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CBSE Class 12 Board Exam 2019 : Mathematics (Series 4)

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SET-1 H$moS> Z . Series BVM/4 Code No. amob Z . 65/4/1 narjmWu H$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 65/4/1 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 $bHw$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 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$m h & Question numbers 1 to 4 carry 1 mark each. 1. do N> AMa A H$mo {dbw V H$aVo h E dH $m| AdH$b g_rH$aU ~ZmBE & y A 5 x Ho$ Hw$b H$mo {Z ${nV H$aZo dmbm Form the differential equation representing the family of curves A y 5, by eliminating the arbitrary constant A. x 65/4/1 2 2. `{X dJ Am `yh A H$s H$mo{Q> 3 h VWm |A| = 9 h , Vmo |2 . adj A| H$m _mZ {b{IE & If A is a square matrix of order 3, with |A| = 9, then write the value of |2 . adj A|. 3. g_Vbm| r . ( ^i 2 ^j `yZ-H$moU kmV H$s{OE & g_Vb ^ 2k) = 1 Am a ^ ^ ^ r . (3 i 6 j + 2 k ) = 0 Ho$ ~rM H$m AWdm 2x + y z = 5 mam x-Aj na H$mQ>o JE A V:I S> H$s b ~mB kmV H$s{OE & ^ ^ ^ Find the acute angle between the planes r . ( i 2 j 2 k ) = 1 and ^ ^ ^ r . (3 i 6 j + 2 k ) = 0. OR Find the length of the intercept, cut off by the plane 2x + y z = 5 on the x-axis. 4. `{X y = log (cos ex) h , Vmo dy dx If y = log (cos ex), then find kmV H$s{OE & dy . dx I S> ~ SECTION B Z g `m 5 go 12 VH$ `oH$ Z Ho$ 2 A H$ h & Question numbers 5 to 12 carry 2 marks each. 5. kmV H$s{OE : 0 Find : 4 0 65/4/1 4 1 tan x dx 1 tan x 1 tan x dx 1 tan x 3 P.T.O. 6. _mZm g {H $`m * : , a * b = 2a + b, a, b go n[a^m{fV H$s JB h & Om M H$s{OE {H$ `m `h EH$ { AmYmar g {H $`m h & `{X hm , Vmo kmV H$s{OE {H$ `m `h gmhM` ^r h & Let * be an operation defined as * : such that a * b = 2a + b, a, b . Check if * is a binary operation. If yes, find if it is associative too. 7. Xmo {~ X Am| X Am a Y Ho$ p W{V g{Xe H $_e: 3 a H$m p W{V g{Xe {b{IE Omo {H$ aoImI S> XY H$mo h & AWdm + b Am a a 3 b h & Eogo {~ X Z 2 : 1 Ho$ ~m AZwnmV _| {d^m{OV H$aVm _mZ br{OE a = ^i + 2 ^j 3 k^ Am a b = 3 ^i ^j + 2 k^ Xmo g{Xe h & Xem BE {H$ g{Xe ( a + b ) Am a ( a b ) na na b ~dV g{Xe h & X and Y are two points with position vectors 3 a + b and a 3 b respectively. Write the position vector of a point Z which divides the line segment XY in the ratio 2 : 1 externally. OR ^ ^ ^ ^ ^ ^ Let a = i + 2 j 3 k and b = 3 i j + 2 k be two vectors. Show that the vectors ( a + b ) and ( a b ) are perpendicular to each other. 8. `{X A Am a B g_{_V Am `yh Bg H$ma h {H$ AB VWm H$s{OE {H$ AB BA EH$ {df_ g_{_V Am `yh h & BA XmoZm| n[a^m{fV h , Vmo {g If A and B are symmetric matrices, such that AB and BA are both defined, then prove that AB BA is a skew symmetric matrix. 9. 