( )( 2 ) n n! n! n! 1 1 = + + C = +

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1 Subject : Mathematics MARKING SCHEME (For Sample Question Paper) Class : Senior Secondary. ( )( )( 9 )( 9 ) L.H.S. ω ω ω. ω ω. ω ( ω)( ω )( ω)( ω ) [ ω ] ω ω ( )( ) ( ) 4 ω +ω +ω. [ 4 ] R.H.S (Since +ω+ω ) n C n n! n! + C + r! ( n r! ) ( r!n ) ( r+! ) r r n! + ( r!n ) ( r )! r n r + n! n + ( r!n ) ( r! ) ( n r + r ) ( n +! ) r!n ( r +! ) n+ C r. L T L T L T L No. of ways of arranging 4 lions in a row 4! 4 No. of ways of arranging tigers in a row as shown in the table! 6 Required number of arrangements The equation of the ellipse can be written as x y + 4 a 4,b ( ) b a e ( ) 4 e e Foci are ( ± ae,) i.e. (, ),(,) MATHEMATICS 4

2 5. We have to find the sum of ( n.7 ) n 4 4 S4 [ 5+ 96] 7 4, 8 9 T ar 8 t t t ar 56 r r ar 8 8a 8 a The required G.P. is,, 8, A Bφ ( A B' ) U A' {,,5,7,9,}, B' {,4,6,8,9,} A' B' U ( A B' ) 7. The given function is f( x) ( x )( x 5) The function is not defined when ( x )( x 5) For domain of f(x), we have ( x )( x 5) > ( x ) > and( x 5) > or ( x ) < and( x 5) < 5 x > or x< L.H.S. tan tan tan tan MATHEMATICS

3 9. Here p α+β and ( p c) tan R.H.S. αβ + ( α+ )( β+ ) αβ+ ( α+β ) + c p + p + } c + i + i + i 5+ 5i z + i i i + i z +. + a a a b b b c c + c abc + R R + R + R a b c a b c a b c abc + b b b + c c c abc a b c b b b + c c c ( ) ab+ bc+ ca+ abc b c ( C C C,C C C ) MATHEMATICS 45

4 ( ab+ bc+ ca+ abc) ab+ bc+ ca+ abc 4 A 4 A+ A' B A A' 4 C 4 4 A' 4 A B + C, where B is symmetric and C is skew-symmetric... x y +...(i) 9 6 Diferentiating (i) w.r.t. x, we get dy 6 x dx 9 y When the tangent is parallel to x-axis, dy dx x From (i), we get y ± 4 The required points are (,4 ),, ( 4) ( ) f x x 9x + 4x+ ( ) ( ) f'x x 6x+ 8 For increasing function, f'x ( ) > x 6x + 8> ( x 4)( x ) > x > 4 or x < For decreasing function < x< 4 I sec xdx secx sec xdx Integrating by parts, we get secxtanx tanx secxtanxdx 46 MATHEMATICS

5 ( ) secxtanx secx sec x dx secxtanx sec xdx+ secxdx I secxtanx+ log( secx+ tanx) + c' I secxtanx + log( secx+ tanx) + c I sinx dx...(i) sinx + cosx sin x dx sin x + cos x cosx I dx sinx + cosx...(ii) (i) + (ii) I dx I 4. 6 P (Black card), P (Red card) 5 P (Both Black cards) 4 P (Both red cards) 4 P (Both Black or Both red) r 4. General term tr Cr( x) + x r...(i) Middle terms are 4th and 5th Putting r and r 4 in (i), we get the middle terms as MATHEMATICS 47

6 or 7 4 C ( x) 5 x 8 x 7 and C ( x) and 5 4 6x x 4 5. Let the intercepts on the axes be 'a' and 'b' respectively, For Figure The equation of line is x + y...() a b The portion of () i.e., AB is bisected at the point (,) and a a 4 b b 6 Equation of line is x + y x+ y The given equation of circle is x + y x y+ The equation of tangent to circle at (x, y) is ( ) ( ) xx + yy + g x+ x + f y+ y + c Puttingx,y and c,g,f 5, we have the equation of tangent as x + y ( x ) 5( y+ ) + 4x+ y+ 6...(i) Equation of normal is x 4y+ k It passes through (,) 9 8+ k k 7 The equation of normal is x 4y x cotx x x lim lim cosx tanx cosx x x x x lim lim x tanx x x sin + 48 MATHEMATICS

7 lim x x sin 4 x secx+ tanx secx+ tanx y secx tanx secx+ tanx + secx + tanx secx+ tanx dy secxtanx sec x dx + [ secx] 8. This is a linear diferential equation. secx tanx + Here P secx,q tanx Pdx secxdx log( sec+ tanx) I.F e e e secx + tanx The solution is y( secx+ tanx) tanx( secx+ tanx) dx 9. Classes xi ( ) secxtanxdx+ sec x dx secx+ tan x+ c fi MATHEMATICS 49 d' i fd' fd' Σ fi Σ fd' i i 65 Σ fd' 5 C S.D NΣfd' i i ( Σfd i i) N 5 ( 65) i i i i i i

