Modul Jom Tanya Sifu_Integrations & Vectors 2014

15 Days Programme: Part 3

JOM TANYA SIFU INTEGRATIONS & VECTORS

JOM TANYA SIFU 15 Days Programme: Part 3

Prepared by : Pn Hayati Aini Ahmad

DAY 01 : INTEGRATIONS QUESTION 1

Given

and

. Find the value

of where is a constant if

QUESTION 2

Given

and

. Find the value

of

QUESTION 3

Given

and

. Find the value

of where is a constant if

QUESTION 4

Given

. Find the value of where is a

constant if

QUESTION 5

Given

. Find

(a) the value of

(b) the value of if

QUESTION 6

Given

and

. Find

(a)

(b)

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DAY 02: INTEGRATIONS QUESTION 1

Given

. Find the value of where is a

constant if

QUESTION 2

Given

and when , express in

terms of .

QUESTION 3

Find

QUESTION 4

Given that

, where and

are constants. Find the value of and of

QUESTION 5

(a) Find the value of

(b) Given that

, find the value

of

QUESTION 6 The gradient function of a curve is . The curve passes through the points (1 , 5) and (2 , k). Find (a) the equation of the curve (b) the value of k

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DAY 03: INTEGRATIONS QUESTION 1

Given that

, find the value of

QUESTION 2

Given that

, find the value of

QUESTION 3

Given that

, such that

, where is a

function of . Evaluate

QUESTION 4

Given that

and

, where is a

function of . Find the value of

QUESTION 5

Find

in term of

QUESTION 6

(a) Given that

, find the value

of

(b) Find the value

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DAY 04: INTEGRATIONS QUESTION 1 The gradient of a curve at a point P(1 , 4) is given by . Find the equation of the curve

QUESTION 2 The gradient of a curve at a point (3 ,-5) is given by . Find the equation of the curve

QUESTION 3

A curve with a gradient function

passes through

point . Find the equation of the curve.

QUESTION 4

A curve has gradient function,

, where is a

constant. The gradient of the normal to the curve at point

is

. Find

(a) the value of (b) the equation of the curve

QUESTION 5

The gradient function of a curve is given by

where is a constant. Given that the tangent to the curve at the point is parallel to the straight line . Find (a) the value of (b) the equation of the curve

QUESTION 6

The gradient function of a curve is given by

, where is a constant. Given that the tangent to the curve at the point is parallel to the x-axis. Find (a) the value of (b) the equation of the curve

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DAY 05: INTEGRATIONS QUESTION 1 Diagram 1 shows part of the curve (a) The gradient of normal to the curve at the point

is

. Find

the value of (b) Find the area of the shaded region (c) The region bounded by the curve, both axes and the line is rotated through about the x-axis. Find the volume of revolution, in terms of

QUESTION 2 Diagram 2 shows part of the curve and a straight line (a) Calculate the area of the shaded region. (b) The region enclosed

by the curve, the x-axis, the y-axis and the straight line is revolved through about the y-axis. Find the volume of revolution, in terms of

QUESTION 3 In Diagram 3, the straight line PQ is a tangent to the curve at the point Find (a) the equation of the tangent at A (b) the area of the shaded region

(c) the volume of revolution, in terms of , when the region is bounded by the curve, the x-axis and the y-axis, is rotated through about the y-axis

QUESTION 4 Diagram 4 shows part of the curve which passes through point . The curve has a

gradient function of

(a) Find the equation of the curve (b) A region is bounded by the curve, the x-axis,

the line x =-5 and the line x = -2. Find (i) the area of the region (ii) the volume generated, in term of when the region is rotated through about the x-axis

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DAY 06: INTEGRATIONS QUESTION 1 Diagram 5 shows the curve and the normal to the curve at the point . Calculate (a) the equation of the normal to the curve at A. (b) the area of the region

enclosed by the curve, the normal and the x-axis. (c) the volume of the revolution, in terms of , if the region in 1(b) is rotated through about the x-axis

QUESTION 2 Diagram 6 shows the straight line intersecting the curve at point and . Find (a) the coordinates of point B

(b) area of the shaded region (c) volume of revolution, in terms of , when the region bounded by the curve, x-axis and y-axis is rotated through about the x-axis.

