Inter (Part-II) Multan Board 2015
Mathematics
Part II (Objective Type)
Time Allowed: 30 Minutes 
Max. 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 circles. Cutting or filling two or more circles will result in zero mark in that question.

Question #1
Circle the correct option i.e. A/B/C/D. Each part carries one mark.

• The solution of differential equation  = sec² x is :    

(A) y = cos x
(B) y = sec x
(C) y = cosec² x
(D) y = tan x

•  equal:

(A) 2
(B) 0
(C) 4
(D) -2

•  is equal to:

(A)
(B)
(C)
(D)

• The distance between (1, 2) and (2,1) is :

(A) 1                                                   
(B) 2        
(C)                                                 
(D) 0

• Intercept from of equation of line is :

(A)  +  = c                                       
(B)  +  = 1                  
(C)  +  = c                                       
(D)  -  = 0       

• The  association equation of inequality x + 2y < 6 is :

(A) x + 2y =6                                     
(B) x - 2y= 6            
(C) x + 2y = -6                                   
(D) x – 2y = -6

• The center of circle x2 + y2 + 12x – 10y = 0 is :

(A) (5,6)                                             
(B) (-6,5)           
(C)(5, 6)                                                         
(D) (6, -5 )

• The end points, of the major axis of the ellipse are called its :

(A) Focii                                            
(B) Vertices        
(C) Covertices                                    
(D) Directrix

• The unit vector of 2i + j is :

(A) 2i – j                                            
(B)        
(C)                                                 
(D)

• If the vector  2i + 4j -7k and 2i + 6j + xk are perpendicular then x equal:

(A)  5                                                  
(B) 4
(C) -4                                                   
(D) 2                              

•  equal :                         

(A)                                                 
(B)    
(C) 180π                                             
(D)1

• Cosh² x-Sinh² x equal:          

(A) 0                                                     
(B) -1     
(C) 2                                                     
(D) 1

•  ax is equal : 

(A)                                                
(B)       
(C) axℓna                                            
(D) ax

• If y =  then y4 equal :               

(A) 16                                           
(B) 8    
(C) 2                                             
(D)   

•  is equal to :

(A) –cos h x                                       
(B) cos h x     
(C) tan h x                                          
(D) sec h x

•  is  equal to :

(A) cosec2x                                        
(B) tan2x             
(C) sec2 x                                           
(D) sec x

• If ƒ’(c) = 0 then ƒ has relative max . value at x = c if :

(A) ƒn(c) > 0                                       
(B) ƒn(c) = 0        
(C) ƒn(c) < 0                                       
(D) ƒn(c) 0

•   equal:                        

(A) x                                                   
(B) ℓnx    
(C)                                                   
(D)

•  equal :

(A) sec x tan x                                    
(B) ℓn(sex x tan x)
(C) ℓn(secx + tan x)                           
(D) ℓn(secx – tan x )

•  equal :  

(A) ℓnx                                               
(B) x
(C) ℓn(ℓnx)                                        
(D)

Inter (Part-II) Multan Board 2015
Mathematics
Part II (Subjective)
Time Allowed: 2.30 Hours 
Max. Marks: 80 

Section I

  
2. Attempt any eight parts                                                       8×2=16                      

• Express the perimeter P of square as a function of its  area A.
• ƒ(x) = -2x + 8, then find ƒ-1(x).
• Evaluate
• Differentiate w.r.t. “x”            x-3 + 2x-3/2 + 3
• Find  if xy + y2 = 2
• Find    if y = tan h(x2)
• Find ƒ’ (x) if ƒ(x) = e
• Find y₂ if y = (2x + 5)^3/2
• Find  if y = a
• Write the productive rule in derivative.
• Differentiate w.r.t. “x” ,          
• Find  if x2 + y2 = 4

3. Attempt any eight parts.                                                      8×2=16

• Use differential to find  when x2 + 2y2 = 16.
• Evaluate  ʃ              (x > o)
• ʃ( + 1)2 dx 
• Evaluate ʃ dx
• Evaluate ʃ co sec x dx
• Evaluate ʃ  dx
• Evaluate
• Find the area between the x – axis and the curve y = cos  From x = - π toπ
• Solve the differential equation  = -y
• Evaluate  (cos x – Sin x) dx
• Graph the solution set of linear inequality      5 – 4y  20 in xy – plane.
• Define optimal solution.

4. Attempt any nine parts.                                                 9×2=18

• Find an equation of the perpendicular bisector of the segment joining the points A(3, 5)  B(9, 8)
• Prove ax + by + c = 0 represents a straight line, where a, b, and c are constants and a and b are not simultaneously zero.
• Check whether the lines 4x – 3y – 8 = 0,       3x – 4y – 6 = 0 and x – y – 2 = 0 are concurrent, if so find the point of concurrency,Find the line represented by 9x² + 24xy + 16y² = 0 also find the angle between them.
• Show that the points A(0, 2), B(, -1) and C(0, 2) are vertices of a right  triangle.
• Find center and radius of the circle x² + y² – 6x + 4y  +13 = 0.
• Find the length of the tangent drawn from the point (-5, 4) to the circle 5x² + 5y² – 10x + 15y – 131 = 0
• Prove the point o a parabola with is closest to the focus, is the vertex of the parabola.
• Find foci and vertices of ellipse 4x² + 9y² = 36        
• Find a so that the vector 2 + a +5 and 3 +  + a are perpendicular.
• A force  = 7 + 4 - 3 is applied  P(1, -2, 3). Find its moment abouts the points Q (2, 1, 1)
• Find a vector whose magnitude is 4 and is parallel to 2 - 3 + 6 .
• Prove that cos² a + cos²β + cos²γ = 1 where a, β, γ are direction angles.

                                                  SECTION-11
Note:   Attempt any three questions.

Question #5
(a)     Derive the formula  = 1
(b)     If y = (cos-1 x)2 prove that (1 – x2) - xy1 -2 = 0
Question #6
(a)     Evalute  dx 
(b)     Find the area of the triangle region whose vertices are A(-5, 3), B(-2, 2), C(4, 2).

Question #7
(a)     Solve the differential equation 2 tan y dx + (1 + ex) sec2 y dx = 0
(b)     Maximize f(x, y) = x + 3y subject to constraints 2x + 5y  30,  5x + 4y  20,  x  0,  y 0

Question #8
(a)     The tangent to a ciecle at any point of the circle is perpendicular to the radial segment at   point.
(b)     Show that mid point of hypotenuse in a right triangle is equidistant from its vertices.

Question #9
(a)     Find focus, vertex and directrix of the parabola (x + 1)2 = 8(y + 2)
(b)     Prove that sin(a – β) = sin a cos β – cos a sin β