T-Leng Tuition · mugcornersg.com
Total: 40 marksSuggested time: 1 hour
Name: ______________________________Class: ____________ Date: ____________
Answer all questions. Show all your working. Give non-exact answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless the question says otherwise.
Section A
1Determine whether x2+2x−2 is always positive, always negative or neither for all real values of x. Answer[2]
2A shop's daily profit, $P, from selling x cakes is P=30x−x2−50. By completing the square, find the greatest daily profit. Answer[2]
3Write down the coordinates of the minimum point of the curve y=3(x−4)2+7. Answer[2]
4(a)Write down the coordinates of the minimum point of the curve y=3(x−1)2−4. Answer (a)[2]
(b)By completing the square, find the maximum value of y=12−6x−x2. Answer (b)[3]
5(a)A ball is thrown upwards. Its height, h m, t seconds after it is thrown is given by h=10+20t−5t2. By completing the square, find the greatest height of the ball. Answer (a)[2]
(b)A farmer has 24 m of fencing to make three sides of a rectangular pen; a long wall forms the fourth side. The two sides at right angles to the wall are each x m long. Find the greatest possible area, in m2, of the pen. Answer (b)[3]
Section B
6(a)Write down the coordinates of the minimum point of the curve y=3(x−1)2−4. Answer (a)[3]
(b)Determine whether −2x−x2 is always positive, always negative or neither for all real values of x. Answer (b)[3]
7It is given that f(x)=3x2+12x+17.(a)Express f(x) in the form a(x−h)2+k, where a, h and k are constants. Answer (a)[3]
(b)Write down the coordinates of the minimum point of the curve y=f(x). Answer (b)[1]
(c)Hence find the range of values of p for which f(x)>p for all real values of x. Answer (c)[2]
(d)Find the greatest value of f(x)20. Answer (d)[2]
8The equation of a curve is y=kx2+2x+k−1, where k is a constant, and the equation of a line is y=−2x−1.(a)Show that the x-coordinates of any points where the line meets the curve satisfy the equation kx2+4x+k=0. [2]
(b)Find the range of values of k for which the curve lies entirely above the line. Answer (b)[4]
(c)Find the positive value of k for which the line is a tangent to the curve. Answer (c)[2]
(d)For this value of k, find the coordinates of the point at which the line touches the curve. Answer (d)[2]
END OF PAPER
Question 1
Answer: NEITHER[2]
x2+2x−2=(x+1)2−1−2=(x+1)2−3(x+1)2≥0x2+2x−2≥−3So it is neither: it is positive for some x and negative for others. Question 2
Answer: Greatest profit=175[2] P=−(x2−30x)−50=−[(x−15)2−225]−50=−(x−15)2+225−50=−(x−15)2+175Greatest profit=175 Question 3
Answer: The minimum point is (4,7).[2] x−4=0 when x=4y=7The minimum point is (4,7). Question 4
(a)Answer: The minimum point is (1,−4).[2] x−1=0 when x=1y=−4The minimum point is (1,−4). (b)Answer: Maximum value=21[3] 12−6x−x2=−(x2+6x)+12=−[(x+3)2−9]+12=−(x+3)2+9+12=−(x+3)2+21(x+3)2≥0Maximum value=21 Question 5
(a)Answer: Greatest height=30 m[2] h=−5(t2−4t)+10=−5[(t−2)2−4]+10=−5(t−2)2+20+10=−5(t−2)2+30Greatest height=30 m (b)Answer: Greatest area=72[3] Length=24−2x(A=x(24−2x)A=24x−2x2A=−2(x2−12x)=−2[(x−6)2−36]=−2(x−6)2+72Greatest area=72 Question 6
(a)Answer: The minimum point is (1,−4).[3] x−1=0 when x=1y=−4The minimum point is (1,−4). (b)Answer: Neither[3]
−2x−x2=−(x2+2x)=−[(x+1)2−1]=−(x+1)2+1(x+1)2≥0−2x−x2≤1So it is neither: it is positive for some x and negative for others. Question 7
(a)Answer: f(x)=3(x+2)2+5[3] f(x)=3(x2+4x)+17=3[(x+2)2−4]+17=3(x+2)2+5 (b)Answer: (−2,5)[1] (x+2)2≥0, so f(x) is least when x=−2.Minimum point (−2,5) (c)Answer: p<5[2] f(x)≥5 for all real x, and f(x)=5 when x=−2.So f(x)>p for all real x when p<5. f(x)20 is greatest when f(x) is least, i.e. when f(x)=5.Greatest value =520=4 Question 8
(a)[2]
kx2+2x+k−1=−2x−1kx2+4x+k=0 (shown) (b)Answer: k>2[4] The curve lies above the line when kx2+4x+k>0 for all real x.So k>0 and b2−4ac<0.(4)2−4k2<0k2−4>0(k−2)(k+2)>0k<−2 or k>2Since k>0, k>2. (c)Answer: k=2[2] Tangent: b2−4ac=0(k−2)(k+2)=0k=2 (as k>0) (d)Answer: (−1,1)[2] 2x2+4x+2=0Equal roots: x=−2(2)4x=−1y=−2(−1)−1y=1