12 H$mS> {OZ na 1 go 12 VH$ H$s g `mE A {H$V h (EH$ H$mS> na EH$ g `m), H$mo EH$ {S> ~o _| aIH$a A N>r Vah go {_bm`m J`m & V~ {S> ~o _ o go EH$ H$mS `m N>`m {ZH$mbm J`m & `{X `h kmV hmo {H$ {ZH$mbo JE H$mS> na H$s g `m 5 go ~ S>r h , Vmo m{`H$Vm kmV H$s{OE {H$ `h EH$ {df_ g `m h & 12 cards numbered 1 to 12 (one number on one card), are placed in a box and mixed up thoroughly. Then a card is drawn at random from the box. If it is known that the number on the drawn card is greater than 5, find the probability that the card bears an odd number. 65/4/1 4 10. EH$ {d mb` Ho$ 8 {d{e Q> {d m{W `m|, {OZ_| 3 b S>Ho$ Am a 5 b S>{H$`m h , _| go EH$ Zmo ma {V`mo{JVm Ho$ {bE 4 {d m{W `m| H$s EH$ Q>r_ H$m M`Z {H$`m OmZm h & m{`H$Vm kmV H$s{OE {H$ 2 b S>H$m| Am a 2 b S>{H$`m| H$m M`Z {H$`m J`m hmo & AWdm EH$ ~h {dH$ nr` narjm _| 5 n Z h , {OZ_| `oH$ Ho$ VrZ g ^m{dV C ma h & BgH$s `m m{`H$Vm h {H$ EH$ {d mWu Ho$db AZw_mZ bJm H$a Mma `m A{YH$ Zm| Ho$ ghr C ma Xo XoJm ? Out of 8 outstanding students of a school, in which there are 3 boys and 5 girls, a team of 4 students is to be selected for a quiz competition. Find the probability that 2 boys and 2 girls are selected. OR In a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing ? 11. {Z Z AdH$b g_rH$aU H$m hb kmV H$s{OE : dy + y = cos x sin x dx Solve the following differential equation : dy + y = cos x sin x dx 12. kmV H$s{OE : x . tan 1 x dx AWdm kmV H$s{OE : 65/4/1 dx 5 4x 2x 2 5 P.T.O. Find : x . tan 1 x dx OR Find : dx 5 4x 2x 2 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 4 x 4 x 4 x 4 x 4 x 4 x 0 h , 4 x 4 x 4 x Vmo gma{UH$m| Ho$ JwUY_m o H$m `moJ H$aHo$ x H$m _mZ kmV H$s{OE & Using properties of determinants, find the value of x for which 14. 4 x 4 x 4 x 4 x 4 x 4 x 0. 4 x 4 x 4 x AdH$b g_rH$aU dy = 1 + x2 + y2 + x2y2 H$mo dx hb H$s{OE, {X`m J`m h {H$ y = 1 h O~ x = 0 h & AWdm AdH$b g_rH$aU dy xy 2 dx x y2 H$m {d{e Q> hb kmV H$s{OE, {X`m J`m h {H$ h O~ x = 0 h & 65/4/1 6 y=1 dy = 1 + x2 + y2 + x2y2, given that y = 1 dx Solve the differential equation when x = 0. OR Find the particular solution of the differential equation dy xy , 2 dx x y2 given that y = 1 when x = 0. 15. _mZ br{OE {H$ A = R {2} Am a B = R {1} h & `{X f : A B, f(x) = x 1 x 2 mam n[a^m{fV \$bZ h , Vmo Xem BE {H$ f EH $H$s VWm Am N>mXH$ h & AV: f 1 kmV H$s{OE & AWdm Xem BE {H$ g_w ` A = {x Z : 0 x 12} _| S = {(a, b) : a, b Z, |a b|, 3 go ^m ` h } mam X m g ~ Y S EH$ Vw `Vm g ~ Y h & Let A = R {2} and B = R {1}. If f : A B is a function defined by x 1 f(x) = , show that f is one-one and onto. Hence, find f 1. x 2 OR Show that the relation S in the set A = {x Z : 0 x 12} given by S = {(a, b) : a, b Z, |a b| is divisible by 3} is an equivalence relation. 