Trending ▼   ResFinder  

CBSE Class 12 Board Exam 2019 : Mathematics (Series 3)

11 pages, 39 questions, 0 questions with responses, 0 total responses,    0    0
cbse12
   		+Fave Message
ProfileTimelineUploadsQ & A
 Home > cbse12 >

Instantly get Model Answers to questions on this ResPaper. Try now!
NEW ResPaper Exclusive!

Formatting page ...

SET-1 H$moS> Z . Series BVM/3 Code No. amob Z . 65/3/1 narjmWu H$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 65/3/1 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 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 $ Hw$boQ>a Ho$ `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 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$m h & Questions number 1 to 4 carry 1 mark each. 1. {X dJ Am yh A H$s H$mo{Q> 3 Am a |A| = 4 h , Vmo | 2A| H$m mZ {b{IE & If A is a square matrix of order 3 with |A| = 4, then write the value of | 2A|. dy dx dy If y = sin 1 x + cos 1 x, find . dx y = sin 1 x + cos 1 x 2. {X 3. AdH$b g rH$aU 2 h , Vmo 2 d4y x dy dx 4 dx kmV H$s{OE & 3 H$s H$mo{Q> d KmV {b{IE & 65/3/1 2 Write the order and the degree of the differential equation 2 3 2 d4y x dy . dx 4 dx 4. {X EH$ aoIm Ho$ {XH $-AZwnmV 18, 12, 4 h , Vmo BgHo$ {XH $-H$mogmBZ `m h ? AWdm {~ X ( 2, 4, 5) go Jw OaZo dmbr Cg aoIm H$m H$mVu g rH$aU kmV H$s{OE Omo aoIm x 3 4 y z 8 3 5 6 Ho$ g m Va h & If a line has the direction ratios 18, 12, 4, then what are its direction cosines ? OR Find the cartesian equation of the line which passes through the point x 3 4 y z 8 ( 2, 4, 5) and is parallel to the line . 3 5 6 I S> ~ SECTION B Z g `m 5 go 12 VH$ `oH$ Z Ho$ 2 A H$ h & Questions number 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 = | * Ho$ gmnoj V g H$ Ad d, {X BgH$m Ap V d h , kmV H$s{OE & a2 b2 If * is defined on the set of all real numbers by * : a * b = find the identity element, if it exists in with respect to *. h & a2 b2 , 2 3a 0 0 A= VWm kA = h , Vmo k, a Am a b Ho$ mZ kmV H$s{OE & 3 4 2b 24 2 3a 0 0 If A = and kA = , then find the values of k, a and b. 3 4 2b 24 6. {X 7. kmV H$s{OE : 65/3/1 sin x cos x dx, 0 x / 2 1 sin 2x 3 P.T.O. Find : 8. sin x cos x dx, 0 x / 2 1 sin 2x kmV H$s{OE : sin (x a) dx sin (x a) AWdm kmV H$s{OE : (log x)2 dx sin (x a) dx sin (x a) Find : OR Find : (log x) 9. 10. 2 dx do N> AMam| m VWm a H$mo {dbw V H$aVo h E dH $m| Ho$ Hw$b {Z ${nV H$aZo dmbm AdH$b g rH$aU ~ZmBE & y2 = m (a2 x2) H$mo Form the differential equation representing the family of curves y2 = m (a2 x2) by eliminating the arbitrary constants m and a . g{Xem| a VWm b , Ohm a = ^i 7 ^j + 7 k^ VWm b = 3 ^i 2 ^j + 2 k^ , XmoZm| Ho$ b ~dV EH$ m H$ g{Xe kmV H$s{OE & AWdm ^ ^ ^ i 3 j + 5 k g_Vbr h & Find a unit vector perpendicular to both the