#cbseclass10maths

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DIVISION OF A LINE SEGENT

DIVISION OF A LINE SEGENT :

This content is part of NCERT Solutions for Class 10 Maths Chapter 11 Constructions

           In order to divide a line segment internally is a given ratio m: n, where both m and n are positive integers,

 we follow the following steps:

Step of construction :

           (i)         Draw a line segment AB of given length by using a ruler.

(ii)         Draw and ray AX making an acute angle with AB.

(iii)        Along AX mark off (m + n) points A1, A2,..., Am+n such that AA1 = A1A2 = ....=Am+n+Am+n.

(iv)        Join B Am+n

(v)        Through the point Am draw a line parallel to Am+n B by making an angle equal to at Am.

Suppose this line meets AB at a point P.

           The point P so obtained is the required point which divides AB internally in the ratio m : n.

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NCERT Solutions for Class 10 Science Chapter 3 Metals and Non-metals

This content is part of NCERT Solutions for Class 10 Science Chapter 3 Metals and Non-metals

GENERAL PROPERTIES OF METALS AND NON-METALS :

 7.3 (a) Electronic Configuration of Metals :

 The atoms of metals have 1 to 3 electrons in their outermost shells. For example, all the alkali metals have one electron in their outermost shells (lithium 2, 1; sodium 2,8,1: potassium 2,8,8,1 etc.)

Sodium, magnesium and aluminum are metals having 1,2 and 3 electrons respectively in their valence shells. Similarly, other metals have 1 to 3 electron in their outermost shells.

It may be noted that hydrogen and helium are exception because hydrogen is a non-metal having only  electron in the outermost shell (K shell) of its atom and helium is also a non-metal having 2 electron in the outermost shell (K shell).

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NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas and Volumes

This content is part of NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas and Volumes

MENSURATION (SOLID FIGURES) :

           If any figure such as cuboids, which has three dimensions length, width and height are height are known as three dimensional figures. Where as rectangle has only two dimensional i.e., length and width. Three dimensional figures have volume in addition to areas of surface from which these soils figures are formed.

           Some of the main solid figures are:

 13.3     (a) Cuboid:

             Total Surface Area (T.S.A.) : The area of surface from which cuboid is formed. There are six faces (rectangular), eight vertices and twelve edges n a cuboid.

           (i)Total Surface Area (T.S.A.)  

             (ii) Lateral Surface Area (L.A.A.) =

             (or Area of 4 walls)                   =

             (iii)       Volume of Cuboid = (Area of base) × height

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CIRCLE

CIRCLE

This content is part of NCERT Solutions for Class 10 Maths Chapter 10 Circles

           A circle is the locus of a points which moves in a plane in such a way that its distance from a fixed point

 remains constant.

 9.2        SECANT AND TANGENT :

                  Secant to a circle is a line which intersects the circle in two distinct points.

                  A tangent to a circle is a line that intersects the circle in exactly one point.

 9.3        THEOREM :

Statement :       A tangent to a circle i perpendicular to the radius through the point of contact.

Given :             A circle C (O, r) and a tangent AB at a point P.

 To prove :        OP AB

 Construction :  Take any points Q, other than P on the tangent AB. Join OQ. Suppose OQ meets the circle at R.

 Proof:              Among all line segments joining the point O to a point on AB, the shorted one is perpendicular to

 AB. So, to prove that OP AB, it is sufficient to prove that OP is shorter than any other segment

 joining O to any point of AB.

Clearly  OP = OR

Now,     OQ  OR + RQ

      OQ > OR

      OQ > OP (OP = OR)

Thus, OP is shorter than any other segment joining O to any point of AB.

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ANGLE OF ELEVATION

ANGLE OF ELEVATION :

This content is part of NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry

           In order to see an object which is at a higher level compared to the ground level we are to look up. The line joining the object and the eye of the observer is known as the line sight and the angle which this line of sight makes with the horizontal drawn through the eye of the observer is known as the angle of elevation. Therefore, the angle of elevation of an object helps in finding out its height (figure)

 12.2      ANGLE OF DEPRESSION :

             When the object is at a lower level tan the observer’s eyes, he has to look downwards to have a view of the object. It that case, the angle which the line of sight makes with the horizontal thought the observer’s eye is known as the angle of depression (Figure).

             ILLUSTRACTIONS :

             In order to see an object which is at a higher level compared to the ground level we are to look up. The line joining the object and the eye of the observer is known as the line sight and the angle which this line of sight makes with the horizontal drawn through the eye of the observer is known as the angle of elevation. Therefore, the angle of elevation of an object helps in finding out its height (figure)

 12.2      ANGLE OF DEPRESSION :

             When the object is at a lower level tan the observer’s eyes, he has to look downwards to have a view of the object. It that case, the angle which the line of sight makes with the horizontal thought the observer’s eye is known as the angle of depression (Figure).

             ILLUSTRACTIONS :

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CONGRUENT AND SIMILAR FIGURES

CONGRUENT AND SIMILAR FIGURES:

This content is part of NCERT Solutions for Class 10 Maths Chapter 6 Triangles

Two geometric figures having the same shape and size are known as congruent figures. Geometric figures having the same shape but different sizes are known as similar figures.

8.2       SIMILAR TRIANGLES:

               Two triangles ABC and DEF are said to be similar if their

               (i)            Corresponding angles are equal.

                               i.e. ÐA = ÐD, ÐB = ÐE, ÐC = ÐF

                               And,

               (ii)           Corresponding sides are proportional i.e.

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MENSRTION

MENSRTION:

This content is part of NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas and Volumes

           Figure lying in a plane is called a plane figure. A plane figure made up of lines or curve or both, is said to be a closed figure if it has on free ends. Closed figure in a plane covers some part of the plane, then magnitude o that part of the plane is called the area of that closed figure. the unit of measurement of that part of the plane is called the area of that closed figure. the unit o measurement of area is square unit (i.e. square centimeter, square metre etc.)

  13.1    (a)Mensuration of a Triangle:

             perimeter = a + b + c

           Area = × Base × Height

                                                  Heron’s formula:

             Area =

             Where’s = semi – perimeter

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RECTANGULAR CO-ORDINATES

This content is part of NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

RECTANGULAR CO-ORDINATES :

Take two perpendicular lines X’OX and Y’OY intersecting at the point O. X’OX and Y’OY are called the co-ordinate axes. X’Ox is called the X-axis, Y’OY is called the Y-axis and O is called the origin. Lines X’OX and Y’OY are sometimes also called rectangular axes.

7.1 (a) Co-ordinates of a Point :

Let P be any point as shown in figure. Draw PL and PM perpendiculars on Y-axis and X-axis, respectively. The length LP (or OM) is called the x - coordinate of the abscissa of point P and MP i called the y - coordinate or the ordinate of point P. A point whose abscissa is x and ordinate is y named as the point (x,y) or P(x,y).