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Q. If x = 1 + 2 ^1/3 + 2 ^2/3 then show that x^3 – 3x^2 – 3x – 1 = 0 Q. If x = 1 + root 2 + root 3 then show that x^4 – 4x^3 – 4x^2 + 16x – 8 = 0 For more questions: http://jsuniltutorial.weebly.com/ixnumbersystem.html 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) 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 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 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. 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.] Coordinate Geometry [ JSUNIL Tutorial] 1. Name the mathematician who developed Coordinate geometry. 2. How many points are required to locate a line segment? 3. The axis divide the plane into four parts. What these four parts called? 4. What are the coordinates of origin? 5. What is the point of intersection of axis called? 6. What is the distance of a point form y axis called? 7. What is the distance of a point from xaxis called? 9th Co Ordinate GeometryTest Paper Solved1
9th Co Ordinate GeometryTest Paper Solved2 Rest of the test paper and Study Materials IX Number System IX Polynomial(Algebra) IX Introduction to Euclid Geometry IX Lines and Angles (Geometry) IX Triangle(Geometry) IX COOrdinate Geometry IX Area of Triangle(Heron Formula) Linear equations in two variables 
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