Sec 3 A-Math Equations and inequalities worksheet with answers
A free Sec 3 A-Math worksheet on equations and inequalities: 9 questions, 40 marks, short questions first and then structured ones, in the school exam style. The answers and full worked solutions are on the last page.
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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
1
Find the range of values of k for which x2+7x+k=0 has no real roots.
Answer[2]
2
Find the x-coordinates of the points where the line y=−x+3 meets the curve y=x2+1.
Answer[2]
3
Solve x2+4x−12>0.
Answer[2]
4
Solve 2x2+10x−11=0 by completing the square, giving your answers correct to 2 decimal places.
Answer[3]
5
Find the range of values of k for which the equation x2+(k+3)x+k+6=0 has no real roots.
Answer[3]
6
(a)Find the x-coordinates of the points where the line y=3x−2 meets the curve y=x2−x+1.
Answer (a)[2]
(b)The line y=x−3 meets the curve y2+x=15 at two points. Find the x-coordinates of these points.
Answer (b)[2]
7
(a)Solve (x+3)(x−4)>0.
Answer (a)[2]
(b)Solve x2≤6x−5.
Answer (b)[2]
Section B
8
(a)Express 3x2−3x−8 in the form a(x+h)2+k.
Answer (a)[4]
(b)The equation kx2+6x+1=0 has two equal roots. Find the value of k.
Answer (b)[3]
(c)The line y=x−2 meets the curve 2x2−xy=3 at two points. Find the x-coordinates of these points.
Answer (c)[3]
9
(a)Show that the equation 3x2+(k+5)x+k+1=0, where k is a constant, has real and distinct roots for all real values of k.
[3]
(b)A curve has equation y=x2−3x+4 and a line has equation y=mx−5, where m is a constant. Find the values of m for which the line is a tangent to the curve.
Answer (b)[3]
(c)Find the range of values of m for which the line meets the curve at two distinct points.
Answer (c)[2]
(d)For the greater value of m found in part (b), find the coordinates of the point at which the line touches the curve.
Answer (d)[2]
END OF PAPER
Answers and worked solutions
Question 1
Answer: k>449[2]
b2−4ac<0(7)2−4(1)(k)<049−4k<0−4k<−49k>449
Question 2
Answer: x=−2 or x=1[2]
x2+1=−x+3x2+x−2=0(x+2)(x−1)=0x=−2 or x=1
Question 3
Answer: x<−6 or x>2[2]
(x+6)(x−2)>0x=−6 or x=2The curve is above the x-axis outside the roots.x<−6 or x>2
Question 4
Answer: x=0.93 or x=−5.93 (2 d.p.)[3]
x2+5x−211=0Half of +5 is 25, so the bracket is (x+25).(x+25)2=x2+5x+425(x+25)2−425−211=0(x+25)2=447x+25=±447x=2−5±447x=0.9278… or x=−5.9278…x=0.93 or x=−5.93 (2 d.p.)
Question 5
Answer: −5<k<3[3]
b2−4ac<0(k+3)2−4(1)(k+6)<0k2+2k−15<0(k+5)(k−3)<0k=−5 or k=3The curve is below the axis between the roots.−5<k<3
Question 6
(a)Answer: x=1 or x=3[2]
x2−x+1=3x−2x2−4x+3=0(x−1)(x−3)=0x=1 or x=3
(b)Answer: x=−1 or x=6[2]
(x−3)2+x=15x2−5x−6=0(x+1)(x−6)=0x=−1 or x=6
Question 7
(a)Answer: x<−3 or x>4[2]
x=−3 or x=4The curve is above the x-axis outside the roots.x<−3 or x>4
(b)Answer: 1≤x≤5[2]
x2−6x+5≤0(x−1)(x−5)≤0x=1 or x=5The curve is below the x-axis between the roots.1≤x≤5
Question 8
(a)Answer: =3(x−21)2−435[4]
3x2−3x−8=3(x2−x)−8Half of −1 is 2−1, so the bracket is (x−21).(x−21)2=x2−x+413x2−3x−8=3((x−21)2−41)−8=3(x−21)2−43−8=3(x−21)2−435
(b)Answer: k=9[3]
b2−4ac=062−4k(1)=036−4k=04k=36k=9
(c)Answer: x=−3 or x=1[3]
2x2−x(x−2)=3x2+2x−3=0(x+3)(x−1)=0x=−3 or x=1
Question 9
(a)[3]
b2−4ac=(k+5)2−4(3)(k+1)=k2−2k+13=(k−1)2−1+13=(k−1)2+12Since (k−1)2≥0 for all real k, b2−4ac≥12, which is positive.So the equation has real and distinct roots for all real values of k (shown).
(b)Answer: m=−9 or m=3[3]
x2−3x+4=mx−5x2+(−3−m)x+9=0Tangent: b2−4ac=0(−3−m)2−4(1)(9)=0(m+3)2=36m+3=6 or m+3=−6m=3 or m=−9
(c)Answer: m<−9 or m>3[2]
Two distinct points: b2−4ac>0(m+3)2−36>0(m+9)(m−3)>0m<−9 or m>3