1.

TOPIC “QUADRATIC EQUATIONS “

KEY POINTS:

1. The polynomial of degree two is called quadratic polynomial and equation corresponding to a

Quadratic polynomial P(x) is called a quadratic equation in variable x.

Thus p(x) =ax2 + bx+c =0 , a≠ 0 , a, b, c ∈ R.

2. Zero of a quadratic polynomial:

The value of x for which the polynomial becomes zero is called zero of a polynomial.

Ex. 1 is zero of a polynomial x2 -2x +1, because it became zero at x = 1.

3. Solution of a quadratic equation by factorisation method:

Roots of a quadratic equation ax2 + bx +c = 0 can be found by factorization method or middle

term split.

4. Discriminant method: The expression b2 -4ac is called discriminant of the equation

ax2 +bx +c =0 and it is usually denoted by “D”. D = b2 -4ac

5. Nature of roots of ax2 +bx +c=0

i) If D >0, then roots are real and unequal.

ii) D =0, Then the equation has real and equal roots.

Iii) D< 0, then the equation has no real roots.

iv) If D>0, and D is a perfect square, then roots are rational and unequal.

v) If D>0 and D is not a perfect square, then roots are irrational.

6. Roots of quadratic Equation:

Let the quadratic equation be ax2 +bx +c =0, a≠ 0 and 𝛼 and 𝛽 are roots

−𝑏−√𝑏2−4𝑎𝑐 −𝑏+√𝑏2−4𝑎𝑐
𝛼= , 𝛽=
2𝑎 2𝑎

−𝑏 𝐶
7. Sum of Roots : 𝛼 + 𝛽 = Product of Roots : 𝛼 × 𝛽 = 𝑎
𝑎

8. Forming quadratic equation ,when the roots 𝛼 and 𝛽 are given by :

X2 – (𝛼 + 𝛽) x +𝛼. 𝛽 =0

9. Method of solving Word Problems:

10. Form the word problems into quadratic equations and solve them

2
 (2 Marks Questions)

Q.1 If ½ is a root of the equation x2 +px -5/4 =0, then find the value of p.

Q.2 Check whether x= -1 is a solution of equation 4x2 -3x -1 =0.

Q.3 If D >0, Then write the roots of a quadratic equation ax2 +bx+c =0

Q.4 Find the Discriminant of x2 +5x +5 =0.

Q.5 Find the sum of roots of a quadratic equation x2 +4x -32 =0

Q.6 Find the product of the roots of the quadratic equation 2x2 +7x -4 =0

Q.7 Find the value of K for which the equation 9 x2 +2Kx+1 =0 have real roots.

Q.8 Find the value of K if the equation x2 -2(K+1) x +K2 = 0 has equal roots.

Q.9 Represent the situation in the form of Quadratic equation:

“The product of Rohan’s age (in years) 5 years ago with his age 9 years later is 15.

Q.10 Find the roots of x2 -3x -10 =0

(3 Marks each)

Q.11 What is the nature of roots of the quadratic equation 2x2 -√5 x +1 =0 ?

Q.12 Find the numerical difference of the roots of equation x2 -7x-18 = 0

Q.13 If the discriminant of the equation 6x2 –bx +2=0 is 1, then find the value of b.

Q.14 The product of two consecutive odd numbers is 483.Find the numbers.
1
Q.15 Solve : x - 𝑋 = 3 ( x≠ 0)

Q.16 The hypotenuse of right angled triangle is 6 meters. more than twice the shortest side.

If the third side is 2 meters. less than the hypotenuse, then find all the sides of the triangle.

Q.17 The sum of the reciprocals of Anjali’s age 3 years ago and 5 years from now is 1/3.

Find the present age of Anjali.

Q.18 Check whether: (x +2)3 = 2x (x2 -1) is a quadratic equation or not.

Q.19 Solve for x : √2𝑥 + 9 + x = 13 .

Q.20 Find the roots of quadratic equation 16 x2 -24 x-1 =0 by using the quadratic formula

3
 (4 Marks each)

Q.21 A passenger train takes 3 Hour less for a journey of 360 km. If its speed is increased by
10km/ h from its usual speed. Find its usual speed.

