The sum of the ages of Anup and his father is 100. When Anup is as old as his father now, he will be five times as old as his son anuj is now. Anuj will be eight years older than Anup is now, when anup is as old as his father. What are their ages now?

The sum of the ages of Anup and his father is 100. When Anup is as old as his father now, he will be five times as old as his son anuj is now. Anuj will be eight years older than Anup is now, when anup is as old as his father. What are their ages now?

Sol :

Let Anup's present age be x years.
His father's present age = 100 - x.
Therefore, Anup's present age = (100-x)/5 years.

Anup becomes as old as his father is now after (100-2x) years.

Therefore after 100-2x years, we have anup's age = {(100-x)/5 (100-2x)} =(600-11x)/5.

It is given that after 100-2x years, Anuj is 8 years older than his father Anup is now.

Therefore, (600-11x)/5 = x + 8. 
     ==>     600 - 11x    = 5x + 40.
     ==>          600 - 40 =  5x + 11x.
    ==>            560       =  16x
    ==>                     x  =  560/16  = 35.

Hence , Anup's present age be 35 years
His father's present age = 100 - 35 = 65

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Magnetism

Magnetism

The word magnetism is derived from Magnesia, the place where magnetic iron "ORE" was first discovered. Magnet is a substance which possesses the property of attracting small pieces of iron and sets its in North-South direction when suspended freely. The Magnetism is a phenomenon by which this attraction takes place.

Magnetic Poles

If a magnet is dipped into iron filings, it is noted that the fillings cling in tufts, usually at two places in particular i.e. near the ends. Few filings are attracted in the middle of the bar. The places in a magnet where the resultant attractive force appears to be concentrated are called the poles. The pole which points towards the north is called North Pole and other one is called South Pole.

Magnetic and non-magnetic substances

Apart from iron, the only other elements which are attracted strongly by a, magnet are cobalt and nickel. These together with certain strongly magnetic alloys are described as ferromagnetic substances such as copper, brass, wood and glass are not attracted by a magnet and are commonly described as non-magnetic.

or

The substances attracted by magnet are called magnetic substances. The substances not attracted by magnet are called non-magnetic substances.

Magnetic Axis

The magnetic line passing through the north (N) and south (S) poles of a magnet is called the magnetic axis of a magnet.

Magnetic meridian

It is a vertical plane containing the magnetic axis of a freely suspended magnet of a rest under the action of the earth's magnetic field.

Magnetic Equator

It is a line, which is perpendicular bisector of the line joining the two poles of the magnet.
Magnetic Length
It is the distance between the two poles of a magnet. The magnetic length is smaller than geometrical length of the magnet as the poles are not exactly at the ends but situated near the ends a little is side the magnet.

Action of one magnet on another

If N pole of a magnet is brought near N pole of a suspended magnet, the repulsion will occur. Similarly, repulsion will occur between two poles. On the other hand, a N and S pole always attract one another. Thus like poles repel and unlike poles attract.

Test for polarity of a magnet

The polarity of a magnet may be tested by bringing both its poles, in turn, near to the known as poles of a suspended magnet. Repulsion will indicate similar polarity. If attraction occurs, no firm conclusion can be drawn since attraction would be obtained between either (a) two unlike poles or (b) a pole and a piece of unmagnetized material. Thus the repulsion is only sure test for polarity

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Solved Question No. 3 of Ex. 4.1 of Class 11th Based on Mathematical Induction

By Using the Principle of Mathematical Induction Prove that
                        1 + 1/ (1+2) + 1/ (1+2+3) +........+ 1/(1+2+3+....+n)  =  2n/(n+1)

Sol :

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SOLVED QUESTION NO 2 OF EX 4.1 OF CLASS 11TH BASED ON MATHEMATICAL INDUCTION

By Using the Mathematical Induction Prove that  1³ + 2³ + 3³ + ……. + n³  = {n²(n+1)²}/4

Sol

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Solved Question No 1 of Ex 4.1 of class 11th based on Mathematical Induction

Using the Principle of Mathematical Induction Prove that 1 + 3 + 3² +…….+ 3^n-1 = (3ⁿ – 1)/2 for all natural numbers. 

