZo H$s m{ H$Vm `m h ? (ii) A{YH$V 3 ~ma {MV AmZo H$s m{ H$Vm `m h ? AWdm Xmo {g H$m| H$mo EH$ ~ma EH$ gmW CN>mbZo na {MVm| H$s g m, X H$m m{ H$Vm ~ Q>Z kmV H$s{OE & A coin is tossed 5 times. What is the probability of getting (i) 3 heads, (ii) at most 3 heads ? OR Find the probability distribution of X, the number of heads in a simultaneous toss of two coins. 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. Om M H$s{OE {H$ m g w A = {1, 2, 3, 4, 5, 6} na n[a^m{fV g ~ Y R = {(a, b) : b = a + 1} dVw , g { V m g H $m H$ h & AWdm mZ br{OE {H$ f : N Y, f(x) = 4x + 3, mam n[a^m{fV EH$ $bZ h , Ohm Y = {y N : y = 4x + 3, {H$gr x N Ho$ {bE} h & {g H$s{OE {H$ f w H $ Ur h & BgH$m {Vbmo $bZ ^r kmV H$s{OE & Check whether the relation R defined on the set A = {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive. OR Let f : N Y be a function defined as f(x) = 4x + 3, where Y = {y N : y = 4x + 3, for some x N}. Show that f is invertible. Find its inverse. 14. 15. 4 2 sin cos 1 tan 1 H$m mZ kmV H$s{OE & 5 3 4 2 Find the value of sin cos 1 tan 1 . 5 3 gma{UH$m| Ho$ JwUY_m] H$m moJ H$aHo$, Xem BE {H$ 65/2/1 3a a b b a 3b c a c b a c b c = 3 (a + b + c) (ab + bc + ca) 3c 5 P.T.O. Using properties of determinants, show that 16. {X 3a a b b a 3b c a c b a c b c = 3 (a + b + c) (ab + bc + ca) 3c x 1 y + y 1 x = 0 Am a x y h , AWdm {X (cos x)y = (sin y)x h , Vmo dy dx Vmo {g H$s{OE {H$ dy 1 . dx (x 1) 2 kmV H$s{OE & If x 1 y + y 1 x = 0 and x y, prove that dy 1 . dx (x 1) 2 OR If (cos x)y = (sin y)x, find 17. {X, {H$gr c > 0 Ho$ dy . dx {bE, (x a)2 + (y b)2 = c2 h , Vmo {g H$s{OE {H$ 2 dy 1 dx d 2y 3/ 2 , a Am a b go dV EH$ p Wa am{e h & dx 2 If (x a)2 + (y b)2 = c2, for some c > 0, prove that 2 dy 1 dx d 2y 3/ 2 is a constant independent of a and b. dx 2 18. dH $ h & x2 = 4y na Cg A{^b ~ H$m g rH$aU kmV H$s{OE, Omo {~ X ( 1, 4) go Jw OaVm Find the equation of the normal to the curve x2 = 4y which passes through the point ( 1, 4). 65/2/1 6 19. : kmV H$s{OE x2 x 1 (x 2) (x 2 1) dx Find : 20. x2 x 1 (x 2) (x 2 1) dx {g H$s{OE {H$ a a 0 0 f (x) dx f (a x) dx AV: / 2 0 x dx sin x cos x H$m _y `m H$Z H$s{OE & Prove that a a 0 0 f (x) dx f (a x) dx and hence evaluate / 2 0 21. x dx sin x cos x AdH$b g rH$aU H$mo hb H$s{OE x : dy y = y x tan dx x AWdm AdH$b g rH$aU H$mo hb H$s{OE : x y cos x dy dx 1 sin x 65/2/1 7 P.T.O. Solve the differential equation : x dy y = y x tan dx x OR Solve the differential equation : x y cos x dy dx 1 sin x 22. g{Xem| ^ ^ ^ b = 2i + 4 j 5k AZw{Xe m H$ g{Xe d g{Xe kmV H$s{OE Am a AV: Am a ^ ^ ^ c = i + 2 j + 3k ^ ^ ^ a = i + j + k b + c Ho$ {bE, g{Xe H$m A{Xe JwUZ $b 1 h & b + c Ho$ H$m mZ Ho$ AZw{Xe m H$ g{Xe ^r kmV H$s{OE & ^ ^ ^ The scalar product of the vector a = i + j + k with a unit vector along ^ ^ ^ ^ ^ ^ the sum of the vectors b = 2 i + 4 j 5 k and c = i + 2 j + 3 k is equal to 1. Find the value of and hence find the unit vector along b + c . 23. {X aoImE Vmo x 1 y 2 z 3 3 2 2 Am a x 1 y 1 z 6 3 2 5 na na b ~dV hm|, H$m mZ kmV H$s{OE & AV: kmV H$s{OE {H$ m o aoImE EH$-X gao H$mo H$mQ>Vr h m Zht & If the lines x 1 y 2 z 3 3 2 2 and x 1 y 1 z 6 3 2 5 are perpendicular, find the value of . Hence find whether the lines are intersecting or not. 65/2/1 8 I S> X SECTION D Z g `m 24 go 29 VH$ `oH$ Z Ho$ 6 A H$ h & Question numbers 24 to 29 carry 6 marks each. 24. {X 1 A = 2 5 3 4 2 1 1 1 h , Vmo A 1 kmV H$s{OE & AV: {Z Z g rH$aU {ZH$m H$m hb kmV H$s{OE : x + 3y + 4z = 8 2x + y + 2z = 5 Am a 5x + y + z = 7 AWdm ma {^H$ $nm VaUm| mam, {Z Z Am yh H$m w H $ kmV H$s{OE 2 A = 5 0 1 If A = 2 5 3 1 1 0 1 1 : 1 0 3 4 2 , find A 1. 