QUESTION 3 Diagram 7 shows part of the curve which passes through point and the straight line . The curve has a gradient function of . Find (a) the equation of the

curve (b) the area of the shaded region P (c) the volume of revolution, in terms of , when the shaded region Q is revolved through about the y-axis.

QUESTION 4 Diagram 8 shows the straight line intersecting the curve at point andpoint Find (a) the coordinates of A and B

(b) area of the shaded region P (c) volume of revolution, in terms of , when the shaded region Q is rotated through about the x-axis.

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DAY 07 : INTEGRATIONS QUESTION 1 Diagram 9 shows part of the curve . The straight line intersects the y-axis at and intersects the x-axis at . Given that the area of the shaded region is

unit2. Find the value of

QUESTION 2 Diagram 10 shows part of the curve . The curve intersects the straight line at point A. Calculate the volume generated, in terms of , when the shaded region is revolved through about

the y-axis.

QUESTION 3 In Diagram 11, the straight line PQ is tangent to the curve at . Find (a) the value of (b) the area of shaded region (c) the volume generated, in

terms of , when the region bounded by the curve, the y-axis and the straight line is revolved through about the y-axis.

QUESTION 4 Diagram 12 shows part of the curve and the tangent to the curve at the point . Find (a) the equation of the tangent at A (b) the area of the shaded region (c) the volume of

revolution, in terms of , when the region bounded by the curve, the x-axis and the y-axis, is revolved through about the x-axis.

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DAY 08 : VECTORS QUESTION 1 Given that and , find

(a) in the form of

(b) the values of if

QUESTION 2 Given that and . Find the

magnitude of

QUESTION 3 The coordinates of point A and point B are (1 ,2) and (6 ,-10) respectively.

(a) Express in the form

(b) Find the unit vector in the direction of

QUESTION 4

Given

and

.Find

(a) (b) the unit vector in the direction of

QUESTION 5 Given O(0 ,0), P(-2, 5) and Q(1 , 1). Express in terms of

and .


QUESTION 6 Given that and , find

(a) 2

(b) the unit vector in the direction of

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DAY 09 : VECTORS QUESTION 1 Given the points O(0 ,0), A(3, -2) and B(-2,10)

(a) Express in the form

(b) Find the unit vector in the direction of

QUESTION 2 Given O(0 ,0), A(5, -2) and B(-3 ,7). Express in terms of

and .


QUESTION 3 Given that and . If ,

find the unit vector in the direction of

QUESTION 4

Given that

and

. Find

(a) the vector


QUESTION 5

Given that

and

. Find

(a) the vector


QUESTION 6

Given that and , find the

coordinates of point

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DAY 10 : VECTORS QUESTION 1

Given

and

.Find the values of

such that the vector is parallel to the vector

QUESTION 2

Given and are collinear. It is given that ,

and . Find the value of

QUESTION 3

Given and are collinear such that

and where is a constant. Find the

value of

QUESTION 4 Given that and , where

is a constant. Find the value of if is parallel to

the y-axis

QUESTION 5 It is given that and

, where is a constant. Find the

possible values of if is parallel to

QUESTION 6 Given that and , where is a

constant. (a) Find the value of if and are parallel

(b) By using the value of , find the value of

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DAY 11 : VECTORS QUESTION 1

It is given that , and

. Find

(a)

(b) the value of if and are parallel

QUESTION 2

A, B and C are collinear such that, and

(a) Find the value of

(b) Hence, find , in terms of and , the vector

QUESTION 3 It is given that , and

, where and are constants.

