05a further maths - fp3 questions v2 (1)
TRANSCRIPT
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
1/38
1. An ellipse has equation16
2 x+
9
2 y
= 1.
(a) Sketch the ellipse. (1)
(b) Find the value of the eccentricit e.(2)
(c) State the coordinates of the foci of the ellipse.(2)
!"# $une 2%%2 &n 1'
2. Find the eact value of the radius of curvature of the curve ith equation y = arcsin xat the point here x = 2
1 √2. (6)
!"# $une 2%%2 &n 2'
3. Solve the equation
l% cosh x + 2 sinh x = 11.
*ive each anser in the for ln a here a is a rational nu,er.(7)
!"# $une 2%%2 &n -'
4. I n = x x x n dcos2
%
⌡⌠
π
n ≥ %.
(a) "rove that I n =n
2
π − n(n / 1) I n / 2 n ≥ 2.
(5)
(b) Find an eact epression for I 6.(4)
!"# $une 2%%2 &n 0'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
2/38
5. (a) *iven that y = arctan - x and assuin3 the derivative of tan x prove that
x
y
d
d=
291
-
x+.
(4)
(b) Sho that
⌡⌠ -
-
%
d-arctan6 x x x = 91 (0π − -√-).
(6)
!"# $une 2%%2 &n 6'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
3/38
6. Figure 1
y
O x
4he curve C shon in Fi3. 1 has equation y2 = 0 x % ≤ x ≤ 1.
4he part of the curve in the first quadrant is rotated throu3h 2π radians a,out the x-ais.
(a) Sho that the surface area of the solid 3enerated is 3iven ,
0π ⌡⌠
+
1
%.d)1( x x
(4)
(b) Find the eact value of this surface area. (3)
(c) Sho also that the len3th of the curve C ,eteen the points (1 −2) and (1 2) is3iven ,
2⌡
⌠
+
1
%
.d1
x x
x
(3)(d ) 5se the su,stitution x = sinh2θ to sho that the eact value of this len3th is
2!√2 + ln(1 + √2)'.(6)
!"# $une 2%%2 &n '
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
4/38
7. "rove that sinh (iπ − θ ) = sinh θ .(4)
!"6 $une 2%%2 &n 1'
8 . A =
-00
0#%
0%1
.
(a) erif that
−
1
2
2
is an ei3envector of A and find the correspondin3 ei3envalue.
(3)
(b) Sho that 9 is another ei3envalue of A and find the correspondin3 ei3envector.(5)
(c) *iven that the third ei3envector of A is
− 21
2
rite don a atri P and a
dia3onal atri D such thatP4AP = D.
(5)
!"6 $une 2%%2 &n #'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
5/38
9. 4he plane Π passes throu3h the points
A(−1 −1 1) B(0 2 1) and C (2 1 %).
(a) Find a vector equation of the line perpendicular to Π hich passes throu3h the
point D (1 2 -). (3)
(b) Find the volue of the tetrahedron ABCD.(3)
(c) 7,tain the equation of Π in the for r.n = p.(3)
4he perpendicular fro D to the plane Π eets Π at the point E .
(d ) Find the coordinates of E .(4)
(e) Sho that DE =-#
-#11 .
(2)
4he point D′ is the reflection of D in ∏ .
( f ) Find the coordinates of D′.(3)
!"6 $une 2%%2 &n 8'
10. Find the values of x for hich
0 cosh x + sinh x =
3ivin3 our anser as natural lo3ariths. (6)
!"# $une 2%%- &n 1'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
6/38
11. (a) "rove that the derivative of artanh x −1 x 1 is 211
x−.
(3)
(b) Find ∫ x x d artanh .
(4)
!"# $une 2%%- &n 2'
12. Figure 1
Fi3ure 1 shos the cross:section R of an artificial ski slope. 4he slope is odelled ,the curve ith equation
#.% )90(
1%
2≤≤
+= x
x y
*iven that 1 unit on each ais represents 1% etres use inte3ration to calculate thearea R. Sho our ethod clearl and 3ive our anser to 2 si3nificant fi3ures.
