RAWALPINDI
BOARD 2014
PAPER MATHEMATICS
PART-II
Time: 30Min
(Objective Part)
Marks: 20
Note: Four Answer are given against esch column A,B,C & D. Select the write answer and only separet answer sheet, fill the circle A,B,C or D with pen or marker in front of that question number.

• is equal to:

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

• () is equal to:

(a)  Sinx
(b)  cosecx
(c)  cosx
(d)  cotx

•  (cosec^-1x) is equal to: 

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

• (Inx) is equal to:

(a)
(b)
(c) x
(d) Inx

• ʃsec5xtan5xdx is equal to:

(a) 5sec5x + c 
(b) +c
(c)  +c
(d) +c       

• ʃcos2xdx is equal to:

(a) -2sin 2x+c 
(b) 2sin 2x +c
(c) +c
(d) +c

• If f(x) = cosx, then f(0) =?

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

• 2sinh x is equal to:

(a) e^x + e^-x
(b) e^x - e^-x
(c) 
(d) )

•  (x³) is equal to:

(a) )
(b) x²
(c) 3x²
(d) 4x⁴

• The lines l1,l2 with slopes m1,m2, are perpendicular if:

(a) m1,m2=-1
(b) m1 = m2
(c) m1 + m2= 0
(d) m1m2 = 1

• Which one is not a solution of in-equality 2x + 3y < 0?

(a) (-1,-2)
(b) (1, +2)
(c) (2,3)
(d) (0,1)

• The centre of the circle x² + y² + 2gx + 2fy + c = 0 is:

(a) (g,f)           
(b) (f,g)           
(c) (-f,-g)        
(d) (-g,-f)

• The set of all the points in a plane which are equidistant from a fixed point and
  fixed the is called:

(a) circle
(b) ellipse        
(c) parabola
(d) hyperboia

• j.(k+i) is equal to:

(a) i     
(b) 1    
(c) -1
(d) j

• [a b c] is equal to: 

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

• ʃ(ax+b)ⁿ dx is equal to:

(a)  + c
(b)  + c
(c)  + c
(d)

• If ¹ʃ₀(4x + k) dx=4, then k will be:

(a)
(b) 0
(c) 1
(d) 2   

• ³ʃ_-1x³dx is equal to:      

(a) 20
(b) 80
(c) 28
(d) 2   

• Degree of differential equation  +  - 3x = 0 is:

(a) 1
(b) 2
(c) 0
(d) 3

• The slope of a line wish inclination 90o is:

(a) 0
(b)   
(c) 1
(d) Undefined

Time: 2:30 Hours
(Subjective Part)
Marks: 80
SECTION-I

2. Attempt any Eight Parts.

• Evaluate lim  .
  x→π
• Evaluate lim  .
  x→∞
• Differentiate ( )² w.r.t.’x’
• If y=  +  , then find
• Differentiate sin³x w.r.t cos²x
• If y = , then find
• If y = Sin^-1 (  ), then find y².
• Define power series.
• If x = at², y = 2at,  find  .
• If y = In (x+  ), find
• Find by definition, the derivative w.r.t ‘x’ of x40
• Determine the interval in which f is increasing or decreasing 
  f (x) = x³ + 3x + 2, x €(-4,1)

3. Attempt any Fight Parts. 

• Evaluate ʃsin2xdx
• Evaluate ʃ
• Evaluate ʃdx
• Evaluate ʃe3x() dx
• Evaluate ʃInxdx
• Evaluate ³ʃ₀dx
• Evaluate ⁶ʃ₀cos³ɵdɵ
• Define the optimal solution
• Solve the differential equation (e^x + e^-x)  = e^x – e^-x
• Find the area above the x-axis, bounded by the curse y² = 3-x from x=-1 to x=2
• Graph the solution set of the linear inequality 2x + 1 ≥ 0.
• Find δy and dy of the function y = x² +2x when x changes from 2 to 1.8.

4. Attempt any Nine Parts.18

• The xy coordinate axes are rotated about origin through an angle of 30". If xy  
  coordinates of a point are (5,7), find its xy coordinates.
• The length of perpendicular from origin to a line is 5 units and inclination of this   
  perpendicular is 120o. Find equation of this line.
• Check whether the origin and the point (5, -8) lie on the same side or opposite side of
  the line 3x + 7y + 15 = 0.
• Transform the equation 5x – 12y  + 39 = 0 into slope-intercept form.
• Find the angle front the line line with slope  to the line with slope .
• Write an equation of the circle with centre (-3,5) and radius 7.
• Write down the equation of tangent and normal to the circle x²+y²=25 at point(4,3)
• write an equation of parabola having focus (-3,1) and direct x-2y-3=0.
• Find foci, directix of ellipse 25x²+9y²= 225.
• Find 'α' if │αi + (α+1)j + 2k│=3
• Prove that: μ.(v x w) + v.(w x μ) + w.(μ x v) =3μ.(v x w)
• Find the projection of a along b when        A – I – p = j
• Find volume of tetrahedron with the vertices
  A(0,1,2), B(3,2,1), C(1,2,1) and D(5,5,6).

SECTION-II

Attempt any THREE questions, Each questions carries (10 marks)

5.(a) Prove that lim   = log_e a
x→0
(b) If y= ex sin x, show that  - 2y  +2y = 0         

6.(a) Evaluate ʃ
(b) Find h such that A(-1, h), B(3,2) and C(7,3) are collinear.

7.(a) Evaluate 3ʃ-1│x-3│dx
(b)Maximize f (xy) = x +3y subject to the constraints
2x+ 5y≤30,5x+ 4y≤ 20,x≥0, y≥ 0.

8.(a) Find the coordinates of the points of intersection of the line
x+ 2y = 6 with the circle x²+y²-2x-2y-39=0
(b)Find a vector of length 5 in the direction opposite to that of V=i-2j+3k:

9.(a) Find an equation of parabola with focus (2,5) and directrix y=1. 
(b) Prove by vector method that  cos (α + β) = cosα cosβ – sinα sinβ