vectors a and b , where ^ ^ ^ ^ ^ ^ a = i 7 j + 7 k and b = 3 i 2 j + 2 k . OR ^ ^ ^ ^ ^ ^ ^ ^ ^ Show that the vectors i 2 j + 3 k , 2 i + 3 j 4 k and i 3 j + 5 k are coplanar. {XImBE {H$ g{Xe 65/3/1 ^ ^ ^ ^ ^ ^ i 2 j + 3k , 2i + 3 j 4k 4 VWm 11. EH$ n[adma H$s $moQ>mo hoVw m , {nVm d ~oQ>o H$mo EH$ bmBZ | `m N>`m I S>m {H$ m OmVm h & {X Xmo KQ>ZmE A Am a B {Z Z $n | n[a^m{fV hm|, Vmo P(B/A) kmV H$s{OE : KQ>Zm A : ~oQ>m EH$ {H$Zmao na, KQ>Zm B : {nVm ~rM | Mother, father and son line up at random for a family photo. If A and B are two events given by A = Son on one end, B = Father in the middle, find P(B/A). 12. x1, x2, x3, x4 Bg 2P(X = x1) = 3P(X = x2) = P(X = x3) = 5P(X = x4). mZ br{OE X EH$ m p N>H$ Ma h {OgHo$ g ^m{dV y X H$m H$ma h : m{ H$Vm ~ Q>Z kmV H$s{OE & AWdm EH$ {g H$m 5 ~ma CN>mbm OmVm h & (i) H$ -go-H$ 4 {MV m V H$aZo H$s m{ H$Vm kmV H$s{OE & 4 {MV, Am a (ii) A{YH$-go-A{YH$ Let X be a random variable which assumes values x1, x2, x3, x4 such that 2P(X = x1) = 3P(X = x2) = P(X = x3) = 5P(X = x4). Find the probability distribution of X. OR A coin is tossed 5 times. Find the probability of getting (i) at least 4 heads, and (ii) at most 4 heads. I S> g SECTION C Z g `m 13 go 23 VH$ `oH$ Z Ho$ 4 A H$ h & Questions number 13 to 23 carry 4 marks each. 13. {XImBE {H$ nyUm H$ g w Z na n[a^m{fV g ~ Y EH$ Vw Vm g ~ Y h & AWdm {X f f(x) = 2 4x 3 ,x 3 6x 4 R = {(a, b) : (a b), 2 go h , Vmo {XImBE {H$ g^r x 2 3 Ho$ {bE, {d^m{OV h } fof(x) = x h & H$m {Vbmo ^r kmV H$s{OE & Show that the relation R on the set Z of all integers, given by R = {(a, b) : 2 divides (a b)} is an equivalence relation. OR 2 4x 3 2 If f(x) = , x , show that fof(x) = x for all x . Also, find the 3 3 6x 4 inverse of f. 65/3/1 5 P.T.O. 14. 1 , x > 0 3 {X tan 1 x cot 1 x = tan 1 2 sec 1 x h , Vmo x H$m mZ kmV H$s{OE Am a AV: H$m mZ kmV H$s{OE & 1 If tan 1 x cot 1 x = tan 1 , x > 0, find the value of x and hence find 3 2 the value of sec 1 . x 15. gma{UH$m| Ho$ JwUY m] H$m moJ H$aHo$, {g H$s{OE {H$ b c a a b c a b c c a b 4abc Using properties of determinants, prove that 16. {X b c a a b c a b c c a b sin y = x sin (a + y) h , 4abc Vmo {g H$s{OE {H$ sin2 (a y ) dy dx sin a AWdm dy kmV H$s{OE dx If sin y = x sin (a + y), prove that {X (sin x)y = x + y h , Vmo sin2 (a y ) dy dx sin a OR If (sin x)y = x + y, find 65/3/1 dy . dx 6 & 17. {X y = (sec 1 x)2, x > 0 hmo, Vmo {XImBE {H$ d 2y 3 x) dy 2 = 0 + (2x dx dx 2 If y = (sec 1 x)2, x > 0, show that x2 (x2 1) x2 (x2 1) 18. dH $ y = d 2y 3 x) dy 2 = 0 + (2x dx dx 2 x 7 (x 2) (x 3) Ohm x-Aj H$mo H$mQ>Vm h , Cg {~ X go dH $ na ne -aoIm d A{^b ~ Ho$ g rH$aU kmV H$s{OE & Find the equations of the tangent and the normal to the curve x 7 y= at the point where it cuts the x-axis. (x 2) (x 3) 19. kmV H$s{OE : (sin x 1) (sin x 3) dx sin 2x 2 2 Find : (sin x 1) (sin x 3) dx sin 2x 2 20. 