mixture

Monday, August 10, 2020

find the coefficient of {x^5} in the expansion of{(1+2x)^6}{(1-x)^7}

ncert cbse chapter 8 binomial theorem miscellaneous exercise

3.find the coefficient of {x^5} in the expansion of{(1+2x)^6}{(1-x)^7}

using binomial expansion

(1+2x)^6 = 1 +C(6,1)(2x)+C(6,2)(2x)^2+C(6,3)(2x)^3 +C(6,4)(2x)^4

+C(6,5)(2x)^5+C(6,6)(2x)^6

C(6,1) = 6 =C(6,5)

C(6,2)=15=C(6,4)

C(6,3)=20

C(6,6)=1

(1+2x)^6 =

1 +6(2x) +15(4(x^2))+20(8(x^3))+15(16(x^4))+6(32(x^5))+1(32(x^6))

=1+12x+60(x^2)+160(x^3)+240(x^4)+192(x^5)+32(x^6)

(1-x)^7=

1-C(7,1)(x)+C(7,2)(x^2)-C(7,3)(x^3)+C(7,4)(x^4)

-C(7,5)(x^5)+C(7,6)(x^6)-C(7,7)(x^7)

C(7,1)=7=C(7,6)

C(7,2)=21=C(7,5)

C(7,3)=35=C(7,4)

C(7,7)=1

(1-x)^7=1 -7x+21(x^2)-35(x^3)+35(x^4)-21(x^5)+7(x^6)-1(x^7)

now multiplying the two expansions and concentrate only on the terms containing

(x^5)

1* {-21(x^5)}+{+12x}*{+35(x^4}+{+60(x^2)}*{-35(x^3)}+{+160(x^3)}{+21(x^2)}+{+240(x^4)}{-7x}+{+192(x^5)}{1}

coefficient of (x^5) is

1*(-21)+(12)*(35)+(60)(-35)+(160)(21)+(240)(-7) +192*1

=(-21)+420-2100+3360-1680+192

=171

chapter 8 binomial theorem miscellaneous exercise

1.Find a , b and n in the expansion of (a+b)^n if the first three terms in the expansion are 729, 7290, 30375

solution

2. Find a if the coefficients of (x^2) & (x^3) in the expansion of {(3+ax)^9} are equal

solution

3.find the coefficient of {x^5} in the expansion of{(1+2x)^6}{(1-x)^7}

solution

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Friday, August 7, 2020

blood tests and injections in early pregnancy

As a woman trying to get pregnant, when you miss your period by a few days. then wait for a few anxious days and then most women are likely to use a urine based pregnancy home test kit for checking if they are pregnant. [or as a joke use the diy pregnancy test based on common salt where you mix your first morning urine with a solution of common salt and water and if it turns milky after some time , it might indicate pregnancy.]

If the woman finds that she is pregnant, it is best to book a prenatal appointment with a doctor especially if the woman has a history of thyroid problems, diabetes or previous loss of pregnancy.

Managing diabetes with pregnancy is hard.

Once you go to the doctor , it will be the start of blood tests and injections. This might be bad news for women who are afraid of syringes and for women who have small veins which are hard to find. Some persons faint while drawing blood. Holding hands of their loved ones like spouses or lying down while drawing blood might be helpful for such women.If allowed drinking water to avoid dehydration might also be helpful.

Most doctors will order a HCG blood test to confirm the pregnancy. HCG blood test refers to a blood test to measure the level (quantitative blood test) of human chorionic gonadotropin [ HCG blood test in short or beta hcg blood test or quantitative hcg blood test]. After conception the level of hcg in the blood increases and can usually be detected after two weeks in the blood.

The doctor might schedule another test to confirm that the HCG levels are increasing to confirm a healthy pregnancy. If the doctor feels that the urine tests used were sensitive , the doctor might wait for the ultrasound tests at 7 to 12 weeks.

There might be problems with insurance companies wanting doctor confirmation of the pregnancy before starting prenatal coverage.

