FAISALABAD BOARD 2013
MATHEMATICS
PART-I
Time: 20 Min.
(Objective Part)
Marks: 17

Note: You have four choice 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.

• Solution set of equation 1 + Cos x = 0 is:

(a) π         
(b) 2π       
(c)          
(d) -

• Cos  =

(a)                   
(b) -                          
(c) 300      
(d) 600

• Δ =    is called:

(a) Law of sines                     
(b) Law of cosine
(c) Law of tangents                
(d) Hero formula

• In any triangle ABC, with usual notation  =

(a) Cos α 
(b) Cos β
(c) Sin α         
(d) Sin β

• Period of Tan function is:

(a) π         
(b) 2π       
(c)          
(d) -                                

• Sin a =

(a) 2Sin   Cos         
(b) Sin    Cos            
(c) 2Sin   + Cos            
(d) Sin   + Cos

• The function of the form    q(x)≠ 0 is called:

(a) Identity                 
(b) Equation   
(c) Fraction                 
(d) Formula

• 5! = :

(a) 140            
(b) -120            
(c) 120            
(d) 120

• If AB exists, then (AB)^-1 is:

(a) A^-1B^-1        
(b) B^-1 A^-1           
(c) AB            
(d) A^-1 B

• A right triangle is in which one angle = :

(a) 45°            
(b) 90°            
(c) 270°          
(d) 360°

• Tan^-1 (-1)=:

(a)          
(b)                    
(c)          
(d) - π

• If a,b,c are sides of a triangle than  =

(a) Cos α              
(b) Cos ẞ        
(c) Sin ϒ         
(d) Sin ẞ

• Sin(-300°)= :

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

• The polynomial 3x² + 2x + 1 has degree: 

(a) 0        
(b) 3         
(c) 2        
(d) 4

• The number of terms in the expansion of (a + b)⁷ is: 

(a) 4        
(b) 6        
(c) 7         
(d) 8

• a > 0 :

 (a) -a > 0       
(b) 2a < 0        
(c)  > 0              
(d)

• The A.M. between x - 3 and x + 5 is: 

(a) x+ 1               
(b) x-1                  
(c) x- 3                
(d) x+ 5

•  rad = :

(a) 3600                
(b) 3350                
(c) 2700                
(d) 2250

• If 4x =   then x = :

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

• The period of tan      is:

(a)π         
(b)2π       
(c)3π       
(d)None

• The rank of is:
  
  (a) 0         
  (b) 1         
  (c) 2         
  (d) 3

• If n is odd, then the expansion of (x + a)n has:

(a) Two middle terms     
(b) 3 middle terms
(c) 4 middle terms           
(d) 0 middle terms

Time: 3:10 Hours
(Subjective Part)Marks: 83

Section II

• Name the properties used in the following equation:
  (a)4 + 9= 9 + 4              
  (b)1000 x l =10000
• Factorize: 9a2 + 16b2
• What is difference between {a,b} and {a,b}?
• Write the two properties of binary operation, when "S" is a non-empty set.
• If    and  A2 =   . find the value of a and b.
• Solve the matrix equation for X : 3x - 2A = B if
• A =            and B =
  
• State the two properties of A square matrix when |A|= 0.
•  If A is symmetric or skew - symmetric matrix, show that A2 is symmetric matrix?
• Solve for x : x(x + 7)= (2x - 1)(x + 4).
• What are the reciprocal equations?
• Evaluate: (1+  - )( 1-  + ).
• If the roots of the equation x2 - px + q = 0 differ by unity, prove than p2 = 4q + 1.

3. Attempt any EIGHT short questions. (8 x 2 = 16)

• Resolve          into partial fraction.
• Resolve          into partial fraction with unknown constants.
• Which term of A.P. 5,2,-1 ………….. is - 85?
• (iv)Find three A. Ms between  and .
• If  y =      +    x1   +    x3    and if 0 < x < 2 then prove that x =   .
• If first term of an H.P is  -   and fifth term is  ,find 9th term.
• Prove that npr = n. n-1Pr-1.
• In how many ways can '5' persons be seated at round table?
• Find the value of n'n if nC10 =  
• Pakistan and India play a cricket match, what is the probability that Pakistan wins.
• Find the coefficient of x5 in the expansion of  .
• Use binomial theorem find value of  up to three decimals places.

4.  Attempt any NINE short questions.   (9 x 2 = 18)

• Prove that sin³ θ + cos³ θ  = (sin θ + cos θ)(1 - sin θ cos θ).
• Prove that cos² θ – sin² θ =   .
• Express sin 5x + sin 3x as product.
• Prove that cos ³a = 4 cos ³a - 3 cos a.
• Prove that  = tan⁵ 60
• Prove that tan (2700 - θ)= cot θ.
• Find the period of tan   .
• Define Hero's formula.
• Define escribed circle.
• Prove that R =   
• The area of triangle is 2437 if a = 70, c = 97, find β .
• Solve the equation 1 + cos x = 0.
• Find the solution of cosec θ = 2 in [0, 2π] .

Attempt any THREE questions.  (8 x 3 = 24)

5. (a) Show that the set {1,ω ,ω²} when ω³ = 1 is an abelian group w.r.t ordinary ultiplication.
(b) Find inverse of the matrix, A =

6. (a) Show that (1 + ω )(1+ ω²)(1 + ω⁴)(1+ ω⁸)……….2n - factors = 1 where   is cube root of unity.
(b) Resolve into partial fraction .

7. (a) Find n so that    may be H.M. between 'a' and 'b'.
(b) If if y =  +    +      +   ……….. prove that y2 +2y -2 = 0.

8. (a) If cosec θ =   , m >0 ,  find value of remaining trigonometric functions?
(b) Prove that sin100 sin300 sin50° sin70° =   .

9. (a) Prove that in an equilateral triangle r: R: r₁: r₂ : r₃ =1: 2: 3 : 3 : 3.
(b) Prove that sin^-1 A + sin^-1  B = sin^-1 (A   + B  ).