20__ Mathematics Past Paper PDF

666 Series PQ1RS/1 Set – 1 àíZ -nÌ H$moS> *PQ1RS/1* Q.P. Code 65/1/1 AZwH«$_m§§H$...

666 Series PQ1RS/1 Set – 1 àíZ -nÌ H$moS> *PQ1RS/1* Q.P. Code 65/1/1 AZwH«$_m§§H$ narjmWu àíZ-nÌ H$moS> >H$mo CÎma-nwpñVH$m Ho$ Roll No. _wI-n¥ð >na Adí` {bIo§ & Candidates must write the Q.P. Code on the title page of the answer-book. · H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV n¥ð> 23 h¢ & · H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| > 38 àíZ h¢ & · àíZ-nÌ _| Xm{hZo hmW H$s Amoa {XE JE àíZ-nÌ H$moS> H$mo narjmWu CÎma-nwpñVH$m Ho$ _wI-n¥ð> na {bI| & · H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go nhbo, CÎma-nwpñVH$m _| àíZ H$m H«$_m§H$ Adí` {bI| & · Bg àíZ-nÌ H$mo n‹T>Zo Ho$ {bE 15 {_ZQ >H$m g_` {X`m J`m h¡ & àíZ-nÌ H$m {dVaU nydm© _| 10.15 ~Oo {H$`m OmEJm & 10.15 ~Oo go 10.30 ~Oo VH$ N>mÌ Ho$db àíZ-nÌ H$mo n‹T>|Jo Am¡a Bg Ad{Y Ho$ Xm¡amZ do CÎma-nwpñVH$m na H$moB© CÎma Zht {bI|Jo & · Please check that this question paper contains 23 printed pages. · Please check that this question paper contains 38 questions. · Q.P. Code 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 write down the serial number of the question in the answer-book 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 KÊQ>o A{YH$V_ A§H$ : 80 Time allowed : 3 hours Maximum Marks : 80 65/1/1-11 Page 1 of 23 P.T.O. 666 gm_mÝ` {ZX}e : {ZåZ{b{IV {ZX}em| H$mo ~hþV gmdYmZr go n{‹T>E Am¡a CZH$m g™Vr go nmbZ H$s{OE : (i) Bg àíZ-nÌ _| 38 àíZ h¢ & g^r àíZ A{Zdm`© h¢ & (ii) `h àíZ-nÌ nm±M IÊS>m| _| {d^m{OV h¡ – H$, I, J, K Ed§ L> & (iii) IÊS> H$ _| àíZ g§»`m 1 go 18 VH$ ~hþ{dH$ënr` VWm àíZ g§»`m 19 Ed§ 20 A{^H$WZ Ed§ VH©$ AmYm[aV 1 A§H$ Ho$ àíZ h¢ & (iv) IÊS> I _| àíZ g§ »`m 21 go 25 VH$ A{V bKw-CÎmar` (VSA) àH$ma Ho$ 2 A§H$m| Ho$ àíZ h¢ & (v) IÊS> J _| àíZ g§»`m 26 go 31 VH$ bKw-CÎmar` (SA) àH$ma Ho$ 3 A§H$m| Ho$ àíZ h¢ & (vi) IÊS> K _| àíZ g§»`m 32 go 35 VH$ XrK© -CÎmar` (LA) àH$ma Ho$ 5 A§H$m| Ho$ àíZ h¢ & (vii) IÊS> L> _| àíZ g§»`m 36 go 38 àH$aU AÜ``Z AmYm[aV 4 A§H$m| Ho$ àíZ h¢ & (viii) àíZ-nÌ _| g_J« {dH$ën Zht {X`m J`m h¡ & `Ú{n, IÊS> I Ho$ 2 àíZm| _|, IÊS> J Ho$ 3 àíZm| _|, IÊS> K Ho$ 2 àíZm| _| VWm IÊS> L> Ho$ 2 àíZm| _| Am§V[aH$ {dH$ën H$m àmdYmZ {X`m J`m h¡ & (ix) H¡$ëHw$boQ>a H$m Cn`moJ d{O©V h¡ & IÊS> H$ Bg IÊS> _| ~hþ{dH$ënr` àíZ h¢, {OZ_| àË`oH$ àíZ 1 A§H$ H$m h¡ & 1. EH$ \$bZ f : R+ ® R (Ohm± R+ g^r G$UoVa dmñV{dH$ g§»`mAm| H$m g_wƒ` h¡) f(x) = 4x + 3 Ûmam n[a^m{fV h¡, Vmo `h \$bZ : (A) EH¡$H$s h¡ naÝVw AmÀN>mXH$ Zht h¡ (B) AmÀN>mXH$ h¡ naÝVw EH¡$H$s Zht h¡ (C) EH¡$H$s VWm AmÀN>mXH$ XmoZm| h¢ (D) Z Vmo EH¡$H$s Am¡a Z hr AmÀN>mXH$ h¡ 2. `{X EH$ Amì`yh Ho$ 36 Ad`d h¢, Vmo BgH$s g§^d H$mo{Q>`m| H$s g§»`m h¡ : (A) 13 (B) 3 (C) 5 (D) 9 65/1/1-11 Page 2 of 23 666 General Instructions : Read the following instructions very carefully and strictly follow them : (i) This question paper contains 38 questions. All questions are compulsory. (ii) This question paper is divided into five Sections – A, B, C, D and E. (iii) In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and questions number 19 and 20 are Assertion-Reason based questions of 1 mark each. (iv) In Section B, Questions no. 21 to 25 are very short answer (VSA) type questions, carrying 2 marks each. (v) In Section C, Questions no. 26 to 31 are short answer (SA) type questions, carrying 3 marks each. (vi) In Section D, Questions no. 32 to 35 are long answer (LA) type questions carrying 5 marks each. (vii) In Section E, Questions no. 36 to 38 are case study based questions carrying 4 marks each. (viii) There is no overall choice. However, an internal choice has been provided in 2 questions in Section B, 3 questions in Section C, 2 questions in Section D and 2 questions in Section E. (ix) Use of calculators is not allowed. SECTION A This section comprises multiple choice questions (MCQs) of 1 mark each. 1. A function f : R+ ® R (where R+ is the set of all non-negative real numbers) defined by f(x) = 4x + 3 is : (A) one-one but not onto (B) onto but not one-one (C) both one-one and onto (D) neither one-one nor onto 2. If a matrix has 36 elements, the number of possible orders it can have, is : (A) 13 (B) 3 (C) 5 (D) 9 65/1/1-11 Page 3 of 23 P.T.O. 666 ìï x 2 + 3, x ¹ 0 3. \$bZ f(x) = í Ho$ {bE, {ZåZ H$WZm| _| go H$m¡Z-gm ghr h¡ ? ïî 1 , x = 0 (A) f(x) g§VV Am¡a AdH$bZr` h¡, g^r x Î ℝ Ho$ {bE (B) f(x) g§VV h¡, g^r x Î ℝ Ho $ {bE (C) f(x) g§VV Am¡a AdH$bZr` h¡, g^r x Î ℝ – {0} Ho$ {bE (D) f(x) AZ§V q~XþAm| na Ag§VV h¡ 4. _mZm f(x) AÝVamb [a, b] _| EH$ g§VV \$bZ h¡ Am¡a AÝVamb (a, b) _| AdH$bZr` h¡ & Vmo `h \$bZ f(x) AÝVamb (a, b) _| {Za§Va dY©_mZ hmoJm, `{X : (A) f ¢(x) < 0, g^r x Î (a, b) Ho $ {bE (B) f ¢(x) > 0, g^r x Î (a, b) Ho $ {bE (C) f ¢(x) = 0, g^r x Î (a, b) Ho $ {bE (D) f(x) > 0, g^r x Î (a, b) Ho$ {bE éx + y 2 ù é6 2 ù æ 24 24 ö 5. `{X ê 5 ú = ê ú h¡, Vmo çç + ÷ H$m _mZ hmoJm : ë xy û ë5 8 û è x y ÷ø (A) 7 (B) 6 (C) 8 (D) 18 b 6. òa f(x) dx ~am~a h¡ : b b (A) ò a f (a – x) dx (B) ò a f (a + b – x) dx b b (C) ò a f (x – (a + b)) dx (D) ò a f ((a – x) + (b – x)) dx ^ ^ 3 7. _mZm Xmo _mÌH$ g{Xem| a Am¡a b Ho$ ~rM H$m H$moU q Bg àH$ma h¡ {H$ sin q = h¡ & Vmo 5 ^ ^ a. b ~am~a h¡ : 3 3 (A) ± (B) ± 5 4 4 4 (C) ± (D) ± 5 3 65/1/1-11 Page 4 of 23 666 3. Which of the following statements is true for the function ìï x 2 + 3, x ¹ 0 f(x) = í ? ïî 1 , x = 0 (A) f(x) is continuous and differentiable ∀ x Î ℝ (B) f(x) is continuous ∀ x Î ℝ (C) f(x) is continuous and differentiable ∀ x Î ℝ – {0} (D) f(x) is discontinuous at infinitely many