Topper Smart Guide-2010 Class-x Math

Chapter : Real Numbers

Key Concepts

1.

An Algorithm is a series of well defined steps which gives a procedure for solving a type of problem.

2.

A lemma is a proven statement used for proving another statement.

3.

Euclid’s Division Lemma: Given positive integers a and b, there where0 ≤ r < b exists unique integers q and r satisfying a = bq + r,

4.

Euclid’s Division Algorithm states that HCF of any two positive integers a and b, with a>b is obtained as follows: Step 1: Apply Euclid’s division lemma, to a and b, to find q and r where a = bq+r, 0 ≤ r 0, the quadratic equation has two distinct real roots If b2 – 4ac = 0, the quadratic equation has two equal real roots If b2 – 4ac < 0, the quadratic equation has no real roots

Top Formulae

1.

Roots of ax 2 + bx + c = 0, a ≠ 0 where

are

b2 − 4ac > 0

2.

Roots of ax 2 + bx + c = 0, a ≠ 0 are

3.

Quadratic identities:

i. ii. iii. 4.

−b + b2 − 4ac −b − b2 − 4ac and 2a 2a −b b and 2a 2a

(a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 - 2ab + b2 a2 - b2 = (a + b) ( a – b)

Discriminant, D = b2-4ac

,where

b2 − 4ac = 0

,

Chapter : Arithmetic Progressions

Top Definitions

1.

An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number d to the preceding term, except the first term.

2.

The difference between the two successive term of an A.P is called the common difference.

3.

Each of the number in the list of arithmetic progression is called a term of an A.P

4.

The arithmetic progression having finite number of terms is called a finite arithmetic progression.

5.

The arithmetic progression having infinite number of terms is called an infinite arithmetic progression.

Top Concepts 1.

A list of numbers a1, a2, a3…… is an A.P, if the differences a2–a1, a3–a2, a4–a3 … give the same value i.e ak+1 – ak is same for all different values of k.

2.

The general form of an A.P is a, a+ d, a+ 2d, a+3d…..

3.

If the A.P a, a+d, a+ 2d………
is reversed to
,
-d,
-2d………a, then the common difference changes to negative of original sequence common difference.

4.

The nth term of an A.P is the difference of the sum to first n terms and the sum to first (n-1) terms of it. i.e an = Sn − Sn−1

Top Formulae 1.

The general term of an A.P is given by: an = a + (n-1)d where a is the first term and d is the common difference.

2.

Sum of n terms of an A.P is given by: n Sn = 2a + (n − 1)d 2 where a is the first term, d is the common difference and n is the total number of terms.

3.

Sum of n terms of an A.P is also given by: n Sn = a +
 2 Where a is the first term and
is the last term.

4.

The general term of an A.P
,
-d,
-2d…….. is given by: a =
+ (n-1)(-d) where
is the last term, d is the common difference and n is the number of terms.

Chapter: Triangles

Top Definitions 1.

Two geometrical figures are called congruent if they superpose exactly on each other that is they are of same shape and size.

2.

Two figures are similar, if they are of the same shape but of different size.

3.

Basic Proportionality Theorem (Thales Theorem): If a line is drawn parallel to one side of a triangle to intersect other two sides in distinct points, the other two sides are divided in the same ratio.

4.

Converse of BPT: If a line divides any two sides of a triangle in the same ratio then the line is parallel to the third side.

5.

A triangle in which two sides are equal is called an isosceles triangle.

6.

AAA

(Angle-Angle-Angle)

similarity

criterion:

If

in

two

triangles,

corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar. 7.

Converse of AAA similarity criterion: If two triangles are similar, then their corresponding angles are equal.

8.

SSS (Side- Side- Side) similarity criterion: If in two triangles, sides of one triangle are proportional to (i.e., in the

same ratio of) the sides of the

other triangle, then their corresponding angles are equal and hence the two triangles are similar. 9.

Converse of SSS similarity criterion: If two triangles are similar, then their corresponding sides are in constant proportion.

10.

SAS (Side-Angle-Side) similarity criterion: If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.

11.

Converse of SAS similarity criterion: If two triangles are similar, then one of the angles of one triangle is equal to the corresponding angle of the other triangle and the sides including these angles are in constant proportion.

12.

Pythagoras Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

13.

Converse of Pythagoras Theorem: If in a triangle, square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Top Concepts 1.

All congruent figures are similar but the similar figures need not be congruent.

2.

Two polygons are similar if • •

3.

If the angles in two triangles are: • • •

4.