The doctor will want to test for blood type and to take precautions in case of

Rh -ve type etc .A CBC might be done to check for anaemia and the doctor might prescribe iron supplements or iron rich foods in case of anaemia.Other screening tests might be for HIV, syphillis, hepatitis B ,genetic disorders , etc .

In some extreme cases with Rh -ve type, the fathers blood test might also be required to screen for complications.

Further diagonostic tests might be ordered based on the results of the screening tests.

Trying to get pregnant or trying to concieve ( TTC )

Pregnancy testing

first trimester of pregnancy

second trimester of pregnancy

Third trimester

labour and child birth

post partum

life of a woman after child birth

Male Fertility

Bipolar disorder medicine during pregnancy.

Changes in women after childbirth

Find a if the coefficients of (x^2) and (x^3) in the expansion of {(3+ax)^9} are equal

ncert cbse chapter 8 binomial theorem miscellaneous exercise

Find a if the coefficients of (x^2) & (x^3) in the expansion of {(3+ax)^9} are equal

using binomial theorem

{(3+ax)^9}= 3^9 + C(9,1)(3^8){(ax)}+ C(9,2)(3^7){(ax)^2}

+ C(9,3)(3^6){(ax)^3} + ...+{(ax)^9}

given coefficients of (x^2) & (x^3) are equal

therfore

C(9,2)(3^7){(a)^2} =C(9,3)(3^6){(a)^3}

36* (3^7){(a)^2}=84*(3^6){(a)^3}

if a cannot be zero

a= {36*(3^7) }/{84*(3^6)}

a =9/7

chapter 8 binomial theorem miscellaneous exercise

1.Find a , b and n in the expansion of (a+b)^n if the first three terms in the expansion are 729, 7290, 30375

solution

2. Find a if the coefficients of (x^2) & (x^3) in the expansion of {(3+ax)^9} are equal

solution

Find a , b and n in the expansion of (a+b)^n if the first three terms in the expansion are 729, 7290, 30375

ncert cbse chapter 8 binomial theorem miscellaneous exercise

1.Find a , b and n in the expansion of (a+b)^n if the first three terms in the expansion are 729, 7290, 30375

(a+b)^n = (a^n) + C(n,1)[a^(n-1)][b]++ C(n,2)[a^(n-2)][b^2] +...+(b^n)

using the given data and also

C(n,1) = n

C(n,2) = {n(n-1)}/{2}

we get

a^n = 729--------------------(1)

n[a^(n-1)][b] = 7290--------------------(2)

{n(n-1)}/{2}*[a^(n-2)][b^2] = 30375---------------(3)

(2)/(1) gives {nb} / {a} = 10 ----------------------(4)

(3)/(2) gives {(n-1)b}/{2a} = {25} / {6}-----------(5)

(5)/(4) gives {n-1}/{2n} = {5}/{12}

therefore 12(n-1)=5(2n)

12n-12 =10n

12n-10n =12

2n=12

n=6

use in (1)

a^6 = 729

a = 3

use these values of n and a in (4)

{6*b}/{3} = 10

gives b =5

Thursday, August 6, 2020

Changes in women after childbirth

Women are usually very self conscious about their own body. Usually many women fret and worry too much about the changes in the appearance of their body after childbirth and try to compare that with their body before pregnancy and they wonder about the changes in their body that motherhood has brought with it. Some women are unduly worried about the stretch marks on the stomach / abdomen and saggy / flabby skin. Some of the new mothers get fooled by some celebrities who magically seem to get back to their pre- pregnant body shape very fast, forgetting that most of the celebrities have access to lots of money, specialist trainers, cosmetic surgery etc.

