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EUCLID’S Postulates
1. A straight line may be drawn from any point to any other point. 2. A terminated line (line segment) can be produced indefinitely. 3. A circle may be described with any centre and any radius. 4. All right angles are equal to one another. 5. 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 the two straight lines if produced indefinitely, meet on that side on which the sum of angles is taken together less than two right angles. Euclid used the term postulate for the assumptions that were specific to geometry and otherwise called axioms. A theorem is a mathematical statement whose truth has been logically established. Visit Page Link to download solved Questions and Guess test Paper CBSE Class IX Introduction to Euclid's geometry Proof of this factor theorem
Let p(x) be a polynomial of degree greater than or equal to one and a be areal number such that p(a) = 0. Then, we have to show that (x – a) is a factor of p(x).Let q(x) be the quotient when p(x) is divided by (x – a). By remainder theorem Dividend = Divisor x Quotient + Remainder p(x) = (x – a) x q(x) + p(a) [Remainder theorem] ⇒ p(x) = (x – a) x q(x) [p(a) = 0]⇒ (x – a) is a factor of p(x) Conversely, let (x – a) be a factor of p(x). Then we have to prove that p(a) = 0 Now, (x – a) is a factor of p(x)⇒ p(x), when divided by (x – a) gives remainder zero. But, by the remainder theorem, p(x) when divided by (x – a) gives the remainder equal to p(a). ∴ p(a) = 0 Proof of remainder theorem. Let q(x) be the quotient and r(x) be the remainder obtained when the polynomial p(x) is divided by (x–a). Then, p(x) = (x–a) q(x) + r(x), where r(x) = 0 or some constant. Let r(x) = c, where c is some constant. Then p (x) = (x–a) q(x) + c Putting x = a in p(x) = (x–a) q(x) + c, we getp(a) = (a–a) q(a) + c ⇒ p(a) = 0 x q(a) + c ⇒ p(a) = c This shows that the remainder is p(a) when p(x) is divided by (x–a).1. [a is –8.] Download study material for polynomial 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
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 Yaxis and 'b' the second number denotes the distance of point from Xaxis. Þ The coordinates of origin are (0,0) Þ Every point on the xaxis is at a distance o unit from the Xaxis. So its ordinate is 0. Þ Every point on the yaxis is at a distance of 0 unit from the Yaxis. So, its abscissa is 0. Þ The coordinates of a point on the xaxis are of the form (x, 0) and that of a point on the yaxis 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. Download links IX Co Ordinate geometry Test Paper1 IX Co Ordinate geometry Test Paper2 IX Co Ordinate geometry Test Paper3 IX Co Ordinate geometry Test Paper4 Assignment IX Co Ordinate geometry 5 Download above File 1. AB is a line segment and line l is its perpendicular bisector. If a point P lies on l, show that P is equidistant from A and B. 2. In DABC , ∠Q > ∠R, PA is the bisector of ∠QPR and PM ^QR. Prove that <∠APM = 1/2(∠< Q – ∠<R). 3. D ABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB. Show that ∠BCD is a right angle. 4. In a right angled triangle, one acute angle is double the other. Prove that the hypotenuse is double the smallest side. 6. Example 5. If the bisector of the vertical angle of a triangle bisects the base, prove that the triangle is isosceles. 7. A triangle ABC is right angled at A. L is a point on BC such that AL ^ BC. Prove that ∠ < BAL = ∠ < ACB 8. Q is a point on the side SR of a Δ PSR such that PQ = PR. Prove that PS > PQ. 9. S is any point on side QR of a Δ PQR. Show that: PQ + QR + RP > 2 PS. 10. D is any point on side AC of a Δ ABC with AB = AC. Show that CD < BD. 11. l  m and M is the midpoint of a line segment AB. Show that M is also the midpoint of any line segment CD, having its end points on l and m, respectively. 12. Bisectors of the angles B and C of an isosceles triangle with AB = AC intersect each other at O. BO is produced to a point M. Prove that ∠MOC =∠ABC. 13. Bisectors of the angles B and C of an isosceles triangle ABC with AB = AC intersect each other at O. Show that external angle adjacent to ∠ABC is equal to ∠BOC. 14. S is any point in the interior of Δ PQR. Show that SQ + SR < PQ + PR. {Produce QS to intersect PR at T} 15. Prove that in a right triangle, hypotenuse is the longest (or largest) side. Related posts: PDF Download CLICK HERE CBSE Sample Questions Papers Maths Ch: Triangle IX Mathematics Congruence of Triangle CBSE Examination Question 201213 CBSE Test sample paper IX Mathematics (Congruent triangle) CBSE TEST PAPER CLASS  IX Mathematics (Congruent triangle) 1. The area of a triangle is 30 cm2. Find the base if the altitude exceeds the base by 7 cm.[ 5 cm , 12 cm.]