Their corresponding angles are equal Their corresponding sides are in same ratio.

Different, the triangles are neither similar nor congruent. Same, the triangles are similar. Same and the corresponding sides are the same size, the triangles are congruent

A line segment drawn through the mid points of one side of a triangle parallel to another side bisects the third side

5.

The ratio of any two corresponding sides in two equiangular triangles is always same.

6.

All circles are similar.

7.

All squares are similar.

8.

All equilateral triangles are similar.

9.

If two triangles ABC and PQR are similar under the corresponding A ↔ P, B ↔Q and C ↔ R, then symbolically, it is expressed as ∆ ABC ∼ ∆ PQR.

10.

If two angles of a triangle are respectively equal to two angles of another triangle, then by the angle sum property of a triangle their third angles will also be equal.

11.

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

12.

The ratio of the areas of two similar triangles is equal to the ratio of the squares of the corresponding medians.

13.

Triangles on the same base and between the same parallel lines have equal area.

14.

In a rhombus sum of the squares of the sides is equal to the sum of squares of the diagonals.

15.

In an equilateral or an isosceles triangle, the altitude divides the base into two equal parts.

3 a. 2

16.

The altitude of an equilateral triangle with side ‘a’ is

17.

In a square and rhombus, the diagonals bisect each other at right angles

18.

If a perpendicular is drawn from the vertex of the right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

Top Formulae 1.

If ∆ ABC ∼ ∆ PQR, then (1) ∠A = ∠P (2) ∠B = ∠Q (3) ∠C = ∠R AB BC AC (4) = = PQ QR P R 2

2

2

2.

 BC  ar(ABC)  AB   CA  If ∆ ABC ∼ ∆ PQR, then =  =  =  ar(PQR)  PQ   RP   QR 

3.

In triangle ABC right angled at B, AB2 + BC2 = AC2

Top Diagrams 1.

∆ ABC ∼ ∆ DEF

2.

∆ ABD ≅ ∆ DEF

Chapter : Coordinate Geometry

Top Definitions 1.

Two perpendicular number lines intersecting at point zero are called coordinate axes. The horizontal number line is the x-axis (denoted by X’OX) and the vertical one is the y-axis (denoted by Y’OY).

2.

The point of intersection of x axis and y axis is called origin and denoted by ‘O’.

3.

Cartesian plane is a plane obtained by putting the coordinate axes perpendicular to each other in the plane. It is also called coordinate plane or xy plane.

4.

The x-coordinate of a point is its perpendicular distance from y axis.

5.

The y-coordinate of a point is its perpendicular distance from x axis.

6.

The point where the x axis and the y axis intersect is coordinate points (0, 0).

7.

The abscissa of a point is the x-coordinate of the point.

8.

The ordinate of a point is the y-coordinate of the point.

9.

If the abscissa of a point is x and the ordinate of the point is y, then (x, y) are called the coordinates of the point.

Top Concepts 1.

The axes divide the Cartesian plane into four parts called the quadrants (one fourth part), numbered I, II, III and IV anticlockwise from OX.

2.

The coordinate of a point on the x axis are of the form (x,0) and that of the point on y axis are (0,y)

3.

Sign of coordinates depicts the quadrant in which it lies. The coordinates of a point are of the form (+, +) in the first quadrant, (-, +) in the second quadrant, (-,-) in the third quadrant and (+,-) in the fourth quadrant.

4.

Three points A, B and C are collinear if the distances AB, BC, CA are such that the sum of two distances is equal to the third.

5.

Three points A, B and C are the vertices of an equilateral triangle if the distances AB = BC = CA.

6.

The points A, B and C are the vertices of an isosceles triangle if the distances AB = BC or BC = CA or CA = AB.

7.

Three points A, B and C are the vertices of a right triangle if AB2 + BC2 = CA2 .

8.

For the given four points A, B, C and D (i) AB = BC = CD = DA; AC = BD ⇒ ABCD is a square. (ii) AB = BC = CD = DA; AC ≠ BD ⇒ ABCD is a rhombus. AB = CD, BC = DA; AC = BD ⇒ ABCD is a rectangle. (iii) (iv) AB = CD, BC = DA; AC ≠ BD ⇒ ABCD is a parallelogram.

9.

Diagonals of a square, rhombus, rectangle and parallelogram always bisect each other.

10.

Diagonals of rhombus and square bisect each other at right angle.

11.

Four given points are collinear, if the area of quadrilateral is zero.

12.

Centroid is the point of intersection of the three medians of a triangle.

