Federal Board HSSC-II (2016)
MATHEMATICS
Time- 25 Minutes
Marks: 20

SECTION - A (Marks 20)

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

• If u = 3i + 2k, v = 2j + k and w = -j + 4k then (u x v). w = ?
  

(A) 25
(B) √25
(C) 5√2
(D) 25a

• What is domain of f-1, when   
  

(A) Real umbers
(B) |1, ∞|
(C) |2, +∞|
(D) |-1, 1|

• For parametric equations x = at², y = 2at represent the equation.
  

(A) x₂ / a₂ – y₂/ b₂ = 1
(B) x₂+ y₂ =1
(C) y₂ = 4ax
(D) x₂ / a₂ + y₂/b₂ = 1

• = ?
  (A) Zero
  (B) e²n
  (C) e²
  (D) en

• Derivative of sin³ x w.r t cos² x is:-
  

(A)
(B)
(C)
(D) 3 sin² x

• d/dx ax = ?

(A) ℓna
(B) axℓna
(C) axℓnx
(D) axℓna

• Notation used for derivative of y=f(x) is:

(A)∫ ydx
(B) dy / dx
(C) f″(x)
(D) D2f/(x)

• If y = (2x + 5)3/2, then y2 will be:
  

(A) 3/ 2x +5
(B) 3(2x 4 5)1/2
(C)
(D) 6(2x + 5) -1/2

• ∫xexdx = ?
  

(A) xex + c
(B) xex
(C) xex – ex + c
(D) xex + ex + c

• ∫21 (x2 + 1)dx = ?
  

(A) X3/3 + x + c
(B) 10/33
(C) 10
(D) 10/3           

• Solution of ydx + xdy = 0 is:
  

(A) xy = 1  
(B) zero
(C) xy = 0 
(D) xy = c

• Two lines ℓ₁ and ℓ₂, with respective slopes m₁ and m₂ are parallel if:
  

(A) m₁ – m₂ = -1
(B) m₁ . m₂ = -1
(C) m₁ + m₂ = -1
(D) m₁ = m₂

• The equation of the straight line whose slope is 2 and y-intercept is 5 is:
  

(A) y-5 / x-2 = m
(B) y= 5x + 2
(C) y = x+2
(D) y = 2x + 5

• If lines are parallel, then solution:
  

(A) Does not exist
(B) Is finite
(C) Exists
(D) Is infinite

• An Expression involving any of the symbols < . > , ≤, ≥ is called:
  

(A) Inequality
(B) Equation
(C) Not inequality
(D) Identity

• The equation of the circle x2 + y2 +2gx + 2 fy + c = 0 has radius:
  

(A)
(B) g2 + f2 - c
(C) g2 + f2
(D) (-g, -f)

• A line that touches the curve without cutting through it is called:
  

(A) Tangent  
(B) Secant
(C) Radius
(D) Normal

• The point of parabola which is closest to the focus is the vertex of the:

(A) Circle
(B) Parabola
(C) Ellipse
(D) Hyperbola

• Unit vector in the same direction of vector v = |3,-4|
  

(A) 3(5), -4(5)
(B) 3i - 4j
(C)
(D)

• Attitudes of a triangle are always:
  

(A) Perfect squares
(B) Parallel
(C) Perpendicular
(D) Concurrent

MATHEMATICS HSSC-II (2016)
Time allowed: 2:35 Hours
Total Marks : 80

Note:  Attempt any ten parts from Section 'B' and any five questions from Section 'C' on the separately provided answer book. Use supplementary answer sheet i.e. Sheet-B if require(D) Write your answers neatly and legibly.

SECTION - B (Marks 40)

2. Attempt any TEN parts. All parts carry equal marks.

• Evaluate
  
• Graph the curve of the following parametric equations x = sec0, y= tanɵ where ɵ is a parameter.
  
• Find dx / dx if x2 -4xy -5y = 0
  
• If y =  
  
• Prove that (2y - 1) dy/dx = sec2x
  
• Find dy/dx if y = x.esinx
  
• Evaluate ∫x (√x + 1)dx (x > 0)
  
• Evaluate ∫0-2 dx
  
• Show that points A (3, 1), B (-2, -3) and C(2, 2) are vertices of an isosceles triangle.
  
• Find an equation of the line through (-4, -6) and perpendicular to a line having slope3/2
  
• Find the equation of the circle whose ends of diameter at (-3 .2) and (5, 6).
  
• Find an equation of the parabola whose focus is F(-3,4) and directrix is 3x – 4y + 5 = 0
  
• Find the point of intersection of the given conic 3x2 – 4y2 and 3y2- 2x2 = 7
  
• Prove that the line segment joining the mid points of two sides of  triangle is parallel to the third side and half as long.
  
• Find area of triangle determined by points P, Q and R P (0,0,0), Q (2,3,2), R (-1,1,4)

SECTION - C

Q.3 If f(x) = 

                                                                               -
Q.4 Show that

Q.5 Solve the differential equation

Q.6 Find the interior angles of the triangle whose vertices are A(-2, 11), B(-6, -3), C(4, -9).

Q.7 Maximize f (x.y) = 2x + 5y subject to the constraints -x ≤ 8; -y ≤ 4 ; x ≥ 0;  y ≥ 0

Q.8 Let a be a positive number and 0 < c < (A) Let F (c, 0) and F’(-c, 0) be two given points prove that the locus of points P (x, y) such that |PF| + |PF| = 2a is an ellipse.

Q-9 Find a unit vector perpendicular to the plane containing a and I, . Also find sine of angle between them.