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CBSE Class XII 2014 : MATHEMATICS

12 pages, 76 questions, 2 questions with responses, 2 total responses,

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H$moS> Z . Series OSR/2 Code No. amob Z . 65/2/1 narjmWu H$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 > 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 minutes 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 : (i) g^r Z A{Zdm` h & (ii) Bg Z n _| 29 Z h Omo VrZ I S>m| _| {d^m{OV h : A, ~ VWm g & I S> A _| 10 Z h {OZ_| go `oH$ EH$ A H$ H$m h & I S> ~ _| 12 Z h {OZ_| go `oH$ Mma A H$ H$m h & I S> g _| 7 Z h {OZ_| go `oH$ N> A H$ H$m h & (iii) I S> A _| g^r Zm| Ho$ C ma EH$ e X, EH$ dm ` AWdm Z H$s Amd `H$Vm AZwgma {XE Om gH$Vo h & (iv) nyU Z n _| {dH$ n Zht h & {\$a ^r Mma A H$m| dmbo 4 Zm| _| VWm N> A H$m| dmbo 2 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 three sections A, B and C. Section A comprises of 10 questions of one mark each, Section B comprises of 12 questions of four marks each and Section C comprises of 7 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 4 questions of four marks each and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions. (v) 65/2/1 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 10 VH$ `oH$ Z 1 A H$ H$m h & Question numbers 1 to 10 carry 1 mark each. 1. _mZm R = {(a, a3) : a, nm M go N>moQ>r A^m ` g `m h } EH$ g ~ Y h & H$s{OE & R H$m n[aga kmV Let R = {(a, a3) : a is a prime number less than 5} be a relation. Find the range of R. 2. 1 1 cos 1 2 sin 1 2 2 H$m _mZ {b{IE & 1 1 Write the value of cos 1 2 sin 1 . 2 2 3. Am `yh g_rH$aU C2 C2 2C1 4 3 2 1 3 0 2 2 3 1 0 1 _| ma {^H$ V ^ g {H $`mAm| H$m `moJ H$s{OE & Use elementary column operations C2 C2 2C1 2 1 2 2 0 4 . equation 3 3 0 3 1 1 4. `{X a 4 8 a 4 If 8 65/2/1 3b 2a 2 6 8 3b 2a 2 6 8 b 2 a 8b h , Vmo a 2b in the matrix H$m _mZ {b{IE & b 2 , write the value of a 2b. a 8b 3 P.T.O. 5. 3 3 `{X A EH$ Eogm {b{IE & Am `yh h {H$ |A| 0 VWm | 3A | = k | A | h , Vmo k H$m _mZ If A is a 3 3 matrix, | A | 0 and | 3A | = k | A |, then write the value of k. 6. _mZ kmV H$s{OE : dx 2 sin x cos2 x Evaluate : 7. dx sin2 x cos2 x _mZ kmV H$s{OE : /4 tan x dx 0 Evaluate : /4 tan x dx 0 8. g{Xe ^ ^ ^ i + j + k H$m g{Xe ^ j Ho$ AZw{Xe jon {b{IE & ^ ^ ^ ^ Write the projection of vector i + j + k along the vector j . 9. g{Xe 2 ^ i 21 _m H$ h & ^ ^ 3 j + 6k Ho$ AZw{Xe EH$ Eogm g{Xe kmV H$s{OE {OgH$m n[a_mU Find a vector in the direction of vector magnitude 21 units. 