13.

Centroid divides the median in the ratio of 2:1.

14.

The incentre is the point of intersection of internal bisector of the angles. It is also the centre of the circle touching all the sides of a triangle.

15.

Circum centre is the point of intersection of the perpendicular bisectors of the sides of the triangle.

16.

Ortho centre is the point of intersection of perpendicular drawn from the vertices on opposite sides (called altitudes) of a triangle and can be obtained by solving the equations of any two altitudes.

17.

If the triangle is equilateral, the centroid, incentre, orthocentre, circum centre coincides.

18.

If the triangle is right angled triangle, then orthocentre is the point where right angle is formed.

19.

If the triangle is right angled triangle, then circumcentre is the midpoint of hypotenuse.

20.

Orthocentre, centroid and circum centre are always collinear and centroid divides the line joining Orthocentre and circumcentre in the ratio of 2:1.

21.

In an isosceles triangle centroid, orthocentre, incentre, circumcentre lies on the same line.

22.

Angle bisector divides the opposite sides in the ratio of remaining sides.

23.

Three given points are collinear, if the area of triangle is zero.

Top Formulae 1.

If x ≠ y, then (x,y)≠(y,x) and if (x,y) = (y,x), then x=y.

2.

The distance between P(x1,y1) and Q(x2,y2) is

(x2 − x1 )2 + (y2 − y1 )2 .

x2 + y2 .

3.

The distance of a point P(x,y) from origin is

4.

Coordinates of point which divides the line segment joining the points (x1,y1) and (x2,y2) in the ratio m : n internally are mx2 + nx1 my2 + ny1 x= and y = m+n m+n

5.

Coordinates of mid-point which divides the line segment joining the x + x1 y + y1 points (x1,y1) and (x2,y2) are x = 2 and y = 2 2 2

6.

If A(x1, y1), B(x2,y2) and C(x3,y3) are vertices of a triangle, then the coordinates of centroid are  x + x2 + x3 y1 + y2 + y3  G=  1 ,  3 3  

7.

If A(x1, y1), B(x2,y2) and C(x3,y3) are vertices of a triangle, then the coordinates of incentre are  ax + bx2 + cx3 ay1 + ay2 + ay3  I =  1 ,  a+b+c a+b+c  

8.

If A(x1, y1), B(x2,y2) and C(x3,y3) are vertices of a triangle, then the area of triangle ABC is given by 1 Area of ∆ABC = x1(y2 − y3 ) + x2 (y3 − y1 ) + x3 (y1 − y2 ) 2

Top Diagrams 1.

Sign of coordinates in various coordinates.

2.

3.

To plot a point P (3, 4) in the Cartesian plane. (i) A distance of 3 units along X axis. (ii) A distance of 4 units along Y axis.

Area of quadrilateral ABCD = Area of ∆ABC + Area of ∆ACD

D (x4, y4)

A(x1, y1)

C (x3, y3)

B (x2, y2)

4.

Centroid (G) of a triangle.

A

E

F G

B 5.

C

D Incentre (I) of a triangle.

A

E

F I

B 6.

C

D

Circumcentre (O) of a Triangle.

A E

F O

B

D

C

7.

Orthocentre (O) of a Triangle.

A

E

B

O

D

F

C

Chapter : Introduction to Trigonometry

Top Definitions 1.

Trigonometry is the study of relationship between the sides and the angles of the triangle.

2.

Ratio of the sides of the right triangle with respect to the acute angles is called trigonometric ratios of the angle.

3.

Pythagoras theorem: In a right triangle, square of the hypotenuse is equal to the sum of the square of the other two sides.

Top Concepts 1.

Trigonometry is the combination of three Greek words ‘Tri (Three) + gon (sides) + metron (measure)’.

2.

When any two sides of a right triangle are given, its third side can be obtained by using Pythagoras theorem.

3.

Angle measured in anticlockwise direction is taken as positive angle.

4.

Angle measured in clockwise direction is taken as negative angle.

5.

The values of the trigonometric ratios of an angle do not vary with the length of the sides of the triangle, if the angles remain the same.

6.

The value of sin A or cos A never exceeds 1, whereas the value of sec A or cosec A is always greater than 1 or equal to 1.

7.

Each trigonometric ratio is a real number. It has no unit.

8.

Only symbol cosine, sine, tangent, cotangent, sec and cosec has no meaning.

9.

( sin θ )

n

is generally written as sinn θ , n being a positive integer.

Similarly other trigonometric ratios can also be written.

