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CBSE Class 12 Board Exam 2020 : Mathematics (Series 3)

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H$moS> Z . Code No. amob Z . 65/3/1 narjmWu H$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. (I) ZmoQ> H $n`m Om M H$a b| {H$ Bg Z-n _o _w{ V n > 15 h & (II) 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| & (III) H $n`m Om M H$a b| {H$ Bg Z-n _| >36 Z h & H $n`m Z H$m C ma {bIZm ew $ H$aZo go nhbo, C ma-nwp VH$m _| 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 & (IV) (V) NOTE (I) Please check that this question paper contains 15 printed pages. (II) 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. (III) Please check that this question paper contains 36 questions. (IV) Please write down the Serial Number of the question in the answer-book before attempting it. (V) 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$V_ A H$ : 80 Time allowed : 3 hours .65/3/1 Maximum Marks : 80 1 P.T.O. gm_m ` {ZX}e : {Z Z{b{IV {ZX}em| H$mo ~h V gmdYmZr go n{ T>E Am a CZH$m g Vr go nmbZ H$s{OE : (i) `h Z-n Mma I S>m| _| {d^m{OV {H$`m J`m h H$, I, J Ed K & Bg Z-n _| 36 Z h & g^r Z A{Zdm` h & (ii) I S> H$ _| Z g `m 1 go 20 VH$ 20 Z h Ed `oH$ Z 1 A H$ H$m h & (iii) I S> I _| Z g `m 21 go 26 VH$ 6 Z h Ed `oH$ Z 2 A H$m| H$m h & (iv) I S> J _| Z g `m 27 go 32 VH$ 6 Z h Ed `oH$ Z 4 A H$m| H$m h & (v) I S> K _| Z g `m 33 go 36 VH$ 4 Z h Ed `oH$ Z 6 A H$m| H$m h & (vi) Z-n _| g_J na H$moB {dH$ n Zht h & VWm{n EH$ -EH$ A H$ dmbo VrZ Zm| _|, Xmo-Xmo A H$m| dmbo Xmo Zm| _|, Mma-Mma A H$m| dmbo Xmo Zm| _| Am a N :-N : A H$m| dmbo Xmo Zm| _| Am V[aH$ {dH$ n {XE JE h & Eogo Zm| _| go Ho$db EH$ hr {dH$ n H$m C ma {b{IE & (vii) BgHo$ A{V[a $, Amd `H$VmZwgma, `oH$ I S> Am a Z Ho$ gmW `Wmo{MV {ZX}e {XE JE h & (viii) Ho$bHw$boQ>am| Ho$ `moJ H$s AZw_{V Zht h & I S> H$ Z g `m 1 go 20 VH$ `oH$ Z 1 A H$ H$m h & Z g `m 1 go 10 VH$ ~h {dH$ nr` Z h & ghr {dH$ n Mw{ZE & 1. `{X R go R na {Z Z $n go X m Xmo \$bZ f Am a g n[a^m{fV f(x) = |x| + x Am a g(x) = |x| x Vmo 2. fog (x), x < 0 Ho$ (A) 4x (B) (C) (D) 2x 0 4x cot 1 ( 3 ) H$m (A) 6 (B) 6 2 (C) 3 5 (D) 6 .65/3/1 {bE hmoJm : _w ` _mZ h 2 h , General Instructions : Read the following instructions very carefully and strictly follow them : (i) This question paper comprises four Sections A, B, C and D. This question paper carries 36 questions. All questions are compulsory. (ii) Section A Questions no. 1 to 20 comprises of 20 questions of 1 mark each. (iii) Section B Questions no. 21 to 26 comprises of 6 questions of 2 marks each. (iv) Section C Questions no. 27 to 32 comprises