Lahore Board 2017
MATHEMATICS-II
Inter (Part-II) 2017
Mathematics
Group-I
PAPER: II
Time: 30 Minutes
(OBJECTIVE TYPE)
Marks: 20
Note: Four possible answers, A, B, C and D to each question are given. The choice which you think is correct, fill that circle in front of that question with Marker or Pen ink in the answer-book. Cutting or filling two or more circles will result in zero mark in that question.

• 2 sin h x = :

(a) ex - e-x
(b) ex + e-x
(c)  
(d)

(a) 0
(b) 1
(c)  
(d) 2

•  cot² 2x=

(a) 4 cot 2x cosec 2x
(b) -4 cot 2x cosec² 2x
(c)  4 cot² 2x cosec 2x
(d) -4 cot 2x

•  ex+h=

(a)  
(b)
(c) ex+h
(d) hex+h

• If f(x) = cos x then f =

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

• (x²+1)² 3=

(a) 1 -  
(b)
(c) 0
(d)

• -

(a) 2(x² + 1)
(b)
(c) 2x(x² + 1)
(d) 4x(x² + 1)

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

(a) ex. + c
(b)
(c)
(d) ex  x+c

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

(a) sec x tan x + c
(b) sect x tan x + c  km
(c) n (sec x - tan x) + c
(d) n (sec x + tan x) + c

• Two lines represented by ax² + 2hxy + by² = 0 are parallel if:

(a) h² - ab = 0
(b) h² + ab = 0
(c) a+b=0
(d) a-b=0

• If n₁ and m₂ are slopes of two lines then lines are perpendicular if:

(a) m₁n₂ = 0
(b) m₁n₂ + 1 = 0 V
(c) m₁n₂ - 1 =0
(d) m₁ + n₂ = 0

• The distance between the points (0, 0) and (1, 2) is:

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

• Two non-parallel lines intersect each other at:

(a) 1 point V
(b) 0 point
(c)  points
(d) 2 points

• (1, 0) is solution of inequality:

(a) 9x + 2y < 8
(b) - x + 3y < 0
(c) 3x + 5y < 6
(d) 3x + 5y > 4

• x = a sec , y = b tan  represents the parametric equations of:

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

• Length of each latusrectum of an ellipse is:

(a)  
(b) 
(c)  
(d)  

• An angle in a semi-circle is:

(a) Right angle
(b) Obtuse angle
(c) Reflexive angle
(d) 0°

• 2  = :

(a) 0
(b) 2
(c) 1
(d) 4

MATHEMATICS-II
Inter (Part-II) 2017
Mathematics
Group-I
PAPER: II
Time:2: 30 Hours
Marks: 80
(SUBJECTIVE TYPE)

1. Express area of circle as a function of its circumference.
2. Show that x=at2,y=2at, are parametric equations of parabola y2=4ax
3. Evaluate
4. Find derivative of
5. Differentiate
6. Define increasing and decreasing function
7. Determine the interval in which f(x)=4-x2 is increasing x  (-2,2)
8. Find , if y=sin h-1
9. Find f(x) if f (x)=ex(a+n x )
10. Find  if y=tan h(x2)
11. Find  if y=sin1
12. Differentiate x2.sec4 x

3. Write short answer to any EIGHT(8) questions

1. Using differentiate, find   in the equation xy +x=4
2. Evaluate
3. Evaluate
4. Evaluate
5. Evaluate
6. Evaluate
7. State fundamental theorem of calculus in definite integral
8. Evaluate
9. What is order of a differential equation?
10. Solve the differential equation sin y cosec x
11. Indicate the solution set of the system of linear inequalities x + y > 5 and –y +x < 1
12. Graph the solution region linear inequality x + y < 5

4. Write short answer to any NINE (9) questions

1. Find the mid-point of the line joining the two points A(-8,3),B(2,1)
2. The points A(-5,-2) and B (5,-4) are ends of  a diameter of a circle, find the centre and radius of the circle.
3. By means of slope, show the points line on the same line A(-1,-3),B(1,5),C(2,9)
4. Find an equation of the vertical line through (-5,3)
5. Convert the equation into two intercepts from 4x+7y-2=0
6. Find the equation of circle with ends of a diameter at (-3,2) and (5,-6)
7. Find the focus and directrix of the parabola y2=12x
8. Find the equation of ellipse when foci (3,0) and minor axis of length 10
9. Write the standard equation of hyperbola
10. Fins the vector from the point A to the origin where  and B Is the point (-2,5)
11. Find , so that
12. 
13. Find the value of 3

Note: Attempt any THREE(3)questions
Q.5(a) Find
(b) Find  if x=a (cos t + sin t), y=a(sin t – t cos t)

Q.6 (a) Evaluate
(b) The vertices of a triangle are A(-2,3),B(-4,1),C(3,5), find the centre of the Circumcircle of the triangle

Q.7 (a) Find area between x-axis and curve y =
(b) Minimize f(x,y) = x+3y subject to constraints 2x+5y<30, 5x+4y<20    x>0,y>0

Q.8(a) Show that the line 3x-2y=0 and 2x+3y-13=0 are tangents to the circle x2+y2+6x-4y=0
(b) Prove that in any triangle ABC b=c cos A + a cos C by using vectors

Q.9 (a) Find the point of intersection of
(b) Fsind the vaues of (i)  (ii)

MATHEMATICS-II
Inter (Part-II)
Mathematics 2017
Group-II
PAPER: II
Time: 30 Minutes
(OBJECTIVE TYPE)
Marks: 20
Note: Four possible answers, A, B, C and D to each question are given. The choice which you think is correct, fill that circle in front of that question with Marker or Pen ink in the answer-book. Cutting or filling two or more circles will result in zero mark in that question.

