CBSE MATH COMPLETE STUDY

CBSE MATH COMPLETE STUDY: Number System

2011-03-27T23:54:00.000-07:00

CBSE MATH COMPLETE STUDY: Number System: "For more math online E-book study visit : Cbse Mathematics "

Number System

2011-03-27T23:25:00.001-07:00

For more math online E-book study visit : Cbse Mathematics

sample paper FIRST TERM EXAMINATION

2011-03-27T17:25:00.001-07:00

For science sample paper visit CBSE ADDA

jsunilclasses.weebly.com JSUNIL CLASSES"A SMART CLASS CENTER"

M.M : - 80 MATHEMATICS(X) -2011-12 TIME : - 3 hrs.

SECTION – A

1. State Euclid’s division lemma.

2. If one zero of polynomial 5x²+13x+a is reciprocal of the other, find the value of other.

3. In ΔABC, DE||BC meeting AB at D and AC at E. If AB/BD= 4and CE = 2 cm,

find the length of AE.

4. If 4 cotA = 3, find the value of (sinA- 4cos) / (sinA+ 4cos)

5. Find the value of 4cot²θ − 4cos ec²θ .

6. Find the value of sin 36 / 2cos54 - 2sec 41/3cos ec49

7. Why is 11/30a non‐terminating decimal number?

8. Find he value of ‘k’ if following system of equations has no solutions: 3x-y-5=0 ; 6x-2y-k=0.

9. If cosA = 3/5, find 9cot² A−1.

10. The point of intersection of o gives is given by ( 20.5, 30.4). What is median ?

SECTION – B

11. Is 7×5×3×2+3 a composite number. Justify your answer.

12. Find the zeros of polynomial 7x² −6−11x and verify the relation between zeros and coefficients of polynomial.

13. Solve for x and y : a/x- b/y = 0; ab²/x - a b /y= a²+ b²

14. Find A if sin(A+36)=cosA,where A+36 is acute angle.

15. In figure EF || DC|| AB, prove that ED/ AE = FC/BF

16. In an equilateral Δ ABC, Ad is altitude drawn from A to BC. Prove that 3AB²=4AD².

17. Find the mode marks from following data:

Marks Obtained 0-10 10-20 20-30 30-40 40-50

No. of Students 6 8 5 4 4

18. Following table gives the scores of 100 candidates in an entrance examination: Find mode.

Marks 100-150 150-200 200-250 250-300 300-350 350-400

No. of Students 16 15 14 32 11 12

SECTIONC

19. Show that any positive odd integer is of the form 6p+1 or 6p+5, where p is some integer.

20. show that 5−2Ö3 is irrational number.OR Show that 5Ö2 / 3

21. A number consists of two digits whose sum is 9. If 27 is added to the number, the digits are interchanged. Find the number.

22. Obtain all zeros of polynomial x⁴+x³−34x²−4x+120 if two of its zeros are 2 and -2.

23. prove that: cos / (1+sinA) + (1+ sinA) /cosA= 2sec

24. If cosθ +sinθ = 2 cosθ , then show that cosθ −sinθ = 2 sinθ .

25. D is any point on the side BC of Δ ABC such that <ADC = <BAC .Prove that

CA /CD=CB/CA.

26. Δ ABC and Δ DBC are on the same base BC. AD and BC intersect at O.

Prove that ar DABC / ar DDBC = AO/ DO.

27. Using step deviation method, calculate arithmetic mean of the following:

Class Interval 0-20 20- 40 40-60 60-80 80-100 100-120

Frequency 20 35 52 44 38 31

OR , Find the value of ‘p’ if mean of following data is 53.

Class 0-20 20-40 40-60 60-80 80-100

Frequency 12 15 32 p 13

28. Find the median of following data :

Class 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70- 80

frequency 7 14 13 12 20 11 15 8

29. On dividing P(x) = 3x³−2x²+5x−5 by a polynomial g(x), we get quotient and remainder as

x² − x + 2 and-7 respectively. Find g(x).