(7)
!"# $une 2%%- &n -'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
x
y
O
R
#
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
7/38
13 . Figure 2
A rope is hun3 fro points P and Q on the sae hori;ontal level as shon in Fi3. 2.4he curve fored , the rope is odelled , the equation
cosh ka xkaa
xa y ≤≤−
=
here a and k are positive constants.
(a) "rove that the len3th of the rope is 2a sinh k .(5)
*iven that the len3th of the rope is a
(b) find the coordinates of Q leavin3 our anser in ters of natural lo3ariths andsurds here appropriate.
(4)
!"# $une 2%%- &n #'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
x
y
ka:ka
Q P
O
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
8/38
14. 4he curve C has equation
y = arcsec e x x < % % y π 21 .
(a) "rove that )1e(
1
d
d
2 −=
x x
y
. (5)
(b) Sketch the 3raph of C .
(2)
4he point A on C has x:coordinate ln 2. 4he tan3ent to C at A intersects the y:ais atthe point B.
(c) Find the eact value of the y:coordinate of B.(4)
!"# $une 2%%- &n 6'
15. I n = ⌡⌠
1
%
andde x x xn J n = ⌡
⌠ ≥−1
%
.%de n x x xn
(a) Sho that for 1≥n
.e 1−−= nn nI I (2)
(b) Find a siilar reduction forula for J n .(3)
(c) Sho that .e
#2
2 −= J
(3)
(d ) Sho that
⌡
⌠ 1
%
dcosh x x x n = 21 ( I n + J n ).
(1)
(e) ence or otherise evaluate ⌡⌠
1
%
2 dcosh x x x 3ivin3 our anser in ters of
e.(4)
!"# $une 2%%- &n 8'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
9/38
16. 4he hper,ola C has equation .12
2
2
2
=−b
y
a
x
(a) Sho that an equation of the noral to C at the point P (a sec t b tan t ) is
ax sin t + by = (a2 + b2) tan t .(6)
4he noral to C at P cuts the x:ais at the point A and S is a focus of C . *iven thatthe eccentricit of C is 2
- and that OA = -OS here O is the ori3in
(b) deterine the possi,le values of t for .2% π
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
10/38
19. .1
-#
111
11-
≠
−= u
u
A
(a) Sho that det A =2(u / 1).(2)
(b) Find the inverse of A . (6)
4he ia3e of the vector
a
b
c
÷ ÷ ÷
hen transfored , the atri
- 1 1 -
1 1 1 is 1 .
# - 6 6
− ÷ ÷ ÷ ÷ ÷ ÷
(c) Find the values of a b and c.
(3)
!"6 $une 2%%- &n 6'
20. 4he plane 1 Π passes throu3h the P ith position vector i + 2 j / k and is perpendicular to the line L ith equation
r = -i / 2k +λ (−
i + 2 j + -k ).
(a) Sho that the artesian equation of 1 Π is x / # y / - z = −6.(4)
4he plane 2 Π contains the line L and passes throu3h the point Q ith position vectori + 2 j + 2k .
(b) Find the perpendicular distance of Q fro 1 Π .(4)
(c) Find the equation of 2 Π in the for r = a + b + t c. (4)
!"6 $une 2%%- &n 8'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
11/38
21. 5sin3 the definitions of cosh x and sinh x in ters of eponentials
(a) prove that cosh2 x − sinh2 x = 1(3)
(b) solve cosech x − 2 coth x = 23ivin3 our anser in the for k ln a here k and a are inte3ers.
(4)
!"# $une 2%%0 &n 1'
22. 0 x2 + 0 x + 18 ≡ (ax + b)2 + c a < %.
(a) Find the values of a b and c. (3)
(b) Find the eact value of
⌡⌠
++−
#.1
#.%2 1800
1
x x d x!
(4)
!"# $une 2%%0 &n 2'
23. An ellipse ith equation09
22 y x+ = 1 has foci S and S ′.
(a) Find the coordinates of the foci of the ellipse.(4)
(b) 5sin3 the focus:directri propert of the ellipse sho that for an point P on theellipse
SP + S ′ P = 6. (3)
!"# $une 2%%0 &n -'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
12/38
24 . *iven that y = sinhn − 1 x cosh x
(a) sho that x
y
d
d = (n − 1) sinhn − 2 x + n sinhn x.
