## CBSE_class X time bound sample paper 4 marks Trigonometry_Solutions

Here are the solutions for your reference. In case of any queries  do drop your questions so that it can be answered in time.

Ans 1 .  given 3x = cosecA, 3/x = cotA

$=3[ x^2 - 1/x^2 ]$

$=3x^2 - 3/x^2$

$=1/3[ 9x^2 - 9/x^2]$    (multiply and divide by 3 )

$=1/3[ {3x}^2 - {3/x}^2 ]$

$=1/3[ cosec^2A - cot^2A ]$    (identity 3)

$=1/3 * 1$

$=1/3$

Ans 2 .  given 6x = secB , 6/x = tanB

$= 9 [ x^2 - 1/x^2 ]$

$= 9 * 36/36 [ x^2 - 1/x^2 ]$    ( multiply and divide by 36)

$= 9 /36 [ (6x)^2 - (6/x)^2 ]$

$= 1/4 [ sec^2B - tan^2B ]$      (substituting the values of 6x and 6/x)

$= 1/4 [ 1]$     ( identity 2 )

$= 1/4$

Ans 3 . Given

$x.sin^3A + y.cos^3A = sinA.cosA ..............eq(1)$

$, x.sinA = y.cosA ..................eq(2)$

$x = y.cosA / SinA$

substituting in eq(1).

$y.cosA/sinA . sin^3A + y.cos^3A = sinA.cosA$

$y.cosA.sin^2A + y.cos^3A = sinA.cosA$

$y.cosA ( sin^2A + cos^2A) = sinA.cosA$

$y.cosA. (1) = sinA.cosA$

$y = sinA$

$x = sinA. cosA/sinA$

$x = cosA$

$L.H.S = x^2 + y^2$

$= sin^2A + cos^2A$  (identity 1)

$= 1$

Hence L.H.S = R.H.S

Ans 4.

$LHS = [ cos^2 A/ cos^2B + cos^2 A / sin^2B ] . cos^2B$

$= cos^2A [ 1/cos^2B + 1/sin^2B ]. cos^2B$

$= cos^2A[ (sin^2B + cos^2B) / (cos^2B.sin^2B) ]. cos^2B$

$= cos^2A. 1. cos^2B / cos^2B.sin^2B$

$= cos^2A / sin^2B$

$= n^2$

$= RHS$

Ans 5.

$LHS = ( m^2 - n^2 )^2$

$= [( tanA + sinA )^2 - ( tanA - sinA )^2 ]^2$

$= [ tan^2A + sin^2A + 2.sinA.tanA - ( tan^2A + sin^2A - 2.sinA.tanA ) ]^2$

$= [ tan^2A + sin^2A + 2.sinA.tanA - tan^2A - sin^2A + 2.sinA.tanA ) ]^2$

$= [ 4.sinA.tanA ]^2$

$= 16.tan^2A.sin^2A$

$= 16.tan^2A.(1 - cos^2A )$

$= 16.[tan^2A - tan^2A. cos^2A ]$

$= 16.[tan^2A - ( sin^2A / cos^2A ). cos^2A ]$

$= 16.[tan^2A - sin^2A )$

$= 16.( tanA + sinA ).( tanA - sinA )$

$= 16.m.n$

$= RHS$

Ans 6.

$LHS = ( p^2 +1 ).cosA$

$= [ ( cosecA + cotA )^2 +1 ].cosA$

$= [ cosec^2A + cot^2A + 2.cosecA.cotA +1 ].cosA$

$= [ cosec^2A + cosec^2A + 2.cosecA.cotA ].cosA$   ( identity 2 )

$= [ 2.cosec^2A + 2.cosecA.cotA ].cosA$

$= [ 2.cosecA( cosecA + cotA ].cosA$

$= [ 2.cosA/sinA ( cosecA + cotA ]$

$= [ 2.cotA ( cosecA + cotA ]$

$= 2.cotA . cosecA + 2.cot^2A$

$= 2.cotA . cosecA + .cot^2A + cot^2A$

$= 2.cotA . cosecA + .cot^2A + cosec^2A - 1$

$= ( cosecA + .cotA )^2 - 1$

$= p^2 - 1$

$= RHS$

Ans 7.