30. Prove that : (1 + sinA -cos A) ¸ (sinA- 1 + cosA ) = (cosA) ¸(1- sinA)

OR (sec. cosec(90-A ) tan .cot(90-A ) + (sin²55+ sin²35) ¸ (tan10. tan 20. tan30. tan70. tan80)

31. Prove that in a triangle the line drawn parallel to one side ,divides the other two sides proportionally.

OR,Prove that in a right angled triangle, the square of hypotenuse is equal to sum of the squares of other two sides.

32. If x = psecθ+qtanθ and y=p tanθ + q secθ , prove that x²−y²= p²−q².

33. On the same axes draw the graph of each of the following equations :

2x – y +1 =0 , x – 5y +14 =0 ; x – 2y +8=0.

Hence shade the region of the triangle so formed.

34. Daily pocket expenses of (in Rs) of 80 students of a school are given in the table below:

Expenses 0-5 5-10 10-15 15-20 20-25 25-30 30-35

No. of students. 5 15 20 10 10 15 5

Change the data into a’ more than type table’ and hence draw ‘more than type ogive.

JSUNIL TUTORIAL Mob. : 9835859669

Also visit Cbse Mathematics - Home and CBSE MATH COMPLETE STUDY

All India Engineering / Architecture Entrance Examination (AIEEE)

2011-03-27T17:24:00.000-07:00

Q. 21. Let and let , where Then

Sol :

Q. 22. A person is to count 4500 currency notes. Let denote the number of notes he counts in the minute. If and are in an AP with common difference –2, then the time taken by him to
count all notes is.

34 minutes
125 minutes
135 minutes
24 minutes

Sol :

Answer: (1)

Q. 23 . Let be a positive increasing function with

Sol : Clearly

Answer: (4)

Q. 24. Let be a function defined on R such that  equals

Sol:

Answer: (1)

Q. 25. The line L given by passes through the point . The line K is parallel to L and has

the equation . Then the distance between L and K is

Sol :

As L is passing through 

Answer: (3)

Q. 26. Let S be a non-empty subset of R. Consider the following statement:
P : There is a rational number such that .
Which of the following statements is the negation of the statement P?

There is no rational number such that .
Every rational number satisfies .
and is not rational.
There is a rational number such that .

Sol : Answer: (2)

Q. 27. Consider the system of linear equations:
The system has.

exactly 3 solutions
a unique solution
no solution
infinite number of solutions

Sol :

Clearly (iii) and (iv) are parallel planes. No solution.

Answer: (3

Q. 28. Let
.

Sol :

Q. 29. The equation of the tangent to the curve that is parallel to the x-axis, is.

Y =1
Y =2
Y =3
Y =0

Sol:

Q. 30. The circle intersects the line at two distinct points if.

Sol:

Centre of circle (2, 4) and radius is 5.

Answer: (1)

All India Engineering / Architecture Entrance Examination (AIEEE)

2011-03-27T17:21:00.000-07:00

AIEEE 2010 Mathematics

Directions: Questions number 12 to 16 are Assertion-Reason type questions. Each of these questions
contains two statements:
Statement-1 : (Assertion) and
Statement-2 : (Reason).
Each of these questions also has four alternative choices, only one of which is the correct answer.
You have to select the Answer.

Q. 12. Four numbers are chosen at random (without replacement) from the set {1, 2, 3, ……,20}.
Statement-1: The probability that the chosen numbers when arranged in some order will form an AP is.
Statement-2: If the four chosen numbers form an AP, then the set of all possible values of common difference is

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
Statement-1 is true, Statement-2 is false.
Statement-1 is false, Statement-2 is true.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Sol :

Required probability =
Statement -2 is false because common difference can also be ±6.

Answer: (2)

Q. 13. Statement-1: The point is the mirror image of the point
in the plane .
Statement-2: The plane bisects the line segment joining and

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
Statement-1 is true, Statement-2 is false.
Statement-1 is false, Statement-2 is true.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1. 

Sol :

Statement I and II both are true but statement II is not the correct explanation.
Plane is not just bisector but it is perpendicular bisector of AB.

Answer: (1)

Q. 14. Let

Statement-1:
Statement-2:

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
Statement-1 is true, Statement-2 is false.
Statement-1 is false, Statement-2 is true.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Sol :

Answer (2)

Q 15. Let A be a 2 x 2 matrix with non-zero entries and let A2 =1 , where I is 2 x2 identity matrix. Define Tr=(A) sum of diagonal elements of A and = determinant of matrix A.

