FAISALABAD BOARD 2013
PAPER MATHEMATICS
PART-II
Time: 30 Min.
(Objective Part)
Marks: 20
Note: you have four choice 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 circles. Cutting or filling two or more circles will result in zero mark in that question.

• ƒ eax  [ aʄ (x) + ʄ (x)] dx

(a) eax ʄ (x)+ c
(b)  eax f (x)+ c
(c)  f (x)+ c
(d)   eax + c

• If a < c < b,    (x)dx =

(a)  (x) dx
(b)  (x) dx
(c)  (x) dx  +  (x) dx
(d)  (x) dx  -  (x) dx

• The solution of  = -y is:

(a) y = cx              
(b) y = ce-x 
(c) y = e-x             
(d) y =cex

• In a plane two mutually perpendicular lines are called:

(a) Coordinate axes                          
(b) Radii
(c) Medians                                      
(d) Altitudes

• Slope of a line ax + by + c = 0 is:

(a)            
(b)                      
(c) -        
(d)

• An expression involving any one of the symbols <, ≤, > ≥  is called:

(a) An equation                            
(b) Non-inequality
(c) Identity                                   
(d) Inequality

• If a plane passes through the vertex of the cone being perpendicular to the axis of the cone , then the intersection is called:

(a) Parabola                
(b) Hyperbola             
(c) Unit circle             
(d) Point circle

• The length of the diameter of the circle x² + y² = a² is:

(a) a                     
(b) 2a                   
(c) 1                     
(d) 2

• The unit vector of a vector v is:

(a)        
(b)           
(c)                   
(d)

• The non-zero vectors a and 5 are parallel if  x  =

(a) 1                       
(b) -1                       
(c) 0                           
(d) ab

• A function A:X → Y difined by A(X) = aⱯ xε X and aεY and fixed is called:

(a) Linear function                             
(b) Identity function
(c) Constant function                         
(d) Non-linear function

• lim x → + ∞  =

(a) 0                     
(b) 1                     
(c)                         
(d) ∞

• The derivalive of   at x = a is

(a)                 
(b)                   
(c)                   
(d)

• If y= In (ʄ(x)), then  =:

(a)              
(b)                  
(c) ƒ(x)                 
(d)  

• If y = cos , then  =

(a) -sin                        
(b)                 
(c)               
(d)

• For a stationary point for a function ʄ, we have ʄ ’(x) =

(a) 0                     
(b) +ve            
(c) -ve                  
(d) ∞

•  [In(Inx)] =

(a)           
(b)          
(c)                 
(d)

• If ’(x) = ʄ (x), then  (x) is _______ called of ʄ (x)

(a) Derivative                         
(b) Integral
(c) Differential coefficient     
(d) Area

•  =

(a) ln |1 + x²| + c              
(b) (1 + x²)² + c          
(c) |Tan^-1 x | + c                
(d) Tan^-1  x + c

• ƒsec² nx dx =

(a)  sec 3nx + c                     
(b) n tan nx +c
(c) tan nx + c                          
(d)  tan nx + c

Time: 3:10 Hours
(Subjective Part)
Marks: 83

Section-II

2. Attempt any EIGHT short questions. (8 x2 = 16)

• Find the domain and range of f(x) =
• Find the limit of  .
• If y =  +1 then find   .
• If x = t²+ 1, y = t², find   ,  .
• Find   cot-1  
• Find  if y = tanh(x)²
• If y = e^-2x sin 2x find  
• If y = In tanh x find  
• If y = x² e^-x then find y1.
• Write maclaurin's series expansion.
• Find the critical points for f(x) = 3x2 - 4x + 5.
• What do you mean by power series.

3. Attempt any EIGHT short questions. (8 x2 16)

• Find  and dy if y = x² - 1 when x changes from 3 to 3.02.
• Find ʄ   (x > 0).
• Find ʄ  dx
• Find ʄ Tan-1 xdx
  
• Find ʄ .
• Find ʄ sec4 xdx
• Find  (x3 + 3x2 ) dx.
•  dx.
• Find the area between x-axis and curve y = x² +1 from x = 1 to x = 2.
• Solve the differential equation  =  .
• Define a convex region.
• Graph the solution of linear inequality 2x + y  6.

4. Attempt any SIX short questions. (6 x2 =12)

• Find h such that points A (h, l), B(2,7) and C(-6,-7) are vertices of a right triangle with right angle at vertex A.
• The xy coordinate axes are rotated through angle  = 30° and new axes are OX & OY. Find x, y coordinates of P with P(x, y) = (-5, 3).
• Find equation of straight line if its slope is 2 and y intercept is 5.
• Find the whether point (5,8) lies above or below the line 2x-3y + 6 = 0.
• Define homogenous equation of degree n.
• Find equation of circle with center at (5, -2) and radius 4.
• Find focus and directrix of y2 = 8x.
• Find foci and vertices of ellips 25x² + 9y² = 225.
• Find equation of hyperbola having center (0, 0), focus ( 6 , 0) , vertex ( 4 ,0).
• Give two properties of vectors.
• If  = 3  -2 + 2 ,  = 5 -  + 3 find ||
• Find cosine of angle between vectors u = 2 –  +  and  = +
• Calculate the projection of  along  when a =  – , =  .

Attempt any THREE questions. (8 x 3 = 14)

5. (a) If f( x) =    Discuss continuity at x = 2 and x = -2.
(b) Differentiate ab -initio with respect to x, y = sin .

6. (a) Show that  ƒ  = In (x +  + c.
(b) Find the equation of the line through (5,-8) and perpendicular to the join of A(-15,-8) , B (10,7).

7. (a) Solve the differential equation  =  ,  where x > 0, y > 0.
(b) Maximize f(x, y) = 2x +5y subject to constraints 2y – x  8, x -y  4, x  0, y  0.

8. (a) Find equation of a passing through the points A (-7, 7), B (5, -1), C (10,0)
(b) If  = 3 -  - 4 ,  = 2- 4 - 3 and  =  + 2 –  find a unit vector parallel to 3 - 2 +4 .

9. (a) Find an equation of the parabola with focus F(1, 2) and vertex (3, 2).
(b) Find the volume of the tetrahedron with vertices (0, 1, 2), (1, 2, 1) and (5, 5, 6)