points 4. Let f(x) be a continuous function on [a, b] and differentiable on (a, b). Then, this function f(x) is strictly increasing in (a, b) if (A) f ¢(x) < 0, ∀ x Î (a, b) (B) f ¢(x) > 0, ∀ x Î (a, b) (C) f ¢(x) = 0, ∀ x Î (a, b) (D) f (x) > 0, ∀ x Î (a, b) éx + y 2 ù é6 2 ù æ 24 24 ö 5. If ê ú = ê ú , then the value of çç + ÷÷ is : ë 5 xy û ë 5 8 û è x y ø (A) 7 (B) 6 (C) 8 (D) 18 b 6. òa f(x) dx is equal to : b b (A) ò a f (a – x) dx (B) ò a f (a + b – x) dx b b (C) ò a f (x – (a + b)) dx (D) ò a f ((a – x) + (b – x)) dx ^ 3 7. Let q be the angle between two unit vectors ^ a and b such that sin q =. 5 ^ Then, ^a. b is equal to : 3 3 (A) ± (B) ± 5 4 4 4 (C) ± (D) ± 5 3 65/1/1-11 Page 5 of 23 P.T.O. 666 dy 8. AdH$b g_rH$aU (1 – x2) + xy = ax, – 1 < x < 1, H$m g_mH$bZ JwUH$ h¡ : dx 1 1 (A) 2 (B) x –1 x2 – 1 1 1 (C) (D) 1 – x2 1 – x2 9. `{X {H$gr EH$ aoIm Ho$ {XH²$-H$mogmBZ 3 k, 3 k, 3k h¢, Vmo k H$m _mZ h¡ : (A) ±1 (B) ± 3 1 (C) ±3 (D) ± 3 10. EH$ a¡{IH$ àmoJ«m_Z BîQ>V_H$mar g_ñ`m g§~{§ YV hmoVr h¡ : (A) bKwJUH$s` \$bZ go (B) a¡{IH$ \$bZ go (C) {ÛKmVr` \$bZ go (D) MaKmVm§H$s` \$bZ go 11. `{X P(A|B) = P(A¢|B) h¡, Vmo {ZåZ _| go H$m¡Z -gm H$WZ ghr h¡ ? (A) P(A) = P(A¢) (B) P(A) = 2 P(B) 1 (C) P(A 3 B) = P(B) (D) P(A 3 B) = 2 P(B) 2 x +1 x –1 12. ~am~a h¡ : x2 + x + 1 x2 – x + 1 (A) 2x3 (B) 2 (C) 0 (D) 2x3 – 2 13. x Ho$ gmnoj, sin (x2) H$m AdH$bO, x = p na h¡ : (A) 1 (B) –1 (C) –2 p (D) 2 p 65/1/1-11 Page 6 of 23 666 dy 8. The integrating factor of the differential equation (1 – x2) + xy = ax, dx – 1 < x < 1, is : 1 1 (A) 2 (B) x –1 x2 – 1 1 1 (C) (D) 1 – x2 1 – x2 9. If the direction cosines of a line are 3 k, 3 k, 3 k, then the value of k is : (A) ±1 (B) ± 3 1 (C) ±3 (D) ± 3 10. A linear programming problem deals with the optimization of a/an : (A) logarithmic function (B) linear function (C) quadratic function (D) exponential function 11. If P(A|B) = P(A¢|B), then which of the following statements is true ? (A) P(A) = P(A¢) (B) P(A) = 2 P(B) 1 (C) P(A 3 B) = P(B) (D) P(A 3 B) = 2 P(B) 2 x +1 x –1 12. is equal to : x2 + x + 1 x2 – x + 1 (A) 2x3 (B) 2 (C) 0 (D) 2x3 – 2 13. The derivative of sin (x2) w.r.t. x, at x = p is : (A) 1 (B) –1 (C) –2 p (D) 2 p 65/1/1-11 Page 7 of 23 P.T.O. 666 2 3 é æ dy ö ù d 2y 14. AdH$b g_rH$aU ê1 + ç ÷ ú = H$s H$mo{Q> Am¡a KmV H«$_e: h¢ : êë è dx ø úû dx 2 (A) 1, 2 (B) 2, 3 (C) 2, 1 (D) 2, 6 15. g{Xe, {OgH$m A§{V_ q~Xþ A (2, – 3, 5) VWm àma§{^H$ q~Xþ B (3, – 4, 7) h¡, h¡ : ^ ^ ^ ^ ^ ^ (A) i – j + 2k (B) i + j + 2k ^ ^ ^ ^ ^ ^ (C) – i – j – 2k (D) – i + j – 2k 16. y-Aj go q~Xþ P(a, b, c) H$s Xÿar h¡ : (A) b (B) b2 (C) a2 + c2 (D) a2 + c2 17. ì`damoYm| x ³ 0, y ³ 0, x + y ³ 4 go {ZYm©[aV gwg§JV joÌ Ho$ H$moZr` q~XþAm| H$s g§»`m h¡ : (A) 0 (B) 1 (C) 2 (D) 3 18. `{X Xmo g_mZ H$mo{Q> dmbo