Expansion of Ideas A FRIEND IN NEED IS A FRIEND INDEED

Expansion of Ideas

A FRIEND IN NEED IS A FRIEND INDEED

Once it was raining very heavily. It didn't seem that it would stop. There was no means of transport and I wondered how I would reach home. Then as if from nowhere, a classmate happened by in a car. He asked me to get in and reached me to my home. From that day onwards he became one of my best friends. Later on I was able to help him out in mathematics, a subject in which he was very weak. Indeed, a friend in need is a friend indeed!

I have found that many people talk about friendship, but when you need them during the time of crisis, somehow they are not seen anywhere. Those who stand by you when you are really in trouble are your true friends. Isn't that what friends are for? To help each other in times of need?

Life is all about being there for others. Life is all about caring and sharing. There are plenty of fair-weather friends. They desert you in your time of need. True friends are steadfast. True friends are forever.

Difficult Words: Indeed - In truth, certainly. Means of transport - Ways of travelling or moving goods. Classmate - A student in the same class. Crisis - A time of intense difficulty, trouble, or danger. Stand by - To support or remain loyal to someone, especially in a difficult situation. Trouble - Difficulty or problems. Fair-weather friends - People who are friends only when things are going well. Desert (verb) - To abandon someone or something in a way considered disloyal or treacherous. Steadfast - Resolutely or dutifully firm and unwavering.

WHERE THERE IS A WILL THERE IS A WAY

Where there is a will there's a way.

Expansion of Ideas School & College Section. 

Read the biographies of great men and women like Albert Einstein, Mother Teresa, Edmund Hillary, Marie Curie and Abraham Lincoln. You will find that it is their strong will which has helped them overcome great difficulties and setbacks and helped them find a way to achieve their goals. 

Every student thinks that securing a first class in an examination is beyond his or her capacity. But if a student resolves firmly to achieve his or her goal and works towards it sincerely and wholeheartedly, he or she is absolutely certain to achieve success. 

There was a time when flying in the air, travelling in space and sending messages and pictures instantly were all regarded as flights of fancy. But the persistent efforts of some strong - willed people have turned all these dreams into reality. 

Man's will-power, determination and dedication are wonderful qualities. They endow the doer with the will to achieve and the means to find out ways to achieve the impossible. All we have to do is to set a goal, have a strong will and make persistent efforts. If we come across an obstacle and can't climb over it, we must walk around it. There is always a way! There is always a situation! So, it is rightly said that, Where there is a will there's a way.

WHERE THERE IS A WILL THERE IS A WAY

WHERE THERE IS A WILL THERE IS A WAY

WHERE THERE IS A WILL THERE IS A WAY

WHERE THERE IS A WILL THERE IS A WAY

WHERE THERE IS A WILL THERE IS A WAY

WHERE THERE IS A WILL THERE IS A WAY

Expansion of Ideas

An Apple a day keeps the Doctor away.

What is special about the fruit, the apple? Why is it said that the apple keeps the doctor away? Why is no other fruit mentioned in a proverb? The answers are very simple. The apple has the most nutritional value amongst all fruits.

A medium sized apple gives 80 calories and makes a good snack. It contains absolutely no fat. It also helps reduce the risk of some cancers. Because it is low in sodium content, it reduces the risk of high blood pressure and heart disease. The apple has a substance that can reduce bad cholesterol and increase good cholesterol in our body. It is also rich in Vitamin A, which strengthens our vision. Apple also contain Vitamin C that boosts the immune system in the body. Research shows that people who eat five apples or more per week have fewer respiratory problems, including asthma.

After eating any kind of food, eat an apple. An apple is a natural toothbrush. It prevents tooth decay by killing 80% of the bacteria in the teeth. Thus, this proverb reminds us of the role of diet in warding off diseases - and how it saves us the trouble of visiting the doctor.

The proverb highlights the importance of healthy eating habits.

Apples are packed with essential vitamins and nutrients.

Its low sodium and beneficial substances support heart health.

Vitamins A and C in apples strengthen vision and the immune system.

Regular consumption is linked to better lung health.

It helps clean teeth and kill harmful bacteria after a meal.

A good diet is the first step in preventing diseases.

Difficult Words and Their Meanings

Proverb

A short, well-known saying that states a general truth or piece of advice.

