Inter (Part-I) Lahore Board 2016
Mathematics
Part I (Objective Type)
Time Allowed: 30 Minutes 
Max. Marks: 20

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.

Question #1
Circle the correct option i.e. A/B/C/D. Each part carries one mark.

•  equals

(A) ef’(x)        
(B) e f’(x).f’(x)
(C) f’(x) / ex   
(D) ef(x) / f’ (x)

• Distance between (1, 2) and (2, 1) is

(A) 1   
(B) 2
(C)           
(D) 0

• d / dx tanh-1 x, equals.

(A)     
(B)     
(C)
(D) 

• :

(A) 1   
(B) 2
(C) ℓn2           
(D) ℓn

• The expression ℓn(x+  ) equals . 

(A) sinh-1x      
(B) cosh-1x
(C) tanh-1x     
(D) cosech-1x 

• ∫(2x + 3)1/2 dx , equals:

(A) 1/2(2x+3)1/2+C    
(B) 2/3(2x+3)3/2 + C
(C) 1/3(2x+3)1/2 + C  
(D) 1/3(2x+3)3/2 + C

• The line y = mx + c, will be tangent to the parabola y2 = 4ax if:

(A) c = -am2
(B) c = a/m
(C) c = a (1 + m2)
(D) c = m/a

• (ỉ x k) x Ĵ equals:

(A) -1 
(B) 0
(C) 1   
(D) ∞

• Solution set of inequality 2x < 3 is : 

(A)     
(B) 
(C) (-∞, ∞)     
(D) (-3/2, 3/2)

• If α, ß, y be the direction angles of a vector then cos2 α + cos2 ß + cos2 y equals :

(A) -1 
(B) 0
(C) 1   
(D) 2

• The differential co-efficient of esin x is :

(A) esin x.cos x
(B) esin x.sinx
(C) ecos x.cos x
(D) sinx .esin x

• Equation of the line parallel to x + 3y - 9 = 0 is .

(A) 3x – y -  9 = 0      
(B) 3x + 9y + 7 = 0
(C) 2x - 6y - 18 =0     
(D) x - 3y + 9 = 0

• Length of latus rectum of ellipse x2/16 + y2 / 9 = 1 is.

(A) 9 / 2
(B) 9 / 4
(C) 16 / 9        
(D) 9 / 16

• The solution of differential equation dy/dx = sec²x , is:

(A) y = cos x + c        
(B) y = sec x + c
(C) y= cos2 x + c        
(D) y= tan x + c

• If f(x) =2x . then f’(x) equals :

(A) 2x-1           
(B) 2xℓn2
(C) 2x / ℓn2     
(D) ℓn2 / 2x  

• If f’(c) = 0 , then f(x) has relative maxima at x = c . if

(A) f”(c) < 0   
(B) f”(c) = 0
(C) f”(c) > 0   
(D) f”(c) ≥ 1

• If y = ℓn (sin x), then dy / dx equals :

(A) tan x         
(B) cot x
(C) -tan x        
(D) - cot x

• Antiderivative of cot x. equals

(A) ℓn (cos x) + c       
(B) ℓn (sin x) + c
(C) -cosec2 x + c        
(D) ℓ (sec x) + c

• ax + by+ c = 0, will represent equation of straight line parallel to y-axis if 

(A) a = 0
(B) b = 0
(C) c = 0
(D) a = 0, c = 0

•  equals :

(A) 40
(B) 60
(C) 80
(D) 120

Inter (Part-II) Lahore Board 2016
Mathematics
Part II (Subjective)
Time Allowed: 2.30 Hours
Max. Marks: 80

Section I

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

• Find fof (x) if f (x) =
• Evaluate  
• Discuss continuity of f (x) at 3 when f(x) = 
  {x - 1, if x < 3 
  {2x + 1, if 3 ≤ x
• Differentiate f(x) = x2 by definition.
• Differentiate  w.r.t x
• Find dy / dx if 3x + 4y + 7 = 0
• Differentiate sin x w.r.t. cot x
• Differentiate cos w.r.t x.
• Find f’ (x) if f (x) = ℓn (ex + e-x
• Find dy / dx if y = e-x (x3 + 2x2 +1)
• Find y4 if y = sin 3x
• Expand cos x by Maclaurin's series expansion 

Q3. Write short notes on any EIGHT (8) questions:                                 16

• Find δy , if y = x2 + 2x when x changes from 2 to 1.08
• Use differential, find the value of (31) 1/5
• Evaluate ∫ x(√x-1)dx
• Evaluate ∫ tan2xdx
• Evaluate ∫ (1 / xℓnx) dx
• Evaluate ∫ e2x (- sinx + 2cosx) dx
• Evaluate     
• Evaluate
• Find the area above the x-axis and under the curve,  y = 5- x2 from x = -1 to x = 2
• Solve the differential equation y dx + x dy = 0
• What is convex region?
• Graph the solution set of linear inequality 3x + 7y ≥ 21 by shading

Q4. Write short answers to any NINE (9) questions:                                            18

• Find h such that A(-1, h), B (3, 2) and C (7, 3) are  collinear
• Find the point three-fifth of the way along line segment from A(-5, 8) to B(5, 3)
• Find slope and inclination of line joining points (4,6), (4,8)
• Find measure of the angle between the lines represented by x2-xy-6y2 = 0
• Find the distance from the point P(6, -1) to the line 6x - 4y + 9 = 0
• Find the centre and radius of the circle x2+y2+12x-10y = 0
• Find the coordinate of the points of intersection of the line x+2y=6 with the circle x2+y2-2x-2y-39 = 0
• Find the focus and vertex of the parabola y2=8x
• Find an equation of the vertical line through (-5, 3)
• If o is the origin and OP = AB, find the point P when A and B are (-3, 7) and (1, 0) respectively?
• Find a vector whose magnitude is 4 and is parallel to 2i - 3 j + 6 k
• Find a vector perpendicular to each of the vectors a = 2i - j - k and b = 4i + 2 i - k
• Find the value 3j . k x i 

SECTION-II

Note: Attempt any THREE questions.

Question #5.
(a) Evaluate
(b) Differentiate w.r.t x  

Question #6
(a) Evaluate
(b) Find distance between 3x + 4y + 3 = 0 and 3x - 4y + = 0 Also find equation of parallel
line lying midway  between them.                                                            5

Question #7
(a) Evaluate                                                                  5
(b) Maximize f(x,y) = x + 3y subject to the constraints                                 5
2x + 5y ≤ 30 , 5x + 4y ≤ 20 , x ≥ 0, y ≥ 0

Question #8
(a) Find the length of the chord cut off from the line 2x+3y= 13 by the circle x2 + y2 = 26                                                                                            5
(b) Prove that in any triangle ABC by vector method a2=b2+c2-2bc cos A   5

Question #9
(a) Show that the equation 9x2-18x+4y2+8y-23 = 0 represents an ellipse. Find its elements (foci vertices, directnces)                                                      5
(b) Find volume of the tetrahedron whose vertices are                                  5
A(2,1,8), B(3,2,9), C(2,1,4), D(3,3,10)