Q.22 The speed of boat in still water is 15 km/h. It can go 30 km upstream and return downstream
to the original point in 4 hour and 30 minutes. Find the speed of stream.
1
Q.23 Two pipes running together can fill a small tank in 33 minutes. If one pipe takes 3 minutes
more than the other to fill it, then find the time in which each pipe would fill the tank.

Q.24 Solve for:

1 1 2
(𝑥−1)(𝑥−2)
+ (𝑥−2)(𝑥−3)
= ,where x≠ 1,2,3
3

Q.25 If the equation (1+m2) x2 +(2mc) x +(c2 –a2) = 0 has equal roots, then prove that

c2 = a2 (1+m2)

CASE –STUDY QUESTIONS (4 MARKS)

Q.26 Riya has a field with a flowerbed and grass land. The grassland is in the shape of rectangle
while flowerbed is in the shape of square. The length of the grassland is found to be 3 meters more
than twice the length of the flowerbed. Total area of the whole land is 1260 m2.

(i) If the length of the flowerbed is x meters, then what is the total length of the field? (2 Marks)

(ii) What is the area of grassland? (2 Marks)

4
Q. 27 Nidhi and Riya are very close friends. Nidhi’s parents have a Maruti Alto. Riya ‘s parents
have a Toyota. Both the families decided to go for a picnic to Somnath Temple in Gujarat by their
own car. Nidhi’s car travels x km/h, while Riya’s car travels 5km/h more than Nidhi’s car. Nidhi’s
car took 4 hours more than Riya’s car in covering 400 km.

(i) What will be the distance covered by Riya’s car in two hours? How much time took Riya
to travel 400 km? (2 Marks)
(ii) Write the quadratic equation describe the speed of Nidhi’s car. What is the speed of
Nidhi’s car? (2 Marks)

VALUE BASED QUESTIONS (4-MARKS EACH)

Q.28 If the price of petrol is increased by Rs.7 per litre, a person has to buy 1 litre less petrol for Rs.
1740.Find the original price of the petrol at that time.

a) Why do you think the price of petrol is increasing day by day?

b) What should we do to save petrol?

Q.29 Ramesh wants to design a rectangular park of perimeter 80 meters and area 400 m2.for
jogging and walk for the people of colony. Is it possible to design the park? If so, find the length and
breadth of the park. Which value of Ramesh is depicted here?

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 SOLUTIONS/ANSWER

(2 marks each)

1. PUT the value of x=1/2 in the given equation

1 1 5
( 2 )2 + 2 p - 4 =0

P=2

2. Check,if x= -1 ,then put the value in the given equation

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

4+3 -1≠ 0 , NO

−𝑏−√𝑏2−4𝑎𝑐 −𝑏+√𝑏2−4𝑎𝑐
3. IF D>0 , THEN , 𝛼= , 𝛽=
2𝑎 2𝑎

4. compare the given equation to ax2 +bx+c =0 a=1, b =5, c =5

D = b2 -4ac, D = 5

5 Sum of roots = -4

6. Product of roots = -2

7. real roots , D ≥0 ,K≥3 OR K≤ −3

8. Compare the given equation from ax2 +bx +c =0 a=1, b= -2(k+1), c = k2

D = 0, then k = -1/2

9. Let the Rahman’s age be x years.

5 years ago = (x- 5), 9 years later = (x+9)

product of his age = (x-5) (x+9) =15, x2 +4x -60 =0

10 x = ( -2, 5)

(3 marks each)

11. Compare to ax2 +bx +c =0 ,a=2 b= - √5 ,c = 1

D = b2 -4ac , D = (- √5 ) 2 -4 X2X1 , D = 25-8=17 , D>0 Real and distinct(unequal)

12. Solve the equation x= 9 and -2

Numerical difference = 9 – (-2) = 9+2 = 11

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13. Compare to ax2 +bx +c =0 a =6, b = -b, c = 2

D = b2 -4ac = 1

D = (-b)2 -4 ×6× 2 =1,

b2-48 =1 , b= 7,-7

14. Let the first odd number be x and consecutive odd number be (x+2)

x.(x+2) =483, x2 +2x -483 =0

x = 21 ,23

1 𝑥2−1 3−√13 3+√13

15. x - =3 , =3 , x2 – 1 =3x , x2 -3x -1 =0 , x = ( , )
𝑥 𝑥 2 2

16. Let the length of the shortest side be x meters.

According to question hypotenuse = (2x+6)

Third side = (2x +6-2) =2x +4

By the Pythagoras theorem (Hypo)2 = B2 + P2

(2x+6)2 = x2 + (2x+4)2

Then equation: x2 -8x -20 =0

x = 10, -2, but the length cannot be negative.