Statement-1: = Tr(A) =0
Statement-2: =

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
Statement-1 is true, Statement-2 is false.
Statement-1 is false, Statement-2 is true.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1

Sol :

Answer: (2)

Q.16. Solution of the differential equation is.

ysecx = ten x + c
ytanx=secx + c
tanx = (secx +c)y
secx =(tanx +c )y

Sol :

Q. 17. The area bounded by the curves

Sol :

Required area =

Answer: (4)

Q. 18.

–1
1
2
2

Sol :

Answer: (2)

Q 19. The number of complex numbers z such that equals.

Sol :

Only one solution and that is circumcentre of triangle formed by (1, 0), (0, 1) and (–1, 0).

Answer: (1)

Q. 20. A line AB in three-dimensional space makes angles 45° and 120° with the positive x-axis and the
positive y-axis respectively. If AB makes an acute angle with the positive z-axis, then equals

Sol :

Answer: (2)

All India Engineering / Architecture Entrance Examination (AIEEE)

2011-03-27T17:18:00.000-07:00

. R be a differentiable function with f(0)=-1and f’(0)=1,Let g(x)= .then

–4
0
–2
4

Sol :

Answer: (1)

Q. 2. There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is.

Sol :

Total number of ways

Answer: (4)

Q. 3. An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colours is.

Sol :

Required probability=

Answer: (1)

Q. 4. For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. The variance of the combined data set is

Sol :

Answer: (1)

Q. 5 . Let be a continuous function defined by . .

Statement-1: , for some . .

Statement-2: for all .

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
Statement-1 is true, Statement-2 is false.
Statement-1 is false, Statement-2 is true.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Sol :

Answer: (4)

Q. 6. For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following is.

There is a regular polygon with
There is a regular polygon with
There is a regular polygon with
There is a regular polygon with

Sol :

If O is centre of polygon and AB is one of the side, then by figure:

Answer : (2)

Q. 7. If two tangents drawn from a point P to the parabola are at right angles, then the locus of p is.

2x +1=0
x = -1
2x +1=0
x = 1

Sol.

Locus of point P is directrix of parabola i.e., x=-1

Answer: (2)

Q. 8. If the vectors and are mutually orthogonal ,
then

(2,-3)
(-2,3)
(3,-2)
(-3,2)

Sol :

Answer (c)

Q. 9. Consider the following relations:

R = {(x, y) | x, y are real numbers and x=wy for some rational number w };

Then

neither R nor S is an equivalence relation
S is an equivalence relation but R is not an equivalence relation
R and S both are equivalence relations
R is an equivalence relation but S is not an equivalence relation

Sol :

R is not equivalence   but
S in equivalence

Answer : (2)

Q. 10. be defined by . If f has a local minimum at x = -1 ,
then a possible value of k is.

0
-1
1

Sol :

Answer : (3)

Q. 11. The number of 3 X 3 non-singular matrices, with four entries as 1 and all other entries as 0, is

5
6
at least 7
less than 4

Sol :

are 6 non-singular matrices because 6 blanks will be filled by 5 zeros and 1 one.

Similarly, are 6 non-singular matrices.

So, required cases are more than 7.

Answer: (3)

Number System

2011-03-27T17:07:00.000-07:00

Probability

2011-03-27T17:01:00.000-07:00

JSUNIL TUTORIAL PANJABI COLONY GALI 01

CBSE Class X (Mathematics) Board Exam - 2012

Q. 1. What is the probability of getting 53 Tuesdays in a year ?

Q. 2. What is probability of getting 53 Mondays in a leap year ?

Q. 3. What is the probability that two people have same birthday ?

Q. 4. What is the probability that two people have different birthday ?

Q. 5. If a card is drawn from a well suffeled deck of 52 cards then what is the probability of getting a face card ?

Q. 6. If probability of happening of an event is 0.09. what wiil be probability of not happening of that event ?

Q. 7. What is the probability of a certain event ?

Q. 8. What is the probability of an impossible event?

Q. 9. A coin is tossed twice . What will be the probability of getting at most two heads ?

Q. 10. A coin is tossed twice . What will be the probability of getting at least one tail.

Q. 11. If a die thrown once, what will be the probability of getting a prime even number ?

Q. 12. If a die is thrown twice, what will be the probability of getting a doublet ?

Q. 13. If a die is thrown twice , what will be the probability of getting a pair of prime numbers ?

Q. 14. If a card is drawn from a deck of 52 cards, what will the probability of getting a black king ?

Q. 15. King and queen of spade are removed from a deck of 52 cards and then a card is drawn, what will be the probability of getting a red king ?

