FAISALABAD BOARD 2016
PAPER MATHEMATICS
PART-II
Time: 30 min
(Objective)
Marks: 20
Note:Four Answers are given against each column A,B,C&D. Select the write answer   and only separate answer sheet, fill the circle A,B,C or with pen or marker in front of that question number.

Question#1

•   2  ×  . =:
  
  (a) 2
  (b) 0
  (c) I
  (d) -2
  
  
• For any two vectors   and  projection  of  on b is:
  
  (a) 
  (b) 
  (c) 
  (d)  .
  
  
• Focus of the parabola y2  = -4ax is:
  
  (a) (a, 0)
  (b) (-a,0)
  (c) (0,a ) 
  (d) (0,-a)
  
  
• Center of the circle 5x2 + 5y2 + 14x + 12y -10 = 0 is at:
  
  (a) (-7,-6)
  (b) 
  (c) (7,6)
  (d)  
  
  
• Point (1, 2) lies in the solution region of the inequality:
  
  (a) 2x + y > 5                         
  (b) x + 3y > 5
  (c) 2x + y < 3
  (d) 2x + y > 6
  
  
• The distance of the line 12x + 5y = 7 from origin is:
  
  (a) 
  (b)
  (c)
  (d) 13
  
  
• If a = 0 then the line ax + by + c = 0 is:
  
  (a) Parallel to x-axis  
  (b) Parallel to y-axis
  (c) Perpendicular to x-axis
  (d) Passes through origin
  
  
• Distance of (-3, 7) from x-axis is:
  
  (a) -3 
  (b) 3 
  (c) 7
  (d) 10
  
  
•  sec²  x dx =:
  
  (a) I  
  (b)
  (c)
  (d) 0
  
  
•  dx =:
  
  (a) tan x
  (b) cot x 
  (c) ℓn cot x
  (d) ℓn tan x
  
  
• ʃ cos sec²  2xdx =:
  
  (a) cos 2x 
  (b)  cot 2x
  (c) 2 tan 2x
  (d)   cot 2x
  
  
• ʃ  (xn) dx=:
  
  (a)  
  (b) Xn 
  (c) nxn-1
  (d) xn-1
  
  
• ʃ tan x dx = :
  
  (a) ℓn sec x
  (b) sec2 x
  (c) ℓn cos x
  (d) in sin x
  
  
•  (=:
  
  (a)
  (b)  
  (c) 2x
  (d) 2x
  
  
• If ʃ’ (c)= 0 and ʃ n(c) < 0 then f (x) will give at x = c:
  
  (a) Maximum value
  (b) Minimum value
  (c) Neither maximum nor minimum value
  (d) Stationary value
  
  
•     = :
  
  (a) –cos sec x cot x
  (b) co sec x cot x
  (c) cos sec² x
  (d) –cos sec² x
  
  
• If ʃ (x) = 3 -  then ʃ (1) = :
  
  (a)  
  (b)
  (c)
  (d)
  
  
• The function f (x) = cos x + sin x is:
  
  (a) Even
  (b) Odd
  (c) Neither even nor odd
  (d) Both even and odd
  
  
• If f (x) = x sec x then f (0):
  
  (a) -1
  (b) 1   
  (c) 0 
  (d) ∞

Time: 2:30 Hours
(Subjective Part)
Marks: 80
SECTION-I

2. Attempt any Eight Parts. 16

• If  f (x) = sin x + cos x . Check whether f  is even or odd, neither f  is even nor odd.
  
• 
•  (x) =  , check whether f is continuous or not continuous at x = 1
• If y =  , find  .
• Differentiate x2  sec4x  w.r.t x.
• Differentiate ℓn (e2 + e-3)   w.r.t. x.
• Differentiate ℓn  w.r.t x.
• Differentiate  y = sinh-1 (x3)  w.r.t x.
  
• Differentiate (ℓnx)2    w.r.t x.
• If x2 +y2 = a2   ,find
  
• Apply Maclaurin series to prove ex = 1+ x +  + ……         
• Differentiate (x+4  w.r.t x.

3. Attempt any Eight Parts. 16

• Using the differential find  in the equation Ay + x = 4.
• Evaluate ʃ   dƟ
  
• Evaluate ʃ   .
  
• Evaluate ʃ  sin2  xdx .
  
• Evaluate ʃ tan2  xdx .
  
• Evaluate ʃ   dx.  
•  Evaluate
  
• Evaluate
                     
  
• Find the area between the x-axis and the curve y = sin2 x, from X = 0 to x =  .
  
• Solve the differential equation   =  .
  
• What is an objective function?
  
• Graph the solution set of the linear inequality  5x - 4y ≤20 by shading.

4. Attempt any Nine Parts. 16

• The point (-5, 3) is the center of circle and point  (7, -2)  lies on it. Find radius of    circle.
  
• Let P (x, y) = (-6, 9) and axis are translated through  O'(-3,2) .Find translated coordinates of P.
• Find the equation of line bisecting first and third quadrant.
  
• Define centroid of triangle.
  
• Find the measure of angle between lines represented by x2 -xy- 6 y2 = 0.
  
• Find the equation of circle with center at (5, -2) and radius 4.
  
• Check the position of point   (5, 6) w.r.t the circle x2 + y2 = 81 .
  
• Find the vertex and focus of parabola  x2 = -16y .
  
• Find the foci and vertices of ellipse x2 +4y2 = 16.
• Find the vector from point A to origin when  = 4 - 2  -  and B is the point (-2, 5).
  
• Find the projection of    along    when  = 3 + - and  = -2 -  +
  
• Find   ×   when  = [2,1,-1] and  = [1,-1,1]
  
• Find the value of 2  × 2 •

SECTION-II

Attempt any THREE questions. Each questions carries 10 'Marks.
Question#5
(a) Prove that
(b) Show that y =  has maximum value at x = e.

Question#6
(a) Evaluate  ʃ
(b) Find the interior angles of the triangle whose vertices are
A(6,1)
 B(2,7)
 C(-6,-7)

Question#7
(a) Evaluate 
(b) Maximize the function f(x,y)= 20x+15y subject to constraints x+ y ≤ 100,
3x+ 2y ≥ 240, x≥ 0  ,y ≥ 0

Question#8
(a) Find equation of tangents to circle x2 +y2 = 2 and perpendicualr to line 3x+ 2y = 6
(b) Prove that in any triangle ABC, using vector method c2 = a2 + b2 -2tab cos C.

Question#9
(a) Write the equation of parabola whose focus (-3,1) , directrix   x = 3.
(b) Find the area of the triangle determined by the points
P (0 ,0 ,0)
Q(2,3,2)
R(-1,1,4).