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Chapter 26 Ellipse Exercise Ex. 26.1

Question 1

Solution 1

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 4

Solution 4

Question 5(i)

Solution 5(i)

Question 5(ii)

Solution 5(ii)

Question 5(iii)

Solution 5(iii)

Question 5(iv)

Solution 5(iv)

Question 5(v)

Solution 5(v)

Question 5(vi)

Solution 5(vi)

Question 5(vii)

Solution 5(vii)

Question 5(viii)

Solution 5(viii)

Question 5(ix)

Solution 5(ix)

Question 5(x)

Solution 5(x)

Question 5(xi)

Solution 5(xi)

Question 5(xii)

Solution 5(xii)

Question 5(xiii)

Solution 5(xiii)

Question 6

Solution 6

Question 7

Solution 7

Question 8(i)

Solution 8(i)

Question 8(ii)

Solution 8(ii)

Question 9(i)

Solution 9(i)

Question 9(ii)

Solution 9(ii)

Question 10(i)

Solution 10(i)

Question 10(ii)

Solution 10(ii)

Question 10(iii)

Solution 10(iii)

Question 10(iv)

Solution 10(iv)

Question 10(v)

Solution 10(v)

Question 10(vi)

Solution 10(vi)

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 3(v)

Find the eccentricity, coordinates of foci, length of the latus-rectum of the ellipse 9x² + 25y² = 225Solution 3(v)

Question 18

A rod of length 12m moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the, which is 3cm from the end in contract with x-axis.Solution 18

Question 19

Find the equation of the set of all points whose distances from (0, 4) are of their distances from the line y = 9.Solution 19

Question 20

Find the equation of the set of all points whose distances from (0, 4) are 2/3 of their distances from the line y = 9.Solution 20

Chapter 13 Complex Numbers Exercise Ex. 13.1

Question 1 (i)

Solution 1 (i)

Question 1 (ii)

Solution 1 (ii)

Question 1 (iii)

Solution 1 (iii)

Question 1 (iv)

Solution 1 (iv)

Question 1 (v)

Solution 1 (v)

Question 1 (vi)

Solution 1 (vi)

Question 1 (vii)

Solution 1 (vii)

Question 1 (viii)

Solution 1 (viii)

Question 2

Solution 2

Question 3 (i)

Solution 3 (i)

Question 3 (ii)

Solution 3 (ii)

Question 3 (iii)

Solution 3 (iii)

Question 3 (iv)

Solution 3 (iv)

Question 3 (v)

Solution 3 (v)

Question 3 (vi)

Solution 3 (vi)

Question 3(vii)

Find the values of the following expressions:

(1 + i)⁶ + (1 – i)³Solution 3(vii)

Chapter 13 Complex Numbers Exercise Ex. 13.2

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

What is the smallest positive integer n for which

(1+i)²ⁿ = (1-i)²ⁿ?Solution 24

Question 1 (i)

Solution 1 (i)

Question 1 (ii)

Solution 1 (ii)

Question 1 (iii)

Solution 1 (iii)

Question 1 (iv)

Solution 1 (iv)

Question 1 (v)

Solution 1 (v)

Question 1 (vi)

Solution 1 (vi)

Question 1 (vii)

Solution 1 (vii)

Question 1 (viii)

Solution 1 (viii)

Question 1 (ix)

Solution 1 (ix)

Question 1 (x)

Solution 1 (x)

Question 1 (xi)

Solution 1 (xi)

Question 1 (xii)

Solution 1 (xii)

Question 2 (i)

Solution 2 (i)

Question 2 (ii)

Solution 2 (ii)

Question 2 (iii)

Solution 2 (iii)

Question 2 (iv)

Solution 2 (iv)

Question 3 (i)

Solution 3 (i)

Question 3 (ii)

Solution 3 (ii)

Question 3 (iii)

Solution 3 (iii)

Question 3 (iv)

Solution 3 (iv)

Question 3 (v)

Solution 3 (v)

Question 3 (vi)

Solution 3 (vi)

Question 4 (i)

Solution 4 (i)

Question 4 (ii)

Solution 4 (ii)

Question 4 (iii)

Solution 4 (iii)

Question 4 (iv)

Solution 4 (iv)

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 16 (i)

Solution 16 (i)

Question 16 (ii)

Solution 16 (ii)

Question 16 (iii)

Solution 16 (iii)

Question 16 (iv)

Solution 16 (iv)

Question 16(v)

Evaluate the following :

Solution 16(v)

