Statistics

Statistics

Points to be Remember:-

1. Pictorial representation of the numerical data using picture

     symbols is called pictograph of the data.

2. A bar graph is a pictorial representation of the numerical data by a

    number of bars of uniform  width, erected horizontally or vertically,

    with equal spacing between them.

3. The uniform width of the bars and the uniform gap between them is

    suitably chosen, keeping in view the given information (data) and

    the space available for the diagram.

4. it is not necessary to make the bar graph on a graph paper, but

    from the point of view of convenience and accuracy, one should do

    so.

5. A bar graph should have a little. The chosen scale should also be

    mentioned below the graph.

Surface Areas and Volumes

Surface Areas and Volumes

Points to be Remember:-

1. A cuboid is a figure in space, formed by three pairs of opposite

    congruent rectangular faces, such that whenever two faces meet,

    they meet in a line segment.

2. A cuboid has 6 faces, 12 edges and 8 vertices.

3. The part of the space enclosed by a cuboid is called its interior.

4. A cuboid along with its interior is called a cuboidal region.

5. The length, breadth and height of a cuboid are called its three

     dimensions.

6. A cuboid whose length, breadth and height of a cuboid are called a 

    cube.

7. The magnitude or measure of a space (solid) region is called its

     volume.

8. A cubic centimeters is the volume of the region formed by a cube

    of side ( or edge ) 1 cm.

9. With usual meaning for V, I, b and h

     (i) Surface area of a cuboid = 2 (lb + bh + hl)

     (ii) Surface area of a cube = 6l²

    (iii) Lateral surface area of a cuboid = 2 (l + b)h

    (iv) Volume of a cuboid (V) = l x b x h

    (v) Volume of a cube (V) = l³

10. Standard units of volume

       (i) 1m³ = 1000000cm³ = 100³cm³

       (ii) 1cm³ = 1000mm³ = 10³mm³

      

       (iii) 1cm³ = 1000l = 1kl

       (iv) 1m3 = 1000l = 1kl

Areas to rectangle

Areas to rectangle

Points to be Remember:-

1. Area of a rectangle  =  length  x  breadth

2. Area of a square  =  ( side )²

3. Area of a rectangular path inside ( or outside) a rectangular field        

     = Area of the outer rectangle - Area of the inner rectangle.

4. Area of a cross path = Area of all the rectangles making the paths

    – Area of the common rectangle ( or square).

Circles

Circles

Points to be Remember:-

1. The region bounded by an arc of the circle and the chord of the arc

     is called a segment of the circle.

2. The part of the plane that consists of a diameter, semi-circle and

     interior of the circle that is enclosed by the semi-circle and the

     diameter is called a semi-circle region.

3. Angle in a semi-circle is a right angle.

4. Angles formed in the same segment are equal.

Quadrilaterals

Quadrilaterals

Points to be Remember:-

1. If we take four points say A,  B, C, and D in a plane such that no

    three of them are collinear and the line segments AB, BC, CD and  

    DA intersect at their ends only, then the figure so formed by the

    four line segments is called a quadrilateral.

2. The points A,  B, C, and D are called the vertices of the

    quadrilateral ABCD.

3. The line segments AB, BC, CD and DA are called the sides of the

     quadrilateral ABCD.

4. The angles of the quadrilateral ABCD are ÐA, ÐB, ÐC, and ÐD.

5. The two line segments joining the opposite vertices are called the

    diagonals of the quadrilateral ABCD.

6. In a quadrilateral, two sides that have a common vertex are known

    as adjacent sides.

7. In a quadrilateral, any two sides that do not have a vertex in

    common are called opposite sides.

8. Two angles of a quadrilateral are adjacent angles, if they have a

    side of the quadrilateral as a common arm.

9. Two angles of a quadrilateral that are not adjacent are called

     opposite angles.

10. The interior of the quadrilateral ABCD, together with its boundary

       (i.e. the quadrilateral itself) is called the quadrilateral region

       ABCD.

11. The sum of the angle of a quadrilateral is 360°.

Congruent Triangles

Congruent Triangles

Points to be Remember:-

1. Two figures are congruent, if they have the same shape and size.

2. Two triangles are congruent, if in a matching of their vertices, the

     three sides and the three angles of one triangle are respectively

     equal to the corresponding parts of the other.

3. Two triangles are congruent, if two sides and the included angle of

     the one triangle are respectively equal to the two sides and the

     included angle of the other (SAS congruence condition)

    SSS congruence condition

   Two triangles are congruent, if the three sides of the triangle are

    respectively equal to the three sides of the other. This is called

    SSS congruence condition.

     RHS congruence condition

  Two right triangles are congruent, if the hypotenuse and one side of  

  the one triangle are respectively equal to the hypotenuse and one

  side of the other. This is called RHS congruence condition.

