ARAB REPUBLIC OF EGYPT
‫ ق‬/ ‫ أ‬/ ‫ع‬.‫ ج 〕ث‬/ 763〔
Ministry of Education
General Secondary Education Certificate Examination, 2015
〕 Old System – First Stage – First Session 〔
〕E.O / 15〔
‫التاريـخ‬
Differential Calculus & Trigonometry〕 Mathematics ]1[ 〔 Time: 2 Hours
‫] 〕باللغة اإلنجليزية‬1[ ‫التفاضل وحساب المثلثات 〔 رياضيات‬
‫التوقيــــــــع‬
‫االســـــم‬
‫التاريخ‬
‫التوقيــــــــع‬
، ‫ُروجع على النص العربى ومطابق لألصل اليدوى ويطبع على مسئولية اللجنة الفنية‬
〕 ‫〔 األسئلة فى صفحتين‬
. ‫ يسلم الطالب ورقة امتحانية باللغة العربية مع الورقة المترجمة‬: ‫تنبيه مهم‬
Remark: Calculators are permitted.
‫ ق‬/ ‫ أ‬/ ‫ع‬.‫ ج 〕ث‬/ 763〔‫تابع‬
Question 3: ( 6 Marks )
dy
at x = 0
dx
(b) ABC is a triangle in which a = 15 cm , b = 12 cm and
(a) If y = ( x 3 + x + 2 ) 9 , find
c = 8 cm . Find m ( B ) and the radius of the circumcircle of
Question 4: ( 6 Marks )
Question 1: ( 7 Marks )
(a) Find:
(a) Find:
x2  25
x 5
〕2〔
the triangle ABC.
I . Answer the following question:
lim
x 5
〕E.O / 15〔
,
lim
x 0
x6  27
2 x2 + x + 1
lim
lim
,
x
x3 + 1
x 3 x4  9
3
5
(b) If cos A =
, tan B =
, where A,B are two acute angles,
5
12
then find the value of each of: cos 2 A , sin ( A ‫ ـــ‬B ) without
sin 3x
x + tan x
dy
dx
(c) Without using calculator prove that: tan 75 ْ = 2 +
(b) If y = x tan 2x , find
3
II . Answer three only of the following questions:
Question 2: ( 6 Marks )
(a) Find the points on the curve y
= x2
+ 3x + 5 at which
the tangent to the curve makes an angle of measure 45 ْ
with positive direction of x ‫ ــ‬axis.
(b) Solve the  ABC in which a = 12 cm , m ( B ) = 65 ْ
‫االســـــم‬
and m( C ) = 40 ْ .
〕 ‫〔 بقية األسئلة فى الصفحة الثانية‬
using calculator.
Question 5: ( 6 Marks )
(a) Find the function of the average rate of change for the function
f (x) = 2x 2 + 5 , then calculate the rate of change of this
function at x = 2
(b) From a window 9 meters high from the ground surface, the
angle of elevation of the top of a tree was measured to be 12 ْ
and the depression angle of its base was measured to be 28 ْ .
Find the height of the tree to the nearest meter.
〕 ‫〔 انتـهـت األســئـلـة‬