i. XI EDR) VaTSAL 1S¢ HANDBOOK of MATHEMATICS Claee "> PREVIOUS YEARS ISC (XII) QUESTIONS ae Question 1. Find the of th (2, 2) 3) and perpendicular to the straight line the equation of the plane passing through, (1, 2; 3) rf Fin rs Oe 4y 4324120. stion 2, Find the cosine ofthe angle between the planes Ay 29-2246 =0, 2xv2y47+8=0 ony) = 2=7 and the plané ex+y +22 =0. 7 05a RTE mst 3 204 SoG me onil bas O=b+t int (4, 2, 4) and is perpendicular to the planes 2x + Sy + 4z + 1 =O and 4x + 7y.6,, : : i (eons) (2005) ough the point (12, 3) and perpendicutar to le planesx +y +27 =3and shsla ba anil nogwy e007] @inie the ineoining the points (1, ~2, 3) and (2, =1, 5) cuts the planex-2 {from the point 6,4, 0). (2008) and is perpendicular to the plane (2009)THREE OIMENSIONAL GeoMeTr! MEIN | 2M-2) + (296-5) + (-39(-4)| Ue . = - 110. Find the equation of pi at | Wi ae zs lane passing thr erendicila tothe plane 3xy-ay 90 ac aah ae aI __ Equation of plane passing through the , i eye 8 041 C0 x yragag ee tei co mts rae X+Y(2-2)45(3432)-400 « M4esof normal to plane (i) are 1, 2~ and 3 +9), i s (ii is perpendicular to.3x+4y—~254629 Mpeir normals are al8O Lo each other ss ‘ @+4(2-2)-20432) <0 GF gia baba cian) 5-10 = act 2 1 (e+ 2y +32-4)+ Cy 435)=0 > 2x+3y+92-8=0 ai Find the equation of plane Passing through (2, '1, ~3)j/B'(-3,2, Lyrand@2\"4; 1), port et the equation of plane is a (ex) +b Y-y)) +6 @— 21) =0 A(2,1,-3) a&-2)+by-D+e@+3)=0 as through B (-3, -2, 1) Sa-3b+4c=0 ew through C (2, 4, -1) 0-a+3b+2c=0 = => — dx’sare-18, 10 and -15. * Eq.) 18 (¢-2)+10V-D- 15 43) =0 [From . 18x-10y+ 155+ 19=0 through te tine'ofinterszetion ofthe planes:xx+ 2y + 35 —5~Oand 3 ‘on the «and a-axes. the intersection ofplanesx+2y+34~S=Oand3x-2y-s+1=0is -s+1)50 @ 5) +2 (3¥- 2 y+ (3-a) 84) (K-8)=0 oh 2 =1 era 5-1) (a) f& TEESE oa ae (er 2y+3e- 20 a4gnyx+-2) beers26,519) lies on it, THREE DIMENSIONAL ceomera| SEI = et tai0 Ans. Din ve stots tance between the ines #2 3h ah ae ie ape Bk nai 8) 44k) and 6 ainajrskrwls i+ (2015) given equations in cartesian form are eal y-2 = {e-6-j4-4+kO-4)=21 a ans. planes fa(2i+3i 40) passing through the line of intersection of te (2016) the plane rsecting y-axis at (0, 3, 0). ine igh the intersection of two given planes is Mx Sy +78 +2)=0 m)++2)=0 WO g-154+5+24=0 Wy sy +7 +2)=0 (ox + 3y -42+ 9) 73 an 2654 144-70 4988 128 oO ox - iy + 468+ 98 =i x-ll 1, 5)imthe line ~~ = 26x + 39y - axe 2016) =a, = (10a + 1Iyr4bn sh 8) line say OF (1), te oSEQ) arsausc Hano800k of MATHEMATICS Clase XII d.t’s of given line are 10, ~4 and 11. Since, PM and | are perpendicular to each other, “+ 10(10A + 9) - 4(-44 = 1) = 11-11 = 13) = 0 = 100A + 90 + 16. +44 121A +143 =0 1 > 237A=-237 => Ae-1 aul —_ Mm (-10+11,4~=2, 11-8), 46, (1, 2,3) - But M is the mid-point of PP” aoe . ae . 2Sit =3 P2-1,5) = a= 0,B=5,y=1 Therefore, Image = (0, 5, 1) a ofthe plane-passing through the paint (1, -2, 1) and perpendicular to the line joing, » 42h m1 ie ‘Solution Since, plane is perpendicular to the line joining A(3, 2, 1) and B(1, 4, 2). _ | “ABS the normal to the plane, whose d's ae 1 - 3, 4-2, 2-1 ie, 2, 2, 1 and plane pases thin PG, -2, 1). gh ~» Equation of plane is L. ri2 / alex) + -y) + e(e- 21) =0 Aaa) > (2G - 1) +27 +2)+1@-1)=0 | 84.4.3 Ans. += intersect. Find the coordinates o¢ Pay (say) : X=1_y+i_s+10_ ky eee ay = st (Sy) i Q=(2u +1, -3y 1, 81-10) If two lines intersect each of ne i ines wil = : thes then one gener ont on bah ines willbe common te, P and @ wil bet ua ht4=el, Ah-3=-3,-1, Tol = SOMEONES? i) and, a 1° Seite 20, handy ed += 1 and 4 =2 in (il), we see that 7(1) - (2) = -9THREE OMENSIONAL GeoMeTAY | RETIN a of normal ine Pig 2= 3. yo2 ai aye She: 5, -2, 1) general point on this ine say Mig 1 TG = Gay) 6. ’ Me QA+3,-2- 2,444) Miles on the plane 3x—y + 45-29, 39.4 3)-(-A-2) +4644) 209 u-yst-2=0| - et P’(o. By 7) be the image of P in the 16 Mis he mid-point of PP a Pa.,¥) 3+a_3 ee 2 oe a=0 +8. 3 2 Queen B=-1 and Htea s Image P’ = (0, -1, -3). Ans. (+ ap- @ = 9 (2018) ren 5-9 =Oand r:(2i-j +k)=3ie, 2x-y+2-3=0is + 3y—2-9) +2 (2x-yt2-3)=0 @ ha iD {From (9 and GD] Ans. 62. Find the vector equation of a line passing through 218) cartesian equations of line are: 2x- $= [parallel to the given line equations oftine are 2¢- 3-29 + 155" fe-2)-a(r73)- 8) sergeants Swe ges J T5o ‘un’s of required line are also 3, 2,~I- Now, vector equation of Lop el wihose dut’s are 3, 2, =1 is