Find the values of and of if

QUESTION 4

Given that P(3 , 6) , Q(-2 , 8) and

, find the

values of and of

QUESTION 5 Given that A(1 , -2), B(3, 4) and C(m , p ), find the value

of and of such that

QUESTION 6 It is given that , and

, where and are constants. Find

the values of and of if

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DAY 12 : VECTORS QUESTION 1 Diagram 1 shows a triangle ABC and the point D lies on the straight line BC. It is

given that

.

Express in terms of and

QUESTION 2 Diagram 2 shows a rectangle OPQR and the point M lies on the straight line PR. It is given that . Express

in terms of and

QUESTION 3 In Diagram 3,

and

. Given

, find

(a) the value of (b) coordinates of B

QUESTION 4 Diagram 4 shows a triangle ABC. The point S lies on the straight line AC. It is given that

, and

. Express in terms of and/or

(a) (b)

QUESTION 5 In Diagram 5, OPQ is an equilateral triangle where R is a midpoint of

PQ. Given that ,

and , find

the value of

QUESTION 6 In Diagram 6, shows a triangle ABC. The point E is the midpoint of AC and D lies on the line BC such .

Given that and

, express in

terms of and/or

(a) (b)

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DAY 13 : VECTORS QUESTION 1 Diagram 7 shows a triangle ABC. It is given that 2CD=3DB, E is the midpoint of

AB, and

. Express

, in terms of and

QUESTION 2 Diagram 8 shows a triangle OPR and the point Q lies on the straight line PR. It is given that

. Express , in terms of and

QUESTION 3 Diagram 9 shows a parallelogram ABCD and BED is a straight line. Given that

,

and . Express in terms of and .

(a) (b)

QUESTION 4 Diagram 10 shows two

vectors, and . It is

given that

Find the values of and of

QUESTION 5 Diagram 11 shows a parallelogram ABCD drawn on a Cartesian plane. It is given that

and

. Find

(a) (b)

QUESTION 6

In Diagram 12, and . On the same

square grid, draw the line that represents the vector

.

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DAY 14 : VECTORS QUESTION 1 In Diagram 13, E and F are midpoints of BC and AB respectively. Given

that and

and CF is extended to D

such that

(a) Express in terms of and

(i) (ii) (iii)

(b) Hence, show that is parallel to

QUESTION 2 It is given that

,

,

and

.

(a) Express in terms of and

(b) Point lies inside the trapezium ABCD such that

and is a constant.

(i) Express in terms of , and

(ii) Hence, if the point and are collinear, find the value of

QUESTION 3 In Diagram 15, the straight lines QT and PS intersect at point R. It is given that

and

.

(a) Express in terms of and :

(i) (ii)

(b) It is given that and , where

and are constants. Express (i) in terms of , and

(ii) in terms of , and

(c) Hence, find the value of and of

QUESTION 4 Diagram 16 shows triangle OPR and QSP is a straight line. It is given that

and

.


(i) (ii)

(b) It is given that and . By using

, find the value of and of

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DAY 15 : VECTORS QUESTION 1 Diagram 17 shows trapezium ABCD. R is the midpoint of BC. It is given that

,

,

and

. (a) Express in terms of and :

(i) (ii)

(b) It is given that and , where and are constants. Find the values of and of

QUESTION 2 Diagram 18 shows triangles OAB and APR. OPA, OQB, PQR and ABR are straight lines. Given

and

. (a) Express in terms of and :

(i) (ii)

(b) It is given that and ,

where and are constants. Express (i) in terms of , and



QUESTION 3 Diagram 19 shows a triangle OAR. The straight lines OP and AB intersect at point Q such that

and

.

It is given that and .


(i) (ii)

(b) It is given that and ,

where and are constants.Express (i) in terms of , and



QUESTION 4 Diagram 20, PQRS is a quadrilateral. The diagonals PR and QS intersect a point T. It is given

and


(i) (ii)

(b) It is given that and , where and are constants. Express

(i) Express in terms of , and

(ii) Express in terms of , and

(c) Using , find the value of and of

Date of completion:

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Modul Jom Tanya Sifu_Integrations & Vectors 2014

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Transcript of Modul Jom Tanya Sifu_Integrations & Vectors 2014