of 6 questions of 4 marks each. (v) Section D Questions no. 33 to 36 comprises of 4 questions of 6 marks each. (vi) There is no overall choice in the question paper. However, an internal choice has been provided in 3 questions of one mark, 2 questions of two marks, 2 questions of four marks and 2 questions of six marks. Only one of the choices in such questions have to be attempted. (vii) In addition to this, separate instructions are given with each section and question, wherever necessary. (viii) Use of calculators is not permitted. SECTION A Question numbers 1 to 20 carry 1 mark each. Question numbers 1 to 10 are multiple choice type questions. Select the correct option. 1. If f and g are two functions from R to R defined as f(x) = |x| + x and g(x) = |x| x, then fog (x) for x < 0 is 2. (A) 4x (B) 2x (C) 0 (D) 4x The principal value of cot 1 ( 3 ) is (A) (B) (C) (D) .65/3/1 6 6 2 3 5 6 3 P.T.O. 3. 4. 2 A 0 0 `{X (A) 64 (B) 16 (C) 0 (D) 8 6. 7. 2 0 0 0 2 h , Vmo y = x3 + 3x2 + 12x 5 dH $ (A) (B) (C) (D) 5. 0 |adj A| H$m _mZ h H$s dUVm H$m A{YH$V_ _mZ h 15 12 9 0 e x (1 x) cos (xe ) dx ~am~a h 2 x (A) tan (xex) + c (B) cot (xex) + c (C) cot (ex) + c (D) tan [ex (1 + x)] + c AdH$b g_rH$aU (A) 1 (B) 2 (C) 3 (D) 6 p H$m (A) dh _mZ {OgHo$ {bE (D) .65/3/1 H$s KmV h ^ ^ ^ p( i + j + k ) EH$ 0 1 3 (B) (C) 3 2 2 d y dy x x y dx 2 dx 1 3 4 _m H$ g{Xe h , h 3. If A (A) (B) (C) (D) 4. 5. 6. 0 0 0 , then the value of |adj A| is 2 2 0 The maximum value of slope of the curve y = x3 + 3x2 + 12x 5 is (A) 15 (B) 12 (C) 9 (D) 0 e x (1 x) cos (xe ) dx 2 x is equal to (A) tan (xex) + c (B) cot (xex) + c (C) cot (ex) + c (D) tan [ex (1 + x)] + c 3 2 2 d y dy The degree of the differential equation x x y is dx 2 dx (A) (B) (C) (D) 7. 2 0 0 64 16 0 8 1 2 3 6 ^ ^ ^ The value of p for which p( i + j + k ) is a unit vector is (A) 0 1 3 (B) (C) (D) .65/3/1 1 3 5 P.T.O. 8. 9. 10. XZ-g_Vb na {~ X ( 2, 8, 7) (A) ( 2, 8, 7) (B) (2, 8, 7) (C) ( 2, 0, 7) (D) (0, 8, 0) XY-g_Vb go S>mbo JE b ~ Ho$ nmX Ho$ {ZX}em H$ h H$m g{Xe g_rH$aU h (A) ^ r . k =0 (B) ^ r . j =0 (C) ^ r . i =0 (D) r . n =1 EH$ a {IH$ moJ m_Z g_ `m H$m gwg JV jo ZrMo {M _| {XIm`m J`m h _mZm (A) (B) (C) (D) .65/3/1 z = 3x 4y EH$ C o ` \$bZ h & z H$m (0, 0) na (0, 8) na (5, 0) na (4, 10) na 6 `yZV_ hmoJm : 8. 9. 