(3)
4he inte3ral I n is defined , I n = ⌡⌠
1arsinh
%
sinhn x d x n ≥ %.
(b) 5sin3 the result in part (a) or otherise sho that
nI n = √2 − (n − 1) I n − 2 n ≥ 2(2)
(c) ence find the value of I 0.
(4)
!"# $une 2%%0 &n #'
25. Figure 1
Fi3ure 1 shos the curve ith paraetric equations
x = a cos-θ y = a sin- θ % ≤ θ 2π .
(a) Find the total len3th of this curve.(7)
4he curve is rotated throu3h π radians a,out the x:ais.
(b) Find the area of the surface 3enerated.(5)
!"# $une 2%%0 &n 8'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
13/38
26. 4he points A B and C lie on the plane Π and relative to a fied ori3in O the have position vectors
a = -i − j + 0k b = −i + 2 j c = #i − - j + 8k
respectivel.
(a) Find AB AC × .(4)
(b) Find an equation of Π in the for r.n = p.(2)
4he point D has position vector #i + 2 j + -k .
(c) alculate the volue of the tetrahedron ABCD.(4)
!"6 $une 2%%0 &n -'
27 . 4he atri M is 3iven ,
M =
−
cba
p%-
101
here p a b and c are constants and a < %.
*iven that MM4 = k I for soe constant k find
(a) the value of p(2)
(b) the value of k (2)
(c) the values of a b and c
(6)(d ) det M.
(2)
!"6 $une 2%%0 &n #'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
14/38
28. 4he transforation R is represented , the atri A here
=
-1
1-A .
(a) Find the ei3envectors of A.(5)
(b) Find an ortho3onal atri P and a dia3onal atri D such that
A = PDP−1.(5)
(c) ence descri,e the transforation R as a co,ination of 3eoetricaltransforations statin3 clearl their order.
(4)
!"6 $une 2%%0 &n 6'
29. (a) Find ⌡⌠
−√+
)01(
12 x
x d x.
(5)
(b) Find to - decial places the value of
⌡⌠
−√+-.%
%2 )01(
1
x
x d x.
(2)
(!"a# 7 $ark%)
!F"2B"# $une 2%%# &n 1'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
15/38
30. (a) Sho that for x = ln k here k is a positive constant
cosh 2 x =2
0
2
1
k
k +.
(3)
*iven that f( x) = px / tanh 2 x here p is a constant
(b) find the value of p for hich f( x) has a stationar value at x = ln 2 3ivin3 our anser as an eact fraction.
(4)
(!"a# 7 $ark%)
!F"2B"# $une 2%%# &n 2'
31. Figure 1
Fi3ure 1 shos a sketch of the curve ith paraetric equations
x = a cos- t y = a sin- t % ≤ t ≤ 2π
here a is a positive constant.
4he curve is rotated throu3h 2π radians a,out the x:ais. Find the eact value of thearea of the curved surface 3enerated.
!F"2B"# $une 2%%# &n -'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
y
O x
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
16/38
32. I n = ⌡
⌠ ≥ .%de2 n x x xn
(a) "rove that for n ≥ 1
I n =2
1( xn e2 x / nI n / 1).
(3)
(b) Find in ters of e the eact value of
⌡⌠
1
%
22de x x
x .
(5)
!F"2B"# $une 2%%# &n 0'
33. Figure 2
Fi3ure 2 shos a sketch of the curve ith equation
y = x arcosh x 1 ≤ x ≤ 2.
4he re3ion R as shon shaded in Fi3ure 2 is ,ounded , the curve the x:ais andthe line x = 2.
Sho that the area of R is
0
8 ln (2 + √-) /
2
-√.
(!"a# 10 $ark%)!F"2B"# $une 2%%# &n 6'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
21
y
O x
R
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
17/38
34. (a) Sho that for % x ≤ 1
ln
−√− x
x )1(1 2
= /ln
−√+ x
x )1(1 2
.
(3) (b) 5sin3 the definition of cosh x or sech x in ters of eponentials sho that for
% x ≤ 1
arsech x = ln
−√+ x
x )1(1 2
.
