SAHIWAL BOARD 2013
PAPER MATHEMATICS
PART-II
TIME ALLOWED: 30 Minute
(Objective Part)
MAXIMUM MARKS: 20
Note: You have four choices for each objective type question as A, B, C and D .The choice which you think is correct, fill that circle in front of that question number. Use marker Or pen to fill the circle. Cutting or filling two or more circles will result in zero mark in that question.

•   =

a. 
b. 
c. 
d. 

a.
b.
c.
d.

•   =
  

a.
b.
c.
d.

a.1
b.-1
c.0
d.x

a.cosec²ax
b. a cosec²ax
c. -a cosec²ax
d.

• A function defined by f(x)=x³ is.
  

a.even function
b.identity function
c.odd function
d.linear function

• x= at²,y=2at are parametric equations of .
  

a.circle
b.parabola
c.hyperbola
d.ellipse

• + = where λ0

a.  +c
b.  +c
c. +c
d. +c

• The unit vector of a vector v is:
  

a.
b.
c.
d.

• The non-zero vectors  and  are parallel if   x  =
  

a.1
b.-1
c.0
d.ab

• A function A:X àY defined by A(X)=aX and aY and fixed is called.
  

a.linear function
b.identify function
c.constant function
d.non linear function

• lim xà+ x =
  

a.0
b.1
c.1/2
d.

• dx=:

a. +c
b. ++ c
c. 
d. -+ c

• The ratio in which y-axis divides the line joining (2,-3)and (-5,6)is:
  

a. 2:3
b. 2:5
c. 1:2
d. 3:5

• Let f(x,y)=0 and f(kx,ky)=kn f (x,y),then is homogenous equation of degree:
  

a.1
b.2
c.0
d.n

• The non negative inequalities are called:
  

a.parameters
b.constants
c.decision variables
d.vertices

• Centre of circle 4x2+4y2-8x+16y-25=0:
  

a.
b.
c.(1,-2)
d.(1,2)

• Fosi of ellipse  +  =1 are:
  

a.()
b.(0,)
c.()
d.(0,)

• Commutative law holds in:
  

a.vector product
b.cross product in three vectors
c.inner product
d.none of these

a. scalar product
b. vector product
c. cross product
d. meaningless

(Subjective Part)
TIME ALLOWED: 3:10 Hours
MAXIMUM MARKS: 83
SECTION II

• Find the domain and range of f(x) =-4.
• Prove that sech2x=1-tanh2x.
• Define derivative of afunction
• Find   if y = x cos y.
• Find   for y =sin x2
• Find f(x)for f(x)=ex(1+inx).
• If 2x5-3x4+4x3+x-2,find y2.
• If x=acos,y=asin,find  , 
• Find the critical points for f(x) = x2-x-2.
• Define the critical point.
• If y =  ,find  .
• If y = x2 in  , find  .

3. Attempt any eight short questions. (8x2=16)

• Find 8y and dy if y=x2-1 when x changes from 3to 3.02.
• Find  (x>0).
• Find  
• Find  
• Find  dx (a>b).
• Find
• Find +).
•  dx.
• Find the area between x-axis and curve y =x2 +1from x =1 to x=2.
• Solve the differential equation  =  .
• Define a convex region.
• Graph the solution of linear in equality 2x +y 6.

 4. Attempt any SIX short questions. (6 x 2 = 12)

• Find the distance between A(-8,3)and B(2,-1).
• Define latusrectum of the parabola.
• The two points P and O are given in XY coordinate system.find XY coordinate of P referred to the coordinate axex O’X  and O’Y where P(-2,6)and O’ (-3,2)
• Find the slope and inclination of the line joining the points (-2,4)and (5,11).
• Find equations of lines represented by 10x2-23xy-5y2=0.
• Find an equation of aline through (-4, -6) and prependicularto a line having slope -3/2.
• Find an equantion of the circle with centre at (5, -2) and radius 4.
• Find centre and foci of the ellipse x2+4y2 = 16.
• Write down equation of the tangent to the circle x2+y2 =25 at (4,3).
• If  =3 + - and  =5 -+ then find .
• Find the cosine of the angle  between  and  where  =3 + - and  =2 - +.
• If  =  +  and  = -  then compute the cross product  x
• Find a vector from the point A to the origin.where  = 4 - and B is the point (-2,5).

Attempt any THREE questions.  (8 x 3 = 24)

5 .(a)= Discuss the continuity of f(x) at x =2 and x = -2
(b) If x = sinθ and y = sin mθ ,show that (1-x²)y₂-xy₁+m²y =0.

6.(a) Evaluate
(b) Find the conditions that the lines y=m1x+c1;y=m2x+c2;y=m3x+c3 are concurrent.

7. (a) Evaluate the definite integral 2 dx.
(b) Graph the feasible region subject to the following constraints:2x-3y  6; 2x+y  2;x  0; y  0.

8. (a) Find equations of the tangents to the circle x2+y2 =2.Perpendicular to the line 3x + 2y = 6.
(b) Prove that the line segment joining the mid points of two sides of a triangle is parallel to the third side an half as long.

9.(a) Find the focus, vertex and directrix of the parabola y=6x2-1.
(b) Find the volume of tetrahedron with vertices (0,1,2),(3,2,1),(1,2,1) and (5,5,6).