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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?

2013-08-10T06:49:00.000-07:00

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

Magnetism

2013-05-21T05:10:00.000-07:00

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.

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

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

2013-05-14T04:21:00.002-07:00

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 :

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

2013-05-11T19:49:00.000-07:00

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

Sol

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

2013-05-11T18:58:00.000-07:00

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

Sol :

Conditions for Maxima and Minima

2013-04-25T23:09:00.001-07:00

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.

Formulae of Differentiation

2013-04-25T22:53:00.002-07:00

Fundamental Formulae of Differentiation

Derivatives of Trigonometrical Functions

Derivatives of Logarithmic and Exponential Functions

Solution of Triangles

2013-04-10T22:33:00.003-07:00

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.

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.

2013-03-30T21:24:00.002-07:00

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 :

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

2013-03-30T21:12:00.005-07:00

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,

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

2013-03-17T05:43:00.002-07:00

This Question is based on Properties of Triangle

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

Very Very Important Question based on Properties of Triangle.

2013-03-09T03:57:00.002-08:00

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,

Some useful things about Lines and Angles

2013-02-26T00:20:00.000-08:00

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.

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

2013-02-16T00:31:00.001-08:00

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.

MEASUREMENT OF ANGLES

2013-01-21T01:32:00.000-08:00

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.

Angles and Measurement of Angles

2013-01-15T01:54:00.000-08:00

ANGLES AND MEASUREMENT OF ANGLES

Generation of angles

In plane geometry, an angle is usually said to be formed by two rays (half-lines), radiating from the same end point. The common point is called the vertex of the angle. All angles-acute, obtuse or reflex, are positive and their measures are less than 360 degree. There is no place for negative angles in geometry. This definition of an angle is not general for the purpose of trigonometry, where an angle may be positive or negative of any magnitude (size).

An angle_,in trigonometry, is formed by the rotation of a ray in a plane, around the end point.

The revolving ray is called the generating line of the angle. The initial position and the final position are respectively known as the initial side (arm) and the terminal side (arm). The end point, around which the revolving ray turns, is called the vertex of the angle and the amount of turn is the measure of the angle. The rotating ray is sometimes called the radius vector. If a revolving ray starts from the initial position OX and rotates in a plane, and gradually comes to the positions OA, OB, OC, etc. then the angles ANGLE XOA, ANGLE XOB, ANGLE XOC, etc. are respectively formed.

POSITIVE AND NEGATIVE ANGLES

By convention, an angle is considered positive, if it is generated by the counter-clockwise motion of the revolving ray and the angle is regarded as negative if the motion is in clockwise direction. The rotating line may revolve several times around a point on it, in anti- clockwise or clockwise sense, before arriving to the terminal position. Thus, we may have a positive or a negative angle of any magnitude (size).

NOTE-Only the Initial position OA and the final position OB are not enough to know the direction and the amount of rotation of an angle.

Curved arrowhead is necessary to indicate completely the direction and the size (magnitude) of the angle.

Let be the measurement of the acute angel AOB. The rotating ray, from its initial position OA, revolving counterclockwisely, arrives to the terminal position OB, the measure of the angle AOB is now if the rotating ray arrives at the position OB after one complete revolution, then the measure of angle AOB is 360 + . if it arrives at OB after two complete revolutions, the measure of angle AOB is n x 360 + .

If the rotating ray, revolving clockwise comes to the position OB, the measures of the corresponding angles are -360 + , -2 x 360 + , -3 x 360 + , etc.

Thus the same terminal side may be the bounding line of unlimited number of angles of different measures.

Properties of Triangle

2013-01-11T03:41:00.000-08:00

PROPERTIES OF TRIANGLES

Parts of a triangle

We know that a triangle has six parts viz. Three angles and three sides.

In any triangle ABC, the measures of the angles BAC, CBA and ACB are denoted respectively by the letters A, B and C, and the sides BC,CA and AB opposite to these angles are represented by a, b and c respectively. The radius of the circumcircle, i.e. Circumradius will be denoted by R and the area of the triangle of . All the six parts of the triangle are not independent; they satisfy certain relations.

