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If the polynomial x4 – 6x3 + 16x2 – 26x + 10 – a is divided by another polynomial x2 – 2x + k, the remainder comes out to be x + a. Find k and a.
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Thales is the first mathematician credited with giving the first known proof “a circle is bisected by its diameter.
One of Thales’ most famous pupils was Pythagoras and his group discovered many geometric properties and developed the theory of geometry to a great extent.
At that time Euclid, a teacher of mathematics at Alexandria in Egypt, collected all the known work and arranged it in his famous treatise, called ‘Elements’. He divided the ‘Elements’ into thirteen chapters, each called a book.
Some definitions from book -1 of Elements are:
1. A point is that which has no part.
2. A line is breathless length.
3. The ends of a line are points.
4. A straight line is a line which lies evenly with the points on itself.
5. A surface is that which has length and breadth only.
6. The edges of a surface are lines.
7. A plane surface is a surface which lies evenly with the straight lines on itself.
Þ An axiom or a postulate is a mathematical statement which is assumed to be true without proof. These assumptions are actually obvious universal truths.
Þ Theorems are statements which are proved, using definitions, axioms, previously proved statements and deductive reasoning.
Þ Some of the Euclid’s axioms are:
(i) Things which are equal to same thing are equal to one another.
(ii) If equals are added to equals, the wholes are equal.
(iii) If equals are subtracted from equals, the remainders are equals.
(iv) Things which coincide with one another are equal to one another.
(v) The whole is greater than the part.
(vi) Things which are double of the same thing are equal to one another.
(vii) Things which are halves of the same thing are equal to one another.
Þ Euclid’s five postulates are:
(i) A straight line may be drawn from any point to any other point.
(ii) A terminated line can be produced indefinitely.
(iii) A circle can be drawn with any centre and any radius.
(iv) All right angles are equal to one another.
(v) If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.
A system of axioms is called consistent, if it is impossible to deduce from these axioms a statement that contradicts any axioms or previously proved statement.
Euclid’s first postulate can also be stated as below: Given two distinct points, there is a unique line that passes through them.
Two distinct lines cannot have more than one point in common.
Playfair’s Axiom: For every line l and for every point P not lying on l, there exists a unique line m, passing through P and parallel to l. [~ 5th Postulate]
Two distinct intersecting lines cannot be parallel to the same line.
CBSE Class IX Introduction to Euclid's geometry full study
Fill in the blanks
1. A force can cause a change in state and direction of an object.
2. An interaction between two objects causes a force.
3. Force applied by direct touching an object is called contact force.
4. Gravitational force is an example of non contact force.
5. Liquid exerts pressure in all direction.
6. Air exerts pressure on the earth due to its weight.
7. Atmospheric pressure is measured by using barometer is made by E. Torricelli.
8. Aneroid barometer does not use any liquid
9. Gases and Liquid are collectively called Fluid.
Match the Coolum
1. Earth revolving around the sun->gravitational force
2. A force applied by touching-> Contact force
3. Force between two charged object->electrostatic force
4. Bloating of the tube of cycle tyre -> air pressure
5. Force applied per unit area-> pressure
State true or false
1. A player kicking football in an example of a non contact force-> F
2. Liquid and gases do not exert pressure-> F
3. Pressure in liquid decreases with depth -> F
4. Like charges repel and unlike chare attract each other-> T
5. Magnetic pole attract magnetic substance like iron-> T
Q. What is the role of air pressure in the filling of a syringe with a liquid medicine, by a doctor?
Answer: The air pressure plays an important role in the filling of syringe as the liquid medicine rushes into the syringe when air pressure inside it decreases on pulling piston out.
Q. Why do people with high blood pressure sweat a lot?
Answer: This is because the pressure of their body fluid becomes more than atmospheric pressure .This forces the water to ooze out easily.
Q. There is a famous saying n Hindi a sword cannot replace a needle .Give reason for the saying in the light of physics
Answer: In the physics, the pressure applied on needle gets concentrated its tip i.e. on very small area and hence it pierces the cloth easily. But to make the sword cut a surface; we have to apply much higher pressure due to large surface area of the cutting edge of the sword. Hence a sword cannot replace a needle.