10.

11.

The value of sin θ increases from 0 to 1 when θ increases from 00 to 900. The value of cos θ decreases from 1 to 0 when θ increases from 00 to 900.

Top Formulae −1

≠ cos−1 θ

1.

sec θ = ( cos θ )

2.

Trigonometric ratio’s si dopposite to ∠A p = (i) sin A = hypotenuse h si de adjacent to ∠A b (ii) cos A = = hypotenuse h si de adjacent to ∠A p (iii) tan A = = si de opposite to ∠A b hypotenuse h (iv) cos ecA = = si de opposite to ∠A p hypotenuse h = (v) s ecA = si de adjacent to ∠A b si de adjacent to ∠A b (vi) co t A = = si de opposite to ∠A p

3.

Relation between trigonometry ratios sin θ (vii) tan θ = cos θ 1 (viii) cos ec θ = sin θ 1 (ix) s ec θ = co s θ 1 co s θ (x) cot θ = = tan θ sin θ

4.

Trigonometric ratios of complementary angles (i) sin (90 – A) = cos A (ii) cos (90 – A) = sin A (iii) tan (90 – A) = cot A (iv) cot (90 – A) = tan A (v) sec (90 – A) = cosec A (vi) cosec (90 – A) = sec A

5.

Trigonometric Identities (i) sin2 θ + co s2 θ = 1

(ii)

1 + tan2 θ = sec2 θ

(iii) 1 + cot2 θ = sec2 θ

Top Diagrams 1.

Sides of right triangle

2.

Values of Trigonometric ratios:

∠A sin A

0o 0

30o 1 2

45o 1 2

60o 3 2

90o 1

cos A

1

3 2

1 2

1 2

0

tan A

0

1 3

1

3

Not defined

cosec A

Not defined

2

2

2 3

1

sec A

1

2 3

2

2

Not defined

cot A

Not defined

3

1

1 3

0

Chapter : Some Application of Trigonometry Top Definitions

1.

The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer.

2.

The angle of elevation of an object viewed is the angle formed by the line of sight with the horizontal when it is above the horizontal level, i.e., the case when we raise our head to look at the object.

3.

The angle of depression of an object viewed is the angle formed by the line of sight with the horizontal when it is below the horizontal level, i.e., the case when we lower our head to look at the object.

4.

Pythagoras theorem: In a right triangle, square of the hypotenuse is equal to the sum of the square of the other two sides.

5.

Ratio of the sides of the right triangle with respect to the acute angles is called trigonometric ratios of the angle.

Top Concepts 1.

The height or length of an object or the distance between two distant objects can be determined by the help of trigonometric ratios.

2.

When any two sides of a right triangle are given, its third side can be obtained by using Pythagoras theorem.

3.

Angle measured in anticlockwise direction is taken as positive angle.

4.

Angle measured in clockwise direction is taken as negative angle.

5.

The values of the trigonometric ratios of an angle do not vary with the length of the sides of the triangle, if the angles remain the same.

6.

Each trigonometric ratio is a real number. It has no unit.

7.

Only symbol cosine, sine, tangent, cotangent, sec and cosec has no meaning.

8.

The two heights above and below the ground level in case of reflection from the water surface are equal.

Top Formulae −1

≠ cos−1 θ

1.

sec θ = ( cos θ )

2.

Trigonometric ratio’s si dopposite to ∠A p (i) sin A = = hypotenuse h si de adjacent to ∠A b = (ii) cos A = hypotenuse h si de adjacent to ∠A p (iii) tan A = = si de opposite to ∠A b hypotenuse h (iv) cos ecA = = si de opposite to ∠A p hypotenuse h (v) s ecA = = si de adjacent to ∠A b si de adjacent to ∠A b (vi) co t A = = si de opposite to ∠A p

3.

Relation between trigonometry ratios sin θ (vii) tan θ = cos θ 1 (viii) cos ec θ = sin θ 1 (ix) s ec θ = co s θ 1 co s θ (x) cot θ = = tan θ sin θ

4.

Trigonometric ratios of complementary angles (i) sin (90 – A) = cos A (ii) cos (90 – A) = sin A (iii) tan (90 – A) = cot A (iv) cot (90 – A) = tan A (v) sec (90 – A) = cosec A (vi) cosec (90 – A) = sec A

5.

Trigonometric Identities (i) sin2 θ + co s2 θ = 1

(ii)

1 + tan2 θ = sec2 θ

(iii) 1 + cot2 θ = sec2 θ

Top Diagrams

1.

