1 - a 1 - b 1 - c a) 1 b) 2 c) -1 d) The projection of OP on a unit vector OQ equals thrice the area of parallelogram OPRQ.

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1 Regter Number MODEL EXAMINATION PART III - MATHEMATICS [ENGLISH VERSION] Time : Hrs. Ma. Marks : 00 SECTION - A 0 = 0 Note :- (i) All questions are ompulsory. (ii) Eah question arries one mark. (iii) Choose the most suitable answer from the given four alternatives. -. The rank of the matri ) ) -l. If the matri k - has an inverse then the values of k 5 k any real number k = - ) k - ) k. The system of equations a y z = 0; by z = 0 ; y z = 0 has a non-trival solution then = - a - b - ) - ) 0. If u = a ( b ) b ( a ) ( a b ), then u a unit vetor u = a b ) u = 0 ) u 0 5. The projetion of OP on a unit vetor OQ equals thrie the area of parallelogram OPRQ. Then POQ tan - / os - ( ) ) sin - ( ) ) sin - ( ) If a, b, are a right hane tria of mutually perpeniular vetors of magnitue a, b, then the value of [ a b ] a b 0 ) ab ) ab - 6 y z - y z 7. The point of intersetion of the lines = = an = = (0, 0, -) (, 0, 0) ) (0,, 0) ) (,, 0) 8. If (m - 5) i(n ) the omple onjugate of (m ) i(n - ) then (n, m) are (-, - 8) (-, 8) ) (, -8) ) (, 8) 9. If the amplitue of a omple number / then the number purely imaginary purely real ) 0 ) neither real nor imaginary www.

2 0. If P represents the variable omple number z an if z - = z then the lous of P the straight line = / the straight line y = / ) the straight line z = / ) the irle y - - = 0. The tane between the foi of the ellipse 9 5y = ) 8 ) The angle between the asymptotes to the hyperbola y - = tan - ( ) - tan - ( ) ) tan - ( ) ) tan - ( ). The iretri of the parabole y = = = - ) = - ) =. The veloity v of a partile moving along a straight line when at a tane from the origin given by a bv = where a an b are onstants. Then the aeleration b a ) b ) a 5. The equation of the normal to the urve θ = /t at the point (-, -/) θ = 7 t θ = 7t - 80 ) θ = 7t 80 ) θ = /t 6. For the urve = e t os t ; y = e t sin t the tangent line parallel to the -a when t equal to - ) 0 ) 7. The urve y ( - ) = ( ) has an asymptote parallel to -a an asymptote parallel to y-a ) aymptotes parallel to both aes ) no asymptotes 8. The perentage error in the th root of the number 8 approimately... times the perentage error in 8. 8 ) ) 8 9. The value of ( - ) 0 ) ) The area of the region boune by the graph of y = sin an y = os between = 0 an = / - ) - ). The urve surfae area of a sphere of raius 5, interepte between two parallel planes of tane an from the entre 0 0 ) 0 ) 0. If os an integrating fator of the ifferential equation / Py = Q then P = -ot ot ) tan ) -tan. The omplementary funtion of (D )y = e (A B)e A os B sin ) (A B)e ) (A B)e -. The ifferential equation satfie by all the straight lines in y plane y y = a onstant = 0 ) y = 0 ) y = 0 www.

3 5. The number of rows in the truth table of ~ [p (~q)] ) 6 ) 8 6. A monoi beomes a group if it also satfies the losure aiom assoiative aiom ) ientity aiom ) inverse aiom 7. The value of [] ([5] [6]) [0] [] ) [] ) [] 8. If f() = k, 0 < < a probability ensity funtion then the value of k { 0, elsewhere ) ) Variane of the ranom variable X. Its mean. then E(X ) ) 6 ) 8 0. If f() a p..f of a normal tribution with mean µ then f() 0.5 ) 0 ) 0.5. If X a rete ranom variable then P( = P( < - P( ) - P( < ) 0. "" not a binary operation on N Z ) C ) Q - {0} y. The orer an egree of the ifferential equation are = y,, ), / ),. The area boune by the urve = f(y) to the left of y-a between the lines y = an y = - ) y ) - y 5. lim 0 tan - ) 0 ) 6. The length of the L.R of y 9 - = / 8/ ) / ) 9/ 7. The number of values of (os θ i sin θ) p/q where p an q are non-zero integers prime to eah other, p q ) p q ) (p - q) 8. The vetor equation of a sphere whose entre origin an raius "a" r = a r - = a ) r = a ) r = a 9. The angle between two vetors a an b if a b = a b / / ) /6 ) / 0. If ρ(a) = ρ[ab] then the system onstent an has infinitely many solution onstant an has a unique solution ) onstent ) inonstent www.