More about Triangles

More about Triangles

Points to be Remember:-

1. If two sides of a triangle are equal, then their opposite angles are

     also equal.

2. If two angles of a triangle are equal, then their opposite sides are

    also equal.

Points to be Remember:-

1. Pythagoras theorem: In a right angled triangle, the square of

    hypotenuse equals the sum of the square of its side

2. In a right angled triangle, the hypotenuse is the longest side.

3. If the square of one side of a triangle is equal to the sum of the

    squares of the remaining two sides, then the angle opposite the

    first side is a right angle.

Points to be Remember:-

1. The perpendicular line segment drawn from any vertex of a triangle 

     into the opposite side is called altitude of the triangle.

2. A triangle has three altitudes. From each vertex one altitude is

    Obtained

3. The altitudes of a triangle are concurrent. The point of

     concurrence of the altitudes of a triangle is called the orthocenter

     of the triangle.

4. The line segment joining any vertex of a triangle to the mid-point of

     its opposite side is known as median of the triangle.

5. A triangle has three vertices and from each vertex one median is

     obtained. Thus a triangle has three medians.

6. The medians of a triangle are concurrent. The point of concurrence

     of the medians of a triangle is called centroid of triangle.

Points to be Remember:-

1. A perpendicular bisector of a side of a triangle is the line that is

    perpendicular to the side and bisects it.

2. A triangle has three sides. Therefore, it has three perpendicular

    bisects.

3. The perpendicular bisects of the sides of a triangle are concurrent.

    The point of concurrence of the perpendicular bisectors of the 

    sides of a triangle is called the circumcentre of the triangle.

4. An angle bisector of a triangle is the line segment which bisects a

     angle of the triangle and has its other end point on the opposite

     side of that angle.

5. A triangle has three angles and each angle has a bisector.

     Therefore a triangle has three angle bisector.

6. The angle bisector of a triangle are concurrent. The point of

     concurrence of the angle bisectors of a triangle is called its

     incentre.

Linear Equations in One Variable

Linear Equations in One Variable

Points to be Remember:-

1. Any value of the variable that satisfies the given equation is called

    a solution or the root of the equation.

2. The equality symbol ( = ) in the equation remains unchanged, if

     we:

     (i) add the same number on both sides of the equation.

    (ii) subtract the same number from both sides of the equation.

   (iii) multiply both sides of the equation by the same non-zero 

         number.

   (iv) divide both sides of the equation by the same non-zero 

         number.

3. By transposing a term we simply means changing its sign and

    carrying it to the other side of the equation.

4. To solve a word problem, denote the unknown by some variable 

     and translate the statements of the problem step / word by

     step / word into a mathematical statement i.e. an equation, and 

     then find the solution of the equation.

Factorization of Algebraic Expressions

Factorization of Algebraic Expressions

Points to be Remember:-

1. A given algebraic expression is the product of some numbers and

    algebraic expressions, these numbers and algebraic expressions

    are called the factors of the given algebraic expression.

2. Highest common factor (H.C.F.) of the given monomials is the

    product of H.C.F. of their coefficients and common literals raised to

    the power their least exponents.

3. Binomial expression can be factorised by taking out H.C.F. from all  

     the terms of the expression.

4. Expressions having three or more terms can be factorised by

     dividing the terms of the expression into group and taking out a

     common factor from each group.

5. Sometimes identities are utilized in factorization of algebraic

    expressions.

Algebraic Expressions

Algebraic Expressions

Points to be Remember:-

1. We having the following thumb-rule for multiplying any number of 

     monomials:

(i)  The coefficient of the product of given monomials is the product of 

      coefficients of these monomials.

(ii)  The literal part of the product contains all the literals occurring in

      the given monomials. The exponents of each literal is the sum of

      the exponents of this literal in the given monomials.

2.  To multiply a monomial and a binomial, we use the following

      properties:

 (i)  a (b + c) = ab + ac; a(b – c) = ab – ac

(ii)  (a +b)c = ac + bc ; (a – b) c = ac – bc

Points to be Remember:-

To multiply binominals, we use the following properties:

          (a + b) (c + d) = a(c +d) + b(c + d)

          (a + b) (c + d) = (a + b)c + (a + b)d.

To multiply binominals with a trinomial we use the following properties:

          (a + b) (c + d + e) = a (c + d + e) + b(c + d + e)

                                       = ac + ad + ae + bc + bd + be

Or     (a + b) (c + d + e) = (a + b)c + (a + b)d + (a + b)e

                                      = ac + bc + ad + bd + ae + be

Points to be Remember:-

1. (a + b)² = a² + 2ab + b²

2. (a - b)² = a² - 2ab + b²

3. (a – b) (a + b) = a² – b²

Percentage and Its Applications

Percentage and Its Applications

Points to be Remember:-

Percentage means per hundred and it is denoted by the sign %.