Angle of elevation.

2.

Angle of depression.

3.

Values of Trigonometric ratios.

∠A sin A

0o 0

30o 1 2

45o 1 2

60o 3 2

90o 1

cos A

1

3 2

1 2

1 2

0

tan A

0

1 3

1

3

Not defined

cosec A

Not defined

2

2

2 3

1

sec A

1

2 3

2

2

Not defined

cot A

Not defined

3

1

1 3

0

Chapter : Circles Top Definitions 1.

A tangent to a circle is a line that intersects the circle only at one point.

2.

The common point of the circle and the tangent is called point of contact.

3.

The length of the segment of the tangent from the external point P and the point of contact with the circle is called the length of the tangent.

4.

Pythagoras theorem: In a right triangle, square of the hypotenuse is equal to the sum of the square of the other two sides.

Top Concepts 1.

A tangent to a circle is a special case of the secant when the two end points of the corresponding chord coincide.

2.

There is no tangent to a circle passing through a point lying inside the circle.

3.

There are exactly two tangents to a circle through a point outside the circle.

4.

At any point on the circle there can be one and only one tangent.

5.

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

6.

The lengths of the tangents drawn from an external point to a circle are equal.

7.

The centre lies on the bisector of the angle between the two tangents.

8.

There can be infinite number of chords passing through a point which is inside the circle.

Top Diagrams 1.

Various lines on a circle.

2.

Two equal tangents (PA = PB) from an external point P.

3.

Radius ⊥ Tangent.

Chapter: Constructions

Top Definitions 1.

The ratio of the side of the triangle to be constructed with the corresponding sides of the given triangle is known as their scale factor.

2.

Reduced scale factor figures are the geometric figure to be constructed is smaller in size.

3.

Enlarged scale factor figures constructed is larger in size.

are

the

geometric

figure

to

be

Top Concepts 1.

To divide a line segment internally in a given ratio m: n, where both m and n are positive integers, we follow the following steps: Step 1: Draw a line segment AB of given length by using a ruler. Step 2: Draw any ray AX making an acute angle with AB. Step 3: Along AX mark off (m + n) points Am+1,………,Am+n, such that AA1 = A1A2 = Am+n-1 Am+n.

A1,

A2,………AM,

Step 4: Join B Am+n Step 5: Through the point Am draw a line parallel to Am+n B by making an angle equal to ∠AAm+n B at Am. Suppose this line meets AB at point P. The point P so obtained is the required point which divides AB internally in the ratio m: n.

2.

Constructions of triangles similar to a given triangle: (a) Steps of constructions when m < n: Step 1: Construct the given triangle ABC by using the given data. Step 2: Take any one of the three side of the given triangle as base. Let AB be the base of the given triangle. Step 3: At one end, say A, of base AB. Construct an acute angle ∠BAX below the base AB. Step 4: Along AX mark off n points A1, A2, A3,………, An such that AA1 = A1A2 = ……… = An-1 An Step 5: Join AnB Step 6: Draw AmB’ parallel to AnB which meets AB at B’. Step 7: From B’ draw B’C’||CB meeting AC at C’.

m Triangle AB’C’ is the required triangle each of whose sides is   n

th

of

the corresponding side of ∆ABC.

(b)

Steps of construction when m > n:

Step 1: Construct the given triangle by using the given data. Step 2: Take any one of the three sides of the given triangle and consider it as the base. Let AB be the base of the given triangle. Step 3: At one end, say A, of base AB. Construct an acute angle ∠BAX below base AB i.e., con the opposite side of the vertex C. Step 4: Along AX mark off m (large of m and n) points A1, A2, A3,………An of AX such that AA1 = A1A2 = ………= Am-1Am. Step 5: Join AnB to B and draw a line through Am parallel to AnB, intersecting the extended line segment AB at B’. Step 6: Draw a line through B’ parallel to BC intersecting by the extended line segment AC at C’. Step 7: ∆AB’C’ so obtained is the required triangle.

3.

Constructions of tangent to a circle: a. To draw the tangent to a circle at a given point on it, when the centre of the circle is known. Given

: A circle with centre O and a point P on it.

Required

: To draw the tangent to the circle at P.

Steps of construction: i.

Join OP

ii. Draw a line AB perpendicular to OP at the point P. APB is the required tangent at P.

4.

To draw the tangent to a circle at a given point on it, when the centre of the circle is not known. Given

: A circle and point on it.

Required

: To draw the tangent to the circle at P.

Steps of construction: i.