In geometry you are familier with the following relations :

(i) A + B + C = 180 degree (ii) a+b > c b + c > a c + a > b

Law of sines or sine rule

a/sin A = b/sin B = c/sin C

Law of cosine or cosine rule

1. a^2 = b^2 + c^2 -2bc cosA or cos A = (b^2 + c^2 – a^2) / 2bc

2. b^2 = c^2 + a^2 – 2ca cos B or cos B = (c^2 + a^2 – b^2) / 2ca

3. c^2 = a^2 + b^2 – 2ab cos C or cos C = (a^2 + b^2 – c^2) / 2ab

Law of tangents or tangent rule

1. tan (B-C)/2 = (b-c)/(b+c) cot A/2

2. tan (C-A)/2 = (c-a)/(c+a) cot B/2

3. tan (A-B)/2 = (a-b)/(a+b) cot C/2

The relation between the three sides and two angles of any triangle ABC are as follows

1. a = b cos C + c cos B

2. b = c cos A +a cos C

3. c = a cos B + b cos A

To find sines of half an angle of a triangle in terms of the sides

1. sin A/2 = (s-b)(s-c)/bc

2. sin B/2 = (s-c)(s-a)/ca

3. sin C/2 = (s-a)(s-b)/ab

To find cosines of half an angle of a triangle in terms of the sides

1. cos A/2 = s(s-a)/bc

2. cos B/2 = s(s-b)/ca

3. cos C/2 = s(s-c)/ab

To find tangent of half an angle of triangle in terms of the sides

1. tan A/2 = (s-b)(s-c) / s(s-a)

2. tan B/2 = (s-c)(s-a) / s(s-b)

3. tan C/2 = (s-a)(s-b) / s(s-c)

Expressions for area of a triangle ABC

1. Area = 1/2 ab sin C

2. Area = 1/2 bc sin A

3. Area = 1/2 ca sin B

Values of sines of angles in terms of sides and area

1. sin A = 2/bc s(s-a)(s-b)(s-c)

2. sin B = 2/ca s(s-a)(s-b)(s-c)

3. sin C = 2/ab s(s-a)(s-b)(s-c)

Values of tangents and cotangents of half an angle in terms of sides and area

1. tan A/2 = (s-b)(s-c) / s(s-a)(s-b)(s-c)

2. tan B/2 = (s-c)(s-a) / s(s-a)(s-b)(s-c)

3. tan C/2 = (s-a)(s-b) / s(s-a)(s-b)(s-c)

4. cot A/2 = s(s-a) / s(s-a)(s-b)(s-c)

5. cot B/2 = s(s-b) / s(s-a)(s-b)(s-c)

6. cot C/2 = s(s-c) / s(s-a)(s-b)(s-c)

LOGARITHMS

2013-01-09T18:50:00.000-08:00

LOGARITHMS

Introduction

By the use of logarithms it is possible to minimize the tedious arithmetical computations, which are found in some trigonometrical problems. The uses of logarithm help us to reduce multiplication and division to the simple work of addition and subtraction respectively; also raising a number to a power (involution) and extracting the roots (evolution) of a given number are reduced to the work of multiplication and division respectively.

Definition of Logarithms

Logarithm of a positive number of the power to which the base is to be raised to obtain the given number.

If a, x, N are three numbers, so that

a^x = N, (N>0, a>0, a=! 1)…………….. (1)

Then the exponent x is the logarithm of N to the base a and this is written as

x = logaN………………………………(2)

This relation is read as , x=logarithm of N to the base a. To emphasize the idea, we repeat that, with respect to a given base, exponents are logarithms.

To remember this statement, the following relation may be noted.

(base)^log= number.

It is clear, from definition, that the relations in (1) and (2) are equivalent i.e. two different ways of expressing the same statement. Being equivalent, any one form may be transformed to the other.

NOTE : If a^x=N then x=log N therefore a^{log N} = N. (where base is a)

Example

Let us consider the number 81.

Now, we have 81=3^4.

Therefore , By definition, 4=log₃81. Thus, for the same number 81, we get different logarithms for different bases. Hence, the base should be stated, otherwise the logarithm of a number is meaningless.

But, for brevity, the base is not indicated when base is understood to be known or if the relation is true for any value of the base.

Some Special Cases

The logarithm of 1, with respect to any finite nonzero base, is equal to zero.

Since a^0=1. therefore log_a1=0.

The logarithm of any number, with respect to itself as the base, is equal to 1.

Since a^1=a therefore log_aa=1.

Laws of Logarithms

If we remember how logarithms are close to indices, we may easily derive the laws of logarithms from the laws of indices.

The logarithm of the product of two numbers with respect to any base is equal to the sum of the logarithms of the numbers with respect to the same base., i.e.

Log_a(m x n) = log_am + log_an

The logarithm of the quotient of two numbers with respect to any base is equal to the difference of the logarithms of the dividend and the divisor with respect to the same base, i.e.

Log_a(m/n) = log_am - log_an

The logarithm of a power of a number with respect to any base is equal to the index of the power multiplied by the logarithm of the number with respect to the same base, i.e.