The breasts of a new mother usually become enlarged with milk inside the milk ducts of the breast for breastfeeding the new born child. But after the mother stops breast feeding her child, the size of the breast may not go back to their pre-pregnant size and in some women, the breasts may appear saggy / less firmer than it was before pregnancy . But breast feeding itself might not be the cause of the sagging of the breasts. Many new mothers wonder how to get back their firm breasts after their breast feeding stage. Push-ups and dumbbell fly exercises might be useful to get firmer breasts. The fat for lactation or breast milk may be stored around the hips and those areas are likely to become bigger.

The legs / feet of a woman may swell during pregnancy, but even after pregnancy, the feet / arch of the feet may not go back to their old size / structure and might be flatter / longer and hence the new mother might need bigger footwear after childbirth. Hormones bring about a lot of changes to the skin and the hair during and after pregnancy. Some areas of the skin may darken.

Some new mothers find a lot of changes in their metabolism after childbirth. Some mothers find that they have new allergies ( breaking out in hives , intolerance to some food or food allergies , celiac disease or intolerance to wheat gluten etc ) after childbirth because being exposed to allergens when the body / immune system is weak might start off the allergies.

Sometimes women with irregular periods may find that they get regular periods after childbirth and sometimes there will be decrease in the menstrual cramps also.

Some women experience urinary incontinence ( or slight urine leakage during stress, orgasm etc ) after childbirth due to excessive stretching / injury of the ligaments and pelvic area during pregnancy. The vagina will also feel more bigger / looser after vaginal childbirth. Some women also experience dryness in the vagina. Kegel exercises or pelvic floor exercises after child birth may be helpful to strengthen the muscles in the pelvic region and may help with the problem of urinary incontinence.

Bipolar disorder medicine during pregnancy.

Bipolar disorder ( manic depression ) refers to experience of extremes of mood swings from high energy levels / manic euphoria to depressive levels usually with cycles over many days, weeks or months . Some famous actors and celebrities like Carrie Fisher (who is famous for her character of Princess Leia in Star wars ) have been very open about their battle with bipolar disorder. Carrie Fisher has publicly disclosed her diagnosis with bipolar disorder and given many interviews about her experience with bipolar disorder. Drinking alcohol is also likely to negatively affect the mood swings of people having bipolar disorder.

People with bipolar disorder are usually prescribed a combination of different drugs called a drug cocktail of mood stabilizers, anti depressants, anti psychotic ,anti convulsant drugs and(examples are lithium, valproic acid, lamotrigine, carbamazepine , ariprazole , clozapine, lamotragine etc ). Many people wonder if bipolar women on medication can give birth to healthy children. There are many examples over the years of bipolar women on medication giving birth to healthy children. Bipolar disorder in pregnant woman usually needs a lot of attention from properly trained doctors. But some of the good psychiatrists may charge expensive fees and may be out of reach of most pregnant women who may have to consult their insurance companies also . Insurance companies sometimes decline them or charge exorbitant amounts for people with bipolar disorder.

One of the main concerns for pregnant women is if the medications for bipolar disorder can be continued during pregnancy. The doctors you consult will usually show you which medications are usually harmful and which are not usually harmul during pregnancy in various medical studies. There may be a lot of trial and error testing with different combinations, before settling on a good combination of drugs to use during pregnancy. So it is better to get the consultation with the doctor and get a stable combination going before trying to conceive ( ttc ). But that is not always possible because of unplanned pregnancies . ( Bipolar disorder may also be associated with irregular menstrual cycles and some other menstrual abnormalities. Some of the medicines for bipolar disorder may cause menstrual irregularities.

Some of the common medications for bipolar disorder may increase the risk of birth disorders and some other birth anomalies like Ebstein’s anomaly ( especially during the first trimester ). Some studies show that medications involving Lithium may be associated with defects of the baby's heart, while some doctors would like the mother to be hydrated so as to control lithium toxicity. Doctors who prescribe lamotrigine (an anti convulsant or anti epileptic drug ) may ask the women to take higher dosage of folic acid supplement than usual.