Nutritional Value

How good a food is for your body, based on its vitamins, minerals, and energy.

Calories

A unit of energy that your body gets from food.

Sodium

A mineral found in salt; too much can lead to health problems like high blood pressure.

Cholesterol

A fatty substance in your blood. There is "good" and "bad" cholesterol; too much "bad" is unhealthy.

Vision

The ability to see; eyesight.

Immune System

The body's defense system that fights off infections and diseases.

Respiratory Problems

Health issues related to breathing and the lungs, such as coughing or difficulty breathing.

Asthma

A medical condition that makes breathing difficult by causing the airways to the lungs to become narrow.

Tooth Decay

The process of teeth getting damaged and developing cavities (holes).

Warding Off

To prevent something, like an illness, from happening or affecting you.

More expansion of ideas 💡

NUMBERS AND SEQUENCES EX. NO. 2.7

2. NUMBERS AND SEQUENCES

EX. NO. 2.7

(1)  Which of the following sequences are in G.P.?

(i) 3, 9, 27, 81,…                     Solution

(ii) 4,44,444,4444,...            Solution 

(iii) 0.5, 0.05, 0.005,…            Solution

(iv) 1/3, 1/6, 1/12,.......           Solution

(v)  1, −5, 25, −125,…             Solution

(vi) 120,60,30,18,…           Solution

(vii)  16, 4, 1, 1/4,..........        Solution

(2)  Write the first three terms of the G.P. whose first term and the common ratio are given below.

(i)  a = 6, r = 3         Solution

(ii)  a = √2, r = √2      Solution

(ii)  a = 1000, r = 2/5    Solution

(3)  In a G.P. 729, 243, 81,… find t7 .     Solution

(4)  Find x so that x + 6, x + 12 and x + 15 are consecutive terms of a Geometric Progression.        Solution

(5)  Find the number of terms in the following G.P.

(i) 4, 8, 16,…,8192 ?         Solution

(ii) 1/3, 1/9, 1/27,................1/2187        Solution

(6)  In a G.P. the 9th term is 32805 and 6th term is 1215. Find the 12th term.          Solution

(7)  Find the 10th term of a G.P. whose 8th term is 768 and the common ratio is 2.         Solution

(8)  If a, b, c are in A.P. then show that 3a, 3b, 3c are in G.P.       Solution

(9)  In a G.P. the product of three consecutive terms is 27 and the sum of the product of two terms taken at a time is 57/2 . Find the three terms.         Solution

(10)  A man joined a company as Assistant Manager. The company gave him a starting salary of ₹60,000 and agreed to increase his salary 5% annually. What will be his salary after 5 years?            Solution

(11)  Sivamani is attending an interview for a job and the company gave two offers to him.

Offer A: ₹20,000 to start with followed by a guaranteed annual increase of 6% for the first 5 years.

Offer B: ₹22,000 to start with followed by a guaranteed annual increase of 3% for the first 5 years.

What is his salary in the 4th year with respect to the offers A and B?       Solution

(12)  If a, b, c are three consecutive terms of an A.P. and x, y, z are three consecutive terms of a G.P. then prove that xb−c × yc−a × za−b = 1.       Solution

எத்தனை கைகள் என்னை தள்ளிவிட்டாலும் என் நம்பிக்கை என்னை கை விடாது

எத்தனை கைகள்
என்னை தள்ளிவிட்டாலும்
என் நம்பிக்கை
என்னை கை விடாது!

எத்தனை கைகள்
என்னை தள்ளிவிட்டாலும்
என் நம்பிக்கை
என்னை கை விடாது

எல்லோரும் பயணிக்கிறார்கள் என்று நீயும் பின்தொடராதே உனக்கான பாதையை நீயே தேர்ந்தெடு...

எல்லோரும் பயணிக்கிறார்கள் என்று நீயும் பின்தொடராதே உனக்கான பாதையை நீயே தேர்ந்தெடு...

எல்லோரும் பயணிக்கிறார்கள் என்றுநீயும் பின்தொடராதே உனக்கான பாதையை
நீயே தேர்ந்தெடு...