65/2/1 4 ^ ^ ^ 2i 3 j + 6k which has 10. ^ ^ ^ ^ ^ ^ r = 2 i 5 j + k + (3 i + 2 j + 6 k ) VWm ^ ^ ^ ^ ^ r = 7 i 6 k + ( i + 2 j + 2 k ) Ho$ ~rM H$m H$moU kmV aoImAm| H$s{OE & ^ ^ ^ ^ ^ ^ Find the angle between the lines r = 2 i 5 j + k + (3 i + 2 j + 6 k ) ^ ^ ^ ^ ^ and r = 7 i 6 k + ( i + 2 j + 2 k ). I S> ~ SECTION B Z g `m 11 go 22 VH$ `oH$ Z 4 A H$ H$m h & Question numbers 11 to 22 carry 4 marks each. 11. _mZm f : W W, f(x) = x 1, `{X x {df_ h VWm f(x) = x + 1, `{X x g_ h , mam n[a^m{fV h & Xem BE {H$ f `w H $_Ur` h & f H$m {Vbmo_ kmV H$s{OE, Ohm W g^r nyU g `mAm| H$m g_w ` h & Let f : W W, be defined as f(x) = x 1, if x is odd and f(x) = x + 1, if x is even. Show that f is invertible. Find the inverse of f, where W is the set of all whole numbers. 12. x Ho$ {bE hb H$s{OE : 3 cos (tan 1 x) = sin cot 1 4 AWdm {g H$s{OE {H$ cot 1 : 7 + cot 1 8 + cot 1 18 = cot 1 3 Solve for x : 3 cos (tan 1 x) = sin cot 1 4 OR Prove that : cot 1 7 + cot 1 8 + cot 1 18 = cot 1 3 65/2/1 5 P.T.O. 13. gma{UH$m| Ho$ JwUY_m] H$m `moJ H$a {g H$s{OE {H$ a x y z x a y z x y a z a 2 (a x y z) Using properties of determinants, prove that a x y2 If y2 15. `{X a y z x `{X z x 14. y y a z x = a cos + b sin 2 d y dx 2 x a 2 (a x y z) VWm y = a sin b cos dy y 0. dx x = a cos + b sin d2y dy x y 0. 2 dx dx xm yn = (x + y)m + n y = a sin b cos , and h , Vmo {g H$s{OE {H$ f(3.02) H$m g{ H$Q> _mZ Xe_bd Ho$ 2 show that dy y . dx x dy y . dx x If xm yn = (x + y)m + n, prove that 16. h , Vmo Xem BE {H$ WmZm| VH$ kmV H$s{OE, Ohm f(x) = 3x2 + 5x + 3 h & AWdm dh A Vamb kmV H$s{OE {OZ_| \$bZ (a) 3 4 x 4x3 45x2 + 51 2 {Za Va dY _mZ h & (b) f(x) = {Za Va mg_mZ h & Find the approximate value of f(3.02), upto 2 places of decimal, where f(x) = 3x2 + 5x + 3. OR 65/2/1 6 Find the intervals in which the function f(x) = (a) 17. strictly increasing. (b) 3 4 x 4x3 45x2 + 51 is 2 strictly decreasing. _mZ kmV H$s{OE x cos 1 x 1 x2 : dx AWdm _mZ kmV H$s{OE : (3x 2) x 2 x 1 dx Evaluate : x cos 1 x 1 x 2 dx OR Evaluate : 18. (3x 2) x 2 x 1 dx AdH$b g_rH$aU O~ x = 1 h , Vmo (x2 yx2) dy + (y2 + x2y2) dx = 0 y=1 H$mo hb H$s{OE, {X`m h {H$ h & Solve the differential equation (x2 yx2) dy + (y2 + x2y2) dx = 0, given that y = 1, when x = 1. 65/2/1 7 P.T.O. 19. dy + y cot x = 2 cos x dx AdH$b g_rH$aU h , Vmo y=0 20. x 2 h & Solve the differential equation when x H$mo hb H$s{OE, {X`m h {H$ O~ . 2 dy + y cot x = 2 cos x, given that y = 0, dx Xem BE {H$ g{Xe a , b , c g_Vbr` h `{X Am a Ho$db `{X VWm c + a g_Vbr` h & AWdm EH$ Eogm _m H$ g{Xe kmV H$s{OE Omo XmoZm| g{Xem| b ~dV h , Ohm ^ ^ ^ a = i + j + k VWm a + b a VWm ^ ^ ^ b = i + 2 j + 3k. + b, b + c a b na Show that the vectors a , b , c are coplanar if and only if a + b , b + c and c + a are coplanar. OR Find a unit vector perpendicular to both of the vectors a + b and ^ ^ ^ ^ ^ ^ a b where a = i + j + k , b = i + 2 j + 3 k . 