compulsory. All parts of this section are to be answered on the separately provided 1st-year Math Paper 2019 Federal Board

_ OMR Answer Sheet which should be completed in the first 25 minutes and handed over to the

” Centre Superintendent. Deleting/overwriting is not allowed. Do not use lead pencil.

## instruction

*[—$————— EE ei SE Owe Lo notuse lead pence
~ Q.1 Choose the correct answer A/B/C / D by filling the refavant bubble for each question on the OMR
* Answer Sheet according to the instructions given there. Each part carries one mark.

— 1) What is the range of y=sin x?

_ A Beyet B. O<y<x c. Fsrst D. Osysa

_ 2) What is the general solution of sinx=0 nR7

a ax San

A {nea} B. {3% nea}

Cc. {ane ine Z} 0. {t2ne:ne Z}

. K)] Under which of the foliowing operations, the set 5 = {-1,0,1} is closed?

~ A Multiplication B. Onision c. Addition D. Subtraction

_ 4) Which of the following sets is equal to {x<Q: x7 =2}

- a {3 B 6 c. {tv2} dD {sl}

7 5) Which of the following binary relations from A = {1,2,3} to B= {(a,4,c} is a function?

_ A {(.a),(2,¢),(2,8)} 8. {0,4),(2,0),0,¢)}

“ c. {G,4),0,8),(2,¢),G,c)} 0. (Ga),(2,a),,c}}

6) Let 4 and B be the square matrices of the same order. Which of the following is true about A and 8?

_ A det( A) = det(B) B. —det( AB) = det((AB)’)

- Cc. det(A+ B) = det A+ det B D. det{ 4B) = det(B4)

7) If two roots of a cubic equation are 0 and /, then the cubic equation is:

~ A @oxe0 BP -teO Hl ae

7 x edse4

tial i ———?

. 8) What could be the partial fractions of: (2M? —8)

: 4 B c A B C+D

_ . e+ es

A ye2* Goa! Pode Bye? Gade adetd

~ A.B. CesD A.B. CxaD

= + St . Stoo os

_ CG 5-2 G2 ea 2ae4 D x-2 (2-2) ede

## 11 1st-year Math Paper 2019

11) It a fair die is rolled, then what is the probability that the top is a prime number?

aA 2 ap 3 c 1 o . 2 ~

5 2 2 : “3 _

x ~i

12) For what values of x, the binomial expansion of (2 -2) is valid? _

A |d>4 B |xf>2. c |af<4 Dd x[<2 ~

13) Haw many lines can be drawn between the five points in a plane? —

A 120 B. 60 Cc. 20 oO. 10 —

2″ –

14) Which term is the middle term in the expansion ox-2) ?

x

hn i * . _

A (n-1)” term B. (3-1) term C. ($+) term DB. (n-+1)” term

18) The radian measurement of the central angle of a circle of radius 6cm which cuts off an arc _

of l2car Jong is: _

A 3 B 4 an DB 2 .

16) Which of the following indentities is TRUE? _

A —*sin3@-=3sin@+4sin’é B. sin30 = 4sin@ + 3sin’ 0 – c, 60838 = 4c08°8 + 30088 D. 00836 = 4.008″ 8 — 30088 ~ 17) Which of the following is equal to ome( 22}? ~ A sinx B. cox c. —cosx D. -sinx *

18) What is primary period of sin 2x? 7

A de B. – GC 4s Dox _

19) Ina right angle triangle ABC , if the tengths of two non-perpendicular sides are § and 3, then what ~

will be the length of the third side? ~

A 4 a (MM 3 Dd 45 ~

## 1st-year Math Paper Federal

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-8 if required. Write your

_ answers neatly and legibly. Graph paper will be provided on request.

~ SECTION – 8 (Marks 40)

_ Q.2 Attempt any TEN parts. All parts carry equal marks. (10×4=40)

_ (} Express the complex number] + iN3 in polar form.