eyÝ`oVa dJ© Amì`yhm| A Am¡a B Ho $ {bE (A + B)2 = A2 + B2 h¡, Vmo : (A) AB = O (B) AB = – BA (C) BA = O (D) AB = BA àíZ g§»`m 19 Am¡a 20 A{^H$WZ Ed§ VH©$ AmYm[aV àíZ h¢ & Xmo H$WZ {XE JE h¢ {OZ_| EH$ H$mo A{^H$WZ (A) VWm Xÿgao H$mo VH©$ (R) Ûmam A§{H$V {H$`m J`m h¡ & BZ àíZm| Ho$ ghr CÎma ZrMo {XE JE H$moS>m| (A), (B), (C) Am¡a (D) _| go MwZH$a Xr{OE & (A) A{^H$WZ (A) Am¡a VH©$ (R) XmoZm| ghr h¢ Am¡a VH©$ (R), A{^H$WZ (A) H$s ghr ì¶m»¶m H$aVm h¡ & (B) A{^H$WZ (A) Am¡a VH©$ (R) XmoZm| ghr h¢, naÝVw VH©$ (R), A{^H$WZ (A) H$s ghr ì¶m»¶m Zht H$aVm h¡ & (C) A{^H$WZ (A) ghr h¡, naÝVw VH©$ (R) µJbV h¡ & (D) A{^H$WZ (A) µJbV h¡¡, naÝVw VH©$ (R) ghr h¡ & 65/1/1-11 Page 8 of 23 666 2 3 é æ dy ö ù d 2y 14. The order and degree of the differential equation ê1 + ç ÷ ú = êë è dx ø úû dx 2 respectively are : (A) 1, 2 (B) 2, 3 (C) 2, 1 (D) 2, 6 15. The vector with terminal point A (2, – 3, 5) and initial point B (3, – 4, 7) is : ^ ^ ^ ^ ^ ^ (A) i – j + 2k (B) i + j + 2 k ^ ^ ^ ^ ^ ^ (C) – i – j – 2k (D) – i + j – 2k 16. The distance of point P(a, b, c) from y-axis is : (A) b (B) b2 (C) a2 + c2 (D) a2 + c2 17. The number of corner points of the feasible region determined by constraints x ³ 0, y ³ 0, x + y ³ 4 is : (A) 0 (B) 1 (C) 2 (D) 3 18. If A and B are two non-zero square matrices of same order such that (A + B)2 = A2 + B2, then : (A) AB = O (B) AB = – BA (C) BA = O (D) AB = BA Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true. 65/1/1-11 Page 9 of 23 P.T.O. 666 é 1 cos q 1 ù 19. A{^H$WZ (A) : Amì`yh A = ê– cos q 1 cos qúú , Ohm± q ∊ [0, 2p] Ho$ {bE, ê êë – 1 – cos q 1 úû |A| ∊ [2, 4]. VH©$ (R) : cos q ∊ [– 1, 1], ∀ q ∊ [0, 2p]. 20. A{^H$WZ (A) : A§V[aj _| EH$ aoIm H$^r ^r EH$ gmW x, y Am¡a z Ajm| Ho $ b§~dV Zht hmo gH$Vr h¡ & VH©$ (R) : {H$gr aoIm Ûmam x, y Am¡a z Ajm| H$s YZmË_H$ {XemAm| Ho $ gmW H«$_e: a, b Am¡a g Ho$ H$moU ~ZmZo na cos2 a + cos2 b + cos2 g = 1 h¡ & IÊS> I Bg IÊS> _| A{V bKw-CÎmar` (VSA) àH$ma Ho$ àíZ h¢, {OZ_| àË`oH$ Ho$ 2 A§H$ h¢ & 21. (H$) Om±M H$s{OE {H$ Š`m \$bZ f(x) = x2 |x|, q~Xþ x =0 na AdH$bZr` h¡ `m Zht & AWdm dy 1 + y4 (I) `{X y = tan x h¡, Vmo {gÕ H$s{OE {H$ x =. dx 4y 22. Xem©BE {H$ \$bZ f(x) = 4x3 – 18x2 + 27x – 7 H$m CƒV_ `m {ZåZV_ _mZ Zht h¡ & 23. (H$) kmV H$s{OE : òx 1 + 2x dx AWdm (I) _mZ kmV H$s{OE : p2 ò 4 sin x dx 0 x 65/1/1-11 Page 10 of 23 666 é 1 cos q 1 ù 19. Assertion (A) : For matrix A = ê– cos q 1 cos qúú , where q ∊ [0, 2p], ê êë – 1 – cos q 1 úû |A| ∊ [2, 4]. Reason (R) : cos q ∊ [– 1, 1], ∀ q ∊ [0, 2p]. 20. Assertion (A) : A line in space cannot be drawn perpendicular to x, y and z axes simultaneously. Reason (R) : For any line making angles, a, b, g with the positive directions of x, y and z axes respectively, cos2 a + cos2 b + cos2 g = 1. SECTION B This section comprises very short answer (VSA) type questions of 2 marks each. 21. (a) Check whether the function f(x) = x2 |x| is differentiable at x = 0 or not. OR dy 1 + y4 (b) If y = tan x , prove that x =. dx 4y 22. Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima. 23. (a) Find : òx 1 + 2x dx OR (b) Evaluate : p2 ò 4 sin x dx 0 x 65/1/1-11 Page 11 of 23 P.T.O. 666 ® ® ® ® ® 24. `{X Xmo eyÝ`oVa g{Xe a Am¡a b Bg àH$ma h¢ {H$ (a + b ) z a Am¡a ® ® ® ® ® (2 a + b ) z b , Vmo {gÕ H$s{OE {H$ | b | = 2 | a |. ® ^ ^ ^ 25. Xr JB© AmH¥${V _|, ABCD EH$ g_m§Va MVw^w©O h¡ & `{X AB = 2 i – 4 j + 5 k VWm ® ^ ^ ^ ® DB = 3 i – 6 j + 2 k h¢, Vmo AD kmV H$s{OE Am¡a BgHo$ à`moJ go g_m§Va MVw^w©O ABCD H$m joÌ\$b kmV H$s{OE & IÊS> J Bg IÊS> _| bKw-CÎmar` (SA) àH$ma Ho$ àíZ h¢, {OZ_| àË`oH$ Ho$ 3 A§H$ h¢ & 26. (H$) g_wƒ` A = {1, 2, 3, 4, 5} _| EH$ g§~§Y R = {(x, y) : |x2 – y2| < 8} Ûmam n[a^m{fV h¡ & Om±M H$s{OE {H$ Š`m `h g§~Y§ R ñdVwë`, g_{_V Am¡a g§H«$m_H$ h¡ & AWdm (I) \$bZ f : R ® R, f(x) = ax + b Ûmam Bg àH$ma n[a^m{fV h¡ {H$ f(1) = 1 Am¡a f(2) = 3. \$bZ f(x) kmV H$s{OE & AV:, Om±M H$s{OE {H$ Š`m \$bZ f(x) EH¡$H$s Am¡a AmÀN>mXH$ h¡ `m Zht & 65/1/1-11 Page 12 of 23 666 ® ® ® ® ® 24. If a and b are two non-zero vectors such that ( a + b ) z a and ® ® ® ® ® (2 a + b ) z b , then prove that | b | = 2 | a |. ® ^ ^ ^ 25. In the given figure, ABCD is a parallelogram. If AB = 2 i – 4 j + 5 k and ® ^ ^ ^ ® DB = 3 i – 6 j + 2 k , then find AD and hence find the area of parallelogram ABCD. SECTION C This section comprises short answer (SA) type questions of 3 marks each. 26. (a) A relation R on set A = {1, 2, 3, 4, 5} is defined as R = {(x, y) : |x2 – y2| < 8}. Check whether the relation R is reflexive, symmetric and transitive. OR (b) A function f is defined from R ® R as f(x) = ax + b, such that f(1) = 1 and f(2) = 3. Find function f(x). Hence, check whether function f(x) is one-one and onto or not. 65/1/1-11 Page 13 of 23 P.T.O. 666 27. (H$) `{X 1 – x2 + 1 – y 2 = a (x – y) h¡, Vmo {gÕ H$s{OE {H$ dy 1 – y2 =. dx 1 – x2 AWdm dy (I) `{X y = (tan x)x h¡, Vmo kmV H$s{OE & dx 28. (H$) kmV H$s{OE : x2 ò (x2 + 4) (x 2 + 9) dx AWdm (I) _mZ kmV H$s{OE : 3 ò 1 (|x – 1|+|x – 2|+|x – 3|) dx dy æ y ö 29. AdH$b g_rH$aU x2 – xy = x2 cos2 ç ÷ H$m {d{eîQ> hb kmV H$s{OE, {X`m dx è 2x ø p hþAm h¡ {H$ y= , `{X x = 1. 2 30. {ZåZ a¡{IH$ àmoJ«m_Z g_ñ`m H$mo AmboIr` {d{Y Ûmam hb H$s{OE : {ZåZ ì`damoYm| Ho$ A§VJ©V x + 2y £ 12 2x + y £ 12 4x + 5y ³ 20 x ³ 0, y ³ 0 z = 500x + 300y H$m A{YH$V_rH$aU H$s{OE & – 31. E Am¡a F Xmo ñdV§Ì KQ>ZmE± Eogr h¢ {OZHo$ {bE P( E ) = 0·6 VWm P(E 4 F) = 0·6 h¡ & – – P(F) Am¡a P( E 4 F ) kmV H$s{OE & 65/1/1-11 Page 14 of 23 666 dy 1 – y2 27. (a) If 1 – x2 + 1 – y 2 = a (x – y), prove that =. dx 1 – x2 OR dy (b) If y = (tan