TISB II

Model Examination2012

MATHEMATICS Max Score: 80

(Maximum time allowed is 2 hours and 45 minutes including cool-off time)

lz r rll. Consider .l =l t 3 I III

L3 -1 -r_l

(i) Is A non singular? Why?

(ii) Find l-'2x+y*z=3

(iiD HenceSolve x+3y*z=6 l+2+2:S3x-y-z=2

l-o' ab o"ltt2. rf n=lba _bl u"l

|," bc -r'l(i) Take a, b, c common from Ct,Cz,C3

(ii) Take a, b, c common from ftr,R2,R3

(iii) P.T A = 4a'b2c' 1+l+Z = 4

3. Considerthe binary operation * defined on the set ofintegers I by a* b = a +b +1

(i) Is * commutative? Justifu(ii) Find the identity element under *(iii) Find the inverse of 3 under {' 1-rl*2:4

4. (a) The value of Cos(Sec-t x + Co sec-' x) is . . .. .

(b) P.T Ztan-t]+ tan-'1 = tun-' 12717l*3:4

6. (a) lfy=Jtanr*ffi fina4dx

fkcosx.- 7Ti-.tl x F -f(x\=)r-2x " 2

5. If ' t^'- l^ ,r is continuous at x =' . find the value of klJ ,llx=- 2t2

(b) x = a(0 + Sin?) and y = a(l- Cos?) then

(i) rind4 uraLde de

(ii) , , *= tunT 3+3:6

OR

7. (a) ! = x' + (togxf then find @dx

(b) rf y =(tur-'r)' p.T (x' +tf n+zx(x, +tr, = 2 3+3:6

8. (i) Verifo Mean Value Theorem for f (x) = x2 +.r - 1 in [0 , 4]

(iD A window is in the shape of a rectangle surmounted by an equilateral triangle.

If the perimeter of the window is l2meters, find the dimensions of the window

that maximize the area of the window.

(iii) Find the points on the circle x2 + y' =25 at which the tangents are parallelto

the line 3x+5y =Q 2*3+2:7

9. Evaluate the following

(i) lSin3xCos4x dx

e sec2 xdx(rr) | _J 4 + 5tanx

rdx(ur) L 2+l+2=Sr x(x5 + l)

OR

10 (i) I#*1T1 t;:-

(ii) l--jlj^'* *I "l Sinx + 4 Cosx6

1 l. (i) Find the value of IV. zW-

(ii) P.T the area enclosed by the ellipse +.+ =l is 6rsq:units 2*3:5'4912. (i) Find the order and degree of the differential equationl y'+2y' =3y,

(ii) Solve (x' +y'h* =2xydy,y(l) = g 2+3=5

2+3:5

13. (a) Considerthe differential equation y'+ytanx =2x + x' tan x

(i) Find the Integrating Factor of the above differential equation

(ii) Hence find the solution

(iii) Find the particular solution at the point (0 , 1)

r2

(b) ff!+=xSinx,xeRfindy 312=5dx'

14 (a) P.T the vectors d =3i - 4j - 4i ,i =zi -j+k and d = i -:j- Sk form the sides of

a right triangle

Lrrr,q ,r 11* ., .rf .i-1, rc-$,=S .r1,j ; *r ;.(b) ' ut'- rr - u r'

2+3:5

15. (a) lf d=3? +2i +Ztc,6=i+4-Zk find a unit vector perpendicular to both

d +6 andd -6(b) Find the area of the parallelogram whose adjacent sides are given

byd=?-j*3i,6=zi-i+t 3+2:5

16. A dietician has to develop a special diet using two foods P and Q. Each packet

(containing 30 gm) of food P contains 12 units of calcium, 4 units of iron, 6 units of

cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Qcontains3 units of calcium,20 units of iron,4units of cholesterol and 3 units of vitamin

A. The diet requires at least 240 units of calcium, at least 460 units of iron, and at most

300 units of cholesterol. How many packets of each food should be used to minimize the

amount of vitamin A in the diet? What is the minimum amount of vitamin A?

1x6:617. Consider the lines t =(t_t\f +Q -2)j +Q*X)i ana

7 = (s + r)i+(zs -1, -(zs+t[ ilren

(i) Find the vectors 6, and 6', which are parallelto the above lines in order

(ii) Evaluate 6,x6,

(iii) Find the shortest distance between 1+2+2:5

18. (a) Find the equation of the plane passing through the intersection of the planes

l^ ^ ^\ I ^ ^ "\i '1,'* l+k)=6 and 7.pi +3j+ak)=-5 at(l.l,l)

(b) Find the distance of the point (3,4,5) from the plane 7 (2i -Sj +:f)= f :3*2=5