(5)
(c) Solve the equation
- tanh2 x / 0 sech x + 1 = %
3ivin3 eact ansers in ters of natural lo3ariths.(5)
(!"a# 13 $ark%)
!F"2B"# $une 2%%# &n '
35. (a) (i) Cplain h for an to vectors a and b a.b × a = %.(2)
(ii) *iven vectors a b and c such that a × b & a × c here a ≠ 0 and b ≠ c shothat
b / c = λ a here λ is a scalar.(2)
(b) A ' and are 2 × 2 atrices.(i) *iven that A' = A and that A is not sin3ular prove that ' = .
(2)
(ii) *iven that A' = A here A =
21
6- and ' =
1%
#1 find a atri
hose eleents are all non:;ero. (3)
!F"-B"6 $une 2%%# &n 2'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
18/38
36. 4he line " 1 has equation
r = i + 6 j / k + λ (2i + -k )
and the line " 2 has equation
r = -i + p j + µ (i / 2 j + k ) here p is a constant.
4he plane 1 Π contains " 1 and " 2.
(a) Find a vector hich is noral to 1 Π .(2)
(b) Sho that an equation for 1 Π is 6 x + y / 0 z = 16. (2)
(c) Find the value of p.(1)
4he plane 2 Π has equation r.(i + 2 j + k ) = 2.
(d ) Find an equation for the line of intersection of 1 Π and 2 Π 3ivin3 our anser inthe for
(r / a) × b = 0.(5)
!F"-B"6 $une 2%%# &n -'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
19/38
37. A &
k 20
2%2
02-
.
(a) Sho that det A = 2% / 0k . (2)
(b) Find A /1.(6)
*iven that k = - and that
−12
%
is an ei3envector of A
(c) find the correspondin3 ei3envalue.
(2)
*iven that the onl other distinct ei3envalue of A is
(d ) find a correspondin3 ei3envector. (4)
!F"-B"6 $une 2%%# &n 8'
38. Cvaluate ⌡⌠
+−√
0
12 )182(
1
x x d x 3ivin3 our anser as an eact lo3arith.
(5)
!F"2B"# $anuar 2%%6 &n 1'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
20/38
39. 4he hper,ola # has equation16
2 x
/0
2 y = 1.
Find
(a) the value of the eccentricit of # (2)
(b) the distance ,eteen the foci of # .(2)
4he ellipse E has equation16
2 x +
0
2 y = 1.
(c) Sketch # and E on the sae dia3ra shoin3 the coordinates of the points
here each curve crosses the aes. (3)
!F"2B"# $anuar 2%%6 &n 2'
40. A curve is defined ,
x = t + sin t y = 1 / cos t
here t is a paraeter.
Find the len3th of the curve fro t = % to t =2
π
3ivin3 our anser in surd for.
(7)
!F"2B"# $anuar 2%%6 &n -'
41. (a) 5sin3 the definition of cosh x in ters of eponentials prove that
0 cosh-
x / - cosh x = cosh - x.(3)
(b) ence or otherise solve the equation
cosh - x = # cosh x
3ivin3 our anser as natural lo3ariths. (4)
!F"2B"# $anuar 2%%6 &n 0'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
21/38
42. *iven that
I n = ⌡⌠ ≥−√
0
%
%d)0( n x x xn
(a) sho that I n = -2+nn I n / 1 n ≥ 1.
(6)
*iven that ⌡
⌠ =−√
0
% -
16d)0( x x
(b) use the result in part (a) to find the eact value of⌡
⌠ −√0
%
2 d)0( x x x .
(3)
!F"2B"# $anuar 2%%6 &n 8'
43. (a) Sho that artanh
0sin
π
= ln (1 + √2).
(3)
(b) *iven that y = artanh (sin x) sho that x ydd = sec x.
(2)
(c) Find the eact value of x x x d)(sinartanhsin0
%
⌡⌠
π
.
(5)
!F"2B"# $anuar 2%%6 &n '
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
22/38
44. A transforation T @ ℝ2 → ℝ2 is represented , the atri
A =2 2
2 1
÷−
here k is a constant.