Some of the women who stop their medication without consulting the doctor during pregnancy may suffer relapse like being unable to sleep, experience anxiety, depression etc . Pregnant women and their partners are usually concerned about what mood stabilizers are safe during pregnancy. Breast feeding after pregnancy is also challenging for women with bipolar disorder on medication. Some of the medications used for bipolar disorders will also be present in traces in the breast milk for the baby . Even after the breast feeding stage, the weaning stage might also present some hormonal changes in the levels of oxytocin and prolactin in the mother which might be a trigger for emotional swings / changes for the mother

Male Fertility

Male Fertility

Usually the tips for ttc ( trying to concieve ) will be written with the woman in mind consisting of tips about FAM ( fertility awareness methods ), bbt charts ( basal body temperature charts ), thickness of the cervical mucus ( CM / EWCM or egg white cervical mucus ) finding the probable ovulation time etc. The idea is that EWCM is likely to be a prelude to ovulation.

Now for a male , the sperm takes nearly 50 to 75 days to mature. In a single ejaculation, millions of sperms are mixed per ml of the semen and ejaculated. When ejaculated into the female vagina, the sperm will move through the cervix and into the uterus and then to the fallopian tubes where hopefully it might fertilize the egg in the female. The sperm may last theoretically from around 3 days to around 5 days in the nutritive supportive cervical fluids ( not in the vagina ) in the woman's body . So the three days around the probable ovulation time is the likely to be the most fertile period for women

When somebody goes for semen analysis / sperm analysis to check the health of his sperm , they usually check for

semen volume (usually over 2 ml per ejaculation ) ,

sperm count / sperm concentration / sperm density ( usually around 20 million to 200 million sperms per ml. ) ,

sperm motility (progressive motility or the ability of the sperm to move in a relatively straight line ),

sperm velocity for propagation,

sperm morphology ( sperm size/shape expressed as a percentage of sperm which appear normal under a microscope ),

Normozoospermic usually refers to normal sperm quality.
If there is no sperm in the semen the condition is known as azoospermia.

Low sperm count/ motility may be caused by exposure to chemicals (like xenoestrogens which imitate estrogens from some components in plastic bottles ) , mobile phone radiation , prolonged exposure to heat in the testicular region like hot baths in tubs, tobacco and alcohol usage.

Sometimes , overweight / obese men may find it hard to impregante their female partners due to problems related to hormone levels . It is good to decrease body weight to normal range in order to boost male sperm count / motility and hence male fertility. High blood pressure and medications for high blood pressure is likely to have detrimental effect on erection in males and also quality of the sperm and male fertility in general. Diabetes is also likely to have bad effects on male fertility due to diabetes causing damage to the nerves ( diabetic neuropathy ) and reduced blood circulation which might lead to problems of erection ( erectile dysfunction ED ), lower ejaculate volume etc .

Regular moderate exercise ( not intense and strenuous exercise ) may boost male fertility. Excess stress levels might reduce sperm quality. reducing stress levels is also good for boosting male fertility. Spme steroids used for muscle development in body building may lead to decrease in sperm quality

mixture

st

Monday, August 10, 2020

find the coefficient of {x^5} in the expansion of{(1+2x)^6}{(1-x)^7}

Friday, August 7, 2020

blood tests and injections in early pregnancy

Trying to get pregnant or trying to concieve ( TTC )

Pregnancy testing

first trimester of pregnancy

second trimester of pregnancy

Third trimester

labour and child birth

post partum

life of a woman after child birth

Male Fertility

Bipolar disorder medicine during pregnancy.

Changes in women after childbirth

Find a if the coefficients of (x^2) and (x^3) in the expansion of {(3+ax)^9} are equal

Find a , b and n in the expansion of (a+b)^n if the first three terms in the expansion are 729, 7290, 30375

Thursday, August 6, 2020

Changes in women after childbirth

Changes in women after childbirth

Bipolar disorder medicine during pregnancy.

Bipolar disorder medicine during pregnancy.

Male Fertility

SAT Mixture & Concentration Problems: The Easy Method That Works

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