2. From a point in the interior of an equilateral triangle, perpendiculars drawn to the three sides are 8 cm, 10 cm and 11 cm respectively. Find the area of the triangle. [486.1 cm2] 3. The difference between the sides at right angles in a rightangled triangle is 14 cm. The area of the triangle is 120 cm2. Calculate the perimeter of the triangle. [24 cm, 10 cm, 60 cm.] 4. Find the percentage increase in the area of a triangle if its each side is doubled? [300%] 5. Calculate the area of the triangle whose sides are 18 cm, 24 cm and 30 cm in length. Also, find the length of the altitude corresponding to the smallest side of the triangle. 6. The sides of a triangle are 10 cm, 24 cm and 26 cm. Find its area and the longest altitude. 7. Two sides of a triangular field are 85 m and 154 m in length, and its perimeter is 324 cm. Find (i) the area of the field, and (ii) the length of the perpendicular from the opposite vertex on the side measuring 154 cm. 8. The sides of a triangular field are 165 cm, 143 cm and 154 cm. Find the cost of ploughing it at 12 paise per sq. m. 9. The base of an isosceles triangle measures 80 cm and its area is 360 cm2. Find the perimeter of the triangle. 10. The perimeter of an isosceles triangle is 42 cm and its base is 11/2 times each of the equal sides. Find (i) the length of each side of the triangle, (ii) the area of the triangle, and (iii) the height of the triangle. 11. The perimeter of a right angle triangle is 40 cm. Its hypotenuse is 17 cm. Find the sides containing the right angle. Also find the area of the triangle. VERY SHORT QUESTIONS WITH ANSWERS (1 MARK) (ii) No residue is left after it has burnt itself out. i) The initial cost is high. ii) It causes displacement of people from large areas of land.
(ii) what type of mirror is used in Box Type Solar cooker? (iii) why it is difficult to use hydrogen as a source of energy? (iv)what is the maximum temperature attained in a concave reflector type solar cooker? (v)Fossil fuels are classified as nonrenewable source of energy. Explain why. (iii) It cannot be stored safely as (a) it burns with explosion and(b) has low ignition temperature. (iv) about 200oC. (v)fossil fuels cannot be replenished. It takes millions of years for their formation. (b) Why is the energy of water flowing in a river considered to be an indirect form of solar energy? (c) Write one advantage of nuclear fission reaction.