Find
(a) the to ei3envalues of A(4)
(b) a cartesian equation for each of the to lines passin3 throu3h the ori3in hich areinvariant under T .
(3)
!DF"-B"6 $anuar 2%%6 &n -'
45. A &
−
−
%19
1%
21
k
k
here k is a real constant.
(a) Find values of k for hich A is sin3ular.(4)
*iven that A is non:sin3ular
(b) find in ters of k A /1.(5)
!F"-B"6 $anuar 2%%6 &n 0'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
23/38
46. 4he plane Π passes throu3h the points
P (/1 - /2) Q(0 /1 /1) and R(- % c) here c is a constant.
(a) Find in ters of c RP RQ× .(3)
*iven that RP RQ× = -i + d j + k here d is a constant
(b) find the value of c and sho that d = 0 (2)
(c) find an equation ofΠ in the for r.n = p here p is a constant.(3)
4he point S has position vector i + # j + 1%k . 4he point S ′ is the ia3e of S under reflection in Π .
(d ) Find the position vector of S ′. (5)
!F"-B"6 $anuar 2%%6 &n 8'
47. Find the values of x for hich# cosh x / 2 sinh x = 11
3ivin3 our ansers as natural lo3ariths. (6)
!F"2 $une 2%%6 &n 1'
48. 4he point S hich lies on the positive x:ais is a focus of the ellipse ith equation
0
2 x + y2 = 1.
*iven that S is also the focus of a para,ola P ith verte at the ori3in find
(a) a cartesian equation for P (4)
(b) an equation for the directri of P . (1)
!F"2 $une 2%%6 &n 2'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
24/38
49 . 4he curve ith equation
y = / x + tanh 0 x x ≥ %
has a aiu turnin3 point A.
(a) Find in eact lo3arithic for the x:coordinate of A.(4)
(b) Sho that the y:coordinate of A is 01 E2√- / ln(2 + √-).
(3)
!F"2 $une 2%%6 &n #'
50. Figure 1
4he curve C shon in Fi3ure 1 has paraetric equations
x = t / ln t
y = 0√t 1 ≤ t ≤ 0.
(a) Sho that the len3th of C is - + ln 0.(7)
4he curve is rotated throu3h 2π radians a,out the x:ais.
(b) Find the eact area of the curved surface 3enerated.(4)
!F"2 $une 2%%6 &n 6'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
y
C
xO
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
25/38
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
26/38
51. Figure 2
Fi3ure 2 shos a sketch of part of the curve ith equation
y = x2 arsinh x.
4he re3ion R shon shaded in Fi3ure 2 is ,ounded , the curve the x:ais and theline x = -.
Sho that the area of R is
9 ln (- + √1%) / 91 (2 + 8√1%).
(10)
!F"2 $une 2%%6 &n 8'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
R
x
y
O -
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
27/38
52. I n = ⌡
⌠ x x x
n dcosh n ≥ %.
(a) Sho that for 2n ≥ ( )1 2sinh cosh 1
n n
n n I x x nx x n n I − −= − + − .
(4)
(b) ence sho that
I 0 = f( x) sinh x + 3( x) cosh x + C
here f( x) and 3( x) are functions of x to ,e found and C is an ar,itrar constant.(5)
(c) Find the eact value of x x x dcosh1
%
0
⌡⌠
3ivin3 our anser in ters of e.
(3)
!F"2 $une 2%%6 &n '
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
28/38
53. 4he ellipse E has equation2
2
a
x +
2
2
b
y = 1 and the line L has equation y = $x + c
here $ < % and c < %.
(a) Sho that if L and E have an points of intersection the x:coordinates of these points are the roots of the equation
(b2 + a2$2) x2 + 2a2$cx + a2(c2 / b2) = %. (2)
ence 3iven that L is a tan3ent to E
(b) sho that c2 = b2 + a2$2.(2)
4he tan3ent L eets the ne3ative x:ais at the point A and the positive y:ais at the point B and O is the ori3in.
(c) Find in ters of $ a and b the area of trian3le OAB.(4)
(d ) "rove that as $ varies the iniu area of trian3le OAB is ab.(3)
(e) Find in ters of a the x:coordinate of the point of contact of L and E hen thearea of trian3le OAB is a iniu.
