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CBSE Class 12 Board Exam 2019 : Mathematics (Series 5)

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SET-1 H$moS> Z . Series BVM/5 Code No. amob Z . 65/5/1 narjmWu H$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 65/5/1 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$m h & Question numbers 1 to 4 carry 1 mark each. 1. aoIm r = (2 ^i ^j + 3 k^ ) + (3 ^i ^j + 2 k^ ) Am a g_Vb ^ ^ ^ r . (i + j + k) = 3 Ho$ ~rM H$m H$moU kmV H$s{OE & AWdm Cg q~X Ho$ {ZX oem H$ kmV H$s{OE Ohm aoIm x 2 1 H$mQ>Vr h & = y 5 3 = z 1 , yz-g_Vb 5 H$mo ^ ^ ^ ^ ^ ^ Find the angle between the line r = (2 i j + 3 k ) + (3 i j + 2 k ) and ^ ^ ^ the plane r . ( i + j + k ) = 3. OR y 5 z 1 x 2 Find the co-ordinates of the point, where the line = = 3 5 1 cuts the yz-plane. 65/5/1 2 2. `{X y = 5e 7x + 6e 7x h , Vmo Xem BE {H$ If y = 5e7x + 6e 7x, show that 3. d 2y dx 2 d 2y 49y h & 49y. dx 2 `{X A H$mo{Q> 2 H$m EH$ dJ Am `yh h Am a H$s{OE Ohm $ A , Am `yh A H$m n[adV h & |A| = 4 h , Vmo |2 . AA | H$m _mZ kmV If A is a square matrix of order 2 and |A| = 4, then find the value of |2 . AA |, where A is the transpose of matrix A. 4. Eogo d mm| Ho$ Hw$b Ho$ AdH$b g_rH$aU H$s H$mo{Q> kmV H$s{OE {OZH$s { `m 3 BH$mB h & Find the order of the differential equation of the family of circles of radius 3 units. I S> ~ SECTION B Z g `m 5 go 12 VH$ `oH$ Z Ho$ 2 A H$ h & Question numbers 5 to 12 carry 2 marks each. 5. Am `yh g_rH$aU x 2 7 5 3 + y 3 1 4 7 = 2 15 6 14 go (x y) H$m _mZ kmV H$s{OE & Find the value of (x y) from the matrix equation 5 6 x 3 4 7 2 + = . 2 7 y 3 1 15 14 6. {Z Z AdH$b g_rH$aU H$mo hb H$s{OE : (y + 3x2) dx x dy Solve the following differential equation : (y + 3x2) 65/5/1 dx x dy 3 P.T.O. 7. kmV H$s{OE : ex (cos x sin x) cosec2 x dx ex (cos x sin x) cosec2 x dx Find : 8. g{Xem| Ho$ `moJ go {g H$s{OE {H$ {~ Xw (2, 1, 3), (3, 5, 1) Am a h & AWdm {H$ ht Xmo g{Xem| a Am a b Ho$ {bE {g H$s{OE {H$ ( a b )2 = a 2 b 2 ( a . b )2 ( 1, 11, 9) . gaoI Using vectors, prove that the points (2, 1, 3), (3, 5, 1) and ( 1, 11, 9) are collinear. OR For any two vectors a and b , prove that ( a b )2 = a 2 b 2 ( a . b )2 9. kmV H$s{OE : x x 1 dx (x 2)(x 3) AWdm Ho$ gmnoj g_mH$bZ H$s{OE : ex 5 4e x e 2x Find : x 1 dx (x 2)(x 3) OR Integrate : ex 5 4e x e 2x 65/5/1 with respect to x. 4 10. `{X P(A) = 0 6, P(B) = 0 5 Am a H$s{OE & P(B|A) = 0 4 h , Vmo P(A B) Am a P(A|B) kmV If P(A) = 0 6, P(B) = 0 5 and P(B|A) = 0 4, find P(A B) and P(A|B). 