Log_am^n=n log_am

Show that : cos(36 - A) cos(36 + A) + cos(54 + A) cos(54 - A) = cos2A

2012-12-28T01:05:00.001-08:00

Show that :

cos(36 - A) cos(36 + A) + cos(54 + A) cos(54 - A) = cos2A

Sol :

L.H.S. = cos(36 - A) cos(36 + A) + cos(54 + A) cos(54 - A)

           = 2/2 cos(36 - A) cos(36 + A) + 2/2 cos(54 + A) cos(54 - A)

          = 1/2{cos(36 -A +36 +A) + cos(36-A-36-A)} + 1/2{cos(54+A+54-A) + cos(54+A-54+A)}

         = 1/2 {cos72 + cos(-2A)} + 1/2 {cos108 + cos2A}

        = 1/2 {cos72 + cos108 +2cos2A}

        = 1/2 {2cos (72+108)/2 .cos(72-108)/2 + 2cos2A}

= 1/2 {2 cos90 . cos16 + 2cos2A}

= 1/2 . 2 cos2A            [cos90 = 0]

= cos2A

        = R.H.S         PROVED

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

The mean and variance of 5 observations are 15 and 6 respectively.If three observations are 11,15,18, find the remaining observations.

2012-12-24T01:17:00.002-08:00

The mean and variance of 5 observations are 15 and 6 respectively.If three observations are 11,15,18, find the remaining observations.

Sol :

Keywords : statistics,CBSE, Geeks Networks, NCERT solved answers, solved answers of mathematics of 11th class,

profiles of users in computer system

2012-12-15T06:02:00.001-08:00

Simply, Profile is used for desktop related settings.
There are three types of profile :-

Local Profile
Roaming Profile
Mandatory Profile

1. Local Profile :- It is automatically provided by Operating System. When we create a user , he gets local profile by default.

2. Roaming Profile :- It is manually created by administrator of Domain Controller (DC).With this profile , If user will change anything in his desktop,he will get that changes in entire domain.

3. Mandatory Profile :- This Profile is also manually created by Network Administrator.The owner/user of this profile changes desktop settings, those settings will not work longer.

How to check Profile

Select my computer ----> Right click--------> Property-------> Advance------------> User profile Setting.

In a parallelogram, The lengths of adjacent sides are 15 cm and 17 cm respectively. If the length of one diagonal is 18 cm, find the length of the other diagonal........

2012-12-15T00:50:00.001-08:00

In a parallelogram, The lengths of adjacent sides are 15 cm and 17 cm respectively. If the length of one diagonal is 18 cm, find the length of the other diagonal........

Sol :

In a parallelogram, the sum of the squares of the diagonals = 2 x ( the sum of the squares of the two adjacent sides).

Or          (D1)^2 + (D2)^2 = 2{(a)^2 + (b)^2}

Or          18^2 + (D2)^2   = 2 { 15^2 + 17^2 }

Or          324 + (D2)^2    = 2 { 225 + 289}

Or             (D2)^2 = 1028 - 324

Or             D2       = sqrt(704)

Or             D2       = 26.53 cm

Find the no. of Diagonals of a Polygon of 15 sides....

2012-12-10T05:43:00.001-08:00

Find the no. of Diagonals of a Polygon of 15 sides....

Sol :

A polygon of n sides will have n vertices . A diagonal or a side of the polygon will be formed by joining any two vertices of the polygon..

No. of Diagonals of the polygon + No. of sides of the polygon = C(n,2) - n

                                                                             = n! / 2!(n-2)! - n

                                                                             = n x (n-1) x (n-2)! / 2!(n-2)! - n

                                                                             = n x (n-1) / 2   - n

                                                                             = {n(n-1) - 2m} / 2

                                                                             = n{n-1-2} / 2

                                                                             = n(n-3) / 2

Now,     Putting n = 15, We get the number of Diagonals   = 15(15-3) / 2

                                                                              = 15 x 12 / 2

                                                                              = 90

Given 7 flags of different colours, how many different signals can be generated if each signal requires the use of two flags, One below the other ?

2012-12-01T23:01:00.003-08:00

Given 7 flags of different colours, how many different signals can be generated if each signal requires the use of two flags, One below the other ?

Sol :

First place can be filled up in seven ways. After filling first place , we have left with 6 flags.
Therefore The second place can be filled up in six ways...

The total number of signals which can be generated = 7 x 6 ways = 42 ways..........

How many 3-letter code words are possible using the first 10 letters of the English alphabet, if no letter can be repeated ?

2012-12-01T22:43:00.002-08:00

How many 3-letter code words are possible using the first 10 letters of the English alphabet, if no letter can be repeated ?

Sol :

First letter of the code word can be selected in 10 ways. After the selection of the first letter, we have nine letters. Hence, the second letter can be selected in nine ways. According to Fundamental principal of Counting(FPC), first two letters can be selected in 10 x 9 ways.

Similarly third letter can be selected in eight ways . According to FPC , number of ways of selecting Three letters out of ten letters of English alphabet = 10 x 9 x 8 ways = 720 ways...........