RAWALPINDI BOARD 2016
PAPER MATHEMATICS
PART -2
Time:30 Min.
(Objective Part)
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 D with pen or maker in front of that question number.

(a)
(b) –
(c) cosec ²x
(d) – cosec x cot x

• d/dx(ℓne^x) =

(a) e^x 
(b) 1
(c) x
(d) 1/x

•  =

(a) π 
(b) π/6
(c) π/4
(d) π/2

•  = 4, then k=

(a) 5                             
(b) 4                      
(c) 2                      
(d) 0

• If g(x)= (x≠0) then gog (x) is equal to:

(a) 1                             
(b) x²                    
(c) x⁴                     
(d)

• The function  is not continuous at:

(a) x = -3                      
(b) x= -                              
(c) x = 0                   
(d) x = 1

•   =

(a) 2cosx²                  
(b) cos x²            
(c) 2x cos² x        
(d) 2 cos x²

•   [tan-1x-cot-1x] =

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

• ƒ (x)=tan-1 x then ƒ′(cot x) =

(a) cos² x                    
(b) sin² x                              
(c) cosec² x        
(d) cot² x

• Mid-point of line segment joining focil of ellips is called its =

(a) Centre                 
(b) Vertex          
(c) Directix         
(d) Major-axis

• A circle touches the two axis at (a, 0) and (0,a) then centre of circle is:

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

• What is the value of [ a b b ] =

(a) 1                             
(b) -1                    
(c) 0                      
(d) 2

• Which of triples can be direction angles of a single vector =:

(a) 90°, 90°, 45°                       
(b) 0°, 0°, 45°                     
(c) 45°, 45°, 90°
(d) 30°, 30°, 30°

(a) sin 2x                   
(b) 2sin 2x         
(c)             
(d) 2 sin x

(a) ℓogx                     
(b)  ℓog(ℓogx) 
(c)            
(d)  

(a) ℓn sin π/4           
(b) 1                      
(c) sec² π/4              
(d) x tan π/4 

(a) sin p sec²x          
(b) sin p tan x    
(c) cos p sec²x                   
(d) sec² x

• Centroid is a poit which divies each median in ratio =

(a) 2:1                         
(b) 1:2                  
(c) 1:1                   
(d) 3:2

• Slope of line is 1 ( one ) and angle made by line with x-axis =

(a) 45°                         
(b) 30°                  
(c) 60°                  
(d) 75°

• The solution set of x<4=

(a) 0 < x < 4               
(b) 10 < x < 15   
(c) -∞ < x < 4     
(d) 4 < x < ∞ 

Time:30 Min.
(Subjective Part)
Marks: 80
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 D with pen or maker in front of that question number.

Section-1

2. Attempt any Eight Parts.

• prove that cosh2 x – sinh2 x = 1.
• Evaluate: .
• Evaluate:  .
• Find  by 1st principle .
• Differentiate w.r.t x .
• Differentiate x2 -  w.r.t x4.
• Differentiate x sin-1 .
• Find  if y = .
• Find y2 if y =  + .
• Expand ax by maclaurin’s series.
• Define critical point.
• Find the interval for which function is increasing and decreasing ƒ(x) = 4-x2 for x

2.  Attempt any Eight Parts.

• Evaluate :
• Evaluate:
• Evaluate:  x > 0.
• Find:
• Evaluate: 
• Define definite integral. Give one example.
• Evaluate:
• Solve the differential equation
• Find the area above the x-axis and the curve y=5=x² from x= -1 to x=2.
• Find δy and dy of the function defined as ƒ(x)= x² when x=2 and dx =0.01.
• Define vertex of the solution region.
• Graph the solution set of the inequality 3x + 7y ≥21.

4. Attempt any nine Parts.

• Find h such that A (-1 , h), B (3,2) and C (7,3) are collinear.
• If (x,y) co-ordinates of a point are (-2,6). Find (x,y) transformed co-ordinates if new origin is O’(-3,2).
• Three points A(7, -1), B(-2,2) and C (1,4) are consecutive vertices of a parallelogram. Find the fourth vertex.
• Find the point intersection of lines x+4y-12=0 and x-3y+3=0.
• Find acute angle between the lines represented by x2 –xy-6y2 = 0.
• Show that the line 3x-2y=0 is tangent to the circle x2+y2+6x-4y=0.
• Find the equation of tangent drawn from (0,5) to circle x2+y2=16.
• Find focus and vertex of parabola y=6x2-1.
• Find an equation of sllipse whose vertices are (0,±5) and eccentricity 3/5.
• Find x so that =3.
• Find unit vector perpendicular to = 2i-6j-3k,
• Constant force moves an object from (3,1,-2) to (2,4,6). Find the work done.
• Find a vector of magnitude >parallel to 2i+2k.

Section-2

Attempt any three questions, each question carries 10 marks.

5. (a) Evaluate .
(b) If y = a cos ( 
6.  (a) Evaluate:
(b) Find h such that the points A(,-1), B(0,2), C(h,-2) are the vertices of a right triangle with right angle at the vertex A.

7. (a) Find the area between the X-axis and the curve y= when  >0
(b) Find the maximum value of ƒ(x) = 4x+6y under the constraints.2x-3y ≤ 6, 2x + y ≤2, 2x+3y ≤12, x ≥y ≥0.

8. (a) Find the equation of the tangents to the circle x2+y2=2 parallel to the         line x-2y+1=0.
(b) Find the number Z, so that the triangle with vertices A(1,-1,0), B(-2,2,1)   and C(0,2,Z) is a right angle triangle with right angle at C.

9. (a) Find an equation of the parabola whose focus is F(-3,4) and directrix is    3x-4y+5=0.
(b) Find the value of α so that  and 2 are complainer.