(A) 2log( tan cot ) [ ], 2 MATHEMATICS. 1. Which of the following is correct?

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1 MATHEMATICS. Which of the following is coect? A L.P.P always has unique solution Evey L.P.P has an optimal solution A L.P.P admits two optimal solutions If a L.P.P admits two optimal solutions then it has infinitely many optimal solutions. Mean maks scoed by the students of a class is 5. The mean maks of the gils is 55 and the mean maks of the boys is 50. What is the pecentage of gils in the school? Let Q be the set of all ational numbes. Define an opeation X on Q { } by a * b = a + b + ab. Then identity element of * on X on Q { } is (C ) 0. Let Then [ ], x x x< f( x) = x, x lim f( x ) is equal to x, whee [ x] denotes the geatest intege function. 0 none of these 5. If I = cosx sinx dx, then I equals cosxsinx log( tan cot ) x x + C log sinx+ cosx+ sinx + C log sinx cosx+ sinxcosx + C ( ) log sin x+ + sinxcosx + C Mathematics (SET A) [ ] P.T.O.

2 6. The solution set of the equation 5 7 x 9 6 = 0 is {0} {6} { 6} {0, 9} 7. If A = and B is a squae matix of ode such that AB=I, then B is equal to Mathematics (SET A) [ ] Contd. 8. Satement : If the pependicula bisecto of the line segment joining P(, ) and Q(K, ) has y-intecept, then K 6 = 0 Statement : Locus of a point equidistant fom two given points is the pependicula bisecto of the line joining the given points Statement is tue, Statement is tue; Statement is a coect explanation fo Statement Statement is tue, Statement is tue; Statement is not a coect explanation fo Statement Statement is tue, Statement is false Statement is false, Statement is tue 9. cot (9) + cosec is equal to 6

3 0. Fist tem of a G.P of n tems is a and the last tem is l. Then the poduct of all tems is. n (a + l) ( ) n a+ l ( al ) ( ) n 8 lim x x 0 x x is equal to log x log log 8 a+ b. a b. If a and b ae two unit vectos, then the value of ( ) ( ) 0 none of these. If A and B ae two independent events, then P (A/B)is equal to P P ( C) P (A/B) P (A /B). x-axis is the intesection of the two planes xy and yz yz and zx xy and zx none of these al n is equal to a bc a 5. b ac b c ab c is equal to 0 abc Mathematics (SET A) [ ] P.T.O.

4 6. a x dx is equal to x a + x a x + sin + C a x a x a x sin + C a x a x a + log x+ a x + C x a x a x + sin + C a 7. The minimum value of P = 6x + 6y subject to the constaints x 0, y x, y 0 is If the vaiance of α, β, γ is 9, then the vaiance of 5α, 5 β and 5γ is Mathematics (SET A) [ ] Contd sin sin 5 is equal to The point of discontinuity of the function f defined by x+, if x< f (x) = 0, if x= is x, if x > 0 R {}. The equation of a cicle which touches the x-axis and whose cente is (,) is x + y + x + y = 6 x + y 6x 8y + 9 = 0 x + y + 8x + 0y + 5 = 0 x + y 9x 6y + 5 = 0 5 0,

5 . If A = {,,,, 5} and B = {,, 6, 7}, then numbe of elements of (A B) (B A) is equal to x 0 0. A = x x and A =, then x is equal to. The anti deivative F of f defined by f (x) = x 6, whee F (0) = is x 6x + x x 6x x 6x 5. The line though the points (, 6) and (, 8) is pependicula to the line though the points ( 8, ) and ( x, ), then the value of x is If A is a singula matix, then A.(adjA) is equal to a null matix a unit matix (C ) a scala matix none of these 7. Two cads ae dawn fom a pack of 5 cads. The pobability of being queens is 8. 6 Mathematics (SET A) [ 5 ] P.T.O. none of these d dx log tan x + is equal to sec x cosec x tan x cot x 9. dy The geneal solution of the diffeential equation dx = ex y is y = xc e y = e x + C e x y = C none of these 0. The aithmetic mean of values 0,, n is n n + n (n +)

