SOLUTION & ANSWER FOR KCET-2009 VERSION A-2

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1 SOLUTION & ANSWER FOR KCET-9 VERSION A- [MATHEMATICS]. cos ec( a) cos ecd si a [ si( a) cos ec] + C Sol. : si[ ( a) sicos(-a) cos si( a) cos ec( a) cos ecd d si a si( a) [ cot( a) cot ] d si a si( a) + C si a si [ si( a) cos ec] + C si a. If f() tdt, the.. ( ) + Sol. : f() tdt tdt + tdt + tdt tdt + ( + ). d +. Sol.: b a f()d b a f() + f(a + b ) Here a, b so aswer. The area bouded betwee the parabola.. [ ] Area d + ( ) sq. uits. 5. The differetial equatio of the famil of. d + d The famil is + λ + λ + ( + ) d + d 6. A populatio grows at the rate of % of the ears If p is the populatio at a time, dp p dt dp dt p t p c + p t, p where P iitial populatio p give p t ears 7. O the set of all atural umbers N, which a b a + b obviousl, ol a b a + b results i closure propert. 8. If f()d 5, the the value of. Questio is icomplete d 9 sq. uits.

2 9. If a + b, where a, b, ad are. (, ) It is ot possible that (, ).. The digit i the uit place of the umber ! eds i zero. Last digit of 7886 is same as last digit of sice last digit repeats i steps of.. If b, the i A. P. + a + c Observe that R + R R for the first two colums. sice determiat is zero, same must be true for colum. a + c b. a, b, c are i A.P.. The value of.. If A z z z determiat z z z z 8. A 9. adja A 8.. If A ad B are square matrices of. z 8. r r r r a.b a b cosθ θ 8. r r r r 6. If a + b + c O, the. ( c a) r r ad 6 ( r b c r ) r r r r a + b + c O a (a + b + a b c a b ( a + b + a b + b c c ( a + b + c) c a b c addig, r r r r r r a b c a + b c a b + b c + c a c a Also, we ca verif that r r 6 b c is also true. ( ) 7. If the volume of the parallelepiped.. 8. r [ b + c, c + a, a + b] [ a,b,c ] I the group G {,,,.}... ( 6 ) 9. Which oe of the followig.. Fourth roots of uit form a additive abelia group. Obviousl, fourth roots of uit form a abelia group uder multiplicatio ad ot uder additio.. The umber of sub groups... The egatio of.. Z will have ol two sub-groups sice it is a group of prime order. B (ABA ) (ABA ) r 5. If a. b r a b, the the.. ~p (q r). questio is prited wrogl. if it is p (q ~ r), the the aswer is ~p (q r).

3 . If, the ---- (... ) We have si is a egative acute agle si cos 9. If + si + si +.. up to If ` is a positive iteger, the + is ---- ( + ) ( + ) [( )(+) + ] ( ) ( + ) + M ().. O the set of itegers Z, defie f : Z Z as ---- surjective but ot ijective Obviousl, f is surjective but ot ijective. 5. If α ad β are the roots of + +, ---- π π, + si si si π π,. + i. The comple umber i α 6 + β 6 ω 6 + (ω ) 6 ω + ω. secod quadrat ( + i)( + i) + i 6. The total umber of terms i the epasio of ( + ) + ( ) There are terms i each epasio. But eve ordered terms will cacel. After simplificatio, 5 terms will remai. 7. cot (. ) + cot (. ) π th term cot ta ta ( + ) ( ) ta + + ( )( ) ta ( + ) ta ( ) So, sum to terms ta ( + ) ta sum to π. 8. If ` takes egative permissible value, ---- cos. If P is the poit i the Argad diagram correspodig to the comple umber i or i π π P is + i cos + i si 6 6 π π π π Q is cos + + i si π π si + i cos i + i π π π π Q is cos + i si 6 6 i.. The smallest positive itegral value of ` such that π π cos i 6 6 i π e π i π 8 cos e 6 6 π π π cos 8 8.