21. aoImAm|, {OZHo$ g{Xe g_rH$aU {Z Z h , Ho$ ~rM H$s `yZV_ X ar kmV H$s{OE ^ ^ ^ ^ ^ r = i + j + (2 i j + k ) VWm : ^ ^ ^ ^ ^ ^ r = 2 i + j k + (3 i 5 j + 2 k ). Find the shortest distance between the lines whose vector equations are ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ r = i + j + (2 i j + k ) and r = 2 i + j k + (3 i 5 j + 2 k ). 22. A N>r H$ma go \|$Q>r JB Vme H$s 52 n mm| H$s J >r _| go VrZ n mo `m N>`m ({~Zm {V WmnZm Ho$) {ZH$mbo JE & {ZH$mbo JE bmb n mm| H$s g `m H$m m{`H$Vm ~ Q>Z kmV H$s{OE & AV ~ Q>Z H$m _m ` kmV H$s{OE & Three cards are drawn at random (without replacement) from a well shuffled pack of 52 playing cards. Find the probability distribution of number of red cards. Hence find the mean of the distribution. 65/2/1 8 I S> g SECTION C Z g `m 23 go 29 VH$ `oH$ Z Ho$ 6 A H$ h & Question numbers 23 to 29 carry 6 marks each. 23. Xmo {d mb` P VWm Q AnZo MwZo h E {d m{W `m| H$mo ghZerbVm, X`mbwVm VWm ZoV d Ho$ _y `m| na nwa H$ma XoZm MmhVo h & {d mb` P AnZo H $_e 3, 2 VWm 1 {d m{W `m| H$mo BZ VrZ _y `m| na H $_e < x, < y VWm < z XoZm MmhVm h O~{H$ BZ nwa H$mam| H$m Hw$b _y ` < 2,200 h & {d mb` Q AnZo H $_e 4, 1 VWm 3 {d m{W `m| H$mo BZ _y `m| Ho$ {bE Hw$b < 3,100 XoZm MmhVm h (VWm {d mb` P O go hr VrZ _y `m| na dhr nwa H$ma am{e XoZm MmhVm h ) & `{X BZ VrZm| _y `m| na {XE JE EH$-EH$ nwa H$ma H$s Hw$b am{e < 1,200 h , Vmo Am `yhm| H$m `moJ H$aHo$ `oH$ _y ` Ho$ {bE Xr JB nwa H$ma am{e kmV H$s{OE & Cn`w $ VrZ _y `m| Ho$ A{V[a $ EH$ A ` _y ` gwPmBE, Omo nwa H$ma XoZo Ho$ {bE em{_b hmoZm Mm{hE & Two schools P and Q want to award their selected students on the values of Tolerance, Kindness and Leadership. The school P wants to award < x each, < y each and < z each for the three respective values to 3, 2 and 1 students respectively with a total award money of < 2,200. School Q wants to spend < 3,100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school P). If the total amount of award for one prize on each value is < 1,200, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award. 24. Xem BE {H$ D$na go Iwbo VWm {XE JE Am`VZ dmbo ~obZ H$m Hw$b n >r` jo \$b `yZV_ hmoJm O~ CgH$s D $MmB , CgHo$ AmYma H$s { `m Ho$ ~am~a h & Show that a cylinder of a given volume which is open at the top has minimum total surface area, when its height is equal to the radius of its base. 65/2/1 9 P.T.O. 25. _mZ kmV H$s{OE : x tan x dx sec x tan x 0 Evaluate : x tan x dx sec x tan x 0 26. XrK d m x2 y 2 1 9 4 VWm aoIm x y 1 3 2 mam {Kao N>moQ>o jo H$m jo \$b kmV H$s{OE & Find the area of the smaller region bounded by the ellipse and the line 27. x y 1. 