• x = at², y = 2at are the parametric equations of:

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

(a) e^1/5
(b) e⁵
(c) e^-5
(d) e^-1/5

• If f(x) = cos h x then f(x)² – f’(x)² =

(a) 0
(b) 1
(c)
(d) 22

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

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

• If y = cosx, u = sin x then

(a) cos x
(b) -cot x
(c) -tan x
(d) -cosec x

• a₀+a₁x+a₂x²+--------a_nxⁿ+--------is

(a) Maclaurin's series
(b) Taylor series
(c) Power series
(d) Binomial series

(a) ln cot x + c
(b) ln cos x + c
(c) ln sin x +c
(d) ln sec x +c

(a) ex tan x + c
(c)  + c
(b) ex sin x  + C
(d) ex tan-1 x + c

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

• The solution of   = -y is:

(a) y = e²x
(b) y ce^-x
(c) y = e^x
(d) ce^x

• Equation of horizontal line through (a, b) is:

(a) y=a
(b) y = b
(c) x = a
(d) x = b

• The two lines a₁x + b₁y = C₁;  a₂x b₂=y = c₂ are parallel if:

(a) a₁ a₂ = 0
(b) a₁ - b₁ =
(c) a₁b1 - a₂b₂ = 0
(d) a₁b₂ a₂b₁ = 0

• The inclination of x = y is:

(a) 30v
(b) 60°
(c) 45°
(d) 180°

• If the line (3x y + 5) + k(2x - 3y -4) = 0 is parallel to y-axis, then k

(a)
(b)
(c)
(d) 0 

• ax + b < c is:

(a) Linear inequality
(b) Identity
(c) Equation
(d) Not inequality

• If the ends of the diameter of the circle are (0, 1) and (2, 3), then its area is:

(a) π 
(b) 2π
(c) 4π
(d) 8π

• The directrix of the parabola x² = -8y is:

(a) x + 2= 0
(b) x-2=0
(c) y +2= 0
(d) y -2 = 0

• If vectors 2 and are perpendicular, then =:

(a) 1
(b) 2
(c) 3
(d)4

• [ a b a] if is equal to :

(a) 1
(b) a
(c) 0  
(d) b

MATHEMATICS-II
Inter (Part-II)
Mathematics 2017
Group-II
PAPER: II
Time: 2:30 Hours  
(SUBJECTIVE TYPE)
Marks: 80

2. Write short answer to any EIGHT(8) questions

1. If f(x)= and g(x)= find g of (x)
2. If f(x)=(-x+9)3 find f-1(x)
3. Evaluate
4. If y=(x2+5)(x3+7) find
5. Find  if y2+x2-4x=5
6. Differentiate  w.r.t.x
7. Differentiaten(x2+2x) w.r.t.x
8. If y=sin h-1 (ax+b)find
9. If y= cos y find
10. Prove that  sin -1x =
11. Define point of inflection
12. Define critical point

3. Write short answer to any EIGHT(8) questions

1. Find dy in y=x2+2x when x changes from 2 to 18
2. Evaluate
3. Find
4. Evaluate
5. Evaluate
6. Evaluate
7. Write two properties f define integration
8. Evaluate
9. Solve the differential equation  
10. Find the area between the x-axis and the curve y=4x-x2
11. Define optimal solution
12. Define decision variable

4. Write short answer to any NINE(9) questions

1. Show that the points A(-1,2),B(7,5) and C (2,-6) are vertices of  a triangle
   
2. Find h, such that A(-1,h), B (3,2) and(7,3) are collinear
   
3. Convert the equation 4x +7y-2=0 into two intercept form
   
4. Find an equation of the perpendicular bisector of the segment joining the points A (3,5) and (9,8)
   
5. Check whether the point (-4,7) is above or below of the line 6x-7y+70=0
   
6. Find an equation of tangent to the circle x2+y2=2 parallel to the line x-2y+1=0
   
7. Define focal chord of parabola
   
8. Find centre and vertices of ellipse =1
   
9. Find vertices and equation of directories of hyperbola x2-y2=9
   
10. Find a vector of length 5, in the direction of opposite that of v=i-2j+3k
    
11. Find a scalar , so that the vectors 2i+j+5k and 3i+j+k, are perpendicular
    
12. If a+b+c=0, ythen prove that a×b=b×c=c×a
    
13. Prove that the vectors i-2j+3k,-2i+3j-4kamd i-3j+5k are coplanar

Section-II

Note : Attempt any THREE questions
Q.5(a) Prove that
(b) If y=tan (p tan-1x),show that (1+x2)y1-p(1+y2)=0

Q.6(a) Show that
(b) find the point three-fifth of the way along the line segment from A(-5,8) to B(5,3)

Q.7(a) Evaluate
(b)Graph the feasible region subject to the following constrains :

Q.8(a) find a joint equation to the pair if tangents drawn from(5,0) to the circle x2+y2=9
(b) prove that the angle in a semi-circle is a right angle

Q.9(a) derive equation of hyperbola in standard form
(b) by vector method, prove in any triangle ABC,