(3)
!F"2 $une 2%%6 &n 9'
54. A =
1%%
11%
211
.
"rove , induction that for all positive inte3ers n
An =
+
1%%
1%
)-(1 221
n
nnn
.
(5)
!F"- $une 2%%6 &n 1'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
29/38
55 . 4he ei3envalues of the atri M here
M =
−11
20
are λ 1 and λ 2 here λ 1 λ 2.
(a) Find the value of λ 1 and the value of λ 2. (3)
(b) Find M /1. (2)
(c) erif that the ei3envalues of M /1 are λ 1 /1 and λ 2 /1. (3)
A transforation T @ ℝ2 →
ℝ2 is represented , the atri M. 4here are to lines passin3 throu3h the ori3in each of hich is apped onto itself under thetransforation T .
(d ) Find cartesian equations for each of these lines. (4)
!F"- $une 2%%6 &n #'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
30/38
56. 4he points A B and C lie on the plane 1 Π and relative to a fied ori3in O the have position vectors
a = i + - j / k b = -i + - j / 0k and c = #i / 2 j / 2k
respectivel.
(a) Find (b / a) (c a).(4)
(b) Find an equation for 1 Π 3ivin3 our anser in the for r.n = p.(2)
4he plane 2 Π has cartesian equation x + z = - and 1 Π and 2 Π intersect in the line " .
(c) Find an equation for " 3ivin3 our anser in the for (r − *) × + = 0. (4)
4he point P is the point on " that is the nearest to the ori3in O.
(d ) Find the coordinates of P . (4)
!F"- $une 2%%6 &n 8'
57. Evaluate ⌡⌠
−+√-
12 )#0(
1
x x d x , giving your answer as an exact
logarithm.(5)
!F"2 $une 2%%8 &n 1'
58. The ellipse D has equation2#
2 x +
9
2 y = 1 and the ellipse E has
equation
0
2 x +
9
2 y = 1.
(a !"etch D and E on the same diagram, showing the coordinates o# the points where each curve crosses the axes.
(3)
The point S is a #ocus o# D and the point T is a #ocus o# E.
(b $ind the length o# ST . (5)
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
31/38
!F"2 $une 2%%8 &n 2'59. The curve C has equation
y = 01 (% x % & ln x , x ' .
$ind the length o# C #rom x = .) to x = %, giving your answer in the#orm a + b ln %, where a and b are rational num*ers.
(7)
!F"2 $une 2%%8 &n -'
60. (a !tarting #rom the denitions o# cosh and sinh in terms o# exponentials, prove that
cosh( A & B = cosh A cosh B & sinh A sinh B.(3)
(b ence, or otherwise, given that cosh ( x & 1 = sinh x , show that
tanh x =1e2e
1e2
2
−++
.
(4)
!F"2 $une 2%%8 &n 0'
61. -iven that In = ⌡
⌠ −
6
%
-
1
d)6( x x x n , n ≥ ,
(a show that In =0-
20
+nn
In & 1, n ≥ 1.
(6)
(b ence nd the exact value o# ⌡
⌠ −+
6
%
-
1
d)6)(#( x x x x .
(6)
!F"2 $une 2%%8 &n 6'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
32/38
62.
Figure 1
$igure 1 shows part o# the curve C with equation y = arsinh (√ x , x ≥ .
(a $ind the gradient o# C at the point where x = .(3)
The region R, shown shaded in $igure 1, is *ounded *y C, the x /axisand the line
x = .
(b 0sing the su*stitution x = sinh% θ , or otherwise, show that thearea o# R is
k ln (% + √) & √),
where k is a constant to *e #ound. (10)
!F"2 $une 2%%8 &n 8'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
33/38
63. -iven that
−11
%
is an eigenvector o# the matrix A, where
A =
−−-11
01
0-
%
p
,
(a nd the eigenvalue o#A corresponding to
−11
%
,
(2)
(b nd the value o# p and the value o# q.(4)
The image o# the vector
n
$
"
when trans#ormed *y A is
−-
0
1%
.
(c 0sing the values o# p and q #rom part (b, nd the values o# theconstants l, m and n.