x)x, then find. dx 28. (a) Find : x2 ò (x2 + 4) (x 2 + 9) dx OR (b) Evaluate : 3 ò 1 (|x – 1|+|x – 2|+|x – 3|) dx 29. Find the particular solution of the differential equation given by dy æ y ö p x2 – xy = x2 cos2 ç ÷ , given that when x = 1, y =. dx è 2x ø 2 30. Solve the following linear programming problem graphically : Maximise z = 500x + 300y, subject to constraints x + 2y £ 12 2x + y £ 12 4x + 5y ³ 20 x ³ 0, y ³ 0 – 31. E and F are two independent events such that P( E ) = 0·6 and – – P(E 4 F) = 0·6. Find P(F) and P( E 4 F ). 65/1/1-11 Page 15 of 23 P.T.O. 666 IÊS> K Bg IÊS> _| XrK©-CÎmar` (LA) àH$ma Ho$ àíZ h¢, {OZ_| àË`oH$ Ho$ 5 A§H$ h¢ & é1 – 2 0ù 32. (H$) `{X A = 2 – 1 – 1ú ê h¡, Vmo A–1 kmV H$s{OE Am¡a BgHo$ à`moJ go, {ZåZ ê ú êë0 – 2 1úû g_rH$aU {ZH$m` H$mo hb H$s{OE : x – 2y = 10, 2x – y – z = 8, – 2y + z = 7 AWdm é – 1 a 2ù é 1 –1 1ù (I) `{X A = êê 1 2 x úú VWm A–1 = ê – 8 7 – 5úú ê h¡, êë 3 1 1úû êë b y 3úû Vmo (a + x) – (b + y) H$m _mZ kmV H$s{OE & 33. (H$) _mZ kmV H$s{OE : p ò 4 sin x + cos x dx 0 9 + 16 sin 2x AWdm (I) _mZ kmV H$s{OE : p ò0 2 sin 2x tan–1 (sin x) dx x2 y2 34. g_mH$bZ {d{Y Ho$ à`moJ go, XrK©d¥Îm + = 1 Ho $ Cg joÌ H$m, Omo aoImAm| x=–2 16 4 Am¡a x = 2 Ho$ ~rM h¡, joÌ\$b kmV H$s{OE & x y –1 z–2 35. `{X aoIm = = _| q~Xþ P(x, y, z) H$m à{V{~å~ P¢ (1, 0, 7) h¡, Vmo q~Xþ 1 2 3 P Ho$ {ZX}em§H$ kmV H$s{OE & 65/1/1-11 Page 16 of 23 666 SECTION D This section comprises long answer type questions (LA) of 5 marks each. é1 – 2 0ù 32. (a) If A = 2 – 1 – 1ú , find A–1 and use it to solve the following ê ê ú êë0 – 2 1úû system of equations : x – 2y = 10, 2x – y – z = 8, – 2y + z = 7 OR é– 1 a 2ù é 1 –1 1ù (b) If A = ê 1 2 x ú and A–1 = ê– 8 7 – 5úú , ê ú ê êë 3 1 1úû êë b y 3úû find the value of (a + x) – (b + y).$ 33. (a) Evaluate : p ò 4 sin x + cos x dx 0 9 + 16 sin 2x OR (b) Evaluate : p ò0 2 sin 2x tan–1 (sin x) dx x2 y2 34. Using integration, find the area of the ellipse + = 1, included 16 4 between the lines x = – 2 and x = 2. x y –1 z–2 35. The image of point P(x, y, z) with respect to line = = is 1 2 3 P¢ (1, 0, 7). Find the coordinates of point P. 65/1/1-11 Page 17 of 23 P.T.O. 666 IÊS> L> Bg IÊS> _| 3 àH$aU AÜ``Z AmYm[aV àíZ h¢, {OZ_| àË`oH$ Ho$ 4 A§H$ h¢ & àH$aU AÜ``Z – 1 36. Q´>¡{ \$H$ nw{bg Zo eha _| {d{^Þ ñWmZm| na Amoda ñnrS> Cëb§KZ {S>Q>oŠeZ (OSVD) àUmbr ñWm{nV H$s h¡ & `o H¡$_ao 300 _rQ>a H$s Xÿar go VoO J{V go MbZo dmbo dmhZ H$s \$moQ>mo bo gH$Vo h¢ Am¡a A±Yoao _| ^r H$m_ H$a gH$Vo h¢ & amS>ma ñnrS> {S>Q>oŠeZ Am¡gV ñnrS> {S>Q>oŠeZ Xÿar amS>ma dmhZm| H$s J{V H$mo gQ>rH$ ê$n ñnrS> (Mmb) = amS>ma go _mnZo Ho$ {bE bm¡Q>r ao{S>`mo Va§Jm| H$s {~ÝXþ A {~ÝXþ B g_` B – g_` A Amd¥{Îm _| n[adV©Z H$mo _mnVm h¡ (S>m°ßba à^md) g_` g_` A B amS>ma Ûmam CËg{O©V ao{S>`mo Va§Jm| Ho$ dmng bm¡Q>Zo