Q.6. A student has set up a solar cooker in a box by using a black painted aluminum sheet, a black cooking vessel, some glass wool, a glass sheet and a mirror plate. What is the role of each item used in the solar cooker? 1. What is magnetic field? Answer: The region around the magnet, where force of attraction or repulsion can be felt by magnetic materials, is called as magnetic field. 2. What will be the frequency of an alternating current, if its direction changes after every 0.05 s? Answer: The time period (T) of one cycle would be = 2 x (0.05 s) = 0.1 s. frequency, f = 1/T. Hence, f = (1 / 0.1) = 10 Hz. 3.What is the principle on which working of electric generator is based? What are its important parts? Answer: The working of electric generator is based on the principle of electromagnetic induction. Important parts of electric generator are armature, slip rings, brushes and field magnets. 4. Why two magnetic lines of force don’t intersect each other? Answer: If so then at the point of intersection there will be two different directions of magnetic field which is not possible. 5. Give two methods by which we can increase the strength of magnetic field produced by a circular coil carrying current? Answer: The two methods by which we can increase the strength of magnetic field are given below: 1. By increasing the number of turns of wire in the coil. 2. By increasing the current flowing through the coil. 6. What are the patterns of magnetic field lines inside and outside of a solenoid? What do they indicate? Answer: (i)The field lines inside the solenoid are parallel straight lines. This indicates that the magnetic field is uniform and is therefore, same at all points inside it. (ii)The field lines outside the solenoid are curved lines. This indicates that the magnetic field is nonuniform. 7. What are the factors which govern the force experienced by a current carrying conductor placed in a uniform magnetic field depends? Answer: The factors which govern the force experienced by conductor which is placed in a uniform magnetic field are: 1. strength of the magnetic field in which conductor is placed. 2. strength of current flowing through the conductor. 3. length of conductor. 8. Distinguish between an electric motor and generator? Answer: The major differences between an electric motor and generator are stated below: Electric Motor 1. It converts electrical energy into mechanical energy. 2. It needs electrical energy for its working. 3. They are used as water pumps, marble grinders e.t.c Generator 1. It converts mechanical energy into electrical energy 2. It needs mechanical energy for its working. 3. They are used as water pumps, marble grinders e.t.c 3. Diesel generator, hydroelectric generator are the examples of it. 9. Why mostly all electrical home appliances like refrigerator, toaster etc. are provided with a wire having green insulation? Answer: Earth wire is the insulated green coloured copper wire which is connected to a metal plate deep inside the earth near the house. It is used as a safety measure. It is kept at zero potential by connecting it to the ground, thus, providing a lowresistance conductive path for the current. When by electric fault, if current flows through the metallic body of electric appliance and we touch it, then current finds the lower resistance path of earth wire than our body. So, current flows through earth wire only and we remain safe. 10. Describe domestic household circuit. Answer: From an electric pole or underground cables, we receive electric supply in our homes. Domestic electric circuit consists of three main wires.The wire with red insulation is called live wire. The wire with black insulation is called neutral wire. Potential difference between the neutral and the live wire is 220 V. The wire with green insulation is called earth wire. It is connected to a metallic body deep inside the earth. According to new International Convention, insulation of live wire should be of brown colour whereas neutral and earth wires should be of light blue and green (or yellow) insulation cover. In our homes, we receive AC electric power of 220 V with a frequency of 50 Hz. Live wire and neutral wire maintains the p.d. of 220 V. These wires pass through the fuse board. Fuse is specially connected with the live wire. Current rating of this fuse depends on house load. From the fuse board, these wires pass through the electric metre. From the metre, the earth wire is locally inserted inside the ground of the house. After the electric metre, these wires pass through the main switches and fuses in different rooms. Depending on the power of electrical appliances, two types of fuse (5A or 15 A) are used. From the switch and fuse board, the power lines are distributed to different electrical appliances. 11. Describe about short circuiting and overloading in detail. Answer: Short Circuiting: Shortcircuiting occurs in a circuit when the live wire comes in direct contact with the neutral wire. This causes much damage to the electric appliances connected with them. Reason of shortcircuiting: When the live wire touches the neutral wire, the contact portion of the two wires behave like load and due to very low resistance of copper wire, it draws very large amount of current. In the presence of this high current, the copper wire catches fire, due to joules heating effect. Overloading: Overloading occurs in a circuit when the current in the circuit increases abruptly. It causes overheating of the wire and might lead to the fire. It can also occur due to accidental increase in the supply voltage or on connecting too many appliances to a single socket. 