Question 25

If z₁, z₂, z₃ are complex numbers such that |z₁| = |z₂| = |z₃| =  = 1, then find the value of |z₁ + z₂ + z₃|.Solution 25

Question 26

Find the number of solutions of z² + |z|² = 0Solution 26

Chapter 13 Complex Numbers Exercise Ex. 13.3

Question 1 (i)

Solution 1 (i)

Question 1 (ii)

Solution 1 (ii)

Question 1 (iii)

Solution 1 (iii)

Question 1 (iv)

Solution 1 (iv)

Question 1 (v)

Solution 1 (v)

Question 1 (vi)

Solution 1 (vi)

Question 1 (vii)

Solution 1 (vii)

Question 1 (viii)

Solution 1 (viii)

Question 1 (ix)

Find the square root of the following complex numbers: -iSolution 1 (ix)

Chapter 13 Complex Numbers Exercise Ex. 13.4

Question 1 (i)

Solution 1 (i)

Question 1 (ii)

Solution 1 (ii)

Question 1 (iii)

Solution 1 (iii)

Question 1 (iv)

Solution 1 (iv)

Question 1 (v)

Solution 1 (v)

Question 1 (vi)

Solution 1 (vi)

Question 1 (vii)

Solution 1 (vii)

Question 1 (viii)

Solution 1 (viii)

Question 2

Write (i²⁵)³ in polar form.Solution 2

Question 3(i)

Express the following complex numbers in the form

 r(cos q + I sin q): 1 + i tan a Solution 3(i)

Question 3(ii)

Express the following complex numbers in the form

 r(cos q + I sin q): tan a – iSolution 3(ii)

Question 3(iii)

Express the following complex numbers in the form

r (cos q + I sin q): 1 – sin a + i cos a Solution 3(iii)

Question 3(iv)

Express the following complex numbers in the form

 r (cos q + I sin q):Solution 3(iv)

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Chapter 33 Probability Exercise Ex. 33.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

An experiment consists of tossing a coin and then tossing it second time if head occurs. If a tail occurs on the first toss, then a die is tossed once. Find the sample space.Solution 10

In this experiment, a coin is tossed and if the outcome is tail then a die is tossed once.

Otherwise, the coin is tossed again.

The possible outcome for coin is either head or tail.

The possible outcome for die is 1,2,3,4,5,6.

If the outcome for the coin is tail then sample space is S1={(T,1),(T,2),(T,3),(T,4),(T,5),(T,6)}

If the outcome is head then the sample space is S2={(H,H),(H,T)}

Therefore the required sample space is S={(T,1),(T,2),(T,3),(T,4),(T,5),(T,6),(H,H),(H,T)}Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

In a random sampling, three items are selected so it could be any of the following:

a) All defective or

b) All non-defective or

c) Combination of defective and non defective.

Sample space associated with this experiment is

S={DDD, NDN, DND, DNN, NDD, DDN, NND, NNN}Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

In this experiment, a die is rolled. If the outcome is 6 then experiment is over. Otherwise, die will be rolled again and again.

Chapter 33 Probability Exercise Ex. 33.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

A card is picked up from a deck of 52 playing cards.

(i)  What is the sample space of the experiment?

(ii) What is the event that the chosen card is back faced card?Solution 9

Chapter 33 Probability Exercise Ex. 33.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

\

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 9

Solution 9

Question 10

Two unbiased dice are thrown. Find the probability that the total of the numbers on the dice is greater than 10.Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 8

A bag contains 8 red and 5 white balls. Three balls are drawn at random. Find the probability that:

(i) All the three balls are white

(ii) All the three balls are red

(iii) One ball is red and two balls are white.Solution 8

Chapter 33 Probability Exercise Ex. 33.4

Question 1(a)

Solution 1(a)

Question 1(b)

Solution 1(b)

Question 1(c)

Solution 1(c)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Suppose an integer from 1 through 1000 is chosen at random, find the probability that the integer is a multiple of 2 or a multiple of 9.Solution 25

Question 26

In a large metropolitan area, the probabilities are 0.87, 0.36, 0.30 that a family (randomly chosen for a sample survey) owns a colour television set, a black and white television set, or both kinds of sets. What is the probability that a family owns either any one or both kinds of sets?Solution 26

Question 27

Solution 27

Question 28

Solution 28

Chapter 32 Statistics Exercise Ex. 32.1

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 2(v)

Calculate the mean deviation about the mean of the following data:

(v) 57, 64, 43, 67, 49, 59, 44, 47, 61, 59Solution 2(v)

Question 4

Solution 4

Question 5(i)

Solution 5(i)

Question 5(ii)

Solution 5(ii)

Question 5(iii)

Solution 5(iii)

Question 6

Solution 6

Question 3

Calculate the mean deviation on the following income groups of five and seven members from their medians:

I Income in Rs.	II income in Rs.
4000	3800
4200	4000
4400	4200
4600	4400
4800	4600
	4800
	5800

Solution 3

Note: Answer given in the book is incorrect.