Similar Triangles,Exercise

2011-03-27T16:57:00.001-07:00

Exercise

1. In the bisector of intersects the side AC at D. A line parallel to side AC intersects line segment AB, DB and CB at points P, R and Q respectively. Prove that

AB X CQ = BC X AP
PR X BQ = QR X BP

2. ABCD is a quadrilateral in which AB = AD. The bisector of intersects the side BC and CD respectively at E and F. Prove that the segment EF is parallel to the diagonal BD.

3. In and the bisector of intersects AC at D. Prove that

4. If the diagonal BD of a quadrilateral ABCD bisects both show that

5. D is the midpoint of side BC of DE and DF are respectively bisectors of such that E and F lie on AB and AC, respectively. Prove that EF || BC.

6. O is a point inside a The bisector of meet the sides AB, BC and CA in points D, E and F respectively. Prove that AD. BE. CF = DB. EC. FA

7. In the adjoining figure, , AD is bisector of Prove that DE X (AB + AC) = AB X AC.

8. If the bisector of an angle of a triangle bisect the opposite side, prove that the triangle is isosceles.

9. BO and CO are respectively the bisectors of AO is produced to meets BC at P. Show that

AP is the bisector of

Similar Triangles,Internal Bisector of an angle of a Triangle

2011-03-27T16:55:00.001-07:00

Internal Bisector of an angle of a Triangle

The internal bisector of an angle of a triangle divides the opposite side in the ratio of the sides containing the angle.
If a line-segment drawn from the vertex of a triangle to its opposite side and divides it in the ratio of the sides containing the angle, then the line segment bisect the angle of the vertex.

Example 17. In the adjoining fig AD is the bisector of If BD = 4cm, DC = 3cm and AB = 6cm, determine AC.

Solution:- In is the bisector of

Example 18. In the adjoining fig, AD is bisector of If AB = 5.6cm, AC = 4cm, DC = 3cm, find BC.

Solution:- In is the bisector of

Similar Triangles-exercise

2011-03-27T16:54:00.001-07:00

Exercise

1. The perpendicular AD on the base BC of a intersects BC at D so that DB = 3CD. Prove that 2AB² = 2AC² + BC².

2. P and Q are points on the side CA and CB respectively of a right angled at C.

Prove that AQ² + BP² = AB² + PQ^2.

3. In , if AD is the median, Show that AB² + AC² = 2(AD² + BD²)

4. PQR is an isosceles right triangle, right angled at R. Prove that PQ² = 2PR².

5. In a is an acute angle and Prove that AC² = AB² + BC² – 2BC. BD.

6. In the adjoining figure, find the length of BD, If

7. Prove that the altitude of an equilateral triangle of side

8. P and Q are the midpoint of the sides CA and CB respectively of right angled at C. Prove that 4(AQ² + BP²) = 5AB²

9. In a triangle ABC, AD is perpendicular on BC. Prove that AB² + CD² = AC² + BD²

10. Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.

11. In adjoining figure, OD, OE and OF are respectively perpendiculars to the sides BC, CA and AB from any point O in the interior of the triangle Prove that
(i) OA² + OB² + OC² – OD² – OE² – OF² = AF² + BD² + CE²
(ii) AF² + BD²+ CE² = AE² + CD² + BF²

12. O is any point in the insertor of a rectangle ABCD. Prove that interior OB² + OD² = OC² + OA²

Answers

(6) 13m

Similar Triangles

2011-03-27T16:53:00.001-07:00

Example 16. ABC is a right triangle, right angled at C. If p is the length of the perpendicular from C to AB and a, b, c have the usual meaning, then prove that

pc = ab

Solution:- (i) Area of taking BC as base = 1/2 X BC X AC

= 1/2ab--------(i)
Area of taking AB as base = 1/2 X AB X CD
= 1/2 cp------(ii)
from (i) and (ii) 1/2 ab = 1/2 cp
Or, pc = ab
Or, c = ab/p---------(iii)
(ii) In right ,

Similar Triangles

2011-03-27T16:52:00.000-07:00

Example 14. In an equilateral triangle PQR, the side QR is trisected at S. prove that

Solution:-

Given:- In an equilateral is trisected at S.