So x= 10 m.

Hypo =26, B =10, P=24

17. Let the Anjali’s age be x years

Anjali’s age 3 years ago = x-3

Anjali’s age 5 years from now = x+5

1 1 1
According to question + 𝑥+5 =
𝑥−3 3

x = 7, -3. but age cannot be negative. so x=7 years

18. After solution we get -x3 +6x2 +14x +8 =0.

It is not in the form of ax2 +bx +c =0

So it is not a quadratic equation.

19. √2𝑥 + 9 + x = 13 .

√2𝑥 + 9 = 13 – x. squiring both sides

7
 (2x +9) = (13-x)2

x2 -28 x +160 =0

x= 20, 8 but x=20 does not satisfy the equation.

So x = 8

3+√10 3−√10
20. x= ,
4 4

(4 marks each)

21. Let the usual speed of the train = x km/h

Total distance =360 km.

Time = Distance /speed.

300
So the time taken by the train = h
𝑥

If the speed is increased by 10 km/h, then the new speed of the train = (x+10) km/h
300
Time taken by the train =𝑥+10

300 300
According to the question = +3
𝑥 𝑥+10

300 300
- =3
𝑥 𝑥+10

After solution x = - 40 km and 30 km, but speed cannot be negative.

x = 30 km.

22. Let speed of the stream = x km/h

Given, speed of boat in still water = 15 km/h

Speed of boat upstream = (15-x) km/h

Speed of boat downstream = (15+x) km/h

According to the question

30 30 1
+15+𝑥 = 4 2 ( TIME = DISTANCE / SPEED )
15−𝑥

X2 -225x +200 =0, x = 5, -5

But speed cannot be negative. So x = 5 km/h.

23 Let faster pipe takes x minutes to fill the tank.

Then. Slower pipe will take (x+3) minutes to fill the tank.
8
 Since, portion of the tank filled by the faster pipe in 1 minute = 1/x
1
And portion of the tank filled by the slower pipe in 1 minute = 𝑥+3

1 40
IN 31 3 minutes =(13 minutes)

Both pipe will fill to tank together

40 1 1
( 𝑥 + 𝑥+3 ) =1
13

After calculation we get 13 x2 -41x -120 =0

−24
x = 5 or x = , but time cannot be negative.
13

So, x = 5 minutes.

Faster pipe takes 5 minutes and slower pipe takes (x+3) =8 minutes to fill the tank.

24. Solution of the given equation

𝑥−3+𝑥−1 2
(𝑥−1)(𝑥−2)(𝑥−3)
= 3

(x-1) (x-3) = 3

solution of x = 0 or x = 4

25. Compare the given equation to A x2 +Bx +C =0

A (1+m2), B =2mc and C= (c2 –a2)

Since the given equation has equal roots.

So, D = b2-4ac =0, then prove it.

CASE –STUDY SOLUTIONS)

26. (i) (3x+3) m.

(ii) 860m2.

27. (i) 2(x+5)km , 16 hours

(ii) (c) x2 +5x -500= 0, 20 km /h

SOLUTIONS OF VALUE BASED QUESTIONS

28 . Let the original price of the petrol be Rs. x per litre.

1740
The amount of petrol that can be purchased = 𝑥

9
 According to question
1740 1740
- =1 ,
𝑥 𝑥+2

1740 (x+2 – x) = x(x+2)

x2 +2x – 3480 =0

x = 58, (-60)-rejected

Original cost of petrol was Rs.58 per litres.

a) Petrol is a natural resource which is depleting day by day. So due to more demand and less

supply, its price is increasing.

b) We should use more of public transport and substitute petrol with CNG or other

renewable resources.

29. Let the Length = L and Breadth = B of the Park.

So, Area = L X B = 400 m2, So L = 400/B

PERIMETER= 2 (L + B) = 80 m.

So, L +B =40, put the value of L = 400/B

We get B2 -40B +400 = 0

B=20 m, L=20 m.