Sol :

          

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Conditions for Maxima and Minima

Conditions for Maxima and Minima

If y = f(x), then for maximum or minimum value of y for a value of x the first differential coefficient of y w.r.t. x should be zero.

i.e.

dy/dx = 0.

Find the value of x from here.

Now, find the 2nd derivative of y w.r.t. x i.e. d²y/dx² and put the value of x.

1. If the value of d²y/dx² is negative, then y is maximum for given value of x.

2. If the value of d²y/dx² is positive, then y is minimum for a given value of x.

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Formulae of Differentiation

Fundamental Formulae of Differentiation

Derivatives of Trigonometrical Functions

Derivatives of Logarithmic and Exponential Functions

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                        Solution of Triangles

In a triangle, there are six elements or parts, three sides and three angles. From plane geometry, we know that if three of the elements are given, at least one of which must be a side, then the other three elements can be uniquely determined. If only three angles are given, then no unique solution of the triangle is possible but unlimited number of equiangular triangles can be found, the three angles of which are equal to the three given angles. The method of determining the unknown parts from the known parts, is called solving a triangle.

Again, it is evident that the three given parts cannot be independent or arbitrary. For instance, in a triangle, two of its angles can not be obtuse or the sum of any two sides cannot be less than or equal to the third.
A triangle is called a right-angled triangle or right triangle if one of its angle is a right angle otherwise, the triangle is called an oblique triangle. We start with the solution of right-angled triangles and the solution of oblique triangle will follow.
Whenever found convenient, logarithms should be used.

Solution of right-angled triangle

In this case, one angle is equal to 90 degree. If any two (at least one of which is a side) of the remaining five parts are given then the other unknown parts of the triangle can be determined.
If ABC be a right-angled triangle, where C = 90 degree then for solving the triangle, we have the following relations:
C = 90 degree, A B = 90 degree,
c^2 = a^2 b^2.
sin A =a/c, sin B = b/c, tan A =a/b = cot B.

                     Solution of oblique triangles

The various cases that may arise in solving oblique triangles are the following:
a) Three angles are given:
In this case, no definite values of the sides can be found but the ratio of the sides is determined. 

b) Three sides are given
In this case, unique solution of the triangle is possible.

c) Two sides and the included angle between them are given
In this case, triangle is unique.

d) Two angles and one side are given
The triangle is also unique in this case.

e) Two sides and the angle opposite to one of them are given
In this case, no triangle or more than one triangle is possible (ambiguous case).

General Remarks
Different trigonometric formulae may be used in different cases. The same problem may also be solved by using different formulae and by different methods. If no additional data are given in the problem, the formula, which is most suitable, should be chosen. If additional data are supplied, then the formula, which will fit in with the given data, must be selected. It has been found that for logarithmic computations tangent formula is most suitable.

The difference in the value of tan x is larger than that of tabular sin x or cos x. Hence, a small error in the tabular tangent will not affect the value of x so much as in the case tabular sine or cosine. In fact, 4-figure logarithmic tangent table gives more accurate result than even the 7-figure logarithmic sine or cosine table.

Note: If the angle is determined by the sine formula, then ambiguity may arise in the result. For example, if sin A = 1/2, then A = 30 degree or 150 degree, both are equally possible for being an angle of a triangle. This difficulty will not arise, when the angle is determined by cosine or tangent formula.

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Properties of Triangles (This question is based on Properties of Triangles), Prove that in a triangle ABC, a^3 . cos(B-C) + b^3 . cos(C-A) + c^3 . cos(A-B) = 3abc.

Prove that in a triangle ABC,
                   a^3 . cos(B-C) + b^3 . cos(C-A) + c^3 . cos(A-B) = 3abc.