To Prove:-

Construction:- is drawn

Proof:- QD = DR = QR/2 ------------(i)

Side QR is trisected at S(given)
In is acute

Example 15. In the given figure, ABC is right angled triangle with the AB = 6cm and AC = 8cm. A circle with centre O has been inscribed inside the triangle. Calculate the value of r, the radius of the inscribed circle.

Solution:- In right

BC² = AB² + AC² [By Pathagoras theorem]
= 6² + 8²
= 36 + 64
= 100
= 1/2 X 6 X 8 = 24cm²
= 24/12

= 2cm

Similar Triangles,Pythagoras Theorem.

2011-03-27T16:50:00.001-07:00

Pythagoras Theorem. (B audhayan Theorem)

Theorem 8.3: - In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Given: - is a right angle of

To Prove:-
Construction:- is drawn
Proof:- In
Or, AB² = AC X AD ---------------------(i)
Similarly
Or, BC² = AC X CD ---------------------(ii)
Adding (i) and (ii) we get
AB² + BC² = AC X AD + AC X CD
= AC X (AD + CD)
= AC X AC
= AC²
Or, AC² = AB² + BC²

Theorem 8.4 (Converse of Pythagoras Theorem): - In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Given:- In

To prove:-
Construction:- A triangle PQR is constructed such that PQ = AB, QR = BC and
Proof:- In
Or, PR² = AB² +BC²--------------(i) [PQ = AB, QR = BC]

But AC² = AB² + BC² ------------(ii) (given)
Or, PR = AC
Or,
Hence

Example 13. Determine whether the triangle having sides (2a – 1) cm, and (2a + 1) cm is a right angled triangle.

Sol:- Let AB = (2a - 1) cm,

AC = (2a + 1) cm
is a right angled triangle.

Similar Triangles-exercise

2011-03-27T16:49:00.001-07:00

Exercise

1. Prove that the area of the equilateral triangles describe on the side of a square is half the are of the equilateral triangle describe on its diagonals.

2. In the given figure Also If BC = 12cm, find QR.

3. ABC is a triangle right angled at A, AD is perpendicular to BC. IF BC = 13cm and AC = 5cm, find teh ratio of the areas of and .

4. The area of two similar triagles are 121cm² and 64cm² respectively. If the median of the first triangle is 12.1cm, find the correstponding median of the other.

5. In an equilateral triangle with side a, prove that the area of the triangles is

6. D and E are points on the sides AB and Ac respectively of such that DE is parallel to BC and AD : DB = 4 : 5. CD and BE intersect each other at F. Find the ratio of the areas of and

Answers
(2) 6cm	(3) 169 : 25	(4) 8.8cm
(6) 16 : 81

Similar Triangles

2011-03-27T16:48:00.001-07:00

Example 12. In th e given fig ABC and PQR are
isosceles triangles in which .

Solution:-

Given:- In and PQR, and PQ = PR
To find:
Proof:- is isosceles with AB = AC
is isosceles with PQ = PR
From (1) and (2) we get
Or,

from (3) and (4)
We know

Similar Triangles

2011-03-27T16:47:00.000-07:00

Example 10. ABC is a triangle, PQ is the line segruent intersecting AB in P and AC in Q such that PQ || BC and divides into two parts equal in area. Find BP : AB

Solution:

Given:  in which PQ || BC, and PQ
divides  into two parts equal in area.
To find: BP : AB
Proof:- In  and ABC

Example 11. In the given figure, ABC and DBC are two
triangles on the same base BC. IF AD intersects BC at O,
Prove that

Solution:

Given: ABC and DBC are two triangles on the same base BC. AD intersect BC at O.
To Prove:
Construction:- are drawn.
Proof:- In and

Similar Triangles

2011-03-27T16:46:00.000-07:00

Theorem 2. The ratio of the ares of two similar triangles is equal to the ratio of the squares of their corresponding sides.