Chapter 32 Statistics Exercise Ex. 32.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 4(iii)

Solution 4(iii)

Question 4(iv)

Solution 4(iv)

Question 5(ii)

Solution 5(ii)

Question 5(iii)

Solution 5(iii)

Question 4(v)

Find the mean deviation from the mean for the following data:

Size:	1 3 5 7 9 11 13 15
Frequency:	3 3 4 14 7 4 3 4

Solution 4(v)

Question 5(i)

Find the mean deviation from the median for the following data:

xi	15 21 27 30
fi	3 5 6 7

Solution 5(i)

Note: Answer given in the book is incorrect.

Chapter 32 Statistics Exercise Ex. 32.3

Question 1

Solution 1

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 7

Solution 7

Question 8

Solution 8

Question 6

Calculate mean deviation about median age for the age distribution of 100 persons given below:

Age:	16-20	21-25	26-30	31-35	36-40	41-45	45-50	50-55
Number of persons	5	6	12	14	26	12	16	9

Solution 6

Chapter 32 – Statistics Exercise Ex. 32.4

Question 1(i)

Solution 1(i)

Question 1(ii)

Find the mean, variance and standard deviation for the data:

6, 7, 10, 12, 13, 4, 8, 12.Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Show that the two formulae for the standard deviation of ungrouped data

Solution 11

Chapter 32 Statistics Exercise Ex. 32.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

(i)

(ii)

Solution 3

Question 4

(ii)

Solution 4

Chapter 32 Statistics Exercise Ex. 32.6

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Calculate the Mean, median and Standard Deviation of the following distribution:
Solution 5

Question 7

Solution 7

Question 8

Mean and standard deviation of 100 observation were found to be 40 and 10 respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27 respectively, find the correct standard deviation.Solution 8

Question 9

While calculating the mean and variance of 10 readings , a student wrongly used the reading of 52 for the correct reading 25. He obtained the mean and variance as 45 and 16 respectively. Find the correct mean and the variance.Solution 9

Question 6

Find the mean and variance of frequency distribution given below:

xi:	1x <3	3 x<5	5x<7	7x<9
Fi:	6	4	5	1

Solution 6

Note: Answer given in the book is incorrect. Question 10

Calculate mean, variance and standard deviation of the following frequency distribution:

Class:	0-10	10-20	20-30	30-40	40-50	50-60
Frequency:	11	29	18	4	5	3

Solution 10

Chapter 32 Statistics Exercise Ex. 32.7

Question 1

Solution 1

We observe that the average monthly wages in both firms is same i.e. Rs. 2500. Therefore the plant with greater variance will have greater variability. Thus plant B has greater variability in individual wages. Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Life of bulbs product by two factories A and B are given below:

Length of life(in hours):Factory A:	550-650 650-750 750-850 850-950 950-1050
(Number of bulbs)Factor B:	10   22 52 20 16
(Number of bulbs)	8  60 24 15 12

The bulbs of which factory are more consistent from the point of view of length of life?Solution 11

Question 12

Following are mark obtained, out of 100, by two students Ravi and Hashina in 10 tests:

Ravi:	25 50 45 30 70 42 36 48 35 60
Hashina:	10 70 50 20 95 55 42 60 48 80

Who is more intelligent and who is more consistent?Solution 12

Chapter 31 Mathematical Reasoning Exercise Ex. 31.1

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 1(ix)

Solution 1(ix)

It is not a statement.

The sentence “This sentence is a statement.” cannot  be assigned a truth value of either true or false, because either assignment contradicts the sense of the sentence.Question 1(x)

Solution 1(x)

Question 1(xi)

Solution 1(xi)

Question 1(xii)

Solution 1(xii)

Question 1(xiii)

Solution 1(xiii)

Question 1(xiv)

Solution 1(xiv)

Question 1(xv)

Solution 1(xv)

Question 1(xvi)

Solution 1(xvi)

Question 2

Solution 2

Chapter 31 Mathematical Reasoning Exercise Ex. 31.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Chapter 31 Mathematical Reasoning Exercise Ex. 31.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Chapter 31 Mathematical Reasoning Exercise Ex. 31.4