[IIT 1970;   IC 1983]

SOLUTION : 

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In a triangle ABC, Prove that asin(B-C) + bsin(C-A) + csin(A-B) = 0

In a triangle ABC, Prove that asin(B-C) + bsin(C-A) + csin(A-B) = 0 

[ISM 1973; BIT(Mesra)]

SOLUTION

BY : V K PANDEY

KEYWORDS :Geeks Networks, mathsmania, NCERT solved answers, trigonometric functions, trigonometry, 

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In any Triangle ABC, Prove that acos(B-C)/2 = (b+c)SinA/2.

 This Question is based on Properties of Triangle

In any Triangle ABC, Prove that  acos(B-C)/2 = (b+c)SinA/2.

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Very Very Important Question based on Properties of Triangle.

Very Very Important Question based on Properties of Triangle.

This Question has been asked in IIT 1970;JEE(WB) 91;Jh 2002...

Keywords : CBSE, Geeks Networks, NCERT solved answers, solved answers of mathematics of 11th class, trigonometry, trigonometric functions,

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Some useful things about Lines and Angles

        Some useful things about Lines and Angles

 

1. Two angles are said to be supplementary if the sum of their measures is 180 degree.

2. Two angles are said to be complementary if the sum of their measures is 90 degree.

3. Two angles with the same vertex, one arm common and other arms lying on opposite sides of the common arm, are called adjacent angles.

4. Two adjacent angles are said to form a linear pair of angles if their non-common arms are two opposite rays.

5. If a ray stands on a line then the sum of the adjacent angles so formed is 180 degree.

6. The sum of all angles formed on the same side of a line at a given point on the line is 180 degree.

7. The sum of all angles around a point is 360.

8. Two angles are called a pair of vertically opposite angles if their arms form two pairs of a opposite rays.

9. If two lines intersect each other then the vertically opposite angles are equal.

Some Examples based on complement and supplement of angles

Example 1.
Find the complement of each of the following angles:
i) 50 degree
ii) 30 degree
iii) 78 degree

Solution

i) The given angle is 50 degree.
Let the measure of its complement be x degree. Then,
x 50 = 90 or x = 90 - 50 = 40.
Hence, the complement of the given angle measures 40 degree.

ii) The given angle is 30 degree. Let the measure of its complement be x degree. Then,x 30 = 90 or x = 90 - 30 = 60.Hence, the complement of the given angle measures 60 degree.

iii) The given angle is 78 degree. Let the measure of its complement be x degree. Then,x 78 = 90 or x = 90 - 78 = 12.Hence, the complement of the given angle measures 12 degree.

Example 2

Find the supplement of each of the following angles:
i) 120 degree
ii) 65 degree
iii) 48 degree

Solution

i) The given angle is 120 degree. Let the measure of its supplement be x degree. Then,x 120 = 180 or x = 180 - 120 = 60.
Hence, the supplement of the given angle measures 60 degree.

ii) The given angle is 65 degree. Let the measure of its supplement be x degree. Then,x 65 = 180 or x = 180 - 65 = 115.
Hence, the supplement of the given angle measures 115 degree.

iii) The given angles is 48 degree. Let the measure of its supplement be x degree. Then,x 48 = 180 or x = 180 - 48 = 132.
Hence, the supplement of the given angle measures 132 degree.

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By using the principle of mathematical induction prove that 1.2.3+ 2.3.4+ 3.4.5 .... +n(n+ 1)(n+ 2) = n(n +1)(n+ 2)(n+ 3) /4

By using the principle of mathematical induction prove that 1.2.3+ 2.3.4+ 3.4.5 .... +n(n+ 1)(n+ 2) = n(n +1)(n+ 2)(n+ 3) /4

Solution

Let P(n) : 1.2.3 + 2.3.4 + 3.4.5 ... + n(n + 1)(n + 2) = n(n + 1)(n + 2)(n + 3) / 4

