D.G KHAN BOARD-2015
Inter Part II
Mathematics
Group I
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.

Q.No.1

• The range of f (x) =x² is:
  

(a)  (-∞,∞)                   
(b) (-∞,0)                    
(c) (0,∞)                      
(d) (-1, 0)

•  =
  

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

• If y=  then  =
  

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

• If f(x)= Sin x then f(0) =
  

(a) Cos x
(b) 1                             
(c) 0                            
(d) -1

•  (tan-1 x) =
  

(a)                       
(b)                          
(c) tan x                                   
(d) Sec x

• If y= e^ax then y⁴ =
  

(a) 4 e^ax
(b) 2                               
(c) 3e^ax                                                                   
(d) xe^ax

•  ax =
  

(a)ax
(b) ax^x-1                           
(c) xe^ax                        
(d)

•  dx =

(a) e^ax
(b) ae^ax                            
(c) xe^ax                       
(d)

•  =
  

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

•  x tan x dx =
  

(a) Sec x tan 2 x         
(b)                             
(c)                           
(d)

•  dx =
  

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

(a) ln(ln x)                
(b) x                            
(c)                            
(d)  

• Solution of differential equation  = -y is:
  

(a) y= Ce-x                 
(b) Ce x                         
(c) e cx                         
(d) xe –x

• Distance between the point (2,3) and (3,2) is:
  

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

• The centroid of triangle divides each median in the ratio:
  

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

• A vertical line divides the plane into:
  

(a) Upper and lower half planes
(b) Upper and right half planes
(c) Left and right half planes
(d) Left and lower half plane

• A circle is called a point circle if:
  

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

• A symptoms are very helpful in graphing:

(a) Circle                
(b) Parabola                     
(c) Hyperbola              
(d) Ellipse

• i.(j x k)=

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

• If α , β ,γ are direction angles of a vector then Cos 2 α + Cos2 β + Cos 2 γ =

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

SUBJECTIVE TYPE:

• Express the area A of a circle as a function of its circumference C.
• Express the limit in terms of e limn→+∞(1+)n/2
• Evaluate limn→0
• Write the Quotient rule in derivative
• Differentiate w. r. t. x,
• Find , if xy + y2 =2
• Find f(x) if f(x) =x3 . e1/x
• Find , if y= cosh2x
• Find y2, if x2 + y2 = a2
• Differentiate w. r. t. x,
• Find  , if y= x.esinx
• Find f(x), if f(x) = Ir. (ex + e-x)

3. Attempt any eight short questions:

• Using differential, find  when x2+2y2=16
  
• Evaluate dx
  
• Compute  x dx
  
• Find  x dx
  
• Evaluate  dx
  
• Compute .In x dx
  
• Find (Cos x – Sin x) dx
  
• Compare
  
• Find area between x- axis and the curve y= Cos x from x =  to x =
  
• Solve Differential equation
  
• Shade the solution region of inequality –y + x ≤  1
  
• Define objective function.

4. Attempt any 9 short questions: (18)

• Find h if A(-1,h), B(3,2), C(7,3) are collinear
  
• Find an equation of line through (-2,1) and (6,-4)
  
• Find an equation of line through A(-6, 5) having slope 7
  
• Find the distance from the point P(6, -1) to the line 6x – 4y +9 =0
  
• Find angle from the line having slopes -7/3 and 5/2
  
• Find an equation of circle having end points of a diameter at (-3, 2) and (5,6)
  
• Write equation of parabola having focus (-3,1), directrix line x=3
  
• Find center and focus of a ellipse x²+4y²=16
  
• Check the position of point (5,6) w.r.t circle x² + y² =81
  
• Find unit vector in the direction of vector  v =2i – j
  
• Find α so that [] =3
  
• Find Cos θ when u = 2i –j k and v = i + j
  
• Find the value of 2i x 2. K

Attempt any three questions from this section:

5. (a) If  f(x) =    ; x  ≠ 2 find k so that f(x) is continuous at x=2;
(b) Differentiate  w.r.t x³

6. (a) Evaluate  dx
(b) One vertex of a parallelogram s (1,4), the diagonals intersect at (2, 1) and the sides has slope 1 and -1/7. Find other three vertices.

7.(a) Solve the different equation y-x  =2 (y2+)
(b) Maximize the function defined by f(x,y) =2x+ 3y subject to the constraints
2x+y ≤8 , x+2y ≤ 4, x ≥0, y ≥ 0

8. (a) Find an equation of the circle passing through the point A(1,2) and B(1, -2) and touching the line x+2y+5=0
(b) By vector method, prove that Cos(α+β)=Cos α  Cos β – Sin α  Sin β

9. (a) A parabolic arch has a 100m base and height 25m. Find the height of the arch at the point 30m from the centre of these.
(b) If a + b + c = 0 then prove that a x b = b x c = c x a

D.G KHAN BOARD-2015
Inter Part II
Mathematics
Group II
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.