Q. Every square centimeter of our body experience atmospheric force equal to mass of 100kg that is enough to grind us. Then why do people cannot crushed by atmospheric pressure?
Answer: this is because there is pressure exerted by our body due to flow of blood that equalizes atmospheric pressure.
Q. A plastic comb when rubbed with hair can attract piece of paper. Name force and its nature?
Answer: Electrostatic Force. It is generated when a charged body meets to another charged or by rubbing two uncharged body.
Q. Why it is not easy to walk wearing high heels shoe?
Ans: High heels shoe distribute a large amount force in a Small area hence making her feel uncomfortable while walking on ground.
Q. Give two uses of fluid pressure?
Ans: Cooling, Cooking, filling LPG in cylinder
Q. Give reason:
(a)A rolling ball stops after moving some distance
Answer: This is because of friction force of the ground that opposes motion.
(b) Every object left above the surface of the earth without a support, fall downwards.
Answer: This is due to gravitational pull of the earth.
(c) If gravitational force act between you and your friend. Then why should not you pull each other?
Answer: Gravity only becomes noticeable when there is a really massive object like a moon, planet or star. Due to small masses no force of gravity is noticed.
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9th Coordinate Geometry: Key concepts
ÞCoordinate Geometry: The branch of mathematics in which geometric problems are solved through algebra by using the coordinate system is known as coordinate geometry.
Þ Coordinate axes: The position of a point in a plane is determined with reference to two fixed mutually perpendicular lines, called the coordinate axes.
Þ Coordinate System, position of a point is described by ordered pair of two numbers.
Þ Ordered pair: A pair of numbers a and b listed in a specific order with 'a' at the first place and 'b' at the second place is called an ordered pair (a, b). Note that (a, b) ¹ (b , a) and (x, y) = (y, x), if x = y.
Þ P(a,b) be any point in the plane. 'a' the first number denotes the distance of point from Y-axis and 'b' the second number denotes the distance of point from X-axis.
Þ The coordinates of origin are (0,0)
Þ Every point on the x-axis is at a distance o unit from the X-axis. So its ordinate is 0.
Þ Every point on the y-axis is at a distance of 0 unit from the Y-axis. So, its abscissa is 0.
Þ The coordinates of a point on the x-axis are of the form (x, 0) and that of a point on the y-axis are (0, y).
Þ A point in the first quadrant will be of the form (+, +). Similarly, a point in the second, third and fourth quadrants will be of the form (–, +), (–, –) and (+, –) respectively.
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Assignment- IX Co -Ordinate geometry -5
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Principal: MR. MD. ARIF
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BIHAR ; Phone 6274 - 222970
Q. Prove that no number of the type 4k+2 can be a perfect square.
Ans: If p is a prime factor of a perfect square, p2 must also be a factor of that perfect square. 4k+2 = 2(2k+1)
2 is a factor of 4k+2 but 2k+1 is odd and cannot have factor 2, so 4k+2 is not divisible by 4, and therefore cannot be a perfect square.
Q. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?
Ans: We need to calculate the LCM to find the answer. 18 = 2x3x3 ;12= 2x2x3 ; LCM = 36
Q. show that only one out of n, n+2 or n+4 is divisible by 3 where n is positive integer.
Solution: When n = 1, exactly one of 1, 1+2, and 1+4 is divisible by 3,
namely 1+2, since 3 is divisible by 3, and the other two 1 and 5 are not.
Suppose for n < k, only one out of n, n+2, n+4 is divisible by 3
For n = k, we consider k, k+2, k+4.
By the induction hypothesis, only one of k-1, k+1 and k+3 is
divisible by 3. We look at the three possible cases.
Case 1: k-1 is the one which is divisible by 3. Then k-1 = 3m, for some positive integer m.
Then add 1 to both sides of k-1 = 3m:
k = 3m+1 then k is NOT divisible by 3
Now add 3 to both sides of k-1 = 3m:
k+2 = 3m+3
k+2 = 3(m+1)
then k+2 IS divisible by 3
Now add 5 to both sides of k-1 = 3m:
k+4 = 3m+5
k+4 = 3m+3 + 2 = 3(m+1)+2
then k+4 is NOT divisible by 3.