10. The coordinates of the foot of the perpendicular drawn from the point ( 2, 8, 7) on the XZ-plane is (A) ( 2, 8, 7) (B) (2, 8, 7) (C) ( 2, 0, 7) (D) (0, 8, 0) The vector equation of XY-plane is (A) ^ r . k =0 (B) ^ r . j =0 (C) ^ r . i =0 (D) r . n =1 The feasible region for an LPP is shown below : Let z = 3x 4y be the objective function. Minimum of z occurs at (A) (0, 0) (B) (0, 8) (C) (5, 0) (D) (4, 10) .65/3/1 7 P.T.O. Z g `m 11 go 15 VH$ Ho$ g^r Zm| Ho$ Imbr WmZ ^[aE & 11. `{X y = tan 1 x + cot 1 x, x R dy = __________ . dx h , Vmo AWdm `{X cos (xy) = k, Ohm k EH$ AMa h VWm xy n , n Z h , Vmo dy = __________ . dx 12. `{X x, f (x ) cos x, mam n[a^m{fV \$bZ 13. dH $ y = sec x Ho$ 14. {dH$Um] ^ 2i `{X `{X x x f, x = na {~ X g VV hmo, Vmo H$m _mZ hmoJm (0, 1) na ne -aoIm H$m g_rH$aU h VWm 3 k^ dmbo g_m Va MVw^w O H$m jo \$b h __________ & __________ & __________ dJ BH$mB & AWdm H$m _mZ {OgHo$ {bE g{Xe h __________ & 15. ^ ^ ^ 2i j + k VWm ^ ^ ^ i +2j k bm {~H$ h , EH$ W bo _| 3 H$mbr, 4 bmb d 2 har J oX| h & `{X VrZ J|X| EH$ gmW `m N>`m W bo go {ZH$mbr JB h , Vmo BZ J|Xm| Ho$ {^ -{^ a Jm| H$s hmoZo H$s m{`H$Vm hmoJr __________ & Z g `m 16 go 20 A{V g {j C ma dmbo Z h & 16. 2 2 H$m 17. x Ho$ 18. dh A Vamb kmV H$s{OE {Og_| h & .65/3/1 Am `yh gmnoj, A = [aij] ~ZmBE, sin2 ( x ) Ohm Ad`d aij = |(i)2 j| mam X m h & H$m AdH$bZ H$s{OE & f(x) = 7 4x x2 8 mam X m \$bZ f, {Za Va dY _mZ Fill in the blanks in question numbers 11 to 15. 11. If y = tan 1 x + cot 1 x, x R, then dy is equal to __________ . dx OR If cos (xy) = k, where k is a constant and xy n , n Z, then equal to __________ . 12. dy is dx The value of so that the function f defined by x, f (x ) cos x, if x if x is continuous at x = is __________ . 13. 14. The equation of the tangent to the curve y = sec x at the point (0, 1) is __________ . ^ ^ The area of the parallelogram whose diagonals are 2 i and 3 k is __________ square units. OR ^ ^ ^ ^ ^ ^ The value of for which the vectors 2 i j + k and i + 2 j k are orthogonal is __________ . 15. A bag contains 3 black, 4 red and 2 green balls. If three balls are drawn simultaneously at random, then the probability that the balls are of different colours is __________ . Question numbers 16 to 20 are very short answer type questions. 16. Construct a 2 2 matrix A = [aij] whose elements are given by aij = |(i)2 j|. 17. Differentiate sin2 ( x ) with respect to x. 18. Find the interval in which the function f given by f(x) = 7 4x x2 is strictly increasing. .65/3/1 9 P.T.O. 19. _mZ kmV H$s{OE : 2 |x|dx 2 AWdm kmV H$s{OE : 20. dx 9 4x 2 EH$ AZ{^ZV {g $m 4 ~ma CN>mbm OmVm h & H$_-go-H$_ EH$ ~ma {MV m H$aZo H$s n m{`H$Vm kmV H$s{OE & I S> I Z g `m 21 go 26 VH$ `oH$ Z 2 A H$m| H$m h & 21. x Ho$ {bE hb H$s{OE : 2 sin 1 4x + sin 1 3x = AWdm cos x 3 , tan 1 x 2 2 1 sin x 22. Am `yh 4 A 2 3 1 H$mo gabV_ $n _| ` $ H$s{OE & H$mo EH$ g_{_V Am `yh VWm EH$ {df_-g_{_V Am `yh Ho$ `moJ\$b Ho$ $n _| ` $ H$s{OE & 1 y 2 cos