(3) Show that (4 — BY = 4’ B'(Demorgan’s Law) Where Aand B are subsets of a universal set UL’.

(aii) \f a,4 are elements of a group G under ihe operation of multiplication. Then show that (a)! = a”

tw) tt asa, bo cand Ae &. then show that 44-4 =(2-I)4

~ iv) Determine whether p — (g > p) is a tautoiogy. a contingency or an absurdity

” iv) —_ Discuss the nature of roots of 2×7 -5x+1=0

_ ive) if a number exceeds its square root by 56. Find the number.

_ (vn) Find the 13¢h term of the sequence x,1, 2~x. 3-2x,…

_ (ix) Find the sum of » terms of the seres whose n™ term is 3n* ++)

{x} Abox contains 10 red, 30 white and 20 black marbles. A marble ts drawn at random. Find the

probability that it is either red or white,

~ — 45

{xi) If x is so small that its square and higher powers can be neglected, then show that > =2+ = x

_ ax

in* (r+ 2s

— (ai) Prove that -— _-— Sin + O)tan(F +9) =cos@

cot’(4s – O}cos* (x – O)cos ec(2:x – @)

— (xii) Fa triangle 48C is with a= V3 -1, b= 3-1 and y =60° then find ¢

~ (xv) Without using calculator or table. prove that 2 tan”! i +tan”! L =

!

— SECTION – C (Marks 40)

= Note: Attempt any FIVE questions. All questions carry equal marks. {5×82 40)

Q.3 > Find the value of 4 for which the system:

aeyez=d

~ Bx+yp-Az=d

_ xe ly-27r=0

has a non-trivial solution, Also solve the system

Q.4 — Show that roots of x* + (mx +c)? =a* will be equal if ¢* = a°(i+m”)

_ 2 a9? pity

_ Q.5 Sum the following series to n terms: Ci bet peer +… tO n terms.

## 2019 Federal Board

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-8 if required. Write your

answers neatly and legibly. Graph paper will be provided on request.

~ SECTION – B (Marks 40)

_ Q.2 Attempt any TEN parts. All parts carry equat marks. (10×4=40)

_ (i) Simplify the following by justifying each step: ae

_ (i) Simplify the following complex number py expressing in the form a+ ib

2

~ VS-N-8

_ (th) Find all the fourtn roots of unity.

(ww) Complete the fotlowing table, to obtain that S = {a,b,c} is a semigroup under the operation *

— *|a bie

ajc 2 b&b

_ bia bc

Cf. . @

7 $2), 2 1G

(v) Find matrix Y if x=

_ 2 1|°7 5 10]

_ (vi) Find the numerical vatue of & if the polynomial x’ + fax? — 7x +6 has a remainder -$ when

divided byx+2 .

## 1st-year Math Paper *Resolve *

(vil) Resolve eo . into partial fractions

(x! -D(Qe+3)

— 2 Pye ‘

(iy yeleteZ4,. then show that x= 2 22)

_ 204 Loy

mont

_ tix} Find n if PiP=91

_ (9 Evaluate {32 correct to three places of decimal, by using binomial expansion.

ix) Prove that (sec@ – tan yé = 88

= J+sind

er 2 4 sii >

_ (ut) Prove that tan 56° = gost? +sin Ul” without using calculator

cost l*- sini]?

_ (xii) A ladder leaning against a vertcai wall makes an angle of 24° with the wall. Its foot is 5m from

the wall. Find its length.

7 (xiv) Without using table / calculator, prove that sin” ¥ + cor! 3 =

N

SECTION — C (Marks 40}

~ Note: Attempt any FIVE questions. All questions carry equal marks. (5×8=40)

_ Q.3 Prove that ~9.(p—> 9) – p is a tautology, where p and q are any two logical statements.

Q.4 Solve the fotlowing system by reducing their augmented matrix to the Echelon form:

— mp my