Question 1

Solution 1

Question 2

Solution 2

Chapter 31 Mathematical Reasoning Exercise Ex. 31.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Chapter 31 Mathematical Reasoning Exercise Ex. 31.6

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Chapter 30 Derivatives Exercise Ex. 30.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7 (i)

Solution 7 (i)

Question 7 (ii)

Solution 7 (ii)

Question 7 (iii)

Solution 7 (iii)

Question 7(iv)

Solution 7(iv)

Chapter 30 Derivatives Exercise Ex. 30.2

Question 1(i)

Solution 1(i)

Question 1 (ii)

Solution 1 (ii)

Question 1 (iii)

Solution 1 (iii)

Question 1 (iv)

Solution 1 (iv)

Question 1 (v)

Solution 1 (v)

Question 1 (vi)

Solution 1 (vi)

Question 1 (vii)

Solution 1 (vii)

Question 1 (viii)

Solution 1 (viii)

Question 1 (ix)

Solution 1 (ix)

Question 1 (x)

Solution 1 (x)

Question 1 (xi)

Solution 1 (xi)

Question 1 (xii)

Solution 1 (xii)

Question 1 (xiii)

Solution 1 (xiii)

Question 1 (xiv)

Solution 1 (xiv)

Question 2 (i)

Solution 2 (i)

Question 2 (ii)

Solution 2 (ii)

Question 2 (iii)

Solution 2 (iii)

Question 2 (iv)

Solution 2 (iv)

Question 2(ix)

Solution 2(ix)

Question 2(x)

Solution 2(x)

Question 2(xi)

Solution 2(xi)

Question 3 (vii)

Solution 3 (vii)

Question 3 (viii)

Solution 3 (viii)

Question 3 (ix)

Solution 3 (ix)

Question 3 (x)

Solution 3 (x)

Question 3 (xi)

Solution 3 (xi)

Question 3 (xii)

Solution 3 (xii)

Question 3 (i)

Solution 3 (i)

Question 3 (ii)

Solution 3 (ii)

Question 3 (iii)

Solution 3 (iii)

Question 3 (iv)

Solution 3 (iv)

Question 3 (v)

Solution 3 (v)

Question 3 (vi)

Solution 3 (vi)

Question 4 (i)

Solution 4 (i)

Question 4 (ii)

Solution 4 (ii)

Question 4 (iii)

Solution 4 (iii)

Question 4 (iv)

Solution 4 (iv)

Question 5(i)

Solution 5(i)

Question 5 (ii)

Solution 5 (ii)

Question 5 (iii)

Solution 5 (iii)

Question 5(iv)

Solution 5(iv)

Question 6 (i)

Solution 6 (i)

Question 6 (ii)

Solution 6 (ii)

Question 6(iii)

Solution 6(iii)

Question 6(iv)

Solution 6(iv)

Question 1(xv)

Solution 1(xv)

Question 2(v)

Differentiate -x using first principles.Solution 2(v)

Question 2(vi)

Differentiate (-x)^-1 using first principles.Solution 2(vi)

Question 2(vii)

Differentiate sin(x + 1) using first principles.Solution 2(vii)

Question 2(viii)

Differentiate cos using first principles.Solution 2(viii)

Chapter 30 Derivatives Exercise Ex. 30.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Defferentiate f (x) = Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Chapter 30 Derivatives Exercise Ex. 30.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Differentiate the following functions with respect to x:

Solution 25

Question 26

Differentiate the following functions with respect to x:

(ax + b)ⁿ (cx + d)^mSolution 26

Question 27

Differentiate in two ways, using product rule and otherwise, the function (1 + 2 tan x) (5 + 4 cos x). Verify that the answer are the same.Solution 27

Question 28(i)

Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.

(3x² + 2)²Solution 28(i)

Question 28(ii)

Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.

(x + 2)(x + 3)Solution 28(ii)

Question 28(iii)

Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.

(3 sec x – 4 cosec x) (-2 sin x + 5 cos x)Solution 28(iii)

Chapter 30 Derivatives Exercise Ex. 30.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Differentiate the following functions with respect to x:

Solution 29

Question 30

Differentiate the following functions with respect to x:

Solution 30

Chapter 29 Limits Exercise Ex. 29.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13(i)

Solution 13(i)

Question 13(ii)

Solution 13(ii)

Question 13(iii)

Solution 13(iii)

Question 13(iv)

Solution 13(iv)

Question 13(v)

Solution 13(v)

Question 13(vi)

Solution 13(vi)

Question 13(vii)

Solution 13(vii)

Question 13(viii)

Solution 13(viii)