4 SECTION - B 0 6 = 60 Note :- (i) Answer any ten questions. (ii) Question No.55 in ompulsory an hoose any nine questions from the remaining. (iii) Eah question arries si marks. -. If A = [ -], verify the result A (aj A) = (aj A) A = A I. Fin the rank of ( Fin the area of the parallelogram etermine by the sie i j k an -i - j k ( The volume of a parallelopipe whose eges are represente by -i λk, j - k, i j - 5k 56. Fin the value of λ.. Fin the oorinates of the entre an the raius of the sphere whose vetor equation r - r. (8i - 6j 0k) - 50 = 0 5. State an prove triangle inequality on omple numbers. 6. If n a positive integer, prove that ( i) n ( - i) n = n os n/6 7. A stanar retangular hyperbola has its verties at (5, 7) an (-, -). Fin its equation an asymptotes. 8. Obtain the Malaurin's Series for log e ( ) 9. Fin the area of the region boune by the ellipse y a b = 50. Fin the ifferential equation that will represent the family of all irles having entres on the -a an the raius unity. 5. Construt the truth table for (p q) (~r) 5. Show that p q ((~ p) q) ((~ q) p) 5. In a Poson tribution if P(X = ) = P(X = ) fin P(X = 5) [given e - = 0.050]. 5. The overall perentage of passes in a ertain eamination 80. If 6 aniates appear in the eamination what the probability that atleast 5 pass the eamination. 55. ( Fin two numbers whose sum 00 an whose prout a maimum. (OR) ( If w = y z an = os t; y = sin t; z = t. Fin w/t. SECTION - C 0 0 = 00 Note :- (i) Answer any ten questions. (ii) Question No.70 ompulsory an hoose any nine questions from the remaining. (iii) Eah question arries ten marks. 56. For what value of µ the equations y z = 0, y µz = 0, y z = 0 have a (i) trivial solution, (ii) non-trivial solution by using rank. 57. Prove that sin (A - B) = sin A os B - os A sin B. 58. Fin the vetor an artesian equations of the plane pasing through the ponts (,, -), (,, ) an (7, 0, 6) 59. P represents the variable omple number z. Fin the lous of P, if arg z - = z www. ( )

5 60. Assume that water suing from the en of a horizontal pipe, 7.5m above the groun, esribes a paraboli path. The verte of the paraboli path at the en of the pipe. At a position.5m below the line of the pipe, the flow of water has urve outwar m beyon the vertial line through the en of the pipe. How far beyon th vertial line will the water strike the groun? 6. The orbit of the planet merury aroun the sun in elliptial shape with sun at a fous. The semi-major a of length 6 million miles an the eentriity of the orbit Fin (i) how lose the merury gets to sun? (ii) the greatest possible tane between merury an sun. lim 6. Evaluate sin 0 6. Trae the urve y = 6. Fin the length of the urve ( ) / y ( a a ) / = 65. Fin the volume of the one obtaine by revolving the area of the triangle whose three verties are (0, 0), (, 0) an (, ) about -a. 66. Solve ( y) = 67. The rate at whih the population of a ity inreases at any time proportional to the population at that time. If there were,0,000 people in the ity in 960 an,60,000 in 990 what population may be 6 antiipate in 00. [log e ( ) = ; e =.5] 68. Show that the set G of all matries of the form [, where R - {0}, a group uner matri ] multipliation. 69. The air pressure in a ranomly selete tyre put on a ertain moel new ar normally tribute with mean value psi an stanar eviation 0. psi. (i) What the probability that the pressure for a ranomly selete tyre ( between 0.5 an.5 psi ( between 0 an psi (ii) What the probability that the pressure for a ranomly selete tyre eees 0.5 psi? 70. (i) If the urve y = an y = k are orthogonal then prove that 8k =. (OR) (ii) Prove that the line 5 y = 9 touhes the hyperbola - 9y = 9 an fin its point of ontat. www. 5

E R K HIGHER SECONDARY SCHOOL ERUMIYAMPATTI PART III- MATHEMATICS (English Version) Time Allowed: 3 hours Maximum Marks: 200

www.paasalai.net E R K HIGHER SECONDARY SCHOOL ERUMIYAMPATTI PART III- MATHEMATICS (English Version) Time Allowe: hours Maimum Marks: SECTION- A NOTE: Choose the most suitable answer from the given four