a 2 x dy dx 23. `{X 24. Xem BE {H$ {H$ ht Xmo ey `oVa g{Xem| h , `{X Am a Ho$db `{X Xem BE {H$ g{Xe h , Vmo a Am a kmV H$s{OE & b a Am a Ho$ {bE |a + b | = |a b | b ~dV g{Xe h & AWdm ^ ^ ^ ^ ^ ^ 2i j + k, 3i + 7 j + k { ^wO H$s ^wOmE {ZYm [aV H$aVo h & .65/3/1 b 10 Am a ^ ^ ^ 5i + 6 j + 2k EH$ g_H$moU 19. Evaluate : 2 |x|dx 2 OR Find : 20. dx 9 4x 2 An unbiased coin is tossed 4 times. Find the probability of getting at least one head. SECTION B Question numbers 21 to 26 carry 2 marks each. 21. Solve for x : sin 1 4x + sin 1 3x = 2 OR cos x 3 , x in the simplest form. Express tan 1 2 2 1 sin x 22. 4 Express A 2 3 as a sum of a symmetric and a skew symmetric 1 matrix. 23. 24. dy 1 If y 2 cos a 2 , then find . dx x Show that for any two non-zero vectors a and b , | a + b | = | a b | iff a and b are perpendicular vectors. OR ^ ^ ^ ^ ^ ^ ^ ^ ^ Show that the vectors 2 i j + k , 3 i + 7 j + k and 5 i + 6 j + 2 k form the sides of a right-angled triangle. .65/3/1 11 P.T.O. 25. ( 1, 1, 8) Am a (5, 2, 10) 26. `{X A Am a B Xmo KQ>ZmE Bg H$ma h {H$ P(A) = 0 4, P(B) = 0 3 VWm P(A B) = 0 6 h , Vmo P(B A) kmV H$s{OE & I S> J go Jw OaZo dmbr aoIm Cg {~ X Ho$ {ZX}em H$ kmV H$s{OE & ZX-Vb H$mo {Og {~ X na H$mQ>Vr h , Z g `m 27 go 32 VH$ `oH$ Z 4 A H$m| H$m h & 27. Xem BE {H$ f (x ) x , x ( , 0) mam 1 |x| n[a^m{fV \$bZ EH$ EH $H$s d Am N>mXH$ \$bZ h & AWdm Xem BE {H$ g ~ Y R g_w ` A = {1, 2, 3, 4, 5, 6} _| {d^m{OV h } EH$ Vw `Vm g ~ Y h & 28. `{X y = x3 (cos x)x + sin 1 29. _mZ kmV H$s{OE x h , Vmo dy dx f : ( , 0) ( 1, 0) R = {(a, b) : |a b|, 2 go kmV H$s{OE & : 5 (|x| + |x + 1| + |x 5|) dx 1 x2y dx (x3 + y3) dy = 0 H$m 30. AdH$b g_rH$aU 31. {Z Z{b{IV a {IH$ moJ m_Z g_ `m H$mo AmboI {d{Y go hb H$s{OE : `mnH$ hb kmV H$s{OE & {Z Z{b{IV `damoYm| Ho$ A VJ V z = 5x + 7y H$m `yZV_rH$aU H$s{OE : 2x + y 8 x + 2y 10 x, y 0 32. EH$ W bo _| Xmo {g o$ h EH$ A{^ZV Am a X gam AZ{^ZV h & A{^ZV {g o$ H$mo CN>mbZo na {MV AmZo H$m g `moJ 60% h & XmoZm| _| go EH$ {g o$ H$mo `m N>`m MwZm OmVm h Am a Cgo CN>mbm OmVm h & `{X {g o$ na nQ> AmVm h , Vmo `m m{`H$Vm h {H$ dh AZ{^ZV {g $m h ? AWdm .65/3/1 12 25. Find the coordinates of the point where the line through ( 1, 1, 8) and (5, 2, 10) crosses the ZX-plane. 26. If A and B are two events such that P(A) = 0 4, P(B) = 0 3 and P(A B) = 0 6, then find P(B A). SECTION C Question numbers 27 to 32 carry 4 marks each. 27. Show that the function f : ( , 0) ( 1, 0) defined by f (x) x , 1 |x| x ( , 0) is one-one and onto. OR Show that the relation R in the set A = {1, 2, 3, 4, 5, 6} given by R = {(a, b) : |a b| is divisible by 2} is an equivalence relation. 