Question 13(ix)

Solution 13(ix)

Question 13(x)

Solution 13(x)

Question 13(xi)

Solution 13(xi)

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 6

Solution 6

Question 22

Solution 22

Chapter 29 Limits Exercise Ex. 29.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Chapter 29 Limits Exercise Ex. 29.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Evaluate the following limits:  Solution 34

Chapter 29 Limits Exercise Ex. 29.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 26

Solution 26

Question 27

Evaluate

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 25

Evaluate the following limits:  Solution 25

Chapter 29 Limits Exercise Ex. 29.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Chapter 29 Limits Exercise Ex. 29.6

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Evaluate the following limits: Solution 25

Question 26

Evaluate the following limits: Solution 26

Chapter 29 Limits Exercise Ex. 29.7

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

Question 55

Solution 55

Question 56

Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59

Question 34

 Evaluate the following limits: Solution 34

Question 51

 Evaluate the following limits: Solution 51

Question 60

Evaluate the following limits: Solution 60

Question 61

Evaluate the following limits: Solution 61

Question 62

Evaluate the following limits: Solution 62

Question 63

Evaluate the following limits: Solution 63

Chapter 29 Limits Exercise Ex. 29.8

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Evaluate Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Evaluate the following limits: Solution 38

Chapter 29 Limits Exercise Ex. 29.9

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Chapter 29 Limits Exercise Ex. 29.10

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Evaluate the following limits: Solution 41

Question 42

Solution 42

Question 43

Evaluate the following limits: Solution 43

Chapter 29 Limits Exercise Ex. 29.11

Question 1

Evaluate the following limits: Solution 1

Question 2

Evaluate the following limits: Solution 2

Question 3

Evaluate the following limits: Solution 3

Question 4

Evaluate the following limits: Solution 4

Question 5

Evaluate the following limits: Solution 5

Question 6

Evaluate the following limits: Solution 6

Question 7

Evaluate the following limits: Solution 7

Question 8

Evaluate the following limits: Solution 8

Question 9

Evaluate the following limits: Solution 9

Question 10

Evaluate the following limits: Solution 10

Chapter 28 Introduction to 3-D coordinate geometry Exercise Ex. 28.1

Question 1(i)

Name the octants in which the following points lie:

 (i) (5, 2, 3)Solution 1(i)

All are positive, so octant is XOYZQuestion 1(ii)

Name the octants in which the following points lie:

(ii) (-5, 4, 3)Solution 1(ii)

X is negative and rest are positive, so octant is X^‘OYZQuestion 1(iii)

Name the octants in which the following points lie:

(4, -3, 5)Solution 1(iii)

Y is negative and rest are positive, so octant is XOY^‘ZQuestion 1(iv)

Name the octants in which the following points lie:

(7, 4, -3)Solution 1(iv)

Z is negative and rest are positive, so octant is XOYZ^‘Question 1(v)

Name the octants in which the following points lie:

(-5, -4, 7)Solution 1(v)

X and Y are negative and Z is positive, so octant is X’OY’ZQuestion 1(vi)

Name the octants in which the following points lie:

(-5, -3, -2)Solution 1(vi)

All are negative, so octant is X^‘OY^‘Z^‘Question 1(vii)

Name the octants in which the following points lie:

(2, -5, -7)Solution 1(vii)

Y and Z are negative, so octant is XOY^‘Z^‘Question 1(viii)

Name the octants in which the following points lie:

(-7, 2, -5)Solution 1(viii)

X and Z are negative, so octant is X^‘OYZ^‘Question 2(i)

Find the image of :

(-2, 3, 4) in the yz-plane Solution 2(i)

YZ plane is x-axis, so sign of x will be changed. So answer is (2, 3, 4)Question 2(ii)

Find the image of :

(-5, 4, -3) in the xz-plane. Solution 2(ii)

XZ plane is y-axis, so sign of y will be changed. So answer is (-5, -4, -3)Question 2(iii)

Find the image of :

(5, 2, -7) in the xy-plane Solution 2(iii)

XY-plane is z-axis, so sign of Z will change. So answer is (5, 2, 7)Question 2(iv)

Find the image of :

(-5, 0, 3) in the xz-plane Solution 2(iv)

XZ plane is y-axis, so sign of Y will change, So answer is (-5, 0, 3)Question 2(v)

Find the image of :

(-4, 0, 0) in the xy-plane Solution 2(v)

XY plane is Z-axis, so sign of Z will change So answer is (-4, 0, 0)Question 3

A cube of side 5 has one vertex at the point (1, 0, -1), and the three edges from this vertex are, respectively, parallel to the negative x and y axes and positive z-axis. Find the value coordinates of the other vertices of the cube. Solution 3