12. What is an electromagnet? On what factors its magnetic field depends? Answer: Electromagnet is a magnet formed by the magnetization of a piece of a magnetic material (such as soft iron) by inserting it into a solenoid. The magnetic field (strength) of an electromagnet depends upon: (i) Amount of current passing through the coil (ii) Number of turns of the coil (iii) Nature of core material present inside coil 13. Mention the differences between an electromagnet and a permanent magnet. Answer: Differences between electromagnet and permanent magnet are: Electromagnet 1.It is temporary magnet and can be demagnetized. 2.Magnetic strength can be changed. . 3.Its polarity can also be changed. 4.It is prepared from soft iron. Permanent Magnet 1.It is a permanent magnet and cannot be demagnetized easily. 2. Its strength is fixed 3.Its polarity is fixed. 4.It is prepared from hard steel. 14. What do you understand by magnetic effect of electric current? Answer: The phenomenon due to which, a wire behaves like a magnet when electric current is passed through it, is called as the magnetic effect of electric current. 15. Explain Maxwell's right hand thumb rule? Answer: According to Maxwell's right hand thumb rule, “If we hold the currentcarrying conductor in our right hand in such a way that the thumb is stretched along the direction of the current, then the curled fingers give the direction of the magnetic field produced by the current”, 16. What are the properties of magnetic field due to a current through a straight wire? Answer: The properties of magnetic field due to a current through a straight wire are: 1. The magnitude of the magnetic field produced at a given point increases as the current through the wire increases. 2. The magnetic field produced by a given current in the conductor decreases as the distance from it increases. 3. The concentric circles representing the magnetic field around a currentcarrying straight wire become larger and larger as we move away from it. 4. If the direction of the current is reversed in the wire, the lines will still be circular, but the directions of the lines will be reversed, which can be verified using the compass needle. 17. Which rule is used find the direction of the force on a conductor in a magnetic field. Explain? Answer: To find the direction of the force on a conductor in a magnetic field, a simple rule known as Fleming's lefthand rule is used. According to Fleming's lefthand rule, if you stretch the thumb, forefinger and middle finger of your left hand such that they are mutually at right angles, If the First finger points in the direction of the field. The second finger represents the direction of the current (in the classical direction, from positive to negative), then the thumb will point in the direction of the force acting on the conductor or in the direction of the resultant motion. This rule is used to know the direction of the induced current 18. Give characteristics of magnetic field lines. Ans: (i) They represent the magnetic field. (ii) They are directed from north to South Pole outside a magnet and from south to North Pole inside a magnet. (iii) The field lines are closed curves. (iv) The strength of magnetic field in a region is determined by closeness of the field lines in that region. (v) The closer the field lines are, greater will be the field strength and vice–versa. (vi) No two field lines ever cross each other as it would mean two different directions of field at point of intersection, which is not possible. (viii) The parallel lines represent the uniform magnetic field whereas converging lines or diverging lines represents the non uniform magnetic field. 19. What do you mean by electromagnetic induction? Answer: "The process, due to which a changing magnetic field in a conductor induces a current in another conductor, is called electromagnetic induction". 20. A current through a horizontal power line flows in north to south direction.What is the direction of magnetic field (i)at a point directly below it and (ii)at a point directly above it? Answer: (i) West to East (ii) East to West 21. A straight wire carrying electric current is moving out of plane of paper and is perpendicular to it. What is the direction and type of induced magnetic field? Ans . Induced magnetic field will be in the form of concentric circles in the plane of paper. 22.. How can it be shown that magnetic field exist around a wire carrying current? Ans:By using magnetic compass which, shows deflection. 23 . How can a solenoid be used to magnetise a steel bar. Answer: By inserting the steel bar inside the solenoid and switching on electric current. 24. Can a 5 A fuse be used in wire carrying 15 A current? Why? Answer: No, because both of them would then be ineffective in controlling the amount of current flowing. 