16. cos (x a) sin (x b) \$bZ H$m, x Ho$ gmnoj, g_mH$bZ H$s{OE & Integrate the function 17. cos (x a) w.r.t. x. sin (x b) x = sin t, y = sin pt `{X h , Vmo {g H$s{OE {H$ (1 x2) d 2y dx 2 x dy + p2y = 0. dx AWdm 1 cos 65/4/1 x 2 Ho$ gmnoj tan 1 x2 1 x2 1 x2 1 1 x 2 7 H$m AdH$bZ H$s{OE & P.T.O. 2 If x = sin t, y = sin pt, prove that (1 x ) d 2y dx 2 x dy + p2y = 0. dx OR Differentiate tan 18. 1 x2 with respect to cos 1 x2. 1 x2 1 x2 1 1 x 2 {g H$s{OE {H$ : 12 3 56 cos 1 + sin 1 = sin 1 13 5 65 Prove that : 12 3 56 cos 1 + sin 1 = sin 1 13 5 65 19. dy dx dy If y = (x)cos x + (cos x)sin x, find . dx `{X y = (x)cos x + (cos x)sin x h , Vmo a 20. {g H$s{OE {H$ kmV H$s{OE & 1 a f (x) dx 0 f (a x) dx, 0 AV: x 2 (1 x) n dx H$m _mZ 0 kmV H$s{OE & a Prove that 1 f (x) dx f (a x) dx, and hence evaluate x (1 x) 2 0 21. a 0 n dx . 0 x H$m dh _mZ kmV H$s{OE {OgHo$ {bE Mma {~ X Am a D( 4, 4, 4) g_Vbr` hm| & A(x, 1, 1), B(4, 5, 1), C(3, 9, 4) Find the value of x, for which the four points A(x, 1, 1), B(4, 5, 1), C(3, 9, 4) and D( 4, 4, 4) are coplanar. 65/4/1 8 22. EH$ 13 m b ~r gr T>r D$ dm Ya Xrdma Ho$ ghmao PwH$s h B h & gr T>r Ho$ ZrMo H$m {gam, O_rZ Ho$ AZw{Xe Xrdma go X a 2 cm/sec H$s Xa go ItMm OmVm h & Xrdma na BgH$s D $MmB {H$g Xa go KQ> ahr h O~ gr T>r Ho$ ZrMo H$m {gam Xrdma go 5 m H$s X ar na h ? A ladder 13 m long is leaning against a vertical wall. The bottom of the ladder is dragged away from the wall along the ground at the rate of 2 cm/sec. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall ? 23. {~ X Am o A(3, 1, 2), B(5, 2, 4) Am a C( 1, 1, 6) go {ZYm [aV g_Vb H$m g{Xe g_rH$aU kmV H$s{OE & AV: Bg H$ma m g_Vb H$s _yb-{~ X go X ar kmV H$s{OE & Find the vector equation of the plane determined by the points A(3, 1, 2), B(5, 2, 4) and C( 1, 1, 6). Hence, find the distance of the plane, thus obtained, from the origin. 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 EH$ XrK d m Am`V H$m jo \$b kmV H$s{OE & x2 a2 y2 b2 1 Ho$ A VJ V ~Z gH$Zo dmbo ~ S>o-go-~ S>o Using integration, find the area of the greatest rectangle that can be inscribed in an ellipse 25. x2 a2 y2 b2 1. EH$ ~r_m H$ nZr 3000 gmB{H$b MmbH$, 6000 Hy$Q>a MmbH$ Am a 9000 H$ma MmbH$m| H$m ~r_m H$aVr h & EH$ gmB{H$b MmbH$, Hy$Q>a MmbH$ d H$ma MmbH$ H$s X K QZm hmoZo H$s m{`H$Vm H $_e: 0 3, 0 05 Am a 0 02 h & ~r_mH $V `{ $`m| _| go EH$ X K Q>ZmJ V hmo OmVm h & Cg `{ $ Ho$ gmB{H$b MmbH$ hmoZo H$s m{`H$Vm `m h ? An insurance company insured 3000 cyclists, 6000 scooter drivers and 9000 car drivers. The probability of an accident involving a cyclist, a scooter driver and a car driver are 0 3, 0 05 and 0 02 respectively. One of the insured persons meets with an accident. What is the probability that he is a cyclist ? 