4 . Which oe of the followig is possible --- cosθ. taθ 5 < taθ <. If oe side of a triagle is double the other ad the agles opposite right agled a a siθ si( θ + 6) siθ si(θ + 6) si θ taθ θ θ 5. (si cos ) + 6 (si + cos ) ( si) + 6( + si) + ( si ). 6. A cow is tied to a post b a rope. The cow moves a the If + si θ si θ si θ 5 metres s rθ r 7 π r si θ si θ si θ siθ cos C C + C θ si θ si θ siθ siθ (siθ ) siθ 8. The locus of the mid poits of the chords of the circle Mid poit of the chord joiig (, ), (, ) subtedig 9 at the origi. Equatio of the locus is The legth of the chord joiig the poits ( cosθ, siθ) [ ( cos( θ + 6) + si( θ + 6) si θ) ] cos 6.. The umber of commo tagets to the circles The circles touches eterall. Hece there will be commo tagets.. The co-ordiates of the cetre of the smallest circle -----, Back substitutio. The legth of the diameter of the circle which cuts g f c g 5f c g + f c 7 g, f, c Diameter For the parabola, the poit P whose focal (6, 8) or (6, 8) Focus (, ) The ol poits distat 7 from (, ) are (6, 8) ad (6, 8).. The agle betwee the tagets draw to the parabola from the ; a + is the directio.

5 (, ) lies o the directio the tagets are the agle betwee the tagets 9 5. The umber of values of `c such that the lie ---- m + c c a m + b (6) + 65 c ± 65 There are two values for c. 6. If the circle + a itersects the hperbola c a c c + + a a + c sum of roots The foot of the perpedicular from the poit (, ) (, ) The foot of the from (, ) upo + is (h, k) ad it is give b h k + + h ; k h, k (, ) is the poit required. 8. The vertices of a triagle are (6, ), (, 6) ad 96, 6).. The give is at right agled oe. Circumcetre is the mid poit of the hpoteuse s (, ) a (, ) sa + 9. The agle betwee the pair of lies ---- π, 6 6, 6 6, coeff + coeff Lies are The fuctio + a f() a + b Q as 5 ( ) ( b) f() is cotiuous at f() f() a f() - + a a ( b) a + b ( ) ( b) ( + a) ( b) ( ) a b ( ) ( )( ) 5. If f () ( ) f() ( + ) f () ( + ) f () ( ) ( + ) f () ( ) 5. if f() ( e ), the --- e f() ( ) ( ) f () f (e). e ( ) 6 ( ). e b ----

6 5. If si cos, the si { cos( + ) } 55. If f() si cos d si cos cos d + si.( si). si { cos cos si si si { cos( + ) } g ( ) + g( ) g' f () + [ h( ) + h( ) ] ---- f() [ g( ) + g( ) ] + [ h( ) + h( ) ] f () [ ( ) g' ( ) ] + [ h' ( ) h' ( ) ] 56. The taget to a give curve f() is ---- d d Coceptual Taget is parallel to ais d d Taget to ais d d 57. The miimum value of 7 cos 8 si --- Let 7 cos si. 8 cos 7 + si Miimum of ( ) ( ) ( ) 5 + ( ) Miimum 58. A stoe is throw verticall upwards from the top of a tower 6 metres ---- m v ds dt 8 t t ( ) S t 6 m (height attaied from the tower) Height attaied from the groud m 59. The legth of the subtaget at `t o the curve -- a sit a ( cos t) a (t + sit) d a si t dt a + cos t t ta ( ) sub taget ' ta 6. e + d ta e. + c 6 m ( cos t) a t ta Put taθ θ Ι + θ θ θ ta e sec d sec θ ( θ + θ) θ θ e sec ta d e θ taθ + c ta e. + c h v GROUND

MATHEMATICS P1 COMMON TEST JUNE 2014 NATIONAL SENIOR CERTIFICATE GRADE 12

Mathematics/P1 1 Jue 014 Commo Test MATHEMATICS P1 COMMON TEST JUNE 014 NATIONAL SENIOR CERTIFICATE GRADE 1 Marks: 15 Time: ½ hours N.B: This questio paper cosists of 7 pages ad 1 iformatio sheet. Please