So, we have proved case 1 for n = k
Case 2: k+1 is the one which is divisible by 3. Then k+1 = 3m, for some positive integer m.
Then add -1 to both sides of k+1 = 3m:
k = 3m-1
then k is NOT divisible by 3
Now add 1 to both sides of k+1 = 3m:
k+2 = 3m+1 then k+2 is NOT divisible by 3
Now add 3 to both sides of k+1 = 3m:
k+4 = 3m+3
k+4 = 3(m+1)
so k+4 IS divisible by 3.
So, we have proved case 2.
Case 3: k+3 is the one which is divisible by 3. Then k+3 = 3m, for some positive integer m.
Then add -3 to both sides of k+3 = 3m:
k = 3(m-1) then k IS divisible by 3
Now add -1 to both sides of k+3 = 3m:
k+2 = 3m-1 then k+2 is NOT divisible by 3
Now add 1 to both sides of k+3 = 3m:
k+4 = 3m+1 then k+4 is NOT divisible by 3.
So, we have proved case 3.
Q.If the H C F of 210 and 55 is expressible in the form 210 × 5 + 55y then find y
Answer: Let us first find the H C F of 210 and 55.
Applying Euclid division lemna on 210 and 55, we get
210 = 55 × 3 + 45 ....(1)
Since the remainder 45 ≠ 0. So, again applying the Euclid division lemna on 55 and 45, we get
55 = 45 × 1 + 10 .... (2)
Again, considering the divisor 45 and remainder 10 and applying division lemna, we get
45 = 4 × 10 + 5 .... (3)
We now, consider the divisor 10 and remainder 5 and applying division lemna to get
10 = 5 × 2 + 0 .... (4)
We observe that the remainder at this stage is zero. So, the last divisor i.e., 5 is the HCF of 210 and 55.
∴ 5 = 210 × 5 + 55y
⇒ 55y = 5 - 1050 = -1045
⇒ y = -19
Q. Finds the H.C.F. of 65 and 117 and express it in the form of 65m+117n.
Answer: Among 65 and 117; 2117 > 65
Since 117 > 65, we apply the division lemma to 117 and 65 to obtain
117 = 65 × 1 + 52 … Step 1
Since remainder 52 ≠ 0, we apply the division lemma to 65 and 52 to obtain
65 = 52 × 1 + 13 … Step 2
Since remainder 13 ≠ 0, we apply the division lemma to 52 and 13 to obtain
52 = 4 × 13 + 0 … Step 3
In this step the remainder is zero. Thus, the divisor i.e. 13 in this step is the H.C.F. of the given numbers
The H.C.F. of 65 and 117 is 13
From Step 2:
13 = 65 – 52× 1 … Step 4
From Step 1: 52 = 117 – 65 × 1
Thus, from Step 4, it is obtained
13 = 65 – (117 – 65 × 1)
⇒13 = 65 × 2 – 117
⇒13 = 65 × 2 + 117 × (–1)
In the above relationship the H.C.F. of 65 and 117 is of the form 65m + 117 n, where m = 2 and n = –1
Q.Find all positive integral values of n for which n2+96 is perfect square.
Answer: Let n2 + 96 = x2
⇒ x2 – n2 = 96
⇒ (x – n) (x + n) = 96
⇒ both x and n must be odd or both even
on these condition the cases are
x – n = 2, x + n = 48
x – n = 4, x + n = 24
x – n = 6, x + n = 16
x – n = 8, x + n = 12
and the solution of these equations can be given as
x = 25, n = 23
x = 14, n = 10
x = 11, n = 5
x = 10, n = 2
So, the required values of n are 23, 10, 5, and 2.
Q. Prove that one of every three consecutive integers is divisible by 3.
Ans: n,n+1,n+2 be three consecutive positive integers We know that n is of the form 3q, 3q +1, 3q + 2
So we have the following cases
Case – I when n = 3q
In the this case, n is divisible by 3 but n + 1 and n + 2 are not divisible by 3
Case - II When n = 3q + 1
put n = 2 = 3q +1 +2 = 3(q +1) is divisible by 3. but n and n+1 are not divisible by 3
Case – III When n = 3q +2
put n = 2 = 3q +1 +2 = 3(q +1) is divisible by 3.
