SARGODHA BOARD --2016
MATHEMATICS
PART-I
(Objective Part)
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.

Q.1

• The multiplicative inverse of complex number (0, -1) is equal to:

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

• The number of subsets of a set of four elements is equal to:

(a) 16
(b) 8
(c) 4
(d) 6

• If A is a square matrix of order 3 x 3 then |KA| is equal to:

(a) K|A|
(b) K2|A|
(c) K3 |A|
(d) K4|A|

• The trivial solution of the homogeneous linear equations ir equal to:

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

• If x3 + 3x2 - 6x + 2 is divided by x+2 then the remainder is:

(a) -18
(b) 9
(c) -9
(d) 18

• If the roots of the equation x2 + px + q = 0 are additive inverse of one another then:

(a) p= 1
(b) p= 0
(c) q= 1
(d) q= 0

• (x-1)(x-2) is:

(a) Proper fraction
(b) Identity
(c) Improper fraction
(d) Equation

• Arithmetic Mean between √2 and 3√2 is equal to:

(a) 2√5
(b) √2
(c) 3√2 
(d) 2

• 1, ½, 1/3, ¼, ……….. is equal to:

(a) H.P
(b) A.P
(c) G.P
(d) Arithmetic Series

• 4P3  is equal to:

(a)4^P1
(b) 4^P2
(c) 4^P4
(d) 5^P4

• P(E) is equal to:

(a) 1 - P (E)
(b) 1 + P (E)
(c) P (E) -1
(d) 2 - P (E)

• An algebraic expression consisting of two terms is called:

(a) Monomial
(b) Trinomial
(c) Polynomial
(d) Binomial

• Number of terms in the expansion of (a +b)n is:

(a) n
(b) n+1
(c) n-1
(d)  n-2

• The vertex of an angle in standard form is at:

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

• 
  Cot  is equal to:

(a) tan θ
(b) – tan θ
(c) cot θ
(d) None of these

• The period of sin2x is equal to:

(a) π
(b) π/2
(c) π  /3
(d) 2 π           

• R is equal to:

(a)∆/abc
(b) abc/∆
(c) a/2sin α
(d) ∆/s

• r1 is equal to:

(a) ∆/s-a
(b) ∆/s-b
(c) ∆/s-c 
(d)s-a/∆

• sin is equal to:

(a) π/3
(b) π/6
(c) -1/2
(d) 1/2

• Solution set of tan2x = 1 in [0,2 π] is equal to:

(a) { π /4, 5 π /4}
(b) { π /8, 5 π /8}
(c) { π /4, 3 π 4}
(d) { π /6, 5 π /6}

(Subjective Part)
SECTION-I

2. Attempt any Eight Parts.

• Define irrational numbers.       
• Simplify (-1)-21/2
• Find multiplicative inverse of (1 , 0).
• Write down Power set of set {9, 11}.
• Find converse, inverse of the conditional ~p→~q .
• Solve the equation a x = b where a, b ϵ G and G is a group.
• If A = find A(A)t
• Find inverse of
• If A   and A2=  find values of a and b.
• Evaluate (1+ -)(1- +)
• State factor theorem.
• Discuss nature of roots of equation x2 — 5x + 6 = 0 .

3. Attempt any Eight Parts.

• Define conditional equation.
• Find three A.Ms between 3 and 11.
• How many terms of -7 + (-5) + (-3) +………..amount to 65.
• If a = 2i, b = 4i show that AH = G2.
• Find the sum of 9/4+ 3/2+1+ 2/3+ ………∞
• Define an Event.
• Write the formula for addition of probabilities.
• Find n if 11 Pn = 11.10.9.
• Evaluate 20C17.
• Prove that +++………+=  is true for n=1, 2.
• Expand  4 by Binomial theorem.
• Expand(4-3x)1/2 up to two terms.

4. Attempt any Nine Parts.

• What is the circular measure of the angle between the hands of watch at 4 O’ clock?
• Prove that 2 sin 45° + ½ cosec 45°=
• Prove that (sec θ — tan θ)2 = 1-sin θ/1+sin θ
• If α, β, γ are the angles of a triangle ABC, then prove the cos=sin γ/2
• Prove that tan (45° + A) tan (45° -A) =1.
• Express sin 5 θ + sin 3 θ as product.
• Find the period of sin x/5.
• Define angle of elevation.
• With usual notations, prove that cos α/2=
• With usual notations, prove that r = .
• Show that cos (sin-1 x) = V1- .
• Find the solutions of cosec θ = 2 which lie in [0,2]
• Solve the equation cosec2 θ = 4/3.

SECTION II

Attempt any THREE questions. Each questions carries 10 marks.

5.(a) Use Cramer's rule to solve the system of equations.

3x1 + x2 — x3 = -4
X1 + X2 — 2X3 =  -4
—x1 + 2x2 —x3 = 1

(b) Show that the roots of x2 + (mx+ c)² =a² will be equal I c²=a2 (1+ m2)

6.(a) Resolve into Partial Fractions

(b) If y = 2/3x + 4/9x2 + 8/27x3 + ….and 0< x< 3/2, then show that x= 3y/2(1+y)

7.(a) Prove that ^n-1Cr + ^n-1 Cr-1= ⁿ Cr .

(b) Use binomial theorem to show that

1+1/4+ 1.3/4.8 + 1.3.5/4.8.12 + ……=

8.(a) If cosec θ = m2+1/2m and m>0 , Find the values of the remaining trigonometric ratios.

(b) Show that (without Tables/calculator) cos 20° cos40° cos 80° = 1/8 .

9.(a) Prove that in an equilateral triangle:

r: R: r1 :r2 :r3 = 1:2: 3: 3: 3
(b) Prove that sin^-1  + cot^-1 3=