NUMBERS AND SEQUENCES EX. NO. 2.6

2. NUMBERS AND SEQUENCES

EX. NO. 2.6

(1)  Find the sum of the following

(i) 3, 7, 11,… up to 40 terms.       Solution 

(ii) 102, 97, 92,… up to 27 terms.       Solution

(iii) 6 + 13 + 20 +...........+ 97          Solution 

(2)  How many consecutive odd integers beginning with 5 will sum to 480?            Solution

(3)  Find the sum of first 28 terms of an A.P. whose nth term is 4n -3.       Solution

(4)  The sum of first n terms of a certain series is given as 2n2-3n . Show that the series is an A.P.     Solution

(5)  The 104th term and 4th term of an A.P. are 125 and 0. Find the sum of first 35 terms.     Solution

(6)  Find the sum of all odd positive integers less than 450.      Solution

(7)  Find the sum of all natural numbers between 602 and 902 which are not divisible by 4.   Solution

(8)  Raghu wish to buy a laptop. He can buy it by paying 40,000 cash or by giving it in 10 installments as 4800 in the first month, 4750 in the second month, 4700 in the third month and so on. If he pays the money in this fashion, find (i) total amount paid in 10 installments. (ii) how much extra amount that he has to pay than the cost?

   Solution

(9)  A man repays a loan of 65,000 by paying 400 in the first month and then increasing the payment by 300 every month. How long will it take for him to clear the loan?

Solution

(10)  A brick staircase has a total of 30 steps. The bottom step requires 100 bricks. Each successive step requires two bricks less than the previous step.  

(i) How many bricks are required for the top most step?

(ii) How many bricks are required to build the stair case?

Solution

(11)  If S1, S2, S3,....Sm are the sums of n terms of m A.P.’s whose first terms are 1,2, 3,...m and whose common differences are 1, 3, 5,..., (2m -1) respectively, then show that S1 + S2 + S3 +............Sm  = (mn/2)(1 + mn)

Solution

(12)  Find the sum

Solution

NUMBERS AND SEQUENCES EX. NO. 2.5

2. NUMBERS AND SEQUENCES

EX. NO. 2.5

(1)  Check whether the following sequences are in A.P.

(i) a - 3, a - 5, a -7,...         Solution

(ii)  1/2, 1/3, 1/4, 1/5........      Solution

(iii) 9, 13, 17, 21, 25,...      Solution

(iv) -1/3, 0, 1/3, 2/3.........      Solution

(v) 1, -1, 1, -1, 1, -1,............      Solution

(2)  First term a and common difference d are given below. Find the corresponding A.P.

(i) a = 5 , d = 6          Solution

(ii) a = 7 , d =−5          Solution

(iii) a = 3/4, d = 1/2         Solution

(3)  Find the first term and common difference of the Arithmetic Progressions whose nth terms are given below

(i) tn = −3 +2n       Solution

(ii) tn = 4 - 7n     Solution

(4)  Find the 19th term of an A.P. -11,-15,-19,...     Solution

(5)  Which term of an A.P. 16, 11, 6, 1,... is -54 ?    Solution

(6)  Find the middle term(s) of an A.P. 9, 15, 21, 27,…,183. Solution

(7)  If nine times ninth term is equal to fifteen times fifteenth term, show that six times twenty fourth term is zero.           Solution

(8)  If 3 + k, 18 - k, 5k + 1 are in A.P. then find k   Solution

(9)  Find x, y and z, given that the numbers x, 10, y, 24, z are in A.P.    Solution

(10)  In a theatre, there are 20 seats in the front row and 30 rows were allotted. Each successive row contains two additional seats than its front row. How many seats are there in the last row?      Solution

(11)  The sum of three consecutive terms that are in A.P. is 27 and their product is 288. Find the three terms.  Solution

(13)  The ratio of 6th and 8th term of an A.P. is 7:9. Find the ratio of 9th term to 13th term.         Solution

(14)  In a winter season let us take the temperature of Ooty from Monday to Friday to be in A.P. The sum of temperatures from Monday to Wednesday is 0° C and the sum of the temperatures from Wednesday to Friday is 18° C. Find the temperature on each of the five days. Solution

(15)  Priya earned ₹15,000 in the first month. Thereafter her salary increased by ₹1500 per year. Her expenses are ₹13,000 during the first year and the expenses increases by ₹900 per year. How long will it take for her to save ₹20,000 per month.                    Solution