But n and n+1 are not divisible by 3
Hence one of n, n + 1 and n + 2 is divisible by 3
More solved Questions:
8th Science Chapter 3 - Synthetic Fibres and Plastics[Solved Questions]
Q. Name some natural fibres?
Answer: Fibres obtained from plants or animals is called natural fibres for examples cotton, wool, silk, etc.
Q. What do you mean by synthetic fibres?
Answer: Fibres made by human beings using chemical substance is called synthetic fibre. A synthetic fibre is a chain of small unit of chemical substance joined together. These small units combine to form a large single unit called a polymer. The word ‘polymer’ comes from two Greek words; poly meaning many and mer meaning part/unit. So, a polymer is made of many repeating units.
Q. Name a polymer occurs naturally?
Answer: Polymers that occur naturally is cellulose. Cellulose is made up of a large number of glucose units.
Q. Name a fibre having properties similar to that of silk.
Q. Although rayon is obtained from a natural source, wood pulp, yet it is a man-made fibre. Why?
Answer: This is because rayon is obtained by chemical treatment of wood pulp.
Q. Why does Nylon become very popular for making clothes for mankind?
Answer: Nylon is the first fully synthetic fibre prepared from coal, water and air in 1931. Nylon fibres are strong, elastic and light. It is lustrous and easy to wash. So, it became very popular for making clothes.
Q. Is nylon fibre really so strong that we can make nylon parachutes and ropes for rock climbing?
Answer: Yes, It is because nylon thread is actually stronger than a steel wire.
Q. Why does polyester fibre quite suitable for making dress material?
Answer: Fabric made from Polyester fibre does not get wrinkled easily. It remains crisp and is easy to wash. So, it is quite suitable for making dress material. Terylene is popular polyester.
Q. Name a form of polyster which is used for making bottles, utensils, films, wires and many other useful products?
Ans: PET: PET (polyethylene terephthalate) is made from two monomers, terephthalic acid and ethylene glycol, by the process called condensation polymerization
Q. Name the chemical used to make polyster?
Answer: Esters are the chemicals used to make polyster.
Q. Why synthetic fibres are more popular than natural fibres?
Answer: Synthetic fibres dry up quickly, durable, less expensive, readily available and easy to maintain which makes them more popular than natural fibres.
Q. Identify the Synthetic fibre appears to resemble wool?
Download full paper by clicking links given below8th Synthetic fiber and Plastic [Solved Questions]
VIII Synthetic fiber and Plastic
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VIII Synthetic fiber and Plastic
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Q. What are crops?
Answer: The same kind of plants grown and cultivated at one place on a large scale is called a crop. For example, wheat, maize, etc
Q.What is the different types of crops grown in India?
Ans: These are:
(i) Kharif Crops : The crops which are sown in the rainy season (from June to September)are called Kharif crops. Paddy, maize, soya bean, groundnut, cotton, etc., are kharif crops.
(ii) Rabi Crops : The crops grown in the winter season (October to March )are called rabi crops.
Examples of rabi crops are wheat, gram, pea, mustard and linseed.
Q. Why is proper agricultural management and distribution of food necessary ?
Answer: In order to provide food for a large population—regular production, proper management and distribution of food is necessary.
Q. Why can paddy not be grown in the winter season?
Answer: Paddy requires a lot of water. Therefore, it is grown only in the rainy season.
Q. List different agricultural practice ?
Answer: The agricultural practices are listed below.
(i) Preparation of soil (ii) Sowing (iii) Adding manure and fertilizers
(iv) Irrigation (v) Protecting from weeds (vi) Harvesting (vii) Storage
Q. Why does the loosening of soil allow the roots to breathe easily?
Answer: The process of loosening and turning of the soil is called tilling or ploughing.This is done by using a plough.
The air enters easily through loosen soil and the root can breathe easily even when they go deep into the soil.
