ASRJC_9649_2025_Prelim_P1_Solutions

1 (a) ff f ff uv x ux vx u v ∂ ∂∂ ∂∂ ∂ ∂ =+= + ∂ ∂∂ ∂∂ ∂ ∂  ff f ff uv y uy vy u v  ∂ ∂∂ ∂∂ ∂∂=+= − ∂ ∂∂ ∂∂ ∂ ∂  Normal vector of tangent plane: f ff f ff 11 x uv y uv ∂ ∂∂   +  ∂ ∂∂  ∂ ∂∂  = −  ∂ ∂∂   −−       Since tangent plane at the point P(a, b, c) is parallel to the plane with equation 1xz−= , ff 1 ff 0 11 uv uv ∂∂ +∂∂ ∂∂  −= ∂∂ − − ff 1uv ∂∂⇒+=∂∂ ff ff 0uv u v ∂∂ ∂∂−= ⇒=∂∂ ∂∂ f f f121 2u uv ∂ ∂∂∴ = ⇒==∂ ∂∂ , shown Alternative Method Since tangent plane at the point P(a, b, c) is parallel to the plane with equation 1xz−= , f 1 f ff 0 1, 0 11 x y xy ∂ ∂ ∂ ∂∂  = ⇒= =∂ ∂∂ − − f f f 1 11 102 22 xy u xu yu ∂ ∂∂ ∂∂     = + =+=     ∂ ∂∂ ∂∂      f f f 1 11 102 22 xy v xv yv ∂ ∂∂ ∂∂      = + = +−=      ∂ ∂∂ ∂∂       f f1 2uv ∂∂⇒==∂∂ , shown (b) xy uzxy zu xy v +=+− ⇒=− −

f1 1 2u uv ∂ = −∂ , ( ) 3 f 2 u v v ∂ =∂ 11 11 22 v uv u − = ⇒= ( ) 2 33 1 1 1 112 12 uu uu u v u =⇒ =⇒ =⇒= ± ⇒=   (reject 1u=− ) 1v∴= ( )11xy+ = −−− ( )12xy− = −−− Solving gives 1, 0xy= = Coordinates of point P is ( )1 ,0 ,0 . 2 (a) Let U be the specified subset. 12 11 1, 1 00 UU      = −∈ = ∈       uu . However, 12 2 0 0 U  += ∉  uu , since 2220≠ . Not closed under vector addition. Therefore, it does not form a linear space. (b) Let W be the specified subset and let 1 11 1 a b c  = w and 2 22 2 a b c  = w be two arbitrary vectors in W, i.e. 111 2 2 22 0 and 2 0abc a bc−+= −+= . For any real constants ,λµ ∈  , 12 1 2 12 12 aa bb cc λµ λ µ λµ λµ + +=+  + ww . Here, ( ) ( ) ( ) ( ) ( )1 2 12 12 1 1 1 2 2 22 2 20a a b b c c abc a bcλµ λµ λµ λ µ+ − + + + = −+ + −+ = . Closed under vector addition and scalar multiplication. W is a subspace. (b) 2 3223 2 3++ = ⇒+ +=M M I0 M M M0 ( ) 32 23 22 3 3 6 (shown) = −− = −− − − = + M MM MI M MI

( ) ( ) ( ) ( ) ( ) ( ) 8 7 6 62 232 2 2 2 4 24 ( 2 3) 6 12 36 2 3 13 36 11 33 11( 3 ) + + += + + + = + + ++ = ++ =+ ++ = −−+ + = + = + M M M MMM M I M M M MIIM M IIM M M IM MI M I MI MI 3 2 23 2 23 d dd d d22d dd d d y wy wywx x xx x x= + ⇒ = +⇒ = ( ) ( ) 2 2 2 dd 4 24 2 8 1edd xww wx xxx −+ −+ − + += 2 2 2 dd 4 4edd xww wxx −+ += Consider Homogeneous Equation: 2 2 dd 4 40dd ww wxx + += Characteristic Equation: 2 4 40mm+ += ( ) 24 4 4(1) 4 22m −± −= =− Complementary Function: ( ) 2e x cw Ax B −= + where A and B are arbitrary constants. Let P.I. be 22e x pw Cx −= 2 22' 2e 2 e xx pw Cx Cx −−= − 222 2 2' ' 2 e 4e 4e 4 exxx x pw C Cx Cx Cx−−− −= −−+ ( ) 2 2 2 22 2 22 22 22 e 4e 4e 4 e 4 2e 2 e 4 e exxx x x x x xC Cx Cx Cx Cx Cx Cx−−− − − − − −−−+ + − + = 2 2 22 2 22 22 22 e 8e 4 e 8e 8 e 4 e ex xx x xx xC Cx Cx Cx Cx Cx− −− −−− −−+ +− + = 222e e xxC −− = Comparing coefficients, 121 2C