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CBSE Class 12 Board Exam 2017 : Mathematics

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SET-1 H$moS> Z . Series GBM Code No. amob Z . 65/1 narjmWu H$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 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 65/1 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 Mma A H$m| dmbo 3 Zm| _| VWm N> A H$m| dmbo 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 3 questions of four marks each and 3 questions of six marks each. 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. 65/1 2 I S> A SECTION A Z g `m 1 go 4 VH$ `oH$ Z 1 A H$ H$m h & Question numbers 1 to 4 carry 1 mark each. 1. `{X {H$gr 2 2 dJ Am `yh A Ho$ {bE, 8 A(adj A) = 0 0 8 h , Vmo |A| H$m _mZ {b{IE & 8 If for any 2 2 square matrix A, A(adj A) = 0 of |A|. 2. k H$m _mZ kmV H$s{OE {OgHo$ {bE {Z Z{b{IV \$bZ (x 3) 2 36 x 3 f(x) = k 0 , then write the value 8 x=3 na g VV h : , x 3 , x 3 Determine the value of k for which the following function is continuous at x = 3 : (x 3) 2 36 x 3 f(x) = k 65/1 , x 3 , x 3 3 P.T.O. 3. kmV H$s{OE : sin2 x cos 2 x dx sin x cos x sin2 x cos 2 x dx sin x cos x Find : 4. g_Vbm| 2x y + 2z = 5 Find the distance 5x 2.5y + 5z = 20. VWm 5x 2.5y + 5z = 20 between the planes Ho$ ~rM H$s X ar kmV H$s{OE & 2x y + 2z = 5 and 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. `{X A H$mo{Q> 3 H$m EH$ {df_-g_{_V Am `yh h , Vmo {g H$s{OE {H$ det A = 0. If A is a skew-symmetric matrix of order 3, then prove that det A = 0. 6. \$bZ f(x) = x3 3x, H$s{OE & [ 3 , 0] Ho$ {bE amobo Ho$ _o` Ho$ `moJ go c H$m _mZ kmV Find the value of c in Rolle s theorem for the function f(x) = x3 3x in [ 3 , 0]. 7. EH$ KZ H$m Am`VZ 9 KZ go_r/go. H$s Xa go ~ T> ahm h & O~ KZ H$s ^wOm Vmo CgHo$ n >r` jo \$b _| ~ T>moVar H$s Xa kmV H$s{OE & 10 go_r h , The volume of a cube is increasing at the rate of 9 cm3/s. How fast is its surface area increasing when the length of an edge is 10 cm ? 65/1 4 8. Xem BE {H$ \$bZ f(x) = x3 3x2 + 6x 100, na dY _mZ h & Show that the function f(x) = x3 3x2 + 6x 100 is increasing on 9. . q~X Am| P(2, 2, 1) VWm Q(5, 1, 2) H$mo {_bmZo dmbr aoIm na p WV EH$ q~X H$m x-{ZX}em H$ 4 h & CgH$m z-{ZX}em H$$kmV H$s{OE & The x-coordinate of a point on the line joining the points P(2, 2, 1) and Q(5, 1, 2) is 4. Find its z-coordinate. 10. EH$ nmgm, {OgHo$ \$bH$m| na A H$ 1, 2, 3 bmb a J _| {bIo h VWm 4, 5, 6 hao a J _| {bIo h , H$mo CN>mbm J`m & _mZm KQ>Zm A h : m V g `m g_ h VWm KQ>Zm B h : m V g `m bmb h & kmV H$s{OE {H$ `m A VWm B dV KQ>ZmE h & A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green, is tossed. Let A be the event number obtained is even and B be the event number obtained is red . Find if A and B are independent events. 