28. If y = x3 (cos x)x + sin 1 x , find 29. Evaluate : dy . dx 5 (|x| + |x + 1| + |x 5|) dx 1 30. Find the general solution of the differential equation x2y dx (x3 + y3) dy = 0. 31. Solve the following LPP graphically : Minimise z = 5x + 7y subject to the constraints 2x + y 8 x + 2y 10 x, y 0 32. A bag contains two coins, one biased and the other unbiased. When tossed, the biased coin has a 60% chance of showing heads. One of the coins is selected at random and on tossing it shows tails. What is the probability it was an unbiased coin ? OR .65/3/1 13 P.T.O. EH$ `m p N>H$ Ma X H$m m{`H$Vm ~ Q>Z, {Z Z h 0 1, `{X x 0 kx 2 , P( X x) kx, 0, : x 1 `{X x 2 AWdm 3 `{X A `Wm Ohm k EH$ AMa h & kmV H$s{OE : (a) k H$m _mZ (b) (c) P(x 2) Ma X H$m _m ` I S> K Z g `m 33 go 36 VH$ `oH$ Z 6 A H$m| H$m h & 33. {Z Z{b{IV g_rH$aU {ZH$m` H$m hb Am `yh {d{Y go kmV H$s{OE : x y + 2z = 7 2x y + 3z = 12 3x + 2y z = 5 AWdm map ^H$ g {H $`mAm| mam {Z Z{b{IV Am `yh 2 A 1 3 34. 35. 1 1 0 A H$m `w H $_ m H$s{OE, Ohm 3 4 . 2 dH $ 9y2 = x3 Ho$ do {~ X kmV H$s{OE, {OZ na dH $ na A{^b ~ XmoZm| Ajm| na g_mZ I S> ~ZmVm h & A{^b~m| Ho$ g_rH$aU ^r kmV H$s{OE & g_mH$bZ {d{Y go, {Z Z{b{IV jo H$m jo \$b kmV H$s{OE : {(x, y) : y |x| + 2, y x2} 36. AWdm g_mH$bZ {d{Y go EH$ Eogo { ^wO H$m jo \$b kmV H$s{OE {OgHo$ erf (3, 1) h & Xem BE {H$ aoImE x 2 y 2 z 3 1 3 1 EH$-X gao H$mo H$mQ>Vr h & VWm (1, 0), (2, 2) VWm x 2 y 3 z 4 1 4 2 na na {V N>oXZ {~ X Ho$ {ZX}em H$ ^r kmV H$s{OE & BZ Xmo aoImAm| H$mo A V{d > H$aZo dmbo g_Vb H$m g_rH$aU ^r kmV H$s{OE & .65/3/1 14 The probability distribution of a random variable X, where k is a constant is given below : if x 0 0 1, 2 if x 1 kx , P( X x ) if x 2 or 3 kx, otherwise 0, Determine (a) the value of k (b) P(x 2) (c) Mean of the variable X. SECTION D Question numbers 33 to 36 carry 6 marks each. 33. Solve the following system of equations by matrix method : x y + 2z = 7 2x y + 3z = 12 3x + 2y z = 5 OR Obtain the inverse of the following matrix using elementary operations : 2 A 1 3 1 1 0 3 4 2 34. Find the points on the curve 9y2 = x3, where the normal to the curve makes equal intercepts with both the axes. Also find the equation of the normals. 35. Find the area of the following region using integration : {(x, y) : y |x| + 2, y x2} OR Using integration, find the area of a triangle whose vertices are (1, 0), (2, 2) and (3, 1). Show that the lines x 2 y 2 z 3 x 2 y 3 z 4 and intersect. 1 3 1 1 4 2 Also, find the coordinates of the point of intersection. Find the equation of the plane containing the two lines. 36. .65/3/1 15 P.T.O.

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