Vertices of cube are

(1, 0, -1) (1, 0, 4) (1, -5, -1)

(1, -5, 4) (-4, 0, -1) (-4, -5, -4)

(-4, -5, -1) (4, 0, 4) (1, 0, 4)Question 4

Planes are drawn parallel to the coordinate planes through the points (3, 0, -1) and (-2, 5, 4). Find the lengths of the edges of the parallelepiped so formed. Solution 4

3-(-2)=5, |0-5|=5, |-1-4|=5

5, 5, 5 are lengths of edgesQuestion 5

Planes are drawn through the points (5, 0, 2) and (3, -2, 5) parallel to the coordinate planes. Find the lengths of the edges of the rectangular parallelepiped so formed. Solution 5

5-3=2, 0-(-2)=2, 5-2=3

2, 2, 3 are lengths of edgesQuestion 6

Find the distances of the point p(-4, 3, 5) from the coordinate axes. Solution 6

Question 7

The coordinate of a point are (3, -2, 5). Write down the coordinates of seven points such that the absolute values of their coordinates are the same as those of the coordinates of the given point. Solution 7

(-3, -2, -5) (-3, -2, 5) (3, -2, -5) (-3, 2, -5) (3, 2, 5)

(3, 2, -5) (-3, 2, 5)

Chapter 28 Introduction to 3-D coordinate geometry Exercise Ex. 28.2

Question 1

Solution 1

Question 2

Solution 2

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 4(iii)

Solution 4(iii)

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20(i)

Solution 20(i)

Question 20(ii)

Solution 20(ii)

Question 20(iii)

Solution 20(iii)

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 20(iv)

Verify the following

(5, -1, 1), (7, -4, 7), (1, -6, 10) and (-1, -3, 4) are the vertices of a rhombus.Solution 20(iv)

Question 24

Find the equation of the set of the points P such that its distances from the points A(3, 4, -5) and B(-2, 1, 4) are equal.Solution 24

Chapter 28 Introduction to 3-D coordinate geometry Exercise Ex. 28.3

Question 1

The vertices of the triangle are A(5, 4, 6), B(1, -1, 3) and C(4, 3, 2). The internal bisector of angle A meets BC at D. Find the coordinates of D and the Length AD. Solution 1

Question 2

A point C with z-coordinate 8 lies on the line segment joining the points A(2, -3, 4) and B(8, 0, 10). Find its coordinates. Solution 2

Question 3

Show that three points A(2, 3, 4), B(-1, 2, -3) and C(-4, 1, -10) are collinear and find the ratio in which C divides AB. Solution 3

Question 4

Find the ratio in which the line joining (2, 4, 5) and (3, 5, 4) is divided by the yz-plane. Solution 4

Question 5

Find the ratio in which the line segment joining the points (2, -1, 3) and (-1, 2, 1) is divided by the plane

x+ y + z = 5. Solution 5

Question 6

If the points A(3, 2, -4), B(9, 8, -10) and C(5, 4, -6) are collinear, find the ratio in which C divides AB. Solution 6

Question 7

The mid-points of the sides of a triangle ABC are given by (-2, 3, 5), (4, -1, 7) and (6, 5, 3). Find the coordinates of A, B and C. Solution 7

Question 8

A(1, 2, 3), B(0, 4, 1), C(-1, -1, -3) are the vertices of a triangle ABC. Find the point in which the bisector of the angle  meets BC. Solution 8

Question 9

Find the ratio in which the sphere x²+y² +z² = 504 divides the line joining the points (12, -4, 8) and (27, -9, 18). Solution 9

Question 10

Show that the plane ax + by + cz + d = 0 divides the line joining the points (x₁,y₁,z₁) and (x₂,y₂,z₂) in the ratio –Solution 10

Question 11

Find the centroid of a triangle, mid-points of whose sides are (1, 2, -3), (3, 0, 1) and (-1, 1, -4). Solution 11

Question 12

The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinate of A and B are (3, -5, 7) and (-1, 7, -6) respectively, find the coordinates of the point C. Solution 12

Question 13

Find the coordinates of the points which tisect the line segment joining the points P(4, 2, -6) and Q(10, -16, 6). Solution 13

Question 14

Using section formula, show that the points A(2, -3, 4), B(-1, 2, 1) and C(0, 1/3, 2) are collinear. Solution 14