go nVm MbVm h¡ {H$ H$moB© dñVw H$m {S>Q>Šo eZ hþAm h¡ & EH$ I§^o na 5 _rQ>a H$s C±MmB© na EH$ H¡$_am ñWm{nV {H$`m J`m h¡ & `h 20 _rQ>a / goH§$S> H$s J{V go I§^o go Xÿa Om ahr EH$ H$ma H$m nVm bJmVm h¡ & I§^o Ho$ nmX go x _rQ>a Xÿar na {H$gr ^r q~Xþ na, H$ma C go ñnrS> H¡$_ao H$m CÞ`Z H$moU q h¡ & Cn`w©º$ gyMZm Ho$ AmYma na, {ZåZ àíZm| Ho$ CÎma Xr{OE : (i) I§^o na ñWm{nV {H$E JE H¡$_ao H$s D±$MmB© Am¡a x Ho$ ê$n _| q H$mo ì`º$ H$s{OE & 1 dq (ii) kmV H$s{OE & 1 dx (iii) (H$) O~ H$ma I§^o go 50 _rQ>a Xÿa hmo, Vmo Cg jU na g_` Ho$ gmnoj CÞ`Z H$moU _| n[adV©Z H$s Xa kmV H$s{OE & 2 AWdm (iii) (I) `{X I§^o Ho$ nmX go 50 _rQ>a H$s Xÿar na Xÿgar H$ma Ho$ g_` Ho$ gmnoj CÞ`Z H$moU _| n[adV©Z H$s Xa 3 ao{S>`Z/goH§$S> h¡, Vmo H$ma H$s J{V kmV 101 H$s{OE & 2 65/1/1-11 Page 18 of 23 666 SECTION E This section comprises 3 case study based questions of 4 marks each. Case Study – 1 36. The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. RADAR SPEED DETECTION AVERAGE SPEED DETECTION Distance RADAR RADAR measures the change in Speed = POINT A POINT B Time B – Time A the frequency of returned radio waves to precisely measure the speed of vehicles (the Doppler TIME TIME A B effect) Radio waves emitted by the RADAR bounce back to confirm an object was detected A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, x m away from the base of the pole, the angle of elevation of the speed camera from the car C is q. On the basis of the above information, answer the following questions : (i) Express q in terms of height of the camera installed on the pole and x. 1 dq (ii) Find. 1 dx (iii) (a) Find the rate of change of angle of elevation with respect to time at an instant when the car is 50 m away from the pole. 2 OR (iii) (b) If the rate of change of angle of elevation with respect to time of another car at a distance of 50 m from the base of the pole 3 is rad/s, then find the speed of the car. 2 101 65/1/1-11 Page 19 of 23 P.T.O. 666 àH$aU AÜ``Z – 2 37. hmb Ho$ emoY Ho$ AZwgma, Obdm`w n[adV©Z Ho$ H$maU Xþ{Z`m ^a Ho$ {d{^Þ joÌm| _| dm`w {djmo^ ~‹T>Vm h¡ & dm`w {djmo^ C‹S>mZ H$mo _wpíH$b ~Zm XoVm h¡ Am¡a AŠga C‹S>mZ _| Xoar H$aVm h¡ & _mZ br{OE {H$ EH$ hdmB© OhmO g_mZ àm{`H$Vm Ho$ gmW J§^ra {djmo^, _Ü`_ {djmo^ `m hëHo$ {djmo^ H$m AZw^d H$aVm h¡ & BgHo$ Abmdm, J§^ra {djmo^, _Ü`_ {djmo^ Am¡a hëHo$ {djmo^ Ho$ H$maU hdmB© OhmO Ho$ J§Vì` na Xoa go nhþ±MZo H$s àm{`H$Vm H«$_e : 55%, 37% Am¡a 17% h¡ & dm`w {djmo^ J§^ra _Ü`_ hëHo$ ! 1 _rQ>a ! 5 _rQ>a ! 