8 . The given system can be written as AX B, where A, 8 x B, X y 4 z X A B A Adjoint 5 A A 7 5 x 5 8 y z x,y,z. t n ( n )( n+ )( n+ ) 4 ( n )( n+ ) ( n+ )( n + ) Putting n,,,... t 4 5 t t tn 4 ( n )( n + ) ( n + )( n+ ) S n 4 ( n+ )( n + ) 4n ( + )( n+ ). ( ) sinα cosα cos α sin α tanα cotα cotα cosα sinα cosαsinα 5 MATHEMATICS

9 tanα cotα cotα Similarly tanα cotα 4cot4α 4tan4α 4cot4α 8cot8α 8cot8α 8cot8α...(A) Adding (A), we get tanα+ tanα+ 4tan4α+ 8cot8α cot α b + c a cosa bc 44 c + a b cosb ca 6 a + b c cosc ab 7 cosa:cosb:cosc : : 7. Let r be the radius and b the height of cylinder 4. r+ h 6 h 6 r 4::6 V f ( r) r ( 6 r) 6r r dv r r dr dv r 4 dr h 6 r Again Showing d V 6 r dr d V dr as negative at r 4 Maximum volume 6 I xdx b a, h nh n n MATHEMATICS 5

10 { } xdx lim h + ( + h) + ( + h ) ( n h ) h ( n n ) lim hn+ h h ( nh h) nh lim nh + h xdx For correct Figure Area of shaded region 4 6 x dx x x 6 x + 8sin sq.units 5. ( ) ( ) OPTIONAL -I (Vectors and Three Dimensional Geometry) PQ a+ b a b a+ 5b QR 7b a + b a+ 5b ( ) ( ) ( ) PQ QR and Q is common P,Q and R are collinear 6. Let θ bethe angle between a i ˆ+ j ˆ kˆ and b i ˆ jˆ 4kˆ + + ( ) + ( ) + ( 4) cosθ ( ) ( ) MATHEMATICS

11 θ 7. The equation of plane passing through (,,) is a( x+ ) + b( y ) + c( z )...(i) This passes through the point (,,) also a b+ c...(ii) since (i) is prependicular to the plane x+ y+ z 5 a+ b+ c...(iii) From (ii) and (iii) a b c or From (i), equation of plane is 8. The lines are coplanar if a b c ( x+ ) + ( y ) ( z ) x+ y z+ x x y y z z l m n l m n Here, x, y, z 5 x,y 4,z 6 l,m 5,n 7;l,m 4,n ( 8) 7 ( 7) + ( 5) Lines are coplanar. Equation of the plane containing the lines is a( x+ ) + b( y+ ) + c( z+ 5) Where a+ 5b+ 7c a+ 4b+ 7c a b c or a b c MATHEMATICS 5

12 The required equation is ( x+) y 6+ z+ 5 x y+ z Centre of sphere (,, ) Radius of sphere OP 4 Equation of plane: x+ y+ z OO' O'P OP OO' O'P 7 Radius of the circle 7 Equation of line through (,, ) and prependicular to the plane is x y z k x k,y k +,z k+ Since it lies on the plane x+ y+ z 5 k+ 4k+ + 4k+ 4 5 k The coordinates of centre of circle are (,, 4) OPTIONAL -II (Mathematics for Commerce, Economics of Business) 5. Rs stock can be purchases for Rs Rs 96 stock can be purchases for Rs. Rs Tabular premium per thousand Rs..5 Rebate for mode of payment % of Rs..5 Rs..9 Rebate for large sum assured Rs.. per thousand sum assured Total Rs. (.5.9 ) Rs. 7.5 Extra premium towards accident benefit Rs.. per thousand Annual premium per thousand Rs MATHEMATICS

13 Annual premium to be paid 8.5 Rs. Rs.85. Wholesalers cost price Rs. ( 5 ) Rs.5 Rs.5 Rs.87.5 Rs.88 Total Rs. 688 Value Rs. 5 per chair Rs. 75 Total Rs. 48 VAT on the added amount Therefore, final selling price is Rs. ( ) Rs. 5 Total VAT paid to the Govt. is Rs. ( ) Rs Commodity Price in 996( P ) Price in ( P ).5 Rs.75 Rs.9.75 Rs.94 P P Wheat 6. Rice 5 5. Suger 6. Ghee Tea Total Price index for taking 996 as base year 8. p 48 x x p 4 ( ) x R x p.x 4x...(i) x AC( x) + 5 ( ) x C x x+...(ii) 5 MATHEMATICS 55

14 ( ) ( ) ( ) 7x P x i ii 4x 6 7x P' ( x) 4 For maximum profit P' ( x) x 6 7 P"x ( ) maximum 6 p 4 Maximum output 6 and price at this output 56 MATHEMATICS