Question 15

Given that P(3, 2, -4), Q(5, 4, -6) and R(9, 8, -10) are collinear. Find the ratio in which Q divides PR. Solution 15

Question 16

Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, -8) is divided by the yz-plane. Solution 16

Chapter 27 Hyperbola Exercise Ex. 27.1

Question 1

Solution 1

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 2(v)

Solution 2(v)

Question 2(vi)

Solution 2(vi)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 3(v)

Solution 3(v)

Question 4

Solution 4

Question 5(i)

Solution 5(i)

Question 5(ii)

Solution 5(ii)

Question 5(iii)

Solution 5(iii)

Question 6(i)

Solution 6(i)

Question 6(ii)

Solution 6(ii)

Question 6(iii)

Solution 6(iii)

Question 7(i)

Solution 7(i)

Question 7(ii)

Solution 7(ii)

Question 7(iii)

Solution 7(iii)

Question 7(iv)

Solution 7(iv)

Question 8

Solution 8

Question 9(i)

Solution 9(i)

Question 9(ii)

Solution 9(ii)

Question 11(i)

Solution 11(i)

Question 11(ii)

Solution 11(ii)

Question 11(iii)

Solution 11(iii)

Question 11(iv)

Solution 11(iv)

Question 11(v)

Solution 11(v)

Question 11(vi)

Solution 11(vi)

Question 11(vii)

Solution 11(vii)

Question 11(viii)

Solution 11(viii)

Question 11(ix)

Solution 11(ix)

Question 7(v)

Find the equation of the hyperbola whose vertices are at (± 6, 0) and one of the directrices is x = 4.Solution 7(v)

Question 7(vi)

Find the equation of the hyperbola whose

foci at (± 2, 0) and eccentricity is 3/2Solution 7(vi)

Question 10

Solution 10

Question 11(x)

Solution 11(x)

Question 12

Solution 12

Question 13

Show that the set of all points such that the difference of their distance from (4, 0) and (-4, 0) is always equal to 2 represents a hyperbola.Solution 13

Chapter 28 Introduction to 3-D coordinate geometry Exercise Ex. 28.1

Question 1(i)

Name the octants in which the following points lie:

 (i) (5, 2, 3)Solution 1(i)

All are positive, so octant is XOYZQuestion 1(ii)

Name the octants in which the following points lie:

(ii) (-5, 4, 3)Solution 1(ii)

X is negative and rest are positive, so octant is X^‘OYZQuestion 1(iii)

Name the octants in which the following points lie:

(4, -3, 5)Solution 1(iii)

Y is negative and rest are positive, so octant is XOY^‘ZQuestion 1(iv)

Name the octants in which the following points lie:

(7, 4, -3)Solution 1(iv)

Z is negative and rest are positive, so octant is XOYZ^‘Question 1(v)

Name the octants in which the following points lie:

(-5, -4, 7)Solution 1(v)

X and Y are negative and Z is positive, so octant is X’OY’ZQuestion 1(vi)

Name the octants in which the following points lie:

(-5, -3, -2)Solution 1(vi)

All are negative, so octant is X^‘OY^‘Z^‘Question 1(vii)

Name the octants in which the following points lie:

(2, -5, -7)Solution 1(vii)

Y and Z are negative, so octant is XOY^‘Z^‘Question 1(viii)

Name the octants in which the following points lie:

(-7, 2, -5)Solution 1(viii)

X and Z are negative, so octant is X^‘OYZ^‘Question 2(i)

Find the image of :

(-2, 3, 4) in the yz-plane Solution 2(i)

YZ plane is x-axis, so sign of x will be changed. So answer is (2, 3, 4)Question 2(ii)

Find the image of :

(-5, 4, -3) in the xz-plane. Solution 2(ii)

XZ plane is y-axis, so sign of y will be changed. So answer is (-5, -4, -3)Question 2(iii)

Find the image of :

(5, 2, -7) in the xy-plane Solution 2(iii)

XY-plane is z-axis, so sign of Z will change. So answer is (5, 2, 7)Question 2(iv)

Find the image of :

(-5, 0, 3) in the xz-plane Solution 2(iv)

XZ plane is y-axis, so sign of Y will change, So answer is (-5, 0, 3)Question 2(v)

Find the image of :

(-4, 0, 0) in the xy-plane Solution 2(v)

XY plane is Z-axis, so sign of Z will change So answer is (-4, 0, 0)Question 3

A cube of side 5 has one vertex at the point (1, 0, -1), and the three edges from this vertex are, respectively, parallel to the negative x and y axes and positive z-axis. Find the value coordinates of the other vertices of the cube. Solution 3

