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#### 4. Study the diagram. The line  is perpendicular to line             (a) Is CE = EG?            (b) Does PE bisect CG?            (c) Identify any two line segments for which PE is the perpendicular bisector.            (d) Are these true?                    (i) AC > FG                    (ii) CD = GH                    (iii) BC < EH.

(a) CE = 5 - 3 = 2 units

EG = 7 - 5 = 2 units

Therefore CE = EG.

(b) CE = EG therefore PE bisects CG.

(c) PE is the perpendicular bisector for line segments DF and BH

(d) (i) AC = 3 - 1 = 2 units

FG = 7 - 6 = 1 unit

Therefore AC > FG

True

(ii) CD = 4 - 3 = 1 unit

GH = 8 - 7 = 1 unit

Therefore CD = GH

True

(iii) BC = 3 - 2 = 1 unit

EH = 8 - 5 = 3 units

Therefore BC < EH

True

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#### 3. There are two set-squares in your box. What are the measures of the angles that are formed at their corners? Do they have any angle measure that is common?

The angles of the two set quares are

(i) 90o, 60o and 30o

(ii) 90o, 45o and 45o

Yes they have the common angle measure 90o

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#### 2.  Let  be the perpendicular to the line segment  . Let  and  intersect in the point A. What is the measure of  ?

PQ and XY intersect at A

Therefore

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#### 1.  Which of the following are models for perpendicular lines :            (a) The adjacent edges of a table top.            (b) The lines of a railway track.            (c) The line segments forming the letter ‘L’.            (d) The letter V.

(a) The adjacent edges of a table top are models for perpendicular lines.

(b) The lines of a railway track are not models for perpendicular lines as they are parallel to each other.

(c) The line segments forming the letter ‘L’ are models for perpendicular lines.

(d) The line segments forming the letter ‘V’ are models for perpendicular lines.

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#### Q18         If u, v and w are functions of x, then show that                                                                                                                   in two ways - first by repeated application of product rule, second by logarithmic differentiation.

It is given that u, v and w are the functions of x
Let
Now, we differentiate using product rule w.r.t x
First, take
Now,
-(i)
Now, again by the product rule

Put this in equation (i)
we get,

Hence, by product rule we proved it

Now, by taking the log
Again take
Now, take log on both sides

Now, differentiate w.r.t. x
we get,

Hence, we proved it by taking the log

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#### Q17 (3)   Differentiate in three ways mentioned below:               (iii)  by logarithmic differentiation.                  Do they all give the same answer?

Given function is

Now, take log on both the sides

Now, differentiate w.r.t. x
we get,

Therefore, the answer is
And yes they all give the same answer

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#### Q17 (2)   Differentiate in three ways mentioned below:                 (ii) by expanding the product to obtain a single polynomial.

Given function is

Multiply both to  obtain a single higher degree polynomial

Now, differentiate w.r.t. x
we get,

Therefore, the answer is

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#### Q17 (1)   Differentiate in three ways mentioned below:                 (i) by using product rule

Given function  is

Now, we need to differentiate using the product rule

Therefore, the answer is

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#### Q16  Find the derivative of the function given by and hence find            f ' (1)

Given function is

Take log on both sides

NOW, differentiate w.r.t. x

Therefore,
Now, the vale of    is

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#### Q15  Find dy/dx of the functions given in Exercises 12 to 15.

Given function is

Now, take  take log on both the sides

Now, differentiate w.r.t  x

By taking similar terms on same side
We get,

Therefore, the answer is

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