11. `{X nyUm H$m| Ho$ g_w ` Z _| a * b = 2a2 + b mam n[a^m{fV * EH$ g {H $`m h , Vmo kmV H$s{OE {H$ (i) `m `h EH$ { AmYmar g {H $`m h `m Zht, VWm (ii) `{X { AmYmar g {H $`m h , Vmo `m `h H $_{d{Z_o` h `m Zht & If an operation * on the set of integers Z is defined by a * b = 2a2 + b, then find (i) whether it is a binary or not, and (ii) if a binary, then is it commutative or not. 12. A N>r H$ma go \|$Q>r JB Vme H$s J r _| go EH$ Ho$ ~mX EH$ Mma n mo {V WmnZm g{hV {ZH$mbo JE & m{`H$Vm kmV H$s{OE {H$ H$_-go-H$_ VrZ n mo B Q> Ho$ AmE & AWdm Xmo {d m{W `m| A Am a B Ho$ {d mb` _| g_` na AmZo H$s m{`H$VmE H $_e: 2 7 Am a 4 7 h & _m{ZE {H$ "A g_` na AmVm h ' Am a "B g_` na AmVm h ' dV KQ>ZmE h , Vmo m{`H$Vm kmV H$s{OE {H$ CZ_| go Ho$db EH$ hr {d mb` _| g_` na AmVm h & Four cards are drawn one by one with replacement from a well-shuffled deck of playing cards. Find the probability that at least three cards are of diamonds. OR The probability of two students A and B coming to school on time are and 2 7 4 , respectively. Assuming that the events A coming on time and B 7 coming on time are independent, find the probability of only one of them coming to school on time. 65/5/1 5 P.T.O. 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. `{X p q p+q x y = (x + y) h , Vmo {g H$s{OE {H$ If xpyq = (x + y)p+q, prove that 14. dy y dx x Am a d 2y dx 2 0 h & dy y d 2y and 0. dx x dx 2 kmV H$s{OE : (sin x . sin 2x . sin 3x) dx (sin x . sin 2x . sin 3x) dx Find : 15. tan 1 x 1 x Ho$ gmnoj 2 tan 1 3x x 3 1 3x 2 , | x | 1 3 H$m AdH$bZ H$s{OE & AWdm `{X 1 x2 + 1 y2 dy dx 1 x2 Differentiate 1 y 2 = a (x y), |x|< 1, |y|< 1 h , Vmo Xem BE {H$ . 3 1 3x x tan 2 1 3x , | x | x 1 w.r.t. tan 1 . 3 1 x2 OR If 65/5/1 1 x2 + 1 y 2 = a (x y), |x|< 1, |y|< 1, show that 6 1 y2 dy . dx 1 x2 16. AdH$b g_rH$aU (1 + e2x) dy {X`m J`m h {H$ y(0) = 1 h & + (1 + y2) ex dx = 0 H$m {d{e Q> hb kmV H$s{OE, AWdm dy y y sin + x y sin = 0 H$m {d{e Q> dx x x {X`m J`m h y(1) = h & 2 Find the particular solution of the differential equation : AdH$b g_rH$aU x hb kmV H$s{OE, (1 + e2x) dy + (1 + y2) ex dx = 0, given that y(0) = 1. OR Find the particular solution of the differential equation : dy y y x sin + x y sin = 0, given that y(1) = . 2 dx x x 17. {g H$s{OE {H$ g_w ` A = {1, 2, 3, 4, 5, 6, 7} _| X m g ~ Y R EH$ Vw `Vm g ~ Y h & AWdm Xem BE {H$ 2 A=R 3 Am N>mXH$ h & AV:, _|, f 1 kmV f (x ) 4x 3 6x 4 R = {(a, b) : |a b| g_ h } mam mam n[a^m{fV \$bZ EH $H$s Am a H$s{OE & Prove that the relation R in the set A = {1, 2, 3, 4, 5, 6, 7} given by R = {(a, b) : |a b| is even} is an equivalence relation. OR 4x 3 2 Show that the function f in A = R defined as f (x) is 6x 4 3 one-one and onto. Hence, find f 1. 