6 . If A and B ae squae matices of ode such that A = and B =, then AB is equal to 8 8 x+, x< 0. Let f (x) =, x= 0. Then on [,] this function has x, 0< x a minimum a maximum a maximum and a minimum neithe maximum no minimum. The value of x fo which the angle between the vectos a =xiˆ ˆj kand ˆ b = xiˆ+ xj ˆ kˆis acute, and the angle between the vecto b and the axis of odinate is obtuse, ae fo all x > 0 fo all x < 0,, sin y + cosy sin y. The value of the expession + is + cosy sin y cosy 0 siny cosy 5. If f (x) = x, g(x) = tan x and h(x) = log x, then ho (gof) 0 Mathematics (SET A) [ 6 ] Contd. log is equal to 6. The value of θ and p, if the equation x cos θ + y sinθ = p is the nomal fom of the line x + y + = 0 ae θ =, 6 p = θ =, 6 p = θ = 7, 6 p = θ = 7, p= 6

7 7. Statement : f (x) = Statement : f (x) = x e x e is diffeentiable fo all x is continuous fo all x Statement is tue, Statement is tue; Statement is a coect explanation fo Statement Statement is tue, Statement is tue; Statement is not a coect explanation fo Statement Statement is tue, Statement is false Statement is false, Statement is tue 8. If n N then.5 n+ + n+ is divisible by If A and B ae two squae matices such that AB=A and BA=B, then A is equal to B A I 0 0. lim x 0 f(x), whee f (x) = x, x 0 x 0, x = 0 is 0 is is does not exist ( ) cos logx. If k = e 007, then value of I = dx is x 0 e k Mathematics (SET A) [ 7 ] P.T.O If one of the oots of the equation x = px + q is the ecipocal of the othe, then the coect elationship is pq = q = q = pq =

8 . Q + is set of all positive ational numbes and g is a binay opeation on Q + ab + defined by a gb = a, b Q. Then the invese of a Q + is equal to a a a. If A =, then (A I) (A I) is equal to A I O 5I Paagaph fo question numbes 5, 6, 7, 8, 9 and 50 Conside the point P(,, ) and the vecto b = iˆ ˆj + kˆ 5. Vecto equation of a line L passing though the point P and paallel to b is = iˆ+ ˆj kˆ + λ iˆ ˆj+ kˆ ( ) ( ) = ( iˆ ˆj+ kˆ) + λ( iˆ+ ˆj kˆ) = ( iˆ+ ˆj kˆ) + λ( iˆ ˆj+ kˆ) none of these 6. Catesian equation of the plane passing though the point P and pependicula to the vecto b is x y + z = 7 x + y z = 7 x y + z = 7 x + y z = 7 7. Catesian equation of a plane passing though the point with position vecto b and pependicula to the vecto OP, O being oigin is x y + z + 7 = 0 x y + z 7 = 0 x + y z + 7 = 0 x + y z 7 = 0 8. Sum of the lengths of the intecepts made by the plane on the coodinate axes is 9/ 9/7 5/7 Mathematics (SET A) [ 8 ] Contd.

9 9. The equation of a plane passing though point P, pependicula to the plane and paallel to the line L is x y + 6z = 0 x 6y z = 0 x y + z = x y + 5z = The angle between the plane and the line L is sin cos sin cos 7 9 Mathematics (SET A) [ 9 ] P.T.O Let S be the set of eal numbes and R be a elation on S defined by arb a + b =. Then R is equivalence elation eflexive but neithe symmetic no tansitive (C ) tansitive but neithe eflexive no symmetic symmetic but neithe eflexive no tansitive 5. If A is a -owed squae matix and A =, then adj (adj A) is equal to A 6 A (C ) 6 A A 5. If a line makes angle 90, 60 and 0 with the positive X,Y and Z-axes espectively, its diection cosines ae,, 0,, undefined,, none of these 5. Thee dice ae thown togethe. The pobability of getting a total of atleast 6 is

10 55. The locus of the points which ae equidistant fom ( a, 0) and the line x = a is y = ax y = ax x + y = a (x a) + (y + a) = sin x cos x dx is equal to 5 sin x sin x cos x sin x + C + C 5 5 cos x cos x + C none of these If the mean and vaiance of a binomial distibution ae and, then the value of P(X=0) is The numbe of pope subsets of the set {,, } is The line x y z = = is paallel to the plane 5 x + y z = 0 x + y + 5z = 7 x + y + z = x + y + z = If f () = and f () =, then xf lim () f ( x ) x x is equal to Mathematics (SET A) [ 0 ] Contd.

11 6. The value of sin 0 sin 0 sin 60 sin 80 is equal to (C ) 6 8 is equal to 6. sin ( ) - cos sin x + cos sin( cos x) 0 6. The eal value of α fo which the expession i siná +i siná is puely eal is (n + ), n Z (n + ), n Z n, n Z none of these 6. If y = cos x, then d y dx is equal to cos y sin y cosecy cot y cosec y cot y none of these 65. The value of a b b c c a is If the points (, ), B(k, ) and C(8, 8) ae collinea, then k is equal to 5 Mathematics (SET A) [ ] P.T.O.