Given:-
To prove:
Construction: and are drawn as in figure
proof:-
Or,
Now in and PSQ,

Hence,

Example 9. ABC and DEF are two similar triangles such that AB = 2DE and area of is 56sq.cm, find the area of

Solution:-

Given:-
To find: Area of
Proof:
and

Similar Triangles-exercise

2011-03-27T16:43:00.000-07:00

Exercise

1. In the adjoing figure, If

AB = x units CD = y units and PQ = Z units, Prove that ,

2. In a and Q are point on the side AB and AC respectively such that PQ is parallel to BC. Prove that median AD drawn from A to BC, bisect PQ.

3. Through the mid-point M of the side CD of a parallelogram AB CD, the line BM is drawn intersecting AC in L and AD produced in E. Prove that EL = 2BL.

4. ABC is a triangle right anlgled at C. If P is the length of perpendicular from C to AB and AB = c, BC = a and CA = b, show that pc = ab

5. Two right angles ABC and DBC are drawn on the same hypoeuuge BC and on the same side of BC. If AC and BD interscta at P, prove that AP X PC = BP X PD

6. The perimeter of two smilar triangles ABC and PQR are respectively 32cm and 24cm.If PQ = 12cm, find AB.

7. In a right triangles ABC, the perpendicular BD on the hypotenuse Ac is drown. Prove that AC X CD = BC²

8. In is aculte, BD and CE are perenducular on AC and AB respectively. Prove that AB X AE = AC X AD

9. Through the vertex D of a parallotogram ABCD, a line is drawn to intersect the sides AB and CB produced at E and F respectively prove that:

10. Two sides and a mediam bisecting one of these sides of a triangle are respectively proportional to the two sides and the corresponding mediam of the other triangle. Prove that the triangles are similar.

11. If the angles of one triangles are respectively equal to the angles of another tranles. Prove that the ratio of their corresponding sides is the same as the ratio of their corresponding.

medians
altitudes
angle bisectors

12. E is a point on side AD produced of a parallelogram ABCD and BE intersects CD at F. prove that

13. If a perpecdicular is drawn from the vertex of the right angles of a right triangles to the hypoteuuse, the triangles on each side of the perpendicular are similar to the whole triangles and to each other.

Theorem 2. The ratio of the ares of two similar triangles is equal to the ratio of the squares of their corresponding sides.

Given:-
To prove:
Construction: and are drawn as in figure
proof:-
Or,
Now in and PSQ,

Hence,

Similar Triangles

2011-03-27T16:42:00.000-07:00

Example 7. In a and D is a point on side AC, such that BC² = AC X CD

Prove that :- BD = BC

Solution:-

Given:- in which AB = AC and D is a point on AC
Such that BC² = AC X CD
To Prove :- BD = BC
Proof:- BC² = AC X CD
Or, BC X BC = AC X CD
Or,
In and BCD, we have
Or,
Or,
Or,

Example 8. D is a point on the side BC of a such that and are equal.

Prove that CA² = DC X CB

Solution:-

Given:- D is a point on the side BC of a such that
To Prove:- CA² = DC X CB
Proof:- In and DAC

Critieria for similarities of two triangles.

2011-03-27T16:40:00.000-07:00

Critieria for similarities of two triangles.

1. If in two triangles, the corresponding angles are equal, then their corresponading sides are proportional (i.e. in the same ratio) and hence the triangles are similar.

This property is referred to as the AAA similarily criterian

In the above property if only two angles are equal, then the third angle will be automatically equal

Hence AAA criteria is same as AA criteria.

2. If the coreponding sides of two trianlgles are proportional (i.e.in the same ratio), their corresponding angles are equal and hence the triabgles are similar.

This property is referredd to as SSS similarily criteria.

3. If one angles of a triangle is equal to one angle of the other and the sides including these angles are proportional, the triagngle are similar.

This proprerty is referred to as SAS critreria.

Example 6. P and Q are pointes on AB and AC respectively of If AP = 1cm, PB = 2cm, AQ = 3cm and QC = 6cm. Show that BC = 3PQ.