VALUE –Jogging and Morning walk are beneficial or our mental and physical health.

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 2. TOPIC – ARITHMETIC PROGRESSION

(2 MARKS QUESTIONS)

Q1 The fee charged from a student every month by a school for the whole session, when the
monthly fee is Rs 400, Is in the given situation do the list of numbers involved form an AP?
If yes then find total fee for the year.
Q 2. Given a=5, d=3,𝑎𝑛 =50, find Sn
Q3 Find 21st term of an AP whose first two terms are − 3 and 4.
Q4 Which terms of an AP 21,42,63 … is 210, Solve.
Q5 Find 4th term from the end of the AP: −11, −,8,− 5 ,… 49.
Q6 Verify whether the given series 2, 22 ,23,24 form an AP. If yes Find common difference.
Q7 If the first term of an AP is − 5 and d=2, then find the sum of first six term.
Q8 Find 𝑎30 − 𝑎20 in the given series −3, −7, −11, …
Q9 Is the given series √3, √12, √27, √48,…form an AP. If yes find common difference.
Q 10 In an AP a=3.5, d=0, n=101 find 𝑎𝑛 .
1
Q 11 Find the total number of terms the series 7 +102 +14 + ⋯ +84.

Q 12 Verify that 𝑎 + 𝑏, {(𝑎 + 1) + 𝑏}, {(𝑎 + 1) + (𝑏 + 1)},…is an AP.

Find Common difference in case of AP.
Q 13 If 18, 𝑎, b, −3 are in AP, then find 𝑎 + b
Q 14 Write down the first four terms of an AP when a=− 5, d −3
Q 15 Is 0 a term of the AP: 31, 28, 25, ….? Justify your answer.

(3 MARKS QUESTIONS)

Q 16 How many two-digits numbers are divisible by 3.

Q 17 Given an =28, S n =144 and there are total 9 terms. Find a
Q 18 Two APs have the same common difference. The difference between their 100th term is
100, Find the difference between their 1000th terms.
Q 19 Find the 31st term of an A.P. whose 11th term is 38 and 16th term is 73.
Q 20 How many terms of the AP 9,17,25 … must be taken to give a sum of 636?
Q 21 Find the sum of the first 15 multiples of 8.
Q 22 Find the sum of the first 40 positive integers divisible by 6.
Q 23 Find the sum of the odd numbers between 0 and 50.
Q 24 If the 9th term of an AP is zero, prove that its 29th term is twice its 19th term.

11
Q25 Determine the AP whose5th term is 19 and the difference of the 8th term from the 13th term is
20.

(4 MARKS QUESTIONS)
Q 26 In a potato race a bucket is placed at the starting point, which is 5m from the first potato and
the other potatoes are placed 3m apart in a straight line. There are ten potatoes in the line. A
competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in
the bucket, runs back to pick up the next potato, runs to the bucket to drop it in and she
continues in the same way until all the potatoes are in the bucket, what is the total distance
the competitor has to run?
Q 27 Ramkali saves Rs 5 in the first week, of a year and increased her weekly savings by Rs 1.75.
If in the nth week her weekly savings became Rs 20.75, find n.
Q 28 In an AP, if S5 + S7 = 167 and S10 = 235, then find the AP, where s, denotes the sum of its
first n terms
Q.29. The digits of a positive number of three digits are in A.P. and their sum is 15. The number
obtained by reversing the digits is 594 less than the original number. Find the number.
Q 30 Find the sum of all multiples of 7 lying between 500 and 900.
Q 31 A thief runs with a uniform speed of 100 m/minute. After one minute a policeman runs after
the thief to catch him. He goes with a speed of 100 m/minute in the first minute and increases
his speed by 10 m/minute every succeeding minute. After how many minutes the policeman
will catch the thief.
Q 32 Divide 56 in four parts in A.P. such that the ratio of the product of their extremes (1st and
4th) to the product of means (2nd and 3rd) is 5: 6.
Q 33 In a school, students decided to plant trees in and around the school to reduce air pollution. It
was decided that the number of trees, that each section of each class will plant, will be double
of the class in which they are studying. If there are 1 to 12 classes in the school and each class
has two sections, find how many trees were planted by the students.
Q 34 A sum of Rs 1600 is to be used to give ten cash prizes to students of a school for their overall
academic performance. If each prize is Rs 20 less than its preceding prize, find the value of
each of the prizes.
Q 35 If the sum of the first n terms of an AP is 4n − n2, what is the first term (that is S1)?
What is the sum of first two terms? What is the second term? Similarly find the 3rd,
the 10th and the nth terms.

