Inter Part II Gujranwala Board 2012
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. Write the letter A, B, C or D in the column (write correct option) against each question. If there is a contradiction in the bubble and hand written answer, bubble option will be considered correct.

• The solution of the differential equation xdy+dyx=0 is: <

(A) y = In x+ c                    
(B) y = in (cx)                    
(C) ln y = x+c             
(D) xy =c

• Useful substitution of  is:

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

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

• …………..++……… is called:
  

(A) Maclaurin Series     
(B) Taylor series              
(C) Binomial Series         
(D) Power Series

• The function f(x) = 2x2+bx+c have minimum value if:
  

(A) a > 0                             
(B) a<0                                 
(C) a<-1                                                
(D) a<-2

• If any two vectors of scalar triple product are equal, then its value is

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

• . The length of latus rectum of the ellipse  =1 is

(A)                                      
(B)                                       
(c)                                        
(D)

• Equation of normal line to the circle x²+ y²=a²at the point (xi, yi) is
  

(A) xy₁=x₁y 
(B) xy₁=-x₁y              
(C) xx₁+yy₁=0                    
(D) xx₁+yy₁=a2

• If f(kx, ky) = kn.f(x,y), then f(x,y)=0 is a homogeneous equation of degree:
  

(A) n+1                                                
(B) n-1                                 
(C) n                                     
(D) k

(A)                   
(B)                     
(C)                          

(D)

• The derivative of tan x with respect to tan x is:
  

(A) Zero                           
(B) I                                      
(C) sec² x                                 
(D) 2 sec² x

• If f(x) = sin x, then f' (cos^-1x) =
  

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

• If   are unit vectors, then x
  

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

• Focus of the parabola x²= 4ay is:
  

(A) (a, 0)                     
(B) (-a,0)                           
(C) (0, a)                                       
(D) (0, -a)

• A function f: X  Y defined by f(x) = l is called:
  

(A) Identity function      
(B) constant function                    
(C) linear function           
(D) odd function

• if  y=ax, then
  

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

• =
  (A)0                                      
  (B)                                                                   
  (C) 2                                      
  (D)
  
• (3, 2) is not in the solution of inequality:
  

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

• Slope of a vertical line is:
  

(A) m=1
(B) m =0
(C) m =  
(D) m = -1

(A) — cosh x+c
(B) cosh x+c
(C) ln |cosh x| +c
(D) cosech x + c

Gujranwala Board 2012 Inter part II
Mathematics Paper: II
Time:2.30 Hour  Marks:80
SUBJECTIVE

Note: Section I is compulsory. Attempt any three questions from section II (Section-I)

• Define an implicit function and give one example.
• Evaluate
• Define derivative of a function f(x).
• Find by definition derivative of x100.
• Differentiate  w.r.t. x
• Find  if x=
• If x= a cos3bsec3, show that a
• Find
• Find y₂ if y=-x⁵-3x₄+4x³+x-2
• Apply Maclaurin’s Series expansion to prove that ++……….
• Show that y=xx has a minimum value at x=
• Differentiate cos x w.r.t. tan x

3. Write short answers to any eight questions:

• Integrate
• Integrate
• Evaluate
• Evaluate
• Evaluate
• Evaluate
• Evaluate
• Find the area between the x-axis and the curve y=cos  from x=
• Find centre and vertices of ellipse 9x²+y²=18
• The points A(-5,-2) and B (5,-4) are ends of a diameter of a circle. Find the centre and radius.
• The two points P and O’ are given in xy- coordinate system. Find XY- coordinates of P referred to O’X, O’Y where P (-6,-8). O’(-4,-6)
• Find equation of horizontal line through (7,-9)

4. Write short answers to any nine questions

• Find coordinates of the point that divides the join of A(-6,3) and B (5,-2) in ratio 2:3 internally.
• Find equation of line through (-4,-6) and perpendicular to line having slope
• Find where point (5,8) lies above or below the line 2x-3y+6=0
• Find the equation of lines represented by 20x²+17xy-24y²=0
• Find centre and radius of circle given by equation 4x²+4y²-8x+12y-25=0
• Determine whether the point P(-5,6) lies outside, on or inside circle x²+y²+4x-6y-12=0
• Derive standard equation of parabola.
• Find centre, foci of ellipse x²+4y²+16x-16y+76=0
• Find equation of hyperbola having foci(0.+9) and directrices y=+ 4
• Find foci. Eccentricity of
• Find sum of vectors  and  , given the four pints A (1,-1), B (2,0), C(-1,3) and D (-2,2)
• Find vector perpendicular to each of vectors a =2i-j+k and b =4i-2j-k
• Prove that ax(b+c)+bx(c+a)+cx(a+b)=0

Section –II

Q.No.5(a) If f(x)={
Find the value of k so that f is continuous at x=2
(b) If x= sin , show that (1-x2)y2-xy1+m2y=0

Q.No.6 (a) Show that  dx=  +
(b) Find the equation of the perpendicular bisector of the segment joining the points A(3,5) and B (9,8)
Q.No.7 (a) Solve the differential equation y-x
(b) Graph the solution region of the following system of linear inequalities and find the corner points.
2x-3y < 6
2x+3y <  12
x> 0

Q.No.8 (a) Write equation of a circle that passes through the giben points. A(4,5), B (-4,-3), C (8,-3)
(b) The position vectors of points A, B, C and D are 2i-j+k, 3i+j, 2i4j-2k and -i-2j+k respectively.
Show that  is parallel to .

Q.No.9 (a) Prove that the latus rectum of the ellipse:

(b) Find the area of the triangle with vertices A (1,-1,1), B (2,1,-1) and C (-1,1,2) also find a unit vector perpendicular to the plane ABC