Points to be Remember:-

1. Profit percent = Profit  x 100

                               C.P.    

2. Loss percent = Loss  x 100

                               C.P.    

3. S.P. = C.P.(100+Profit percent or – Loss percent

                                             100    

4. C.P. =                             S.P. x 100

                  (100 + Profit percent or – Loss percent) 

Points to be Remember:-

If interest is denoted by I, then we get the following results:

(i)  I = P x R x T

              100

(ii)  P = 100 x I

              R x T

(iii)  R = 100 x I

              P x T

(iv)  T = 100 x I

              P x R

      If amount is denoted by A, then we get A = P + I

                                   

Direct and Inverse Variations

Direct and Inverse Variations

Points to be Remember:-

Two quantities x and y are said to be in direct proportion or vary directly. If an increase (decrease) in x is followed by a corresponding increase (decrease) in y in such a manner that the ratio x remains

                                                                                           y

constant and positive

Points to be Remember:-

1. Two quantities x and y are said to be in inverse proportion or vary   inversely. If an increase in x is followed by a corresponding decrease in y (or vice versa) in such a manner that the product of the corresponding values of two quantities remains constant and positive. It means x and y vary inversely, if there exists a relation of the type xy = k between them, k being a fixed positive number.

2. Relation between time, distance and speed is written below:

           Speed = Distance 

                           Time

Exponent

Exponent

Points to be Remember:-

1.   If a rational number then a x a x a x ….. x a (n times) = aⁿ. Here

     is the base and n is the exponent.        

2.   We can write the exponent of a rational number as the 

     exponents of numerator and denominator separately. For  

     example ( p )ⁿ = pⁿ

                    q        qⁿ

Points to be Remember:-

1.   If a rational number then

a^m x aⁿ  =  (a x a x …. M times)  x  (a x a x …. N times)

                =  (a x a x a …… m + n times) = a^{m + n}

Or     a^m x aⁿ  = a^{m + n}

2.   If a ( ≠ 0 ) is a rational number, and m and n are two positive integers such that:

 (i) m > n then a^m ¸ aⁿ = a^{m – n}

 (ii) m < n then a^m ¸ aⁿ =   1          

                                     a^{n – m}

    3.  (a^m)ⁿ = a^{m x n} = a^mn

Points to be Remember:-

1. a^-m  = ( 1 )^m

                a        or

     ( a ) ^–m  =  ( b )^m 

       b              a

2.  x⁰ = 1

3. If a and b are any two integers or rational numbers ( ≠ 0 ) and m is 

    an integer, then (a x b)^m = a^m x b^m and

   (a x b)^-m = a^-m x b^-m

Points to be Remember:-

Large and small numbers can be expressed in the form of k x 10n, where k is a terminating decimal number and n is an integer (positive for large number and negative for small number). In general we choose k in such a way that 1£ k < 10.

Decimal Representation of Rational Numbers

Decimal Representation of Rational Numbers

Points to be Remember:-

1.   If in a rational number numerator is a natural number and 

     denominator is power of 10 like 10, 100, 1000 etc., then to 

     represent such rational number in decimal form. We count the

     number of zeros just coming after one in denominator and we 

     put decimal in numerator after leaving same number of digit

     from right side to left side.

2.   The rational number whose denominator is a power of 2 or 5 or both like, 2, 4, 8, 16 ……, or 5, 25, 125 …. Then its denominator is changed into the power of 10 by multiplying with a suitable number and then it can be written in decimal form.

3.   The rational number whose denominator is not a power o 10 or it is not possible to write denominator in such form, then numerator is divided by denominator to represent rational number in the decimal form. After dividing upto unit place of numerator we put decimal after one’s place and divide by putting zeros after decimal.

4.   While changing the rational number to its decimal form by division method, if the remainder obtained is zero then the decimal number is known as terminating decimal or a rational number whose denominator contains powers of 2 or 5 or both, can be represented as a terminating decimal.

5.   While changing the rational number to its decimal form by division method, if the division process does not come o any and i.e., we get number (other then zero) as remainder then decimal number so obtained is known as non-terminating decimal.

6.   To change terminating decimal to rational number following steps are followed:

Step 1:  Write the given number

Step 2:  Count the number of digits after decimal in given number.

Step 3: Remove decimal from the decimal number and take all the  

            digits as numerator. Now after one put zeros which are in 

            number equal to the number of digits on the right side of 

            decimal and take it as denominator.

Step 4:  Find H.C.F. of numerator and denominator. Divide numerator

              and denominator by H.C.F. to find the rational number in 

              simplest form.