12
 CASE STUDY QUESTIONS
Q 36 A road roller (sometimes called a roller-compactor, or just roller) is a compactor-type
engineering vehicle used to compact soil, gravel, concrete, or asphalt in the construction of
roads and foundations. Similar rollers are used also at landfills or in agriculture. Road rollers
are frequently referred to as steamrollers, regardless of their method of propulsion. RCB
Machine Pvt Ltd started making road roller 10 year ago. Company increased its production
uniformly by fixed number every year. The company produces 800 rollers in the 6th year and
1130 rollers in the 9th year.

On the basis of the above information, answer any four of the following questions:
(i) Find the company’s production in first year.
(ii) In which year the company’s production was 1350 rollers?
(a) 5th (b) 6th (c) 11th (d) 9
Q 37 Aditya is celebrating his birthday. He invited his friends. He bought a packet of
toffees/candies which contains 120 candies. He arranges the candies such that in the first row
there are 3 candies, in second there are 5 candies, in third there are 7 candies and so on.

On the basis of the above information, answer any four of the following questions:
(i) Find the total number of rows of candies.
(ii) Find the difference in number of candies placed in 7th and 3rd rows.

13
Q 38 In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato,
and the other potatoes are placed 3 m apart in a straight line. There are 12 potatoes in the line
(see Fig.).

A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in
the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she
continues in the same way until all the potatoes are in the bucket.
(i) Find the total distance covered by the competitor after placed the second potato in the
bucket?
(ii) Calculate the total distance covered by the competitor?
Q 39.

(i) Find the minimum number of days he needs to practice till his goal is achieved
(ii) If nth term of an AP is given by an = 2n + 3 then find common difference of AP

Q 40

14
(i). Find the amount paid by him in the 30th installment.
(ii). Find the total installments paid by him.

ANSWERS AND HINTS OF ARITHMETIC PROGRESSION

(1) yes, Rs.4800
𝑛
(2) 440 [find n=16 by last term = a+(n-1)d, 50 = 5+(n-1)3, then apply 𝑆𝑛 = 2 {𝑎 + 𝑙} ]

(3) 137 [a=-3 and d = 4-(-3) = 7 then find 21st term = -3+(21-1)7]
(4) 10 [a=21, d=21 then use formula of last term 210 = 21+(n-1)21 & div.by 21]
(5) 40 [rewrite AP in reverse order a = 49, d = -11+8 = -3
then 4th term from the end = 49+(4-1)(-3) = 40
(6) No because common difference is not same
6
(7) 0 [a=-5, d=2 n=6 then use 𝑠6 = 2 {2(−5) + (6 − 1)2} = 3(0)

(8) -40, [a=-3, d=-4

then 𝑎30 − 𝑎20 = {−3 + 29 × (−4)} − {−3 + 19 × (−4)} = {10} × (−4)]
(9) yes, [because common difference is same =√3 = 2√3 − √3 = 3√3 − 2√3 = ⋯]
(10) 3.5 [ a=3.5 and because common difference is zero so AP will not increase or
decrease i.e. nth term=0]
7 7
(11) 23 [ a=7 d= 2use last term l=a+(n-1)d, 84=7+(n-1)2 ]

(12) Yes, [ because common difference is same=1 ]

(13) 15, [common difference= a-18 = b-a = -3-b now take a-18 = -3-b gives a+b =15]
(14) −5, −8, −11, −14 [use a, a+d, a+2d, a+3d where a=-5 & d=-3]
(15) No, [∴a=31, d=-3, Let nth term=0, then 31+(n-1)×(-3)=0
gives n=34/3 which is not a Positive Integer.]
(16) 30, [ because a = 12 d = 3 and l = 99 then use formula of last term l = a + (n − 1)d]
15
 𝑛 9
(17) 4, [ use 𝑆𝑛 = 2 {𝑎 + (𝑎 + (𝑛 − 1)𝑑)}, 144 = 2 {𝑎 + 28}]