3 2 x2 y 2 1 9 4 Cg g_Vb H$m g_rH$aU kmV H$s{OE {Og_| {~ X (1, 1, 2) p WV h VWm Omo g_Vbm| 2x + 3y 2z = 5 VWm x + 2y 3z = 8 na b ~dV h & AV D$na kmV {H$E JE g_Vb go {~ X P( 2, 5, 5) H$s X ar kmV H$s{OE & AWdm {~ X Am| A(2, 1, 2) VWm B(5, 3, 4) H$mo {_bmZo dmbr aoIm VWm g_Vb Ho$ {V N>oXZ {~ X H$s {~ X P( 1, 5, 10) go X ar kmV H$s{OE & x y+z=5 Find the equation of the plane that contains the point (1, 1, 2) and is perpendicular to both the planes 2x + 3y 2z = 5 and x + 2y 3z = 8. Hence find the distance of point P( 2, 5, 5) from the plane obtained above. OR Find the distance of the point P( 1, 5, 10) from the point of intersection of the line joining the points A(2, 1, 2) and B(5, 3, 4) with the plane x y + z = 5. 65/2/1 10 28. EH$ Hw$Q>ra C moJ {Z_m Vm n S>o Q>b b n VWm bH$ S>r Ho$ eoS> ~ZmVm h & `oH$ Ho$ {Z_m U _| EH$ aJ S>Zo/H$mQ>Zo H$s _erZ Am a EH$ o`a H$s Amd `H$Vm hmoVr h & EH$ n S>o Q>b b n Ho$ {Z_m U _| 2 K Q>o aJ S>Zo/H$mQ>Zo H$s _erZ Am a 3 K Q>o o`a H$s Amd `H$Vm hmoVr h O~{H$ EH$ eoS> Ho$ {Z_m U _| 1 K Q>m aJ S>Zo/H$mQ>Zo H$s _erZ Am a 2 K Q>o o`a H$s Amd `H$Vm hmoVr h & o`a {V{XZ A{YH$V_ 12 20 K Q>o Am a aJ S>Zo/H$mQ>Zo H$s _erZ {V{XZ A{YH$V_ K Q>o Ho$ {bE Cnb Y h & EH$ b n H$s {~H $s na < 25 bm^ VWm EH$ eoS> H$s {~H $s na < 15 bm^ hmoVm h & `h _mZVo h E {H$ {Z_m Vm {Z{_ V g^r b n VWm eoS> ~oM boVm h , Vmo ~VmBE {H$ dh {V{XZ {Z_m U H$s H $gr `moOZm ~ZmE {H$ Cgo A{YH$V_ bm^ hmo & Cnamo $ H$mo EH$ a {IH$ moJm _Z g_ `m ~Zm H$a J m\$ mam hb H$s{OE & A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on the grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The profit from the sale of a lamp is < 25 and that from a shade is < 15. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximise his profit. Formulate an LPP and solve it graphically. 29. EH$ ~r_m H $nZr 2000 Hy$Q>a MmbH$m|, 4000 H$ma MmbH$m| VWm ~r_m H$aVr h {OZHo$ X K Q>ZmJ V hmoZo H$s m{`H$VmE H $_e 6000 Q >H$ MmbH$m| H$m 0.01, 0.03 VWm 0.15 h & ~r_mH $V `{ $`m| (MmbH$m|) _| go EH$ X K Q>ZmJ V hmo OmVm h & Cg `{ $ Ho$ Hy$Q>a MmbH$ AWdm H$ma MmbH$ hmoZo H$s m{`H$Vm kmV H$s{OE & AWdm 65/2/1 11 P.T.O. A N>r H$ma go \|$Q>r JB Vme H$s EH$ J >r _| go nm M n mo, EH$-EH$ H$aHo$ {V WmnZm g{hV, {ZH$mbo OmVo h & m{`H$Vm kmV H$s{OE {H$ (i) g^r {ZH$mbo JE n mo B Q> Ho$ h & (ii) Ho$db 3 n mo B Q> Ho$ h & (iii) H$moB ^r n mm B Q> H$m Zht h & An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probabilities of an accident for them are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver or a car driver ? OR Five cards are drawn one by one, with replacement, from a well shuffled deck of 52 cards. Find the probability that (i) (ii) only 3 cards are diamonds. (iii) 65/2/1 all the five cards are diamonds. none is a diamond. 12 2,800

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