12 67. The value of tan 5 + cot 5 is not defined 68. A die is olled twice and the sum of the numbes appeaing is obseved to be 7. The conditional pobability that the numbe has appeaed atleast once is The statement if x is divisible by 8, then it is divisible by 6 is false if x equals Mathematics (SET A) [ ] Contd A vecto a can be witten as ( ˆ) ˆ. (. ˆ) ˆ + + (. ) ai i a j j akˆ kˆ ( ˆ) ˆ ( ˆ) ˆ (. ˆ) ak i ai j a j kˆ 6 ( ˆ) ˆ ( ˆ) ˆ (. ˆ) a j i ak j ai kˆ ( aa. ) ( iˆ+ ˆj+ kˆ) 7. If A and B ae two sets of the univesal set U, then (A B) equals A B C A C B (C ) A B U A 7. The pobability of the safe aival of one ship out of 5 is. What is the pobability 5 of the safe aival of atleast ships out of 5? The foci of the ellipse 9x + y = 6 ae ( 5, 0) ( ± 5,0) ( 0, ± 5) (0, 5)

13 7. Aea between the cuve y = x (x ) and the x-axis fom x = 0 to x = 5 is 5 0 sq units sq units sq units none of these 75. Range of the function f (x) = x, when x 0 x 0, when x= 0 R {,0,} [,] R { 0} x+ 76. The numbe of integal solutions of > is x + 0 (C ) If (!)! = k(n!) then (n + k) equals Ode of the diffeential equation whose solution y = ae x + be x + ce x (whee a, b, c ae abitay constants) is α + β tan 79. If sin α= 5 sin β, then is equal to α β tan 80. If f (x) = x e loge (x ), then dom f is equal to (, ] [, ) (, ) (, ] (, ) Mathematics (SET A) [ ] P.T.O. is

14 8. The slope of nomal to the cuve y = x + sin x at x = 0 is 8. The locus of a point such that the diffeence of its distances fom (, 0) and (, 0) is always equal to is the cuve 5x y = 5 y 5x = 5 5x + y = 5 6y = x + i 8. If ω + =, then ( + ω+ ω ) is equal to ω 6 ω 8. Two cads ae dawn fom a well shuffled deck of 5 cads one afte the othe without eplacement. The pobability of fist cad being a spade and the second a black king is 0 0 Mathematics (SET A) [ ] Contd x+ y + x y afte 85. The total numbe of tems in the expansion of ( ) ( ) simplification is Let f (x) = sin x, g(x) = x and h(x) = log x. If u(x) = h (f (g(x))) then 87. If d u cos x cot x x cosec x x cot x cosec x ( + x) f() tdt = x, then f () is equal to x 0 / / /5 dx is

15 88. The numbe of ways in which a student can choose 5 couses out of 9 couses in which couses ae compulsoy is The maximum numbe of points of intesection of 8 staight lines is Thee ae two boxes. One box contains white and black balls. The othe box contains 7 yellow balls and black balls. If a box is selected at andom and fom it a ball is dawn, the pobability that the ball dawn is black is (C ) 0 9. The function f (x) = a x is inceasing on R if Mathematics (SET A) [ 5 ] P.T.O < a < a > a < a > 0 9. The integating facto of dy + y cot x = cos x is dx cos x tan x cot x sin x 9. The geatest coefficient in the expansion of ( + x) n +, n N is ( n)! n! ( n+! ) ( n! ) + 9. If fo the matix A = I, then A is equal to A A ( n+ ) n ( n+ )!!! ( n+ ) nn ( + )! A none of these!

16 95. The diffeential equation of all non-hoizontal lines in a plane is dy 0 dx = d y 0 d y dx = dx = 5 dx = 96. An unbiased die is tossed twice. The pobability of getting a, 5 o 6 on the fist toss and a,, o on the second toss is 97. The equation e x + x = 0 has one eal oot two eal oots thee eal oots fou eal oots 98. One of the two events must occu. If the chance of one is of the othe, then the odds in favou of the othe is : : : : 99. The aea in the fist quadant and bounded by the cuve x + y = and the lines x = 0 and x = is 00. Thee ae fou lette boxes in a post office. The numbe of ways in which a man can post 8 distinct lettes is 8 8 dy P Mathematics (SET A) [ 6 ] Contd.

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