Inter (Part-II)
Gujranwala Board 2013
Mathematics
Paper: II
Time: 30 Minutes
Marks: 20
OBJECTIVE
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 with marker or pen on the answer book provided. Cutting or filling two or more' circles Will
result in zero mark in that question. Attempt as many questions as given in objective type question paper and leave others blank.

• If the vector u = 2i + 4 j - 7k and v = 2i + 6j + x k are perpendicular, then x =_____:
  

(A) - 4
(B) 4
(C) 28 
(D) 0

• Any chord passing through the focus of the parabola is called the:
  

(A) Vertex of the parabola
(B) Axis of the parabola
(C) Latus-rectum of the parabola
(D) Focal chord of the parabola

• The feasible solution which maximize or minimize the objective function is called the:
  

(A) Feasible region     
(B) Optimal solution
(C) Convex region     
(D) Feasible solution set

• The point of concurrency of the medians of a triangle is called:
  

(A) in-centre   
(B) centroid
(C) e-centre    
(D) circumcentre

• The area bounded by cos x function from x = - -7c to
  

(A) 1 sq unit
(B) 2 sq unit
(C) 3 sq unit
(D) 4 sq unit

• sec x tan x dx =____.        
  

(A)
(B)
(C) +1
(D) 1

• ax=_____:
  

(A) ax  
(B) ax . ℓn a
(C)
(D)         

• sin-1=______:

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

• cosh² x+ sinh² x=_______:
  

(A) 1
(B) cosh 2x
(C) sinh 2x
(D) 2cosh 2x

• The solution of differential equation =cosec x cot x is:
  

(A) y = cosec x + c     
(B) y = sec x + c
(C) y = - cosec x + c   
(D) y = -cot x + c

• The perpendicular distance of the line 3x + 4y + 10 = 0 fromihe origin is:
  

(A) 0   
(B) 1
(C) 2   
(D) 10

• The length of the tangent drawn from the point (1,1) to the circle x² + y² - 3x + 9y + 8 = 0 is:
  

(A) 1   
(B) 8
(C) 4   
(D) 16

• Projection of vector u = ai + b j + ck along i is:
  

(A) c   
(B) a
(C) b   
(D) 0

• If the expression involves  then the suitable substitution is:
  

(A) x = a sin θ
(B) x = a sec θ
(C) x = a cos θ
(D) x = sin θ

• If  12xdx = 12, then k =_____:   
  

(A) 2, -2         
(B) 2, 6
(C) 4. -4          
(D) 4, 2

• =______:
  

(A)       
(B)  sec2x
(C)    x. sec2 x    
(D)

• The derivative of cos x w.r.t. cos x is:
  

(A) 0   
(B) 1
(C) sin x         
(D) cos x

•  loga x =______:

(A)     
(B) log a.
(C)           
(D)           

• The function f(x) =is discontinuous at:

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

Inter (Part-II)
Gujranwala Board 2013
Mathematics
Paper: II
Time: 2.30 Hours
Marks: 80
SUBJECTIVE
Note: Section I is compulsory. Attempt any three (3) questions from Section II.

(Section - I)

2. Write short answers to any EIGHT questions: (2x8 = 16)

• Find the domain and range of f (x) =
• Prove that sech²x = 1 - tanh² x.
• Define derivative of a function.
• Find  if y = x cos y.
• Find  for y = sin x².
• Find f’ (x) for f (x) = ex (1 + ℓn x ).
• If y = x² ℓn  , find  .
• If y = 5 e^3x-4 +, find .
• If y = 2x⁵ - 3x⁴ + 4x³ + x - 2, find Y₂.
• If x = a cos θ , y = a sin θ . find ,.
• Find the critical points for f(x) = x² - x - 2.
• Define the critical point.

3. Write short answers to any EIGHT questions: (2x8 = 16)

• Find  and dy if y =  when x changes from 4 to 4.41.
• Evaluate dx,(x > -2)
• Evaluate sin2 x dx .
• Find .
• Find x sinx dx.
• Evaluate ex (cosx + sinx) dx.
• Find dx.
• Write two properties of definite integral.
• Find dx.
• Solve the differential equation x dy + y (x-1) dx = 0.
• Define feasible region and feasible solution of system of linear inequalities.
• Draw the graph of 3x + 2y > 6.

4. Write short answers to any NINE questions: (2x9 = 18)

• Show that A (3, 1) , B(-2 , -3), C (2, 2) are vertices of an isosceles triangle.
• Find h such that A (-1 , h), B (3,2), C (7,3) are collinear.
• Find the distance between parallel lines 2x - 5y + 13 =0;-2+ 5y - 6 = 0.
• Find equation of perpendicular bisector of the segment joining the points A (3, 5), B (9,8).
• Find p such that the lines 2x - 3y - 1 = 0, 3x –y-5=0 and 3x + py + 8= 0 are concurrent.
• Find the equation of a circle of radius a and lying in the second quadrant such that it is tangent to both the axes.
• Show that 5x² + 5y² + 24x + 36y + 10 = 0 represents a circle.
• Find focus and equation of directrix of the parabola x2- 4x - 8y + 4 = 0.
• Prove that the laths rectum of the ellipse +=1 is .
• Find a vector of length 5 in the opposite direction of v = i - 2 j + 3k.
• State two properties of dot product.
• Show that the vectors 3i -2j+ k , i -3j+ 5k and 2i + j + 4k form a right triangle.
• If a x b = 0 and a.b = 0, what conclusion can be drawn about a or b?

(Section - II)

6.a) Show that =ℓn  (5)
b) Find h such that the points A(,-1), B(0,2) and C (h, -2) are the vertices of a right triangle with right angle at the vertex A. (5)

7.a) Evaluate x+dx.  (5)
b) Graph the feasible region and find the corner points of linear inequalities 2x - 3y 6, 2x + 3y < 12, x 0, y>0. (5)

8.a) Find an equation of a circle passing through A(-3 , 1) with radius 2 and centre at 2x - 3y + 3 = 0. (5)
b) If a = 3i - j - 4k ,b= -2i -4j-3k,c=i+2j-k Find a unit vector parallel to 3a - 2b + 4c. (5)

9.a) Find focus, vertex and directrix of the parabola x2 - 4x - 3y + 13 = 0. (5)
b) Prove by vector method that cos (α -β) = cos α cosβ + sin a sigβ. (5)