Math booklet for class 12th, Essays (high school) of Mathematics

Download Math booklet for class 12th and more Essays (high school) Mathematics in PDF only on Docsity! 65/6/1 Page 1 of 7 P.T.O. narjmWu àíZ-nÌ H$moS> >H$mo CÎma-nwpñVH$m Ho$ _wI-n¥ð >na Adí` {bIo§ & Candidates must write the Q.P. Code on the title page of the answer-book. Series A6BAB/C Set No. 1 àíZ-nÌ H$moS> Q.P. Code AZwH«$_m§H$ Roll No. J{UV MATHEMATICS : 2 : 40 Time allowed : 2 hours Maximum Marks : 40 7 14 15 10.15 10.15 10.30 Please check that this question paper contains 7 printed pages. 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 check that this question paper contains 14 questions. 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. 65/6/1 65/6/1 Page 2 of 7 : : (i) (ii) (iii) 6 I 2 (iv) 4 II 3 (v) 4 4 (vi) (vii) 14 2 IÊS> H$ 1 6 2 1. _mZ kmV H$s{OE : 2 dx xcot1 1 2/5 2/ 0 2. `{X a = î + ĵ 2 k̂ , b = î + 2 ĵ + 2 k̂ Am¡a c = î + 2 ĵ k̂ VrZ g{Xe h¢, Vmo g{Xem| ( a + b ) Am¡a ( b c ) XmoZm| Ho$ bå~dV² EH$ g{Xe kmV H$s{OE & 2 3. EH$ W¡bo _| 1 go 25 VH$ g§»`m§{H$V H$mS>© h¢ & EH$ Ho$ ~mX EH$, {~Zm à{VñWmnZm Ho$, `mÑÀN>`m Xmo H$mS>© {ZH$mbo OmVo h¢ & àm{`H$Vm kmV H$s{OE {H$ àË òH$ H$mS>© na g§»`m 7 H$m JwUO hmo & 2 4. EH$ W¡bo _| 4 g\o$X Am¡a 5 H$mbr J|X§o h¢ & Xÿgao W¡bo _| 6 g\o$X Am¡a 7 H$mbr J|X| h¢ & nhbo W¡bo go EH$ J|X `mÑÀN>`m {ZH$mb H$a Xÿgao W¡bo _| S>mbr OmVr h¡ Am¡a {\$a Xÿgao W¡bo go EH$ J|X `mÑÀN>`m {ZH$mbr OmVr h¡ & àm{`H$Vm kmV H$s{OE {H$ {ZH$mbr JB© J|X g\o$X h¡ & 2 5. `{X a , b VWm c Eogo _mÌH$ g{Xe h¢ {H$ a + b + c = 0 h¡, Vmo a . b + b . c + c . a H$m _mZ kmV H$s{OE & 2 65/6/1 Page 5 of 7 P.T.O. 6. (a) Find the general solution of the differential equation x cos y dy = (x log x + 1) ex dx. 2 OR (b) Find the value of (2a 3b), if a and b represent respectively the order and the degree of the differential equation 0 dx dy x y dx dy x dx yd yx 23 2 2 . 2 SECTION B Question numbers 7 to 10 carry 3 marks each. 7. (a) Find the area of the region {(x, y) : x2 + y2 9, x + y 3}, using integration. 3 OR (b) Using integration, find the area of the region bounded by the parabola y2 = 4x, the lines x = 0 and x = 3 and the x-axis. 3 8. Find : 3 dx )a2x(sin xsin 9. Find the equation of the plane passing through three points whose position vectors are ĵ , 3 î + 3 ĵ and î + ĵ + k̂ . 3 10. (a) Find the distance between the following parallel lines : 3 r = (2 î + ĵ k̂ ) + ( î + ĵ k̂ ) r = ( î 2 ĵ + k̂ ) + ( î + ĵ k̂ ) OR (b) Find the coordinates of the point where the line through the points ( 1, 1, 8) and (5, 2, 10) crosses the ZX-plane. 3 65/6/1 Page 6 of 7 IÊS> J 11 14 4 11. g_Vbm| r . (2 î + 2 ĵ 3 k̂ ) = 7 Am¡a r . (2 î + 5 ĵ + 3 k̂ ) = 9 H$s à{VÀN>oXZ aoIm H$mo A§V{d©îQ> H$aZo dmbo Am¡a {~ÝXþ (2, 1, 3) go JwµOaZo dmbo g_Vb H$m g_rH$aU kmV H$s{OE & 4 12. (H$) kmV H$s{OE : 4 cos x . tan 1 (sin x) dx AWdm (I) kmV H$s{OE : 4 dx )3e()1e( e xx x 13. AdH$b g_rH$aU xlogxy2 dx dy x 2 , {X`m J`m h¡ {H$ y(1) = 1, H$m {d{eîQ> hb kmV H$s{OE & 4 àH$aU-AÜ``Z AmYm[aV àíZ 14. EH$ njnmVr (biased) nmgm CN>mbm OmVm h¡ Am¡a CgHo$ \$bH$m| na {d{^Þ g§»`mAm| Ho$ àH$Q> hmoZo H$s àm{`H$VmE± {ZåZ h¢ : \$bH$ 1 2 3 4 5 6 àm{`H$Vm 0·1 0·24 0·19 0·18 0·15 K Cn ẁ©º$ gyMZm Ho$ AmYma na, {ZåZ{b{IV àíZm| Ho$ CÎma Xr{OE : (H$) K ? 2 (I) `{X nmgo H$m \$bH$ g_ g§»`m Xem©Vm h¡, Vmo BgHo$ \$bH$ na 2 `m 4 Ho$ hmoZo H$s àm{`H$Vm ? 2 65/6/1 Page 7 of 7 P.T.O. SECTION C Question numbers 11 to 14 carry 4 marks each. 11. Find the equation of the plane passing through the intersection of the planes r . (2 î + 2 ĵ 3 k̂ ) = 7 and r . (2 î + 5 ĵ + 3 k̂ ) = 9 and through the point (2, 1, 3). 4 12. (a) Find : 4 cos x . tan 1 (sin x) dx OR (b) Find : 4 dx )3e()1e( e xx x 13. Find the particular solution of the differential equation xlogxy2 dx dy x 2 , given y(1) = 1. 4 Case-Study Based Question 14. A biased die is tossed and respective probabilities for various faces to turn up are the following : Face 1 2 3 4 5 6 Probability 0·1 0·24 0·19 0·18 0·15 K Based on the above information, answer the following questions : (a) What is the value of K ? 2 (b) If a face showing an even number has turned up, then what is the probability that it is the face with 2 or 4 ? 2