Vertices of cube are

(1, 0, -1) (1, 0, 4) (1, -5, -1)

(1, -5, 4) (-4, 0, -1) (-4, -5, -4)

(-4, -5, -1) (4, 0, 4) (1, 0, 4)Question 4

Planes are drawn parallel to the coordinate planes through the points (3, 0, -1) and (-2, 5, 4). Find the lengths of the edges of the parallelepiped so formed. Solution 4

3-(-2)=5, |0-5|=5, |-1-4|=5

5, 5, 5 are lengths of edgesQuestion 5

Planes are drawn through the points (5, 0, 2) and (3, -2, 5) parallel to the coordinate planes. Find the lengths of the edges of the rectangular parallelepiped so formed. Solution 5

5-3=2, 0-(-2)=2, 5-2=3

2, 2, 3 are lengths of edgesQuestion 6

Find the distances of the point p(-4, 3, 5) from the coordinate axes. Solution 6

Question 7

The coordinate of a point are (3, -2, 5). Write down the coordinates of seven points such that the absolute values of their coordinates are the same as those of the coordinates of the given point. Solution 7

(-3, -2, -5) (-3, -2, 5) (3, -2, -5) (-3, 2, -5) (3, 2, 5)

(3, 2, -5) (-3, 2, 5)

Chapter 28 Introduction to 3-D coordinate geometry Exercise Ex. 28.2

Question 1

Solution 1

Question 2

Solution 2

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 4(iii)

Solution 4(iii)

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20(i)

Solution 20(i)

Question 20(ii)

Solution 20(ii)

Question 20(iii)

Solution 20(iii)

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 20(iv)

Verify the following

(5, -1, 1), (7, -4, 7), (1, -6, 10) and (-1, -3, 4) are the vertices of a rhombus.Solution 20(iv)

Question 24

Find the equation of the set of the points P such that its distances from the points A(3, 4, -5) and B(-2, 1, 4) are equal.Solution 24

Chapter 28 Introduction to 3-D coordinate geometry Exercise Ex. 28.3

Question 1

The vertices of the triangle are A(5, 4, 6), B(1, -1, 3) and C(4, 3, 2). The internal bisector of angle A meets BC at D. Find the coordinates of D and the Length AD. Solution 1

Question 2

A point C with z-coordinate 8 lies on the line segment joining the points A(2, -3, 4) and B(8, 0, 10). Find its coordinates. Solution 2

Question 3

Show that three points A(2, 3, 4), B(-1, 2, -3) and C(-4, 1, -10) are collinear and find the ratio in which C divides AB. Solution 3

Question 4

Find the ratio in which the line joining (2, 4, 5) and (3, 5, 4) is divided by the yz-plane. Solution 4

Question 5

Find the ratio in which the line segment joining the points (2, -1, 3) and (-1, 2, 1) is divided by the plane

x+ y + z = 5. Solution 5

Question 6

If the points A(3, 2, -4), B(9, 8, -10) and C(5, 4, -6) are collinear, find the ratio in which C divides AB. Solution 6

Question 7

The mid-points of the sides of a triangle ABC are given by (-2, 3, 5), (4, -1, 7) and (6, 5, 3). Find the coordinates of A, B and C. Solution 7

Question 8

A(1, 2, 3), B(0, 4, 1), C(-1, -1, -3) are the vertices of a triangle ABC. Find the point in which the bisector of the angle  meets BC. Solution 8

Question 9

Find the ratio in which the sphere x²+y² +z² = 504 divides the line joining the points (12, -4, 8) and (27, -9, 18). Solution 9

Question 10

Show that the plane ax + by + cz + d = 0 divides the line joining the points (x₁,y₁,z₁) and (x₂,y₂,z₂) in the ratio –Solution 10

Question 11

Find the centroid of a triangle, mid-points of whose sides are (1, 2, -3), (3, 0, 1) and (-1, 1, -4). Solution 11

Question 12

The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinate of A and B are (3, -5, 7) and (-1, 7, -6) respectively, find the coordinates of the point C. Solution 12

Question 13

Find the coordinates of the points which tisect the line segment joining the points P(4, 2, -6) and Q(10, -16, 6). Solution 13

Question 14

Using section formula, show that the points A(2, -3, 4), B(-1, 2, 1) and C(0, 1/3, 2) are collinear. Solution 14

Question 15

Given that P(3, 2, -4), Q(5, 4, -6) and R(9, 8, -10) are collinear. Find the ratio in which Q divides PR. Solution 15

Question 16

Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, -8) is divided by the yz-plane. Solution 16