11. Xmo XOu, A VWm B, {V{XZ H $_e: < 300 VWm < 400 H$_mVo h & A EH$ {XZ _| 6 H$_r O| VWm 4 n Q>| {gb gH$Vm h O~{H$ B {V{XZ 10 H$_r Oo VWm 4 n Q>o {gb gH$Vm h & `h kmV H$aZo Ho$ {bE {H$ H$_-go-H$_ 60 H$_r O o VWm 32 n Q>| {gbZo Ho$ {bE `oH$ {H$VZo {XZ H$m` H$ao {H$ l_ bmJV H$_-go-H$_ hmo, a {IH$ moJ m_Z g_ `m Ho$ $n _| gy ~ H$s{OE & Two tailors, A and B, earn < 300 and < 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP. 12. kmV H$s{OE : dx 5 8x x 2 Find : 65/1 dx 5 8x x 2 5 P.T.O. I S> g SECTION C Z g `m 13 go 23 VH$ `oH$ Z Ho$ 4 A H$ h & Question numbers 13 to 23 carry 4 marks each. 13. `{X If tan 1 14. x 3 x 3 tan 1 x 4 x 4 4 tan 1 h , Vmo x H$m _mZ kmV H$s{OE & x 3 x 3 tan 1 , then find the value of x. x 4 x 4 4 gma{UH$m| Ho$ JwUY_m] H$m `moJ H$a, {g H$s{OE {H$ a 2 2a 2a 1 1 2a 1 a 2 1 (a 1)3 3 3 1 AWdm Am `yh A kmV H$s{OE {H$ 1 1 0 A 1 9 4 2 1 3 8 2 22 Using properties of determinants, prove that a 2 2a 2a 1 1 2a 1 a 2 1 (a 1)3 3 3 1 OR 65/1 6 Find matrix A such that 2 1 3 15. `{X 1 1 0 A 1 9 4 xy + yx = a b h , Vmo dy dx 8 2 22 kmV H$s{OE & AWdm `{X ey(x + 1) = 1 h , Vmo Xem BE {H$ If xy + yx = ab, then find d 2y 2 dy . dx dx 2 dy . dx OR If ey(x + 1) = 1, then show that 16. kmV H$s{OE d 2y 2 dy . dx dx 2 : cos (4 sin2 ) (5 4 cos 2 ) d Find : 17. cos 2 (4 sin ) (5 4 cos 2 ) _mZ kmV H$s{OE d : x tan x dx sec x tan x 0 AWdm 65/1 7 P.T.O. _mZ kmV H$s{OE : 4 | x 1| | x 2| | x 4| dx 1 Evaluate : x tan x dx sec x tan x 0 OR Evaluate : 4 | x 1| | x 2| | x 4| dx 1 18. AdH$b g_rH$aU (tan 1 x y) dx = (1 + x2) dy H$mo hb H$s{OE & Solve the differential equation (tan 1 x y) dx = (1 + x2) dy. 19. Xem BE {H$ q~X VWm 3 ^i 4 ^j H$s{OE & A, B, C {OZHo$ p W{V ^ 4 k h , EH$ g_H$moU g{Xe H $_e: 2 ^i ^j + ^k , ^i 3 ^j 5 ^k { ^wO Ho$ erf h & AV: { ^wO H$m jo \$b kmV ^ ^ ^ Show that the points A, B, C with position vectors 2 i j + k , ^ ^ ^ ^ ^ ^ i 3 j 5 k and 3 i 4 j 4 k respectively, are the vertices of a right-angled triangle. Hence find the area of the triangle. 20. ^ ^ ^ H$m _mZ kmV H$s{OE Vm{H$ Mma q~X {OZHo$ p W{V g{Xe 3 i + 6 j + 9 k , ^ ^ ^ ^ ^ ^ ^ ^ ^ i + 2 j + 3 k , 2 i + 3 j + k VWm 4 i + 6 j + k g_Vbr` h & ^ ^ ^ Find the value of , if four points with position vectors 3 i + 6 j + 9 k , ^ ^ ^ ^ ^ ^ ^ ^ ^ i + 2 j + 3 k , 2 i + 3 j + k and 4 i + 6 j + k are coplanar. 65/1 8 21. 4 H$mS> h {OZ na g `mE 1, 3, 5 VWm 7 A {H$V h , EH$ H$mS> na EH$ g `m & Xmo H$mS> {V WmnZm {H$E {~Zm `m N>`m {ZH$mbo JE & _mZm X {ZH$mbo JE Xmo H$mS>m] na {bIr g `mAm| H$m `moJ\$b h & X H$m _m ` VWm gaU kmV H$s{OE & There are 4 cards numbered 1, 3, 5 and 7, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean and variance of X. 22. EH$ {d mb` Ho$ {d m{W `m| Ho$ {bE kmV h {H$ 30% {d m{W `m| H$s 100% Cnp W{V h VWm 70% {d mWu A{Z`{_V h & {nN>bo df Ho$ n[aUm_ gy{MV H$aVo h {H$ CZ g^r {d m{W `m|, {OZH$s Cnp W{V 100% h , _| go 70% Zo dm{f H$ narjm _| A J oS> nm`m VWm A{Z`{_V {d m{W `m| _| go 10% Zo A J oS> nm`m & df Ho$ A V _|, {d mb` _| go EH$ {d mWu `m N>`m MwZm J`m VWm `h nm`m J`m {H$ CgH$m A J oS> Wm & m{`H$Vm `m h {H$ Cg {d mWu H$s 100% Cnp W{V h ? `m {Z`{_VVm Ho$db {d mb` _| Amd `H$ h ? AnZo C ma Ho$ nj _| VH $ Xr{OE & Of the students in a school, it is known that 30% have 100% attendance and 70% students are irregular. Previous year results report that 70% of all students who have 100% attendance attain A grade and 10% irregular students attain A grade in their annual examination. At the end of the year, one student is chosen at random from the school and he was found to have an A grade. What is the probability that the student has 100% attendance ? Is regularity required only in school ? Justify your answer. 