Draw any chord PQ and join P and Q to a point R in major arc


(or PQ ii.

minor arc PQ).

Draw ∠QPB equal to ∠PRQ and on opposite side of chord PQ. The line BPA will be a tangent to the circle at P.

5.

To draw the tangent to a circle from a point outside it (external point) when its center is known. Given

: A circle with center O and a point P outside it.

Required

: to construct the tangents to the circle from P.

Steps of construction:

6.

i.

Join OP and bisect it. Let M be the mid point of OP.

ii.

Taking M as centre and MO as radius, draw a circle to intersect C (O,r) in two points, say A and B.

iii.

Join PA and PB. These are the required tangents from P to C (O,r).

To draw tangents to a circle from a point outside it (when its centre is not known) Given

: P is a point outside the circle.

Required

: To draw tangents from a point P outside the circle.

Steps of construction: i.

Draw a secant PAB to intersect the circle at A and B.

ii.

Produce AP to a point C, such that PA = PC.

iii.

With BC as a diameter, draw a semicircle.

iv.

Draw PO ⊥ CB, intersecting the semicircle at O.

v. Taking PO as radius and P as centre, draw arcs to intersect the circle at T and T’. vi.

7.

Join PT and PT’. Then PT and PT’ are the required tangents.

Two tangents can be drawn to a circle through a point outside the circle and pair of these tangents are always equal in length.

Chapter : Area Related to Circles

Top Definitions 1.

A circle is a collection of all points in a plane which are at a constant distance from a fixed point in the same plane.

2.

A line segment joining the centre of the circle to a point on the circle is called its radius.

3.

A line segment joining any two points of a circle is called a chord. A chord passing through the centre of circle is called its diameter.

4.

A part of a circle is called an arc.

5.

A diameter of circle divides a circle into two equal arcs, each known as a semicircle.

6.

The region bounded by an arc of a circle and two radii at its end points is called a sector.

7.

A chord divides the interior of a circle into two parts, each called a segment.

8.

An arc of a circle whose length is less than that or a semicircle of the same circle is called a minor arc.

9.

An arc of a circle whose length is greater than that of a semicircle of the same circle is called a major arc.

10.

Circles having the same centre but different radii are called concentric circles.

11.

Two circles (or arc) are said to be congruent if we can superpose (place) one over the other such that they cover each other completely.

12.

The distance around the circle or the length of a circle is called its circumference or perimeter.

Top Concepts 1.

The mid – point of the hypotenuse of a right triangle is equidistant from the vertices of the triangle.

2.

Angle subtended at the circumference by a diameter is always a right angle.

3.

Angle described by minute hand in 60 minutes is 360°.

4.

Angel described by hour hand in 12 hours is 360°

Top Formulae 1.

Circumference (perimeter) or a circle = πd or 2πr, where r is the radius of the circle and π =

22 . 7

2.

Area of a circle = πr2

3.

Area of a semi circle =

4.

Perimeter of a semi circle or protractor = πr + 2r

5.

Area of a ring or an annulus = π (R + r) (R-r)

6.

Length of arc AB =

7.

πr θ 1 Area of a sector = Or Area of sector = (
× r ) 360° 2

8.

Area of minor segment =

9.

Area of major segment = Area of the circle – Area of minor segment

1 2 πr 2

2

2πrθ πr or 360° 180° 2

2

πr θ 1 2 − r sin θ 360° 2

πr² - Area of minor segment. 10.

If a chord subtend a right angle at the centre, then

 π 1 2 − r 4 2

Area of the corresponding segment = 

11.

If a chord subtend an angle of 60° at the centre, then

π

Area of the corresponding segment =  

6

12.

π 3

14.

−

3 2 r 4 

Distance moved by a wheel in 1 revolution = Circumference of the wheel. Number

of

revolutions

Dis tan ce moved in 1 minute Circumference 15.

3 2 r 2 

If a chord subtend an angle of 120° at the centre, then Area of the corresponding segment =  

13.

−

Perimeter of sector =

πrθ + 2r 180°

in

one

minute

=

Chapter : Surface Areas and Volumes Top Definitions 1.

A Cube is a special type of cuboids in which length = breadth = height. Also called an edge of a cube.

2.

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball.

3.

A cylinder is a solid or a hollow object that has a circular base and a circular top of the same size.

4.

A hemisphere is half of a sphere.

5.

If a right circular is cut off by a plane parallel to its base, then the portion of the cone between the plane and the base of the cone is called a frustum of the cone.