18. kmV H$s{OE {H$ `m \$bZ f(x) = cos (2x + ); 4 A Vamb 3 5 <x< 8 8 _| dY _mZ h `m mg_mZ h & Find whether the function f(x) = cos (2x + in the interval 65/5/1 3 5 <x< . 8 8 7 ); is increasing or decreasing 4 P.T.O. 19. q~X ( 1, 3, 2) go Jw OaZo dmbo VWm g_Vbm| x + 2y + 3z = 5 Am a go `oH$ na b ~ dmbo g_Vb H$m g_rH$aU kmV H$s{OE & 3x + 3y + z = 0 _| Find the equation of the plane passing through the point ( 1, 3, 2) and perpendicular to the planes x + 2y + 3z = 5 and 3x + 3y + z = 0. 20. {g H$s{OE {H$ : 4 5 63 + tan 1 + cos 1 = 5 12 65 2 Prove that : 4 5 63 sin 1 + tan 1 + cos 1 = 5 12 65 2 sin 1 21. _mZ kmV H$s{OE : 5 (|x 1| + |x 2| + |x 4|) dx 1 Evaluate : 5 (|x 1| + |x 2| + |x 4|) dx 1 22. g{Xem| Ho$ `moJ go, x H$m Eogm _mZ kmV H$s{OE {H$ Mma {~ X C(4, 5, 5) VWm D(4, 2, 2) g_Vbr` hmo OmE & A(x, 5, 1), B(3, 2, 1), Using vectors, find the value of x such that the four points A(x, 5, 1), B(3, 2, 1), C(4, 5, 5) and D(4, 2, 2) are coplanar. 23. `{X x, y, z {^ h VWm `moJ H$aHo$, Xem BE {H$ x x2 x3 1 y y2 y3 1 0 z z2 z3 1 xyz = 1 h h , Vmo gma{UH$m| Ho$ JwUY_m o H$m & x x2 x3 1 If x, y, z are different and y y2 y 3 1 0 , then using z z2 z3 1 properties of determinants, show that xyz = 1. 65/5/1 8 I S> X SECTION D Z g `m 24 go 29 VH$ `oH$ Z Ho$ 6 A H$ h & Question numbers 24 to 29 carry 6 marks each. 24. g_mH$bZ Ho$ `moJ go {Z Z{b{IV jo H$m jo \$b kmV H$s{OE : {(x, y) : x2 + y2 16a2 Am a y2 6ax} AWdm g_mH$bZ Ho$ n `moJ go Eogo { ^wO ABC H$m jo \$b kmV H$s{OE Omo aoImAm| 4x y + 5 = 0, x + y 5 = 0 Am a x 4y + 5 = 0 go n[a~ h & Using integration, find the area of the following region : {(x, y) : x2 + y2 16a2 and y2 6ax} OR Using integration, find the area of triangle ABC bounded by the lines 4x y + 5 = 0, x + y 5 = 0 and x 4y + 5 = 0. 25. (2, 1, 1) go Jw OaZo dmbr aoIm ^ ^ ^ ^ ^ r = ( i + j ) + (2 i j + k ) Ho$ {~ X H$m g{Xe g_rH$aU kmV H$s{OE Omo {H$ aoIm g_m Va h & AV: BZ XmoZm| aoImAm| Ho$ ~rM H$s X ar ^r kmV H$s{OE & AWdm q~X P(1, 3, 4) go g_Vb 2x y + z + 3 = 0 na ItMo JE b ~ Ho$ nmX Q Ho$ {ZX oem H$ kmV H$s{OE & b ~dV X ar PQ VWm g_Vb H$mo Xn U boVo h E Bg q~X P H$m {V{~ ~ ^r kmV H$s{OE & Find the vector equation of the line passing through (2, 1, 1) and ^ ^ ^ ^ ^ parallel to the line r = ( i + j ) + (2 i j + k ). Also, find the distance between these two lines. OR Find the coordinates of the foot Q of the perpendicular drawn from the point P(1, 3, 4) to the plane 2x y + z + 3 = 0. Find the distance PQ and the image of P treating the plane as a mirror. 