23. Z = x + 2y H$m A{YH$V_rH$aU H$s{OE {Z Z AdamoYm| Ho$ A VJ V x + 2y 100 2x y 0 2x + y 200 x, y 0 Cn`w $ a {IH$ moJ m_Z g_ `m H$mo AmboI H$s ghm`Vm go hb H$s{OE & Maximise Z = x + 2y subject to the constraints x + 2y 100 2x y 0 2x + y 200 x, y 0 Solve the above LPP graphically. 65/1 9 P.T.O. I S> X SECTION D Z g `m 24 go 29 VH$ `oH$ Z Ho$ 6 A H$ h & Question numbers 24 to 29 carry 6 marks each. 24. JwUZ\$b 4 7 5 g_rH$aU {ZH$m` H$s{OE & 4 1 3 4 3 1 1 1 2 1 1 2 3 2 1 kmV H$s{OE VWm BgH$m `moJ x y + z = 4, x 2y 2z = 9, 2x + y + 3z = 1 4 Determine the product 7 5 H$mo hb H$aZo _| 4 1 1 1 1 3 1 2 2 and use it to 3 1 2 1 3 solve the system of equations x y + z = 4, x 2y 2z = 9, 2x + y + 3z = 1. 25. f : 4 4 R , 3 3 Omo 4 f(x) = 4x 3 3x 4 mam X m h , na {dMma H$s{OE & Xem BE {H$ f EH $H$s VWm Am N>mXH$ h & f H$m {Vbmo_ \$bZ kmV H$s{OE & AV: f 1(0) kmV H$s{OE VWm x kmV H$s{OE Vm{H$ f 1(x) = 2. AWdm _mZm A = (i) A _| (ii) A A na EH$ { AmYmar g {H $`m h Omo (a, b) * (c, d) = (ac, b + ad) mam n[a^m{fV h , g^r (a, b), (c, d) A Ho$ {bE & kmV H$s{OE {H$ `m * H $_{d{Z_o` VWm ghMmar h & V~, A na * Ho$ gmnoj 65/1 VWm * V g_H$ Ad`d kmV H$s{OE & Ho$ `w H $_Ur` Ad`d kmV H$s{OE & 10 Consider f : 4x 3 4 4 R given by f(x) = . Show that f is 3x 4 3 3 bijective. Find the inverse of f and hence find f 1(0) and x such that f 1(x) = 2. OR Let A = and let * be a binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a, b), (c, d) A. Determine, whether * is commutative and associative. Then, with respect to * on A 26. (i) Find the identity element in A. (ii) Find the invertible elements of A. Xem BE {H$ EH$ ~ X KZm^, {OgH$m AmYma dJm H$ma h VWm Am`VZ {X`m J`m h , H$m n >r` jo \$b `yZV_ hmoJm, O~ `h EH$ KZ h & Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube. 27. g_mH$bZ {d{Y Ho$ `moJ go Cg { ^wO ABC H$m jo \$b kmV H$s{OE {OgHo$ erfm] Ho$ {ZX}em H$ A (4, 1), B (6, 6) VWm C (8, 4) h & AWdm gab aoIm H$s{OE & 3x 2y + 12 = 0 VWm nadb` 4y = 3x2 Ho$ ~rM {Kao jo H$m jo \$b kmV Using the method of integration, find the area of the triangle ABC, coordinates of whose vertices are A (4, 1), B (6, 6) and C (8, 4). OR Find the area enclosed between the parabola 4y = 3x2 and the straight line 3x 2y + 12 = 0. 28. AdH$b g_rH$aU (x y) y = 0 O~ x = 1 h & dy = (x + 2y) dx H$m {d{e Q> hb kmV H$s{OE, {X`m J`m h {H$ Find the particular solution of the differential dy (x y) = (x + 2y), given that y = 0 when x = 1. dx 65/1 11 equation P.T.O. 29. Cg q~X Ho$ {ZX}em H$ kmV H$s{OE Ohm q~X Am| (3, 4, 5) VWm (2, 3, 1) go hmoH$a OmVr aoIm, q~X Am| (1, 2, 3), (4, 2, 3) VWm (0, 4, 3) mam ~Zo g_Vb H$mo H$mQ>Vr h & AWdm EH$ Ma g_Vb, Omo _yb-q~X go 3p H$s AMa X ar na p WV h , {ZX}em H$ Ajm| H$mo A, B, C na H$mQ>Vm h & Xem BE {H$ { ^wO ABC Ho$ Ho$ H$ H$m q~X nW 1 x2 1 y2 1 z2 1 p2 h & Find the coordinates of the point where the line through the points (3, 4, 5) and (2, 3, 1), crosses the plane determined by the points (1, 2, 3), (4, 2, 3) and (0, 4, 3). OR A variable plane which remains at a constant distance 3p from the origin cuts the coordinate axes at A, B, C. Show that the locus of the centroid of 1 1 1 1 triangle ABC is . 2 2 2 x y z p2 65/1 12

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