65/4/1 9 P.T.O. 26. ma {^H$ n { $ $nm VaUm| mam Am `yh 2 3 1 3 2 1 5 4 2 H$m `w H $_ kmV H$s{OE & AWdm {Z Z{b{IV a {IH$ g_rH$aU {ZH$m` H$mo Am `yhm| Ho$ `moJ go hb H$s{OE : x + 2y 3z = 4 2x + 3y + 2z = 2 3x 3y 4z = 11 Using elementary row transformations, find the inverse of the matrix 5 2 3 2 4 . 3 1 1 2 OR Using matrices, solve the following system of linear equations : x + 2y 3z = 4 2x + 3y + 2z = 2 3x 3y 4z = 11 27. g_mH$bZ Ho$ `moJ go nadb` kmV H$s{OE & y2 = 4x Am a d m 4x2 + 4y2 = 9 go {Kao jo H$m jo \$b AWdm g_mH$bZ {d{Y H$m Cn`moJ H$aVo h E Eogo jo H$m jo \$b kmV H$s{OE Omo {H$ aoImAm| 3x 2y + 1 = 0, 2x + 3y 21 = 0 Am a x 5y + 9 = 0 go {Kam h Am h & Using integration, find the area of the region bounded by the parabola y2 = 4x and the circle 4x2 + 4y2 = 9. OR Using the method of integration, find the area of the region bounded by the lines 3x 2y + 1 = 0, 2x + 3y 21 = 0 and x 5y + 9 = 0. 65/4/1 10 28. EH$ Amhma{dX Xmo H$ma Ho$ ^mo `m| H$mo Bg H$ma {_bmZm MmhVm h {H$ {_lU _| {dQ>m{_Z A H$s H$_-go-H$_ 8 _m H$ Am a {dQ>m{_Z C H$s $H$_-go-H$_ 10 _m H$ hm| & ^mo ` I _| {dQ>m{_Z A H$s _m m 2 BH$mB /{H$J m h Am a {dQ>m{_Z C H$s _m m 1 BH$mB /{H$J m h & ^mo ` I Ho$ C nmXZ Ho$ {bE < 50 {V {H$J m H$s bmJV AmVr h & ^mo ` II _| {dQ>m{_Z A H$s _m m 1 BH$mB /{H$J m Am a {dQ>m{_Z C H$s _m m 2 BH$mB /{H$J m h Am a BgHo$ C nmXZ Ho$ {bE < 70 {V {H$J m H$s bmJV AmVr h & Bg Z H$mo EH$ Eogo {_lU {Og_| dm {N>V nmofH$ V d hm|, H$m `yZV_ _y ` kmV H$aZo Ho$ {bE a {IH$ moJ m_Z g_ `m _| ~Xb| & {_lU H$m `yZV_ _y ` ^r kmV H$s{OE & A dietician wishes to mix two types of food in such a way that the vitamin contents of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. It costs < 50 per kg to produce food I. Food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C and it costs < 70 per kg to produce food II. Formulate this problem as a LPP to minimise the cost of a mixture that will produce the required diet. Also find the minimum cost. 29. (2, 3, 2) go Jw OaZo dmbr aoIm H$m g{X m g_rH$aU kmV H$s{OE Omo {H$ ^ ^ ^ ^ ^ r = ( 2 i + 3 j ) + (2 i 3 j + 6 k ) Ho$ g_m Va h & AV: BZ Xmo aoImAm| Ho$ {~ X aoIm ~rM H$s X ar ^r kmV H$s{OE & AWdm q~X P(3, 2, 1) go g_Vb 2x y + z + 1 = 0 na ItMo JE b ~ Ho$ nmX Q Ho$ {ZX oem H$ kmV H$s{OE & b ~dV X ar PQ ^r kmV H$s{OE VWm Bgr g_Vb H$mo EH$ Xn U boVo h E Bg {~ X P H$m {V{~ ~ ^r kmV H$s{OE & Find the vector equation of a line passing through the point (2, 3, 2) and ^ ^ ^ ^ ^ parallel to the line r = ( 2 i + 3 j ) + (2 i 3 j + 6 k ). Also, find the distance between these two lines. OR Find the coordinates of the foot of the perpendicular Q drawn from P(3, 2, 1) to the plane 2x y + z + 1 = 0. Also, find the distance PQ and the image of the point P treating this plane as a mirror. 65/4/1 11 P.T.O.

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