25. Give the factors that affect strength of magnetic field at a point due to a straight conductor carrying current. Answer: Magnitude of electric current, perpendicular distance between that point and conductor. ====================more to study========================= Q. Where do we connect a fuse: with live wire or with neutral wire? Ans: It is always connected with live wire. Q. Give two uses of electromagnets. Ans: (i) It is used in cranes for lifting heavy loads. (ii) used in electric bells. Q. Name any two devices which use permanent magnets. Ans: Loudspeakers, Galvanometer, voltmeter. Q.A currentcarrying straight conductor is placed in the eastwest direction. What will be the direction of the force experienced by this conductor due to earth’s magnetic field? How will this force get affected on? (a) reversing the direction of floe of current (b) doubling the magnitude of current. Answer: The direction of earth’s magnetic field is from Gsouth to Gnorth. Let current is from west to east. Therefore force is vertically upwards. (a) By reversing the direction of current, the direction of will be reversed i.e. vertically downwards. (b) The magnitude of the force is doubled. Q. An electron enters a magnetic field at right angles to it as shown in fig. The direction of the force acting on the electron will be: (a) to the right (b) to the left (c) out of the page (d) into the page Answer: When a conductor carrying current is placed perpendicular to the direction of magnetic field, the acting on it is given by Fleming’s left hand rule. Since the direction of current is the same as that of the motion of a positive charge, the direction of force acting on it when moving perpendicular to the direction of magnetic field is the same as that acting on a currentcarrying conductor placed perpendicular to the direction of magnetic field. Obviously, the force acting on an electron is opposite to that. Therefore in this case it is into the page. Q. A coil of insulated copper wire is connected to a galvanometer. What would happen if a bar magnet is (i) Pushed into the coil? (ii) Withdrawn from inside the coil? (iii) Held stationary inside the coil? Answer: (i) Due to change in magnetic flux linked with coil, the galvanometer shows deflection (say towards right). (ii) Due to change in magnetic flux linked with coil, the galvanometer shows deflection (say towards left opposite to that in case one). ( ii i) As it is stationary no change in magnetic flux linked with coil, so galvanometer shows no deflection. Q.A magnetic compass needle is placed in the plane of paper near point A as shown in Figure 13.6. In which plane should a straight current carrying conductor be placed so that it passes through A and there is no change in the deflection of the compass? Under what condition is the deflection maximum and why? Answer: In the plane of the paper itself. The axis of the compass is vertical and the field due to the conductor is also vertical. It could result in a dip of compass needle which is not possible in this case (dips result only if axis of compass is horizontal). The deflection is maximum when the conductor through A is perpendicular to the plane of paper and the field due to it is maximum in the plane of the paper Q. Under what conditions permanent electromagnet is obtained if a current carrying solenoid is used? Answer: (i) The current through the solenoid should be direct current. (ii) The rod inside is made of a magnetic material such as steel Q. It is established that an electric current through a metallic conductor produces a magnetic field around it. Is there a similar magnetic field produced around a thin beam of moving (i) alpha particles, (ii) neutrons? Justify your answer. Answer: (i) Yes, Alpha particles being positively charged constitutes a current in the direction of motion. (ii) No. The neutrons being electrically neutral constitute no current. Q. Meena draws magnetic field lines of field close to the axis of a current carrying circular loop. As she moves away from the centre of the circular loop she observes that the lines keep on diverging. How will you explain her observation. Answer: Strength of the magnetic field falls as distance increases. This is indicated by the decrease in degree of closeness of the lines of field. Q. What does the divergence of magnetic field lines near the ends of a current carrying straight solenoid indicate? Ans: The divergence, that is, the falling degree of closeness of magnetic field lines indicates the fall in strength of magnetic field near and beyond the ends of the solenoid. Q. What is the role of the two conducting stationary brushes in a simple electric motor? Ans: The brushes are connected to the battery and touch the outer side of two halves of the split ring whose inner sides are insulated and attached to the axle. Q. What is the difference between a direct current and an alternating current? How many times does AC used in India change direction in one second? Ans: Direct current always flows in one direction but the alternating current reverses its direction periodically. The frequency of AC in India is 50 Hz and in each cycle it alters direction twice. Therefore AC changes direction 2 × 50 = 100 times in one second. Q.What is the role of fuse, used in series with any electrical appliance? Why should a fuse with defined rating not be replaced by one with a larger rating? Ans: Fuse is used for protecting appliances due to shortcircuiting or overloading. The fuse is rated for a certain maximum current and blows off when a current more than the rated value flows through it. If a fuse is replaced by one with larger ratings, the appliances may get damaged while the protecting fuse does not burn off. This practice of using fuse of improper rating should always be avoided 
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