1 Hence solve the system of equations x + 3y + 4z = 8 2x + y + 2z = 5 and 5x + y + z = 7 OR 65/2/1 9 P.T.O. Find the inverse transformations : 2 A = 5 0 25. {g H$s{OE {H$ EH$ 2R 3 of 0 following matrix, using elementary 1 0 3 1 1 R the { m Ho$ Jmobo Ho$ A VJ V A{YH$V Am VZ Ho$ ~obZ H$s D $MmB h & A{YH$V Am VZ ^r kmV H$s{OE & Show that the height of the cylinder of maximum volume that can be 2R inscribed in a sphere of radius R is . Also find the maximum volume. 3 26. g mH$bZ {d{Y go Cg { ^wO H$m jo $b kmV H$s{OE {OgHo$ erf (3, 1) h & (1, 0), (2, 2) Am a AWdm g mH$bZ {d{Y go, Xmo d mm| jo $b kmV H$s{OE & x2 + y 2 = 4 VWm (x 2)2 + y2 = 4 Ho$ ~rM {Kao jo H$m Using method of integration, find the area of the triangle whose vertices are (1, 0), (2, 2) and (3, 1). OR Using method of integration, find the area of the region enclosed between two circles x2 + y2 = 4 and (x 2)2 + y2 = 4. 27. {~ X Am|, {OZHo$ p W{V g{Xe ^i + ^j 2 k^ , 2 ^i ^j + k^ VWm ^i + 2 ^j + k^ h , go Jw OaZo dmbo g Vb H$m g{Xe d H$mVu g rH$aU kmV H$s{OE & Cn`w $ g Vb Ho$ g m Va g Vb, Omo {~ X (2, 3, 7) go Jw OaVm h , H$m g rH$aU ^r {b{IE & AV:, XmoZm| g m Va g Vbm| Ho$ ~rM H$s X ar kmV H$s{OE & AWdm {~ X Am| (2, 1, 2) VWm (5, 3, 4) go Jw OaZo dmbr aoIm H$m g rH$aU kmV H$s{OE VWm {~ X Am| (2, 0, 3), (1, 1, 5) VWm (3, 2, 4) go Jw OaZo dmbo g Vb H$m g rH$aU ^r kmV H$s{OE & aoIm d g Vb H$m {V N>oXZ {~ X ^r kmV H$s{OE & 65/2/1 10 Find the vector and cartesian equations of the plane passing through the ^ ^ ^ ^ ^ ^ ^ ^ ^ points having position vectors i + j 2 k , 2 i j + k and i + 2 j + k . Write the equation of a plane passing through a point (2, 3, 7) and parallel to the plane obtained above. Hence, find the distance between the two parallel planes. OR Find the equation of the line passing through (2, 1, 2) and (5, 3, 4) and of the plane passing through (2, 0, 3), (1, 1, 5) and (3, 2, 4). Also, find their point of intersection. 28. VrZ {g Ho$ {XE JE h & EH$ {g Ho$ Ho$ XmoZm| Amoa {MV hr h & X gam {g H$m A{^ZV h {Og | {MV 75% ~ma H$Q> hmoVm h Am a Vrgam AZ{^ZV {g H$m h & VrZm| | go EH$ {g H$m m N> m MwZm J m Am a Cgo CN>mbm J m h & {X {g Ho$ na {MV H$Q> h Am hmo, Vmo m m{ H$Vm h {H$ dh XmoZm| Va $ {MV dmbm {g H$m h ? There are three coins. One is a two-headed coin, another is a biased coin that comes up heads 75% of the time and the third is an unbiased coin. One of the three coins is chosen at random and tossed. If it shows heads, what is the probability that it is the two-headed coin ? 29. EH$ H $nZr Xmo H$ma H$m gm mZ, A Am a B ~ZmVr h , {Og | gmoZo d Mm Xr H$m Cn moJ hmoVm h & H$ma A H$s oH$ BH$mB | 3 g Mm Xr d 1 g gmoZm, VWm H$ma B H$s oH$ BH$mB | 1 g Mm Xr d 2 g gmoZm moJ | AmVm h & H $nZr mXm-go- mXm 9 g Mm Xr d 8 g gmoZo H$m hr moJ H$a gH$Vr h & {X H$ma A H$s EH$ BH$mB go < 40 H$m bm^ d H$ma B H$s EH$ BH$mB go < 50 H$m bm^ H$ m m OmVm h , Vmo A{YH$V bm^ A{O V H$aZo hoVw H $nZr H$mo XmoZm| H$mam| H$s {H$VZr-{H$VZr BH$mB m ~ZmZr Mm{hE ? Cn`w $ g m H$mo a {IH$ moJ m Z g m | n[ad{V V H$aHo$ AmboI {d{Y go hb H$s{OE VWm A{YH$V bm^ ^r kmV H$s{OE & A company produces two types of goods, A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of gold while that of type B requires 1 g of silver and 2 g of gold. The company can use at the most 9 g of silver and 8 g of gold. If each unit of type A brings a profit of < 40 and that of type B < 50, find the number of units of each type that the company should produce to maximize profit. Formulate the above LPP and solve it graphically and also find the maximum profit. 65/2/1 11 P.T.O.

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