30 _rQ>a Cn`w©º$ gyMZm Ho$ AmYma na, {ZåZ àíZm| Ho$ CÎma Xr{OE : (i) hdmB© OhmO Ho$ J§Vì` na Xoa go nhþ±MZo H$s àm{`H$Vm kmV H$s{OE & 2 (ii) `{X hdmB© OhmO AnZo J§Vì` na Xoa go nhþ±MVm h¡, Vmo àm{`H$Vm kmV H$s{OE {H$ Eogm _Ü`_ {djmo^ Ho$ H$maU hþAm h¡ & 2 65/1/1-11 Page 20 of 23 666 Case Study – 2 37. According to recent research, air turbulence has increased in various regions around the world due to climate change. Turbulence makes flights bumpy and often delays the flights. Assume that, an airplane observes severe turbulence, moderate turbulence or light turbulence with equal probabilities. Further, the chance of an airplane reaching late to the destination are 55%, 37% and 17% due to severe, moderate and light turbulence respectively. On the basis of the above information, answer the following questions : (i) Find the probability that an airplane reached its destination late. 2 (ii) If the airplane reached its destination late, find the probability that it was due to moderate turbulence. 2 65/1/1-11 Page 21 of 23 P.T.O. 666 àH$aU AÜ``Z – 3 38. `{X \$bZ f : X ® Y Bg àH$ma n[a^m{fV h¡ {H$ f(x) = y EH¡$H$s VWm AmÀN>mXH$ hmo, Vmo h_ EH$ A{ÛVr` \$bZ g : Y ® X Bg àH$ma n[a^m{fV H$a gH$Vo h¢ {H$ g(y) = x, Ohm± x Î X VWm y = f(x), y Î Y h¡ & \$bZ g H$mo \$bZ f H$m à{Vbmo_ H$hm OmVm h¡ & sine \$bZ H$m àm§V R Am¡a \$bZ sine : R ® R Z Vmo EH¡$H$s h¡ Am¡a Z hr AmÀN>mXH$ h¡ & {ZåZ AmH¥${V _| sine \$bZ H$m AmboI {XIm`m J`m h¡ & _mZ br{OE sine \$bZ g_wƒ` A go [– 1, 1] Bg àH$ma n[a^m{fV h¡ {H$ sine \$bZ Ho$ à{Vbmo_ H$m ApñVËd h¡, `m{Z sin–1 x : [– 1, 1] ® A na n[a^m{fV h¡ & Cn`w©º$ gyMZm Ho$ AmYma na, {ZåZ àíZm| Ho$ CÎma Xr{OE : (i) `{X A _w»` _mZ emIm Ho$ Abmdm AÝ` A§Vamb h¡, Vmo Eogo EH$ A§Vamb H$m CXmhaU Xr{OE & 1 (ii) `{X sin–1 (x) H$mo [– 1, 1] go BgH$s _w»` _mZ emIm _| n[a^m{fV {H$`m J`m hmo, æ 1ö Vmo sin–1 –1 ç – 2 ÷ – sin (1) H$m _mZ kmV H$s{OE & 1 è ø (iii) (H$) [– 1, 1] go _w»` _mZ emIm VH$ Ho$ {bE sin–1 x H$m AmboI ~ZmBE & 2 AWdm (iii) (I) f(x) = 2 sin–1 (1 – x) H$m àm§V Am¡a n[aga kmV H$s{OE & 2 65/1/1-11 Page 22 of 23 666 Case Study – 3 38. If a function f : X ® Y defined as f(x) = y is one-one and onto, then we can define a unique function g : Y ® X such that g(y) = x, where x Î X and y = f(x), y Î Y. Function g is called the inverse of function f. The domain of sine function is R and function sine : R ® R is neither one-one nor onto. The following graph shows the sine function. Let sine function be defined from set A to [– 1, 1] such that inverse of sine function exists, i.e., sin–1 x is defined from [– 1, 1] to A. On the basis of the above information, answer the following questions : (i) If A is the interval other than principal value branch, give an example of one such interval. 1 (ii) If sin–1 (x) is defined from [– 1, 1] to its principal value branch, find æ 1ö the value of sin–1 ç – ÷ – sin–1 (1). 1 è 2ø (iii) (a) Draw the graph of sin–1 x from [– 1, 1] to its principal value branch. 2 OR (iii) (b) Find the domain and range of f(x) = 2 sin–1 (1 – x). 2 65/1/1-11 Page 23 of 23 P.T.O.