Mathematics Lahore Board 2015
Objective Type
Time:30 Min
Inter Part-I
Group - I
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.

• If in is a negative integer than  is :
  

(A) 1  
(B) Not defined
(C) Zero         
(D) n

• Expansion of (1 + x) /4 is valid only if :
  

(A) |x|>1         
(B) |x|<1
(C) |x|<-1        
(D) |x|>-1

• For any triangle ABC, with usual notations r1 is equal to:

(A)  
(B)  
(C)  
(D)

• One radian is equal to :
  

(A) 57.296° 
(B)  57°
(C) 56°
(D) 0175°

• Types of rational fractions are :
  

(A) 3  
(B) 2
(C) 4   
(D) 1

• Sin 2α is equal to :
  

(A) 1- 2 sin² α            
(B) 2cos² α -1
(C) 2 sin α cos α        
(D) sin α

• The roots of x² + x - 6 = 0 are
  

(A) Real         
(B) Equal
(C) Complex  
(D) Trivial

• The power set of the empty set is :
  

(A) Empty set
(B) Non empty set
(C) Proper set 
(D) Improper set

• Pakistan and India play a hockey match, probability that Pakistan will win:
  

(A)  
(B)  
(C)  
(D) 1

• The middle term in the expansion of (1 + 2x)6 is :
  

(A) Third        
(B) Fourth
(C) Fifth         
(D) Sixth

• If an-an-1 = n + 1 and and a4 =14 then a⁵ is equal to :
  

(A) 9   
(B) 14
(C) 20 
(D) 5

• The domain of relation f = Ra,1),(b,1),(c,1)} is:
  

(A) {a, b, c}   
(B) {a}
(C) {b}
(D) {1}

• The sum of roots of equation axe + bx + c = 0, a ≠0 is:
  

(A)    
(B)  
(C)  
(D)

• Inverse of a square matrix A does not exist if A is
  

(A) Singular   
(B) Non singular
(C) Unit          
(D) Diagonal

• cos^-l (- x) is equal to :
  

(A) cos^-l x        
(B) π cos^-l x
(C) π-cos^-l x     
(D) sin^-l x

• The range of y = sin x is equal to:
  

(A) -1 ≤ y ≤ 1       
(B) -1< y <1
(C) -1≤x≤1 
(D) -1< -y<1

• Solution of equation tan x =  1 is:

(A) -1≤y≤1         
(B) -1<y<1
(C) -1≤x≤1         
(D) -1≤y<1

• Rank of matirx  is
  

(A) 1    
(B) 2
(C) 3   
(D) 4

• The geometric mean between 1 and 16 is :
  

(A) 4    
(B) -4
(C) ±4
(D)

• (-i)¹⁹ is equal to:

(A) -i   
(B) i
(C) 1   
(D) -1

Mathematics Lahore Board 2015
Subjective Type
Time:2:30 hrs
Inter Part-I
Group - I
Marks - 80

SECTION - I

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

• Write any two properties of inequalities.
• Show that  is a real number.
• Construct the truth table for (p ^ ~ p) à q .
• For A = {1, 2, 3, 4}, find the relation {(x, y)| y = x} in A.
• Define semi-group.
• If a, b being elements of a group G, then solve the equation ax = b.
• Find x and y if    
• Find the inverse of A=
  
• Without expansion verify that
• Prove that  = -16
• Show that (x - 2) is a factor of x⁴ - 13x² + 36
• Show that the roots of px² - (p - q) x - q = 0 are rational.

3. Write short answers to any EIGHT (8) questions. (16)

• Resolve  into partial fractions.
  
• Find the sequence if a_n – a_n-1= n + 1 ; a₄ = 14
  
• If are in A.P show that b = 
  
• Find G.M between -21 and 8i
  
•      show that 2ab=a+b
  
• Find the value of n when 11Pn =11.10.9
  
• Evaluate ²⁰C₁₇ .
  
• A die is rolled. What is the probability that dots on the top are greater than 4?
  
• Find the number of diagonals of a 6 sided figure.
  
• Prove that n! > 2n -1 for n = 4 , 5
  
• If x is so small that its square and higher powers neglected, then show that  
  
• Expand (1 + 2x)-1 upto four terms.

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

• Define radian.
• Prove that sec θ cosec θ sin θ cos θ =1
• Prove that sin 75°. (Without using calculator)
• Prove that
• Find the value of cos 2α if cos
• Prove that
• Find the period of 3 cos
• Find the area of the triangle ABC, if b = 37c=45,c=45α=30° 50'
• Prove that rricrr₁r₂r₃ = Δ²
• Define angle of elevation.
• Prove, that tan^-1  =2 cos^-12  
• Find the solution of 2 sin θ - sin θ = 0
• Find the solution cot θ =  which lie in [0,2,π]
  

SECTION-II

Note: Attempt any THREE questions.

5. (a) Solve the system of linear equation by Cramer's rule (05)
2x1 - x2 + x3 = 5
4x1 + 2x2 + 3x3 = 8
3x1 - 4x2 - x3 = 3
(b) Solve the equation 4x -3.2x+3 + 128 = 0 (05)

6. (a) Resolve  into partial fractions (05)
(b) For what value of n, is the positive geometric mean between 'a' and 'b'? (05)

7. (a)How many 6-digit numbers can be formed without repeating any digit from the digits , 1, 2, 3, 4, 5? Inhow many of them will 0 be at the tens place?       (05)
(b) If x is very nearly equal 1, then prove that pxp-qxqxp+q Prove the identity: sine 0 + cos6 0 = 1 - 3 sin 20 cos 20 (05)

8. (a) Prove the identity sin⁶ θ+cos⁶ θ=1-3sin² θ
(b) Prove the identity

9. (a) solve the triangle ABC in which a=7,b=3,=38o13 (05)
(b) Prove that tan-1 (05)