Top Concepts 1.

The total surface area of the solid formed by the combination of solids is the sum of the curved surface area of each of the individual parts.

2.

A solid is melted and converted to another, volume of both the solids remains the same, assuming there is no wastage in the conversions. The surface area of the two solids may or may not be the same.

3.

A frustum can be obtained by cutting a cone by a plane, parallel to the base of the cone.

4.

The solids having the same curved surface do not necessarily occupy the same volume.

Top Formulae 1.

Cuboids: Lateral surface area

Or Area of four walls = 2(ℓ + b) h

Total surface area = 2(ℓb + bh + hℓ) Volume = ℓ x b x h Diagonal of a cuboids =

2

2

2


+b +h

2.

Cube Lateral surface area

Or Area of four walls

= 4 x (edge)2

Total surface area = 6 x (edge)² Volume

= (edge)²

Diagonal of a cube = 3.

3 x edge.

Right circular cylinder: Area of each end or Base area = πr² Area of curved surface or lateral surface area = perimeter of the base x height = 2π r (h + r) Total surface area (including both ends) = 2 πrh + 2πr² = 2πr (h + r) Volume = (Area of the base0 x height = πr²h

4.

Right circular hollow cylinder: Area of curved surface = (External surface) + (Internal surface) = (2πRh + 2πrh) = 2 (πR² - πr²) = [2πh(R+ r) + 2π (R² - r²)] = [2π(R + r) (h + R – r)] Volume of the material = (External volume) – (Internal volume) = (πR²h - πr²h) = πh (R² - r²)

5.

Right circular cone: Slant height (ℓ) =

2

h +r

2

Area of curved surface = πrℓ = πr

2

h +r

2

Total surface area = Area of curved surface + Area of base

= πrℓ + πr² = πr (ℓ + r) Volume

6.

=

1 2 πr h 3

Sphere: Surface area = 4 πr² Volume =

7.

4 πr² 3

Spherical shell: Surface area (outer) = 4πR²

8.

Volume of material =

4 4 πr³ − πr³ 3 3

=

4 π (R³ − r³) 3

Hemisphere: Area of curved surface = 2 πr²h Total surface Area = Area of curved surface + Area of base = 2 πr² + πr² = 3πr² Volume

9.

=

2 πr² 3

Frustum of a cone: Total surface area = π[R² + r² + ℓ (R + r)] Volume of the material =

1 πh R² + r² + Rr  3 

Top Diagrams 1.

Cuboid

2.

Cube

3.

Right circular cylinder:

4.

Right circular hollow cylinder:

5.

Right circular cone:

6.

Sphere:

7.

Spherical shell:

8.

Hemisphere:

9.

Frustum of a cone:

Chapter : Statistics Top Definitions 1.

When a frequency distribution is obtained by dividing an ungrouped data in a number of strata, according to the value of variety under study, such information is called grouped data or classified data.

2.

The cumulative frequency of a class is the frequency obtained by adding the frequencies of all the classes preceding the given class to the frequency of the class.

3.

In a less than ogive the upper limit of a class is plotted against its cumulative frequency as a point on the ogive.

4.

In a ‘more than ogive’ the upper limit of a class is plotted against its cumulative frequency as a point on the ogive

5.

The mode for ungrouped data is the value that occurs most often.

6.

In the case of a grouped frequency distribution a class with maximum frequency is called as the modal class.

7.

A distribution in which the values of mean, median and mode coincide (i.e. mean = median = mode) is known as a symmetrical distribution.

8.

Distribution for which values of mean, median and mode are not equal is known as asymmetrical or skewed distribution.

Top Concepts 1.

A cumulative frequency distribution can be represented graphically by means of an ‘ogive’.

2.

The ogives can be drawn only when the given class intervals are continuous and if this is not the case then you don’t need to worry. All you need to do is simply make the class intervals continuous.

3.

The ‘less than ogive’ is a rising curve.

4.

The ‘more than ogive’ is a falling curve.

5.

Direct Method of finding Mean Step 1: First we find the mid values (also called class marks) of the intervals, denoted by ‘x’ or ‘m’ LowerLimit + UpperLimit x= 2 Step 2: Multiply frequency with corresponding mid values obtained in step1. Step 3: Mean is calculated by using the following formula

Arithmetic mean = x = 6.

Short Cut Method/ Assumed Mean Method Step1: Find the class marks Step2: Find the assumed mean (A) from the mid values Step 3: Calculate deviation (d), d = x – A Step 4: find the product of frequency with the corresponding deviations Step 5 : Calculate mean by using the following formula

x = A+ 7.