2 {g H$s{OE {H$ b f (x) dx a b f (a b x) dx a AV: /3 /6 dx 1 tan x H$m _y `m H$Z H$s{OE & Prove that b a /3 /6 65/3/1 f (x) dx b f (a b x) dx and hence evaluate a dx . 1 tan x 7 P.T.O. 21. AdH$b g rH$aU dy x y dx x y AWdm AdH$b g rH$aU hb H$s{OE H$mo hb H$s{OE & : (1 + x2) dy + 2xy dx = cot x dx Solve the differential equation : dy x y dx x y OR Solve the differential equation : (1 + x2) dy + 2xy dx = cot x dx 22. mZ br{OE |c | = 3 a , b Am a c h & {X g{Xe Eogo VrZ g{Xe h {OZHo$ {bE b H$m g{Xe jon EH$-X gao Ho$ ~am~a h VWm g{Xe b a Am a | a | = 1, | b | = 2 VWm c H$m g{Xe a na |3 a 2 b + 2 c | na jon Am a g{Xe c b ~dV hm|, Vmo H$m mZ kmV H$s{OE & Let a , b and c be three vectors such that | a | = 1, | b | = 2 and | c | = 3. If the projection of b along a is equal to the projection of c along a ; and b , c are perpendicular to each other, then find |3 a 2 b + 2 c | . 23. H$m mZ kmV H$s{OE {OgHo$ {bE {Z Z{b{IV aoImE na na b ~dV h x 5 2 y 1 z ; 5 2 5 1 x 1 : 1 2 z 1 2 3 y AV: kmV H$s{OE {H$ m o aoImE EH$-X gao H$mo H$mQ>Vr h m Zht & Find the value of for which the following lines are perpendicular to each other : x 5 2 y 1 z ; 5 2 5 1 x 1 Hence, find whether the lines intersect or not. 65/3/1 8 1 2 z 1 2 3 y I S> X SECTION D Z g `m 24 go 29 VH$ `oH$ Z Ho$ 6 A H$ h & Questions number 24 to 29 carry 6 marks each. 24. {X 1 A = 0 1 1 1 3 1 1 2 h , Vmo A 1 kmV AV: {Z Z g rH$aU {ZH$m H$mo hb H$s{OE H$s{OE & : x + y + z = 6, y + 3z = 11 VWm x 2y + z = 0 AWdm ma {^H$ $nm VaUm| mam, {Z Z{b{IV Am yh H$m w H $ kmV H$s{OE 2 A = 2 3 1 If A = 0 1 1 1 2 3 4 7 : 1 1 2 1 3 , find A 1. 1 Hence, solve the following system of equations : x + y + z = 6, y + 3z = 11 and x 2y + z = 0 OR Find the inverse transformations : 2 3 A = 2 4 3 7 65/3/1 of the following matrix, using elementary 1 1 2 9 P.T.O. 25. {XImBE {H$ A{YH$V Am VZ Ho$ Am a {XE JE n R>r` jo $b Ho$ ~obZ ({OgH$m D$nar ^mJ Iwbm hmo) H$s D $MmB , ~obZ Ho$ AmYma H$s { m Ho$ ~am~a hmoJr & Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base. 26. g mH$bZ Ho$ `moJ go, Cg { ^wO H$m jo $b kmV H$s{OE {OgHo$ erf ( 1, 1), (0, 5) VWm (3, 2) h & AWdm g mH$bZ Ho$ `moJ go dH $m| (x 1)2 + y2 = 1 VWm x2 + y2 = 1 go n[a~ jo H$m jo \$b kmV H$s{OE & Find the area of the triangle whose vertices are ( 1, 1), (0, 5) and (3, 2), using integration. OR Find the area of the region bounded by the curves (x 1)2 + y2 = 1 and x2 + y2 = 1, using integration. 