D.G KHAN MATH 2012 PART-I
(OBJECTIVE) 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. Attempt as many questions as given in objective type question paper and leave others blank. Write the letter A, B, C or D in the column (Write coma option) against each question. If there is a contradiction in the bubble and band written answer, bubble option will be considered correct
QUESTION NO (1)

•   R (Set 7 real numbers) ab  R is called
  

  (A) Closure law          
  (B) Commutative law
  (C) Associative law    
  (D) Distributive Law
  

•  (2) -10 is equal to   
  (A) 1   
  (B) - i  
  (C) -1  
  (D) i
  
  
• If A and B are any two sets, then logical form of (AUB)is
  (A)  (P a ^ q)
  (B) P a ^ q      
  (C) P V q        
  (D)(P V q)
  
  
• Function which is on to is called:
  (A) Objective
  (B) Subjective
  (C) Injection
  (D) Bijective
  


• The value of  is      
  (A) 0
  (B) 2
  (C) -1
  (D) -3
  
  
• Four fourth roots of 81 are:          
  (A) 4 ,4i
  (B) 2 , + 2 i
  (C) 3 , 3i
  (D) +9, +9 i
  
  
• Partial fractions of will A form the
  (A) +
  (B) +
  (C) +
  (D) ++
  
  
• If a1=5 an other consecutive terms are 23 , 26, 29 , then 3rd  term is
  (A) 8
  (B) 11
  (C) 14
  (D) 17
  


• Fifth term of G-P  3 , 6 , 12, …… is:       
  (A) 15
  (B) 48 
  (C) 2
  (D) 36
  
  
•  n different objects can be arranged taken all at a time
  (A) (n-l) ! Ways
  (B) (n+1) ! Ways
  (C)n! Way
  (D) (2n) ! Ways
  
  
• When one dice is rolled, the probability of even number :
  (A)  
  (B)  
  (C)  
  (D)
  
  
• ++…..) equal to:

1. )
2.  )
3. ) 
4. )        

  MATHEMATICS TIME: 2.30 HOURS
SUBJECTIVE  MARKS : 80   
  SECTION-I

• Differentiate between rational number and irrational number
• Factorize 9 a2 + 16 b2
• From suitable properties of union and intersection , prove An (AUB) = AU(A
• B)
• Write the inverse and contra positive of p                                                              
• Define function
• If (G , * ) is a group and aG, there is a unique inverse of a in G. Prove it
• Define singular and non singular matrices
• Without expansion, show that =0                                                 
• If the matrices A and B are symmetric and AB = BA , show that AB is symmetric
• Derive quadratic formula
• Solve the equation 5x5- 5x = 0
• The difference between the cubes of two consecutive even numbers is 296. Find them ,

QUESTION NO. 3 Attempt any eight (8) short questions                              16
                                                                                                                                        

• Resolve   nto partial fraction
• What is a rational fraction?
• Find the next two terms of the sequence 1 , 3 , 5 , - , 9 , -11, ………
• If an - 2 = 3n - 11 Find the nth term of the sequence
• Find two A.Ms between  and 3
• Find 5th  term of G.P . 3, 6 , 12,……
• If 5 is H.M. between 2 and b Find b
• Write n(n-I)(n-2) in factorial form
• What is the probability that a slip of numbers divisible by 4 are picked from the slips bearing numbers 1,2,3………10?
• Find the number of the diagonals of a 6 - sides figure
• Show by Mathematical induction that for all positive integer n  n2 + n is divisible by 2                                                                                        
• Expand (8 - 2n)-1 up to three terms

QUESTION NO, 4 Attempt any Nine (9) short questions           18 

• Convert the system into radian
• Prove that Sin2:::=1:2:3:4
• If α,β,ϒ be the angle of triangle ABC, Prove that Cos= sin
• Prove that Sin(45 )= (Sin +Cos )
• Prove that Cot α-tan α=2Cot2 α
• Express the sum or difference as product . Cos 6 + Cos 3
• Find the Range of Sin
• Find the period of tan 17
• Solve the Right triangle, ABC in which    β=50° 10’ C = 0.832.
• Solve the triangle ABC in which b= 95 c = 34 and α = 52°
• Find the area of triangle ABC a =18 b = 24 c = 30
• Find the solution of equation lie in [ 0 , 2 π] sec x= - 2
• Solve the trigonometric equation Sec2 = 4/3

SECTION-II Note: Attempt any three questions from this section

5.(a) If (G,* ) is a group , then prove that
(i) Identity element is unique        (ii) Inverse of each element is unique
(b) If A= then prove that.
(i) If A+  is skew Hermitian                                    (ii) A- is Hermitian

6.(a) Show that roots of x2 + (mx +c)2 = a2 will be equal if c2= a2 (1+m2 )
(b) Which term of the sequence x2-y2,x+y,   ,……… is   =?

7.(a) A die is thrown , find the probability that the dots on the top are Prime Numbers or Odd Numbers
(b) Find the co-efficient of xn in the expansion of                                                                     

8. (a) Prove that Si + Co                                                     
(b) lf α β, ϒ,  are angle of a triangle ABC.
Show that   Cot +Cot +Cot = Cot Cot Cot

9. (a) Prove that  +  + +  =
(b) Prove that tan-1 + tan-1 = tan-1+tan-1