05a further maths - fp3 questions v2 (1)

8/19/2019 05a Further Maths - FP3 Questions v2 (1)

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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)

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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)

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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)

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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)

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F"- questions fro old "0 "# "6 and F"1 F"2 F"- papers / ersion 2 / arch 2%%9


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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)

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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)

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7. "rove that sinh (iπ − θ ) = sinh θ .(4)

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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)

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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)

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10. Find the values of x for hich

0 cosh x + sinh x =

3ivin3 our anser as natural lo3ariths. (6)

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11. (a) "rove that the derivative of artanh x −1 x 1 is 211

x−.

(3)

(b) Find ∫ x x d artanh .

(4)

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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)

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x

y

O

R

#


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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)

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x

y

ka:ka

Q P

O


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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)

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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)

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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


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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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%)

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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%)

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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.

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y

O x


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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)

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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

-√.

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21

y

O x

R


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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%)

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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)

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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)

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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)

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38. Cvaluate ⌡⌠

+−√

0

12 )182(

1

x x d x 3ivin3 our anser as an eact lo3arith.

(5)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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47. Find the values of x for hich# cosh x / 2 sinh x = 11

3ivin3 our ansers as natural lo3ariths. (6)

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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)

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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)

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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)

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y

C

xO


25/38



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)

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R

x

y

O -


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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)

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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)

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54. A =

1%%

11%

211

.

"rove , induction that for all positive inte3ers n

An =

+

1%%

1%

)-(1 221

n

nnn

.

(5)

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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)

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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)

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57. Evaluate ⌡⌠

−+√-

12 )#0(

1

x x d x , giving your answer as an exact

logarithm.(5)

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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)



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)

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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)

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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)

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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)

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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)

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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)

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65. Sho that

xd

d!ln(tanh x)' = 2 cosech 2 x x < %.

(4)

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66. Find the values of x for hich

cosh x / 0 sinh x = 1-

3ivin3 our ansers as natural lo3ariths.(6)

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67. Sho that6

2

#

-

( 9)

x

x

⌠

⌡

+√ − d x = - ln

√+-

-2

+ -√- / 0.(7)

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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)

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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)

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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)

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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)

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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)

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73.



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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)

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05a further maths - fp3 questions v2 (1)

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