$cosA = 2m/(m^2 + 1)$

$sin^2A = 1 - cos^2A$

$sin^2A = 1 - [ 2m/(m^2 + 1 ) ]^2$

$sin^2A = [ (m^2 + 1)^2 - 4m^2 ] / ( m^2 + 1 )^2$

$sin^2A = [ (m^2 + 1 +2m) ( m^2 + 1 -2m ) ] / ( m^2 + 1 )^2$

$sin^2A = [ (m + 1 )^2 ( m - 1 )^2 ] / ( m^2 + 1 )^2$

$sinA =$$[ (m + 1 )^2 ( m - 1 )^2 ] / ( m^2 + 1 )^2$

$sinA = ( m^2 - 1 ) / ( m^2 + 1 )$

$tanA = sinA / cosA$

$tanA = [(m^2 -1) / (m^2 + 1 ) ] / [ 2m / (m^2 + 1) ]$

$tanA = (m^2 - 1) / 2m$

$cosecA = (m^2 + 1) / (m^2 - 1 )$

$cotA = 2m / (m^2 - 1)$

Ans 8.

given tanA = p/q

LHS = (psinA – qcosA ) / (psinA + qcosA)

dividing both numerator and denominator by qcosA

$= [ p.sinA/q.cosA - 1 ] / [ p.sinA / q.cosA + 1 ]$

$= [ p.tanA/q - 1 ] / [ p.tanA/ q + 1 ]$

$= [ p^2/q^2 - 1 ] / [ p^2/ q^2 + 1 ]$

$= [ (p^2-q^2) / q^2 ] / [ (p^2 + q^2) / q^2 ]$

$= (p^2-q^2) / (p^2 + q^2)$

$= RHS$

hence proved.

Ans 9.

Given tan B = a/b

= (a sinB + bcosB) / (a sinB – bcosB)

dividing both numerator and denominator by bcosB

$= (a/b .tanB + 1) / (a/b tan B - 1)$

$= (a^2/b^2 + 1) / (a^2/b^2 - 1)$

$= (a^2 + b^2 ) / (a^2 - b^2 )$

Ans 10.

given x = a Cos³A  y = b Sin³A

$LHS = [ x / a ] ^2/3 + [ y / b ] ^ 2/3$

$= [ acos^3A/ a ] ^2/3 + [ bsin^3A / b ] ^ 2/3$

$= cos ^2A + sin ^ 2A$     ( identity 1)

$= 1$

## CBSE_class X time bound sample paper 4 marks Trigonometry

Students must ensure that below questions must be attempted in the limited time of 40 mins.

Else Answer sheet will not be corrected.  This is the time bound test to ensure your understanding

about the chapter and thorough revision.

Q 1. If 3x = Cosec A and 3/x = Cot A, find the value of  $3[x^2 - 1/x^2]$.

Q 2. If  6x = Sec B and 6/x = tan B , find the value of $9[x^2 - 1/ x^2]$.

Q 3. If $x.sin^3 A + y. cos^3 A = Sin A. Cos A$ and x.sin A = y.cos A show that

$x^2 + y^2 = 1$.

Q 4. If cos A / Cos B = m , Cos A / Sin B =  n show that

$(m^2 + n^2 ) cos^2 B = n^2$

Q 5. If tan A + Sin A = m  and tan A – sin A = n prove that

$(m^2 -n^2)^2 = 16 mn$

Q 6. If cosec A + cot A = p , prove that $(p^2 + 1)cos A = p^2 - 1$

Q 7. Find the value of the other trigonometric ratios if cos A = $2 m/ (m^2 +1)$

Q 8. If tan A = p/q , show that

$(p sin A - q cos A) / (p sin A + q cos A) = (p^2 - q^2 )/ (p^2 + q ^2)$

Q 9. If tan B = a/b , find the value of

( a Sin B + b cos B)/ (a sin B – b cos B)

Q 10. if $x = a. cos^3 A , y = b. sin ^3 A$

Prove that  $[ x / a ]^2/3 + [ y/b ]^2/3 = 1$

Note : Ensure methods to be written clearly. Marks given for method shown also.