Q. Why earthworms are called farmers friends?
Ans: The loosened soil helps in the growth of earthworms and microbes present in the soil. These organisms are friends of the farmer since they further turn and loosen the soil and add humus to it.
Q.why does the soil need to be turned and loosened before seeds are sown?
Answer: Since only a few centimeters of the top layer of soil supports plant growth, turning and loosening of soil brings the nutrient-rich soil to the top so that plants can use these nutrients.
Q. What are the purposes of the leveling of soil?
Answer: The ploughed field may have big pieces of soil called crumbs. It is necessary to break these crumbs with a plank. The field is levelled for sowing as well as for irrigation purposes. The levelling of soil is done with the help of a leveller.
Q.Why we sometimes add manure to the soil before tilling?
Answer: This helps in proper mixing of manure with soil.
Q. Name some agricultural Implements?
Answer: The main tools used for agriculture purpose are the plough, hoe and cultivator.
Q.How can we separate good, healthy seeds from the damaged ones?
Answer: We put seeds into bucket full of water. Damaged seeds being hollow became lighter therefore they float on water and remaining good seeds sink at bottom.
Q. Why sowing seeds with seed drill is better than by hand?
Answer: Practice of sowing seeds by hand is called broadcasting. The seed drill is used for sowing with the help of tractors. This sows the seeds uniformly at proper distances and depths. It ensures that seeds get covered by the soil after sowing. This also prevents damage caused by birds.
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(i) The rational number that does not have a reciprocal.
(ii) The rational numbers that are equal to their reciprocals.
(iii) The rational number that is equal to its negative.
Answer: (i) 0 (ii) 1 and (–1) (iii) 0
Q. 2. Fill in the blanks.
(i) Zero has ________ reciprocal.
(ii) The numbers ________ and ________ are their own reciprocals
(iii) The reciprocal of – 5 is ________.
(iv) Reciprocal of 1/x , where x ≠ 0 is ________.
(v) The product of two rational numbers is always a _______.
(vi) The reciprocal of a positive rational number is ________.
Answer: 11. (i) No (ii) 1, –1 (iii) -1/5 (iv) x (v) Rational number (vi) positive
Q.3. Write any 3 rational numbers between –2 and 0.
Q.4. Find any ten rational numbers between -5/6 and 5/8
Q.5. Represent these numbers on the number line. (i) 7/4 (ii) -5/6(iii) 36/7(iv) 21/3
Q.6. Multiple Choice Questions
1. A rational number p/q is said to be in the simplest form if the HCF of p and q is
(a) 2 (b) 1 (c) 0 (d) 3 [A]
2. Between any two distinct rational numbers there exist
(a) Finite rational numbers (b) Infinite rational numbers (c) No rational number (d) none of the above [B]
3. A rational number a/b is greater than c/d if
(a) ad > bc (b) ad < bc (c) ad = bc (d) ad ≠ bc [A]
4. Is zero a rational number
(a) Yes (b) No (c) Can’t say [A]
5. Rational numbers are not closed under
(a) Addition (b) Multiplication (c) Division (d) Subtraction [C]
7. What number should be added to −78to get 49?
8. The sum of two rational numbers is −12. If one of the numbers is 56, find the other.
9. What number should be subtracted from −23 to get −12?
10. Divide the sum of 135and−127by the product of −317and1−2.
11. The product of two rational numbers is −169. If one of the numbers is −43, find the other.
12. A drum full of rice weights 4016kg. If the empty drum weights 1334kg, find the weight of rice in the drum.
13. A car is moving at an average speed of 4025km/h. How much distance will it cover in 712h?
14. Raju earns Rs16000/m. He spends 14of his income on food; 310of the remainder on house rent and 521of the remainder on education of children. How much money is still left with him?
15. After reading 79of a book, 40pages are left. How many pages are there in the book?
16. If you subtract 1/2 from a number and multiply the result by 1/2, you get 1/8. What is the number?
17. Three consecutive integers are such that when they are taken in increasing order and multiplied by 2, 3, and 4 respectively, they add up to 74. Find these numbers.
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