∑fx ∑f

∑ fd ∑f

Step Deviation Method Step1: Find the class marks Step2: Find the assumed mean (A) from the mid values Step 3: Calculate deviation (d). d = x – A Step 4: After calculating deviations (d), we make one more column of values by dividing ‘d’ by ‘h’ Step 5: This is new value called step deviation (d’ or u) is multiplied with corresponding frequencies. Step 6: Calculate mean by using the formula

x = A+

∑ fd ' × h ∑f

8.

The mode may be greater than, less than or even equal to the mean.

9.

For finding the median we must arrange the given information i.e. the given data in increasing or decreasing order.

10.

The last of the cumulative frequencies will be always equal to the total of all frequencies.

11.

If the number of observations, n is even, so the median is the average of the (n/2)th observation and the (n/2+1)th observation.

12.

The step deviation method will be convenient to apply if all the deviations (d’s) have a common factor

13.

If class marks so obtained are in decimal form, then step deviation method is preferred to calculate mean.

14.

The median of a grouped data can be obtained graphically as the x coordinate of the point of intersection of two ogives for the data.

15.

The most commonly used method of central tendency is the mean. The biggest problem with mean is that it is effected by the extreme values one large or small number can distort the average. In that case the median is a better measure of central tendency while when the most repeated value or the most wanted one is required, and then mode is used.

16.

The most frequently used measure of central tendency is the mean, because the mean is calculated by taking into account all the observations of a given data. And it lies between the smallest and the largest value of the data.

17.

In general, Mean median and mode could be connected as follows • Mean=Mode • ModeMedian

Top Formulae 1.

Direct Method Mean =

2.

Assumed Mean Method/ Short Cut Method x = A+

3.

∑ fd ∑f

Step Deviation Method x = A+

4.

∑fx ∑f

∑ fd' × h ∑f

Mode for a grouped data is given by



f1 -f 0   ×h  2f1 -f 0 -f 2 

Mode= l + 

l = lower limit of the modal class h = size o f the class interval f1 = frequency of the modal class

f 0 =frequency of the class preceding the modal class f 2 = frequency of the class succeeding the modal class

5.

Formula for median of a grouped data

n  -cf Median= l +  2  f 

  × h  

Where, l= the lower limit of median class. cf = the cumulative frequency of the class preceding the median class. f = the frequency of the median class. h =the class size 6.

3 Median = Mode + Median

Top Diagrams 1.

Less than Ogive

2.

More than Ogive

3.

Median calculated graphically.

4.

Symmetric Distribution

5.

Asymmetrical or skewed distribution

Chapter : Probability Top Definitions 1.

Probability is a quantitative measure of certainty.

2.

Any activity associated to certain outcome is called a random experiment. e.g. (i) tossing a coin (ii) throwing a dice (ii) selecting a card.

3.

Outcome associated with an experiment is called an event. E.g (i) Getting a head on tossing a coin (ii) getting a face card when a card is drawn from a pack of 52 cards.

4.

The event whose probability is one are called sure events/ certain event.

5.

The event whose probability is zero are called impossible events.

6.

An event with only one possible outcome is called an elementary event.

7.

In a given experiment, if two or more events are equally likely to occur or have equal probabilities, then they are called equally likely events.

Top Concepts 1.

Probability of an event lies between 0 and 1.

2.

Probability can never be negative.

3.

A pack of playing cards consist of 52 cards which are divided into 4 suits of 13 cards each. Each suit consists of one ace, one king, one queen, one jack and 9 other cards numbered from 2 to 10. Four suits named spades, hearts, diamonds and clubs.

4.

King, queen and jack are face cards.

5.

The sum of the probabilities of all elementary events of an experiment is 1.

6.

Two events A and B are said to be complements of each other if the sum of their probabilities is 1.

Top Formulae 1.

Probability of an event E denoted as P(E) is given by: P(E) =

Number of outcomes favourable to E TotalNumber of Outcomes

2.

For an event E, P(E) = 1 − P(E) , where the event E representing ‘not E” is the complement of event E.

3.

For A and B two possible outcomes of an event. (i) If P(A) > P(B) then event A is more likely to occur than event B. (ii) If P(A) = P(B) then events A and B are equally likely to occur.

4.

a

Top Diagrams 1.

Suits of Playing Card

Heart

2.

Spades

Diamond

Club

Face Cards

A King of diamond

A Queen of club

A Jack of Clubs