## Oh ! the loud voices

I hate my parents when they give me advice,

Especially when they do it more

Than thrice.

I get it , I get it

I get the point

Now I know more than

My favorite food joint

The advice comes in like swords

Of fire.

You aren’t  liar

I don’t want their voice

To go higher,

Silence right now

Is only the thing I  desire.

But they will never stop,

The day will never come,

A perfect human they want

You to become

Then you realize your

The advice is not to make

Eyes drool

No, not to make you look like

A fool .

Its because they have love

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A perfect human they want you to be.

Written by

Ashwin

Drawings by

The smile

There is a Smile,

Just a far a mile

I  want  to try ,

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The people who cry

That to make our nation

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Smile should  be

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Do your  Best to get

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Written by

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Class V

Euroschool.

## How to utilize your summer vacation to maximum.

Summer vacation is around the corner for all school students from CBSE , ICSE and state board.  Generally if somebody doesn’t plan that itself is plan  which gets executed without you knowing it. So its better to know the plan before and execute it later than its instant execution without aware of planning . So here we list down some of the best plans students can decide to do this summer and after vacation they are looking forward for the next best future learning

1. Make habit of getting up early in morning by 7 am or before. Utilize 1 hour time for your body exercise. It can be anything which person likes. May be playing their favorite games also.

2. Make sure you learn something new this summer apart from what you learnt in school. This can happen only when you know what you want to learn It can be something new. Like new programming language C++ or java . Python,    Or cooking / singing /drawing/ painting / drama / dancing / if you haven’t been public speaker practice extempore and do public speaking .

3. Play your outdoor games hard. Every day make sure you play your sports.

4. Learn 1 song every week. It doesn’t mean you need to do singing. If you like singing well its great. If not still remember it , it challenges your mind to know something new.

5. Talk to your parents for 30 mins every day.

6, Eat food prepared by mom every day at home. During these holidays no goals should be made for eating junk food.

7. Have fruits at night most of the time after dinner.

8. Play online game fortnite only for 1 hour. It might be difficult but yes its needs to be time bound else it will eat whole day.

9. Challenge yourself to read  15 min your favorite book every night.

10. Make sure you study for 1 hour your school text books and it must be productive. Its difficult but can be manageable once you put mind and soul in it.

Wish you wonderful summer holidays and Have fun 🙂

Who has to develop reading habit ?
Why some body has to read books ?
Am done with reading all kinds of books why I should read books?
When we are in the childhood days we learn everyday new things by going to school, learning something new each day and discussing with parents and this is driven by automatic process.
Our mind enables us to learn new words , new ideas , new topics and new ways to work.
As when education gets completed, do people have the habit of reading , I suppose no
unless some body is in the professional field where you need to learn new programming language new project etc.
the problem with the grown ups is that they don’t read at all. They are in the complacent zone.
They have studied every thing they don’t need to know any further.

It seems like soil saying I don’t need to be fertile any more I have already grown paddy and wheat in my field.
When nothing is grown on the land , what automatically grows it just weeds. Natures law of proving every time. When soil is not utilized continuously for crop rotation and left barren for long time, it attracts
growth of weeds automatically.

We are all humans. Many times we have come across very creative ideas when we are shifted to all together different thing after being busy in same routine work. New idea emerges when we read books. How to develop habit of reading books right from childhood. Today in the current scenario there is no limit to how much you can see youtube , online movies, all www. so on.. Even 24hrs is too less to browse through the information which is available online and watching TV. How to induce habit of reading books ?There is no hard work in doing this, lets start with 15 mins of reading daily. which book to pick up ? let parents decide for the children. It can be story book, animation, fiction , non fiction, mythology etc.. Once habit of reading is cultivated in the children at the formative stage. It brings out lot of creative skills to them. They communicate new ideas. show respect, mingle well and do good in studies sports extra curricular activities. They start thinking in terms of doing something which they like to do and how they can excel and bring about wonderful change in the them, and hence happy home to wonderful society to awesome state to great country.

Thanks

www.Tuitionrocks.com