For n = 1
L. H. S. = 6 and R. H. S. = 6.
Therefore, L. H. S. = R. H. S. i.e., P(1) is true.
Let us suppose P(k) is true.
Therefore, putting n = k, we have,
1.2.3 + 2.3.4 + 3.4.5 .... +k(k + 1)(k + 2) = k(k + 1)(k + 2)(k + 3) / 4. ................................................(1)
Adding (k + 1)(k + 2)(k+ 3) to both sides of (1), we have,
1.2.3 + 2.3.4 + 3.4.5 ... + k(k + 1)(k + 2) + (k + 1)(k + 2)(k + 3) = {k(k + 1)(k + 2)(k + 3) /4}  + (k 1)(k 2)(k 3).
                                                                                                     =(k + 1)(k + 2)(k + 3) {(k/4) + 1}.
                                                                                                     =(k + 1)(k + 2)(k + 3)(k + 4) / 4.
                                                                                                    =(k+ 1)(k+ 1+ 1)(k +1 +2)(k+ 1+ 3) / 4.
Here P(n) is true for n = k+ 1 i.e. P(k+ 1) is true.
Therefore, By the principle of mathematical induction, P(n) is true for all natural numbers n.

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MEASUREMENT OF ANGLES

      MEASUREMENT OF ANGLES: DIFFERENT SYSTEMS

Having defined an angle, it is now necessary to find a convenient unit for measuring angles. For different practical purpose, different suitable units are used for the measurement of angles.

Since a right angle is an angle of constant magnitude, it can be conveniently used as our unit for the measurement of angles. But for many purposes right angle is too big a unit, hence smaller unites are obtained from aright angle.

THERE ARE THREE SYSTEMS FOR MEASURING ANGLES, VIZ.,

(1) SEXAGESIMAL SYSTEM,

(2) CENTESIMAL SYSTEM,

(3) CIRCULAR SYSTEM.

                     SEXAGEMISAL OR ENGLISH SYSTEM

The most commonly used system for measuring angles is the sexagesimal system, which is also known as the common system or English system or degree system. the principal unit in this system is degree which is one-ninetieth part of right angle. A degree is divided into 60 equal parts and each part is called a minute (‘). A minute is again divided into 60 equal parts and each part is called a second (“).thus

                                               

                                                1 right angle (rt. Degree) = 90 degrees

                                                1 degree =60 sexagesimal minutes (60’)

                                                1’=60 sexagesimal seconds (60”)

Note Right angle is not a unit of the system.

         CENTESIMAL OR FRENCH SYSTEM

The principal unit of this system is grade (&), which is one-hundredth part of a right angle. One grade is equal to 100 minutes (‘) and one minute is equal to 100 seconds (“). Thus

1rt. Angle =100 grades (100&)

1& =100 centesimal minutes (100’)

1’ = 100 centesimal seconds (100”)

This system was first proposed in France but the attempt to introduce the system was not successful. 

Note : Minute and seconds of sexagesimal system are not the same as the centesimal minutes and seconds. For example,

                        1 rt.  Angle = 90*60 sexagesimal minutes

                                            =100*100 centesimal minutes.

             CIRCULAR SYSTEM OR REDIAN SYSTEM

In advanced mathematics, the most convenient system for the measurement of angles is the circular or radian measure.

      RELATION BETWEEN SEXAGESIMAL AND CENTESIMAL SYSTEMS

To convert the measure of an angle in degree to grades:

 Let D be the number of degrees in an angle.

since                90 degree = 1 rt. Angle, and 100& =1 rt. Angle,

So                        90 degree = 100 grades

Or                          1 degree =100/90 grades =10/9 grades.

Hence           D degrees = 10/9 D grades.

To convert the measure of an angle in grades to degrees:

LET G be the number of grades in an angle.

Since              100 grades = 1 rt. Angle, and 90 degree =1rt. Angle,

So                  100grades = 90 degrees

Or                       1 grade =90/100degree = 9/10 degree

Hence          G degree = 9/10g degrees.

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