(4)
!F"- $une 2%%8 &n -'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
34/38
64. The points A, B and C have position vectors, relative to a xed originO,
a = %i & j,
b = i + % j + k ,
c = %i + j + %k ,
respectively. The plane Π passes through A, B and C.
(a $ind AB AC × .(4)
(b !how that a cartesian equation o# Π is x & y + % z = 7.(2)
The line l has equation (r & )i & ) j & k 2 (%i & j & %k = 0. The line land the plane Π intersect at the point T .
(c $ind the coordinates o# T .(5)
(d !how that A, B and T lie on the same straight line.(3)
!F"- $une 2%%8 &n 8'
65. Sho that
xd
d!ln(tanh x)' = 2 cosech 2 x x < %.
(4)
!F"2 $une 2%% &n 1'
66. Find the values of x for hich
cosh x / 0 sinh x = 1-
3ivin3 our ansers as natural lo3ariths.(6)
!F"2 $une 2%% &n 2'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
35/38
67. Sho that6
2
#
-
( 9)
x
x
⌠
⌡
+√ − d x = - ln
√+-
-2
+ -√- / 0.(7)
!F"2 $une 2%% &n -'
68. 4he curve C has equation
y = arsinh ( x-) x ≥ %.
4he point P on C has x:coordinate G2.
(a) Sho that an equation of the tan3ent to C at P is
y = 2 x / 2G2 + ln (- + 2G2).(5)
4he tan3ent to C at the point Q is parallel to the tan3ent to C at P .
(b) Find the x:coordinate of Q 3ivin3 our anser to 2 decial places.(5)
!F"2 $une 2%% &n 0'
69 . *iven that
I n = ⌡
⌠ π
%
dsine x xn x
n ≥ %
(a) sho that for n ≥ 2
I n =1
)1(2 +−
n
nn I n / 2 .
(8)
(b) Find the eact value of I 0.(4)
!F"2 $une 2%% &n #'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
36/38
70.
Figure 1
Fi3ure 1 shos the curve C ith equation
y =1%
1cosh x arctan (sinh x) x ≥ %.
4he shaded re3ion R is ,ounded , C the x:ais and the line x = 2.
(a) Find ⌡
⌠ x x x d)(sinharctancosh .
(5)
(b) ence sho that to 2 si3nificant fi3ures the area of R is %.-0.(2)
!F"2 $une 2%% &n 6'
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
37/38
71. 4he hper,ola # has equation
16
2 x
/9
2 y
= 1.
(a) Sho that an equation for the noral to # at a point P (0 sec t - tan t ) is
0 x sin t + - y = 2# tan t .(6)
4he point S hich lies on the positive x:ais is a focus of # . *iven that PS is parallelto the y:ais and that the y:coordinate of P is positive
(b) find the values of the coordinates of P .(5)
*iven that the noral to # at this point P intersects the x:ais at the point R
(c) find the area of trian3le PRS .(3)
!F"2 $une 2%% &n 8'
72.
M =
12
-%
21
p
%
p
here p and % are constants.
*iven that
1
2
1
is an ei3envector of M
(a) sho that % = 0 p.
(3)
*iven also that & = # is an ei3envalue of M and p % and % % find
(b) the values of p and %(4)
(c) an ei3envector correspondin3 to the ei3envalue & = #.(3)
!F"- $une 2%% &n 2'
73.
F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9
-
8/19/2019 05a Further Maths - FP3 Questions v2 (1)
38/38
Figure 1
Fi3ure 1 shos a praid PQRST ith ,ase PQRS .
4he coordinates of P Q and R are P (1 % /1) Q (2 /1 1) and R (- /- 2).
Find
(a) PQ PR×(3)
(b) a vector equation for the plane containin3 the face PQRS 3ivin3 our anser inthe for r . n = d .(2)
4he plane Π contains the face PST . 4he vector equation of Π is r . (i / 2 j / #k ) = 6.
(c) Find cartesian equations of the line throu3h P and S .(5)
(d ) ence sho that PS is parallel to QR.(2)
*iven that PQRS is a parallelo3ra and that T has coordinates (# 2 /1)
(e) find the volue of the praid PQRST .(3)
!F"- $une 2%% &n 8'