Mathematics Grade 12 KZN March 2015 Memo

Basic Education KwaZulu-Natal Department of Basic Education REPUBLIC OF SOUTH AFRICA QP RR RS RSS ES ER GRR RSS SSSR SR RR RR SRE R RRR RR ERE SEE E SS | foes _. MATHEMATICS ~ oe Cae COMMON TEST | ‘ () NATIONAL SENIOR CERTIFICATE Marks: 100 Time: 2 hours N.B. This question paper consists of 6 pages and an information sheet. Copyright Reserved Please Turn Over

Downloaded from hlayiso.cqm Mathematics arch 2015 Common Test NSC INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. lL. This question paper consists of 6 questions. 2. Answer ALL the questions. 3. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answers. 4. Answers only will not necessarily be awarded full marks. 5. An approved scientific calculator (non-programmable and non-graphical) may be used, unless stated otherwise. 6. If necessary, answers should be rounded off to TWO decimal places, unless stated ¢ ) otherwise. 7. Diagrams are NOT necessarily drawn to scale. 8. Number the answers correctly according to the numbering system used in this question paper. Write neatly and legibly. Copyright Reserved Please Turn Over

Downloaded from hlayiso.com Mathematics March 2015 Common Test NSC QUESTION 1 Given: g(x)=-x* + 4x? +11x—30 =~(x—2)(x—5)(x +3) 1.1 Write down the y ~ intercept of g. qd) 1,2 Write down the x ~ intercepts of g . (2) 1.3. Determine g’(x). QB) 14 Hence, or otherwise, calculate the coordinates of the turning points of g. (5) 1.5 Sketch a neat graph of g . Clearly indicating the intercepts with the axes and the coordinates of the turning points. (3) 1.6 Determine the value(s) of x for which g is concave up. (3) [17] QUESTION 2 21 Given: f(x)=—-x? +x, determine /’(x) from first principles, (5) 2.2 Determine:- 2.2.1 Sit y Aer -3, (4) 2.2.2 f'(a) if £0) (4) [13] QUESTION 3 Given the equation of the curve p(x) =x° +3x? +4x-8 3.1 Determine the gradient of the tangent to p at x =—1. (2) 3.2. Determine the equation of the tangent to p at x =—1. (4) 3.3. Show that the function p is increasing for all real value(s) of x. “) f11] Copyright Reserved Please Turn Over

Downloaded, from hlayiso.CQIM, 3915 common Test NSC Mathematics QUESTION 4 A container shaped in the form of a cylinder with no top has a volume of 340 ml. It has a radius of x cm and a height of A.cm. Note: 1 ml =1 cm’. 4.1 Write down the height (#) in terms of x. (2) 4.2 Show that the surface area (S) of the cylinder with no top is given by S = TH? + 680 . (2) x 4.3 Calculate the value of x for which the surface area of the cylinder will be a minimum. (5) 19] ' Copyright Reserved Please Turn Over

Downloaded from hlayiso.com NSC Mathematics QUESTION 5 In the diagram below, tangent RP, having equation y — x — 2 = 0, touches a circle with centre M(4 ;.4) at P and cuts the x-axis at R. PT is a diameter of the circle. V is a point on another tangent to the circle that passes through T such that MV = 10. Determine: 5.1 The equation of PT, the diameter of the circle. 5.2 The coordinates of P, the point of contact of tangent PR and citcle M. 5.3. The equation of the circle. 5.4 The coordinates of T . 5.5 Calculate the length of TV. Copyright Reserved March 2015 Common Test (4) (3) (3) (3) (3) [16] Please Turn Over

Downloaded from hlayiso.cor Mathematics rch 2015 Common Test NSC QUESTION 6 ' 6.1 Make use of sin (x+y) = sin xcos y + cos x sin y to derive an expansion of cos (x — y) in terms of sines and cosines of x and y. (4) 62 Ifcos 62° = m, express cos 2° in terms of m. (5) 6.3 Prove that: sin2(360° — A) + cos(90° + 2A) +cos*(180°—A) _ cos A sin A = 9 sin(90° — 2A) cos A + sin A ) ' 64 Given: 4sinx+6sin x cos x—3 cos x = 2. 6.4.1 Show that 4sin x + 6sin xcos x —3cos x =2 can be written as (2 sin x — 1) (2 +3 cos x) = 0 (2) 6.4.2. Hence or otherwise find the general solution of 4sin x +6sin x cos x —-3.cos x = 2. (5) 6.5 Inthe figure below, D, B and C are three points in the same horizontal plane. AD isa vertical tower, The angle of elevation from B to A is x. In addition, AB = AC, ACB = y and BD=q. A B 6.5.1 Express AB in terms of q and a trigonometric ratio of x. Q) 6.5.2. Determine, with reasons, the size of BACin terms of ¥y. (3) 6.5.3 Hence, show that BC. = 2g 608 ¥ (4) cos x [34] TOTAL MARKS: [100] Copyright Reserved Please Turn Over

Downloaded from hlayiso.com NSC Mathematics March 2015 Common Test INFORMATION SHEET: MATHEMATICS INLIGTINGSBLAD: WISKUNDE xe -b4Vb? -4ac 2a A=P(ltni) A= P(l=ni) A= P(l-iy’ A= P(L+i)" Yen yi T, =a+(n-Nd 8, =" @a+(n—Da) i=l fl T,, =ar""' S _ a” =I ; r#] S, =—-;-1le<r<l . r-1l l-r poe 1+)’ -1 purl d+] a t P= tim LEAD-L@) h->0 h xX, +x. + d=y(%)—¥)" + -94) my “2 1 Ye 2 2 y=mete yry, =m(x—-%x,) m=22 FL m=tand 2 7X] (x-a) +(y—b) =r? inaasce; =? _© a? =b? 467 —2be.cosA area ABC =Lab.sinC sind sinB sinc 2 sin(a: + B) = sina@.cos # + cosa.sin 8 sin(a -p) = sina.cos 6 —cosa.sin B cos(a+ f)= cos a.cos 8 —sin asin B cos(a— #)= cosa.cos # +sin a.sin B cos’ a-sin’ a cos 2a =41—2sin’ a sin 2a = 2sin a.cos @ 2cos* a —1 (x; y) > (xcos 6 + ysin@; ycos 0 —xsind) (x3 y) > (xcosO— ysind ; ycosO +xsin 0) x= o n n P(A) = WS) P(A or B) = P(A) + P(B)— P(A and B) nh peatbx b= LE-HO-H) Vi@-x) Copyright Reserved

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