Solution:-

Given:- in which P and Q are points on AB and AC such that AP = 1cm, AQ = 3cm, PB = 2cm, QC = 6cm.
To Prove:- BC = 3PQ
Proof:-
Hence PQ || BC
and
But AB = AP + PB = 1 + 2 = 3cm
Hence BC = 3PQ.

Similar Triangles

2011-03-27T16:39:00.000-07:00

Exercise

1. In the given figure, PQ || BC, AP = 2.4cm, AQ = 2cm, QC = 3cm and BC = 6cm. Find AB and PQ.

2. The diagonals AC and BD of a quadrilatereal ABCD intersect each other at O such that

prove that the quadrilateral ABCD is traperzium.

3. In and if AC = 4.8cm, find AE

4. In the given figurer, PQ || BC and PR || CD,

prove that

5. In is parallel to base BC, with D on AB and E on AC. If find

6. In the given figure, PQ || AB and PR || AC. prove that QR || BC.

7. If three or more parallel lines, are intersected by two transversals, prove that the intercepts made by them on the trans versals are proportional.

8. In the given figure, DE || AC and DC || AP, prove that

9. In the given figure, and DE || AB prove that AD = BE.

10. In the given figure AB || CD. If OA = 3x - 19, OB = x - 4, OC = x - 3 and OD = 4cm, determine x.

Answers
(10). (x = 11 cm or 8 cm)

Similar Triangles

2011-03-27T16:37:00.000-07:00

Example 2. In the given figure, and Find x.

Solution:-

In (given)
Or, (4x - 3) (5x - 3) = (8x - 7) (3x - 1)
Or, 20x² - 12x - 15x + 9 = 24x² - 8x - 21x + 7
Or, 4x² - 2x - 2 = 0
Or, 2x² - x - 1 = 0
Or, 2x² - 2x + x - 1 = 0
Or, 2x(x - 1) + 1 (x - 1) = 0
(x - 1) (2x + 1) = 0
x = 1, -1/2
But sides of a triangle cannot be negative

Example 3. ABCD is a trapezium such that Its diagonals AC and BD intersest each other at o.

prove that .

Solution:- Given AC and BD intersectato

To Prove:
Construction: OP || AB || CD is drawn
Proof:- In
In
From (i) and (ii) we get

Example 4. Prove that any line parallel to parallel sides of a trapezium divides the non-parallel sides proportionally (i.e. in the same ratio)

Solution:-

Given:- PQRS is a trapezium in which PQ || SR || XY
To Prove:-
Construction:- PR is joined which intersects XY at A.
Proof:- In
[ Thales Theorem]
( Taking reciprocals)
From (i) and (ii) we get

Example 5. In the given figure DE || AQ and DF || AR

Prove that EF || QR

Solution:-

In
from (1) and (2) we get

Similar Triangles

2011-03-27T16:36:00.000-07:00

similar triangle

Similar Triangles

Whenever we talk about two congruent figures then they have the ‘same shape’ and the ‘same size’. There are figures that are of the ‘same shape but not necessarily of the ‘same size’. They are said to be similar. Congruent figures are similar but the converse is not true

All regular polygons of same number of sides are similar. They are equilateral triangles, squares etc. All circles are also similar.

Two polygons of the same number of sides are similar if their corresponding angles are sides are proportional.

Two triangles are similar if their corresponding are equal and corresponding sides are proportional.

Basic Proportionality Theorem or Thales Theorem.

Theorem-1

If a line is drawn parallel to one side of a triangle, to interest the other two sides indistinct points, the other two sides are divided in the same ratio.

Given: - In

To prove:-
Construction:- BE and CD are joined. and are drawn.
Proof:-
But
as they are on the same base DE and
from (1), (2) and (3) we get

Corollary: In then

Proof:- We know

(Taking reciprocals)
Multiplying (1) and (2) we get

Property - 1. If a line divides any two sides of a triangle in the same ratio, the line is parallel to the third side.

Example 1. In and Find AB.

Solution:-

(given)
(Thales Theoram)
= 3.6cm
AB = AD + DB
= 2.4 + 3.6
= 6.0cm

Polynomials

2011-03-27T16:18:00.000-07:00