Mathematics
Inter Part-II
Group - I
Lahore Board 2015
Time: 30 Min.
Objective Type
Marks = 20
Note: Four possible answers A, B, C and D to each question are given. The choice which you think is correct, fill that circle in front of that question with Marker or Pen ink in the answer-book. Cutting or filling two or more circles will result in zero mark in that question.

• :
  

(A) tan x         
(B) sec2x
(C) tan2 x        
(D) sec x

• The point of intersection of medians of a triangle is called :
  

(A) Centroid              
(B) Orthocentre
(C) Circumcentre       
(D) Incentre

• The parabola x² = y passes through point :
  

(A)
(B)
(C)
(D)     

• The distance of point P (6, -1) from the line 6x - 4y + 9 =0 is:
  

(A)49
(B)
(C)
(D)

• The integration is the reverse process of :
  

(A) Induction 
(B) Differentiation
(C) Tabulation
(D) Sublimation

• (1, 0) is the solution of inequality:
  

(A) 7x + 2y < 8          
(B) x - 3y < 0
(C) 3x + 5y < 6          
(D) -3x + 5y > 2

•  (sin^-1x) is equal to:
  

(A)          1
(B)
(C)
(D)

• If f(x)=x²- x then f(-2) is equal to :
  

(A) 2   
(B) 6
(C) 0   
(D) -6

• dx is equal to :
  

(A) 4  
(B) -4
(C)  
(D)

• Solution of differential equation, - y is :
  

(A) ce^x
(B) ce^-x
(C) e^x  
(D) e^-x

 dx is equal to :

(A) cos x        
(B) sin x
(C) -sin x        
(D) - cos x

•  is equal to:
  

(A) f (0)          
(B) f ' (a)
(C) f ' (x)        
(D) f ' (0)

• If f (x) = e^ax then f '(x) is equal to:
  

(A)            
(B)
(C)
(D)

• If f (x) = cos x then f (0) is equal to:
  

(A) 0   
(B) -1
(C) 1   
(D)

• A vector with magnitude 1 is called :
  

(A) Null vector
(B) Unit vector
(C) Zero vector
(D) Constant vector

• The set of all points in the plane that are equally distant from a fixed point is called:

(A)Ellipse
(B) Parabola
(C)Hyperbola
(D) Circle

• (k x i) is equal to
  

(A) 0   
(B) -1
(C) 1   
(D) 2

•  is equal to
  

(A)       
(B)
(C)           
(D)

(A)           
(B)
(C)      
(D) f(x)

•   is equal to
  

(A)   
(B)
(C)   
(D)

Mathematics
Lahore Board 2015
Time: 2.30 hrs
Essay Type
Inter-II
Marks=80

SECTION - I

2. Write short answers to any EIGHT (8) questions: (16)

• Show that f (x) = x2/3 + 6 is even function.
• Find the domain and range of f-1(x) where f (x) = 2
• Evaluate
• Find  when y = (x - 5)(3 - x)dx
• Compute when y =
• Differentiate sec-1 x w.r.t. x
• If y = tan (p tan-l x)find
• Find f’ (x) when f (x) = x3e1/x
• Find  when y = a cos (nx) + b sin (nx)
• Using Maclaurin's series expansion, write first two terms of f (x) =
• Find  when y = sin-1
• Find critical values of f(x) = sin x + cos x

3. Write short answers to any EIGHT (8) questions : (16)

• Using differential, find  when xy - nx =
• Evaluate
• Evaluate
• Find
• Evaluate
• Evaluate
• Evaluate
• Evaluate
• Find the area bounded by y = cos x from x = -  to x=
• Solve the differential equation x2(2y + 1)  -1=0
• Graph the solution region of 2x + 1  0
• What are problem constraints?

4. Write short answers to any NINE (9) questions : (18)

• Find the point P on the join of A (1, 4) and B (5, 6) that is twice as far from A as B is from A and lies on the same side of A as B does.
• Two lines  and  with respective slopes ml and m2 are parallel if m1 = m2
• If length of perpendicular from origin to a line is 5 units and its inclination is 120°, find slope and y-intercept of the line?
• Find the lines represented by x2-xy - 6y2 = 0, also find the angle between them.
• Find an equation of the line through (5, -8) and perpendicular to the join of A (-15, -8), B (10, 7)
• Find the coordinates of the points of intersection of the line 2x + y = 5 and x2 + y2 + 2x - 9 = 0
• Find an equation of a circle of radius 'a' and lying in the second quadrant such that it is tangent to both the axes.
• Find the focus and vertex of parabola y2 = - 8 (x - 3)
• Find equation of hyperbola with foci (±5, 0) and vertex (3, 0)
• Calculate projection of  along  when  =  + ,  = j +
• Find the angle between the vectors  = 2 -  +  and =-
• Find the area of the triangle with vertices A (1, -1, 1) B (2, 1, -;1) C (-1, 1, 2)
• Find the direction cosines for , where P (2, 1, 5), Q (1, 3, 1)

SECTION-II

Note: Attempt any THREE questions.
5. (a) Find the values m and n, so that given function f(x) is continuous f (x)= 
(b) Differentiate cos  from the first principles.     
6. (a) Show that
(b) Find equations of two parallel lines perpendicular to 2x - y' + 3 = 0 such that the product of the x – intercept and y - intercept of each is 3

7. (a) Evaluate
(b) Graph the feasible region and also find the corner points : 2x - 3y  6, 2x + 3y  12,x 0 , y 0

8. (a) Show that 2x + 3y - 13 = 0 is tangent to circle x2 + y2 +6x - 4y = 0
(b) Using vector prove that b = c cos A + a cos C

9. (a) Find an equation of the parabola whose focus is F (-3,4), and directrix line is 3x - 4y + 5 = 0
(b) Find the volume of the tetrahedron with the vertices A(0, 1, 2), B (3, 2, 1), C (1,2,1), D (5,5,6)