27. {~ X Am| (2, 5, 3), ( 2, 3, 5) Am a (5, 3, 3) go Jw OaZo dmbo g Vb Ho$ g{Xe d H$mVu g rH$aU kmV H$s{OE & h g Vb, EH$ aoIm, Omo {~ X Am| (3, 1, 5) VWm ( 1, 3, 1) go Jw OaVr h , H$mo {Og {~ X na H$mQ>Vm h Cgo ^r kmV H$s{OE & AWdm g Vbm| r . ( ^i + ^j + k^ ) = 1 VWm r . (2 ^i + 3 ^j k^ ) + 4 = 0 Ho$ {V N>oXZ go hmoH$a OmZo dmbo Cg g Vb H$m g rH$aU kmV H$s{OE, Omo x-Aj Ho$ g m Va hmo & AV: Bg g Vb H$s x-Aj go X ar kmV H$s{OE & Find the vector and cartesian equations of the plane passing through the points (2, 5, 3), ( 2, 3, 5) and (5, 3, 3). Also, find the point of intersection of this plane with the line passing through points (3, 1, 5) and ( 1, 3, 1). OR Find the equation of the plane passing through the intersection of the ^ ^ ^ ^ ^ ^ planes r . ( i + j + k ) = 1 and r . (2 i + 3 j k ) + 4 = 0 and parallel to x-axis. Hence, find the distance of the plane from x-axis. 65/3/1 10 28. Xmo {S> ~o I Am a II {XE JE h & {S> ~o I | 3 bmb d 6 H$mbr J|X| h & {S> ~o II | 5 bmb d n H$mbr J|X| h & XmoZm| {S> ~m| I Am a II _| go EH$ {S> ~o H$mo m N> m MwZm OmVm h Am a Cg | go m N> m EH$ J|X {ZH$mbr OmVr h & {X {ZH$mbr JB J|X bmb h Am a CgHo$ {S> ~o II go AmZo H$s m{ H$Vm 3 5 hmo, Vmo n H$m mZ kmV H$s{OE & There are two boxes I and II. Box I contains 3 red and 6 black balls. Box II contains 5 red and n black balls. One of the two boxes, box I and box II is selected at random and a ball is drawn at random. The ball drawn is found to be red. If the probability that this red ball comes out from box II 3 is , find the value of n . 5 29. EH$ H $nZr bmBdwS> Ho$ Xmo H$ma Ho$ AZyR>o {V-{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 < 50 Am a oH$ B H$ma Ho$ {V-{M na < 60 H$m bm^ hmoZm h & kmV H$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 & Bg 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 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 and 20 minutes available for cutting and 4 hours available for assembling. The profit is < 50 each for type A and < 60 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize profit ? Formulate the above LPP and solve it graphically and also find the maximum profit. 65/3/1 11 P.T.O.

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

 

  Print intermediate debugging step

Show debugging info


 


Tags : cbse, cbse papers, cbse sample papers, cbse books, portal for cbse india, cbse question bank, central board of secondary education, cbse question papers with answers, prelims preliminary exams, pre board exam papers, cbse model test papers, solved board question papers of cbse last year, previous years solved question papers, free online cbse solved question paper, cbse syllabus, india cbse board sample questions papers, last 10 years cbse papers, cbse question papers 2017, cbse guess sample questions papers, cbse important questions, specimen / mock papers 2018.  

© 2010 - 2025 ResPaper. Terms of ServiceContact Us Advertise with us

 

cbse12 chat