65/5/1 9 P.T.O. 26. EH$ H$ nZr bmBdwS> Ho$ Xmo H$ma Ho$ AZyR>o _ {V{M m| H$m {Z_m U H$aVr h & A H$ma Ho$ {V _ {V{M Ho$ {Z_m U _| 5 {_ZQ> H$mQ>Zo Am a 10 {_ZQ> Omo S>Zo _| bJVo h & B H$ma Ho$ {V _ {V{M Ho$ {bE 8 {_ZQ> H$mQ>Zo Am a 8 {_ZQ> Omo S>Zo _| bJVo h & {X`m J`m h {H$ H$mQ>Zo Ho$ {bE Hw$b g_` 3 K Q>o 20 {_ZQ> VWm Omo S>Zo Ho$ {bE 4 K Q>o Cnb Y h & `oH$ A H$ma Ho$ _ {V{M na < 100 Am a `oH$ B H$ma Ho$ _ {V{M na < 120 H$m bm^ hmoVm h & kmV H$s{OE {H$ bm^ Ho$ A{YH$V_rH$aU Ho$ {bE `oH$ H$ma Ho$ {H$VZo -{H$VZo _ {V{M m| H$m H $nZr mam {Z_m U hmoZm Mm{hE & a {IH$ moJ m_Z g_ `m ~Zm H$a Bgo J m\$ mam hb H$s{OE & A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit for type A souvenirs is < 100 each and for type B souvenirs, profit is < 120 each. How many souvenirs of each type should the company manufacture in order to maximise the profit ? Formulate the problem as a LPP and then solve it graphically. 27. EH$ ~hw{dH$ nr Z H$m C ma XoZo _|, O~{H$ `oH$ Z Ho$ Mma {dH$ n h , {OZ_| go Ho$db EH$ ghr h , EH$ {d mWu `m Vmo AZw_mZ bJmVm h AWdm ZH$b H$aVm h , `m Z H$m C ma OmZVm h & CgHo$ AZw_mZ bJm H$a C ma XoZo H$s m{`H$Vm C ma XoZo H$s m{`H$Vm ^r 3 4 1 4 1 4 VWm ZH$b H$aHo$ h & Bg {d mWu mam ZH$b H$aHo$ ghr C ma XoZo H$s m{`H$Vm h & m{`H$Vm kmV H$s{OE {H$ dh Z H$m C ma OmZVm h , O~{H$ {X`m J`m h {H$ CgZo ghr C ma {X`m h & In answering a question on a multiple choice questions test with four choices in each question, out of which only one is correct, a student either guesses or copies or knows the answer. The probability that he makes a 1 1 guess is and the probability the he copies is also . The probability 4 4 3 that the answer is correct, given that he copied it is . Find the 4 probability that he knows the answer to the question, given that he correctly answered it. 65/5/1 10 28. { `m a dmbo d m Ho$ A Xa EH$ g_{ ~mh { ^wO ~Zm h {OgH$m erf H$moU {H$ { ^wO H$m jo \$b A{YH$V_ hmoJm O~ 6 2 h & Xem BE hmoJm & An isosceles triangle of vertical angle 2 is inscribed in a circle of radius a. Show that the area of the triangle is maximum when . 6 29. map ^H$ n { $ $nm VaUm| mam Am `yh 3 2 0 0 3 4 1 0 1 H$m `w H $_ kmV H$s{OE & AWdm Am `yhm| H$m `moJ H$a {Z Z{b{IV a {IH$ g_rH$aU {ZH$m` H$mo hb H$s{OE : 2x + 3y + 10z = 4 4x 6y + 5z = 1 6x + 9y 20z = 2 Using elementary row transformations, find the inverse of the matrix 3 2 0 0 3 4 1 0 . 1 OR Using matrices, solve the following system of linear equations : 2x + 3y + 10z = 4 4x 6y + 5z = 1 6x + 9y 20z = 2 65/5/1 11 P.T.O.

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