Q.No:1

• The range of function f(x) =  x≠ -2 is:

(A) {1}
(B) {-1,1}
(C) {-1}
(D) R-{-1,1}

• f (x) x3 + sin x then f(x) is:

(A) Constant function
(B) Even function
(C) Odd function
(D) Neither even nor odd

• If f (x) = -sin x then (Cos^-1 x)

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

• Instantaneous rate of change of y with respect to x is given by:

(A) Sy/sx
(B) Sx/sy
(C) Dy/dx
(D) Dx/dy

• f (x) =  then f(2)

(A) 1
(B) 0
(C) -1
(D) ∞

• f (x) Cotx then f(π/6)

(A) -4
(B) 4
(C) ¼
(D) -1/4

• f (x) sec^-1 x then f(secx)

(A)
(B) Secx tanx
(C) cos 2x cosec x
(D) -cos 2x cosec x

•  lna dx=

(A) a x + c
(B)
(C) lnax + c
(D) lna ax + c

(A) 1
(B) 0
(C) Constant
(D) x

• y= c e^x2 is solution of:

(A) 1/x  -2y=0
(B) x-2 =0
(C) -2y=0
(D)

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

• sin y cosec y

(A) cosy = cosx +c
(B) y = x+c
(C) cosy = cosy+ c
(D) -Cosy= -Cosy + c

•  sinx dx =

(A) ecos x + c
(B) sin x ecos x + c
(C) e sinx + c
(D) e sinx cos x + c

• The x- intercept of the line 2x + 3y -1=0 is:

(A) ½
(B) 2
(C) 3
(D) 1/3

• What is the nature of line ax + by + c= 0 and a=0 and b≠ 0, c≠ 0

(A) Line is parallel to y-axis
(B) Line is parallel to x-axis
(C) Line passes through origin
(D) Line is perpendicular

• x = 0 is not in the solution in inequality:

(A) 2x+3>0
(B) x +4>0
(C) x+ 5 >0
(D) 2x + 3 <0

• Focal distance of point P(x, y) on x2=4y ay is:

(A) X+2
(B) Y+2
(C) X-a
(D) Y-a

• The equation of tangent to the circle x2+y2 2gx+2fy=0:

(A) X=0
(B) Y=0
(C) Fx+ gy=0
(D) Gx+fy=0

• The direction cosines of x-axis are:

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

• The moment of a force F acting P about C is:

(A) F x CP
(B) CP x F
(C) CP x F
(D) OP x F

SUBJECTIVE TYPE

2. Attempt any 8 questions: (16)

• f (x)= x3-ax2 +bx + 1 If f(+2) = -3 and f (-1) = 0 find the values of a and b.
  
• 
  
• 
  
• Differentiate  w. r. t x
  
• If y= x⁴ +2x² +2 prove that dy/dx =4x √y-1
  
• Find dy/dx if x =θ + 1/ θ , y= θ+1
  
• Differentiate sin³x w.r.t Cos²x
  
• Prove that d/dx [tan-1 x] =
  
• Find dy/dx if x=y siny
  
• Find dy/dx if y = (logx)logx
  
• Find y₂ if x = at², y=bt⁴
  
• If y=(Cos^-1 x)² Prove that (1-x²) y₂-xy₁ -2 =0
  

3. Attempt any eight short questions:

• Use differentiate to approximate the value of (31)1/5
  
• Evaluate
  
• Evaluate
  
• Evaluate
  
• Evaluate
  
• Find   dx
  
• Evaluate (Cos < - Sinx) dx
  
• Evaluate         
  
• Solve the differential equation dy/dx = 1-x/y
  
• Indicate the solution set of the inequality 3x + 7y ≥ 21
  
• Graph the solution region of the linear inequalities  x+y ≤ 5 ; -2x + y ≤2 ; y ≥ 0
  
• Evaluate

4. Attempt any nine short questions:

• Find the midpoint of the line joining the two points A(3,1), B(-2,-4)
  
• The points A(-5, -2) and B(5, -4) are ends of a diameter of a circle. Find centre and radius of the circle.
  
• Find the point trisecting the join of A(-1,4) and B(6,2)
  
• Let P(x₁, y₁) and Q( x₂, y₂) are two points on a line. Find the slope of the PQ
  
• Write the point- slope from of equation of straight line.
  
• Write the standard form of equation of circle.
  
• Find the center and the radius of the circle x²+ y² – 6x 4y +13= 0
  
• Find the length of the tangent from the point P(-5,10) to the circle 5x² + 5y² + 14x+ 12y -10=0
  
• Find the focus, vertex and directrix of parabola y2= 8x
  
• Write the vector PQ in the form xi + yj P(0,5) Q(-1, -6)
  
• Find α so that | α I + (α +1) j + 2k| = 3
  
• Calculate the projection of b along a
  a= 3i + j –k ,          b= -2i -j + k
  

Attempt any three questions from this section:

5. (a) Evaluate
(b) If y = tan (2tan -1 x/2), show that dy/dx =4

6. (a) Solve 2 ex tany dx + (1-ex) Sec2y dy =0
(b) If two lines with slopes m1 and m2 are perpendicular, then prove that m₁.m₂=-1

7. (a) Evaluate
(b) Maximize the function f(x,y) = 2x +3y subject to the constraints:
2x+y ≤8
x+2y≤14
x≥0 ; y≥0

8. (a) Write equation of circle which pass through A(4,5), B(-4,-3), C(8,-3)
(b) prove that in any triangle ABC, b=c CosA + aCosC

9. (a) Find the focus, vertex and directrix of the parabola x+8 –y2 +2y=0
(b) A force of magnitude 6 units acting parallel to 2i + 2j + k displaces the point of application from (1,2,3) to (5,3,7). Find the work done.