(18) 100, [ for first AP: a100 = a1+99d & a1000 = a1+999d and for Second AP: a100 = a2+99d &
a1000 = a2+999d then (a1+999d) − (a2+999d) = a1−a2 & then (a1+99d) − (a2+99d) = 100
hence a1−a2 = 100]
(19) 178, [a+10d=38 & a+15d=73, then a=-32 & d=7 so 31st term = -32+30(7)]
𝑛 𝑛
(20) 12, [𝑆𝑛 = 2 {2𝑎 + (𝑛 − 1)𝑑}, 636 = 2 {2(9) + (𝑛 − 1)(8)},

636 = 𝑛{5 + 4𝑛} 𝑔𝑖𝑣𝑒𝑠 𝑛 = 12]

𝑛(𝑛+1) 15(15+1)
(21) 960, [8×{sum of 15 natural Nos,}, 8× { }, 8× { },8{120}]
2 2

(22) 4920, [Sum of first 40 positive integers which are divisible by 6 = 6,12,18,24,…. 40
40(40+1)
terms = 6× {𝑠𝑢𝑚 𝑜𝑓 𝑓𝑖𝑟𝑠𝑡 40 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑁𝑜𝑠. } = 6 × { } =4920]
2

(23) 625, [𝑢𝑠𝑒 𝑆𝑢𝑚 𝑜𝑓 𝑜𝑑𝑑 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 = 𝑛2 ],

(24) do yourself
(25) 3,7,11,15 [a+4d=19 & {(a+12d)-(a+7d)}=20, d=4 & a= 3 then use a, a+d, a+2d, a+3d]
10
(26) 370 m [s = 2{5+8+11+…up to 10 terms} = 2× {2 × 5 + (10 − 1) × 3} = 370 m]
2

(27) 10 [nth term = a+(n-1)d, 20.75=5+(n-1)X1.75, n = 10]

(28) A.P. is 1, 6, 11…[𝑠5 + 𝑠7 = 167 𝑚𝑒𝑎𝑛𝑠 12𝑎 + 31𝑑 = 167
& 𝑠10 = 235 𝑚𝑒𝑎𝑛𝑠 2𝑎 + 9𝑑 = 47 𝑡ℎ𝑒𝑛 𝑎 = 1 & 𝑑 = 5
(29) 852,
Soln: Let hundred’s place digit = (a – d), ten’s place digit = a, and unit’s place digit = a + d
ATQ, a – d + a + a + d = 15 ⇒ 3a = 15 ⇒ a = 5
Original number = 100(a – d) + 10(a) + 1(a + d) = 100a – 100d + 10a + a + d = 111a – 99d
Reversed number = 1(a – d) + 10a + 100(a + d) = a – d + 10a + 100a + 100d = 111a + 99d
Now, Original no. – Reversed no. = 594
111a – 99d – (111a + 99d) = 594; -198d = 594 ⇒ d = -3
∴ The Original no. = 111a – 99d = 111(5) – 99(-3) = 555 + 297 = 852
(30) 39900 [AP: 504, 511, 518, …,896, apply last term=a+(n-1)d, 896=504+(n-1)X7
57
we get n = 57 then sum of 57 terms = {504 + 896} = 39900]
2

(31) 5 minutes, Let the police catch the thief in n min

As the thief ran 1 min before the police
Time taken by the thief before being caught =(n+1) min
Distance travelled by the thief in (n+1) min =100(n+1) m
Speed of police in 1st min =100m/min
Speed of police in 2nd min=110m/min Speed of police in3rd min =120m/min. and so on

16
 ∴100, 110, 120, ... this forms an AP
𝑛
Total distance travelled by the police in n min = (2×100+(n−1)10)
2

On catching the thief by police, distance travelled by thief= distance travelled by the police
𝑛
⇒100(n+1)= 2(2×100+(n−1)10)

⇒100n+100=100n+n(n−1)5 ⇒100=n(n−1)5 ⇒n2−n−20=0 ⇒(n−5)(n+4)=0

⇒n−5=0,n+4=0 ⇒n=5 OR n=−4(but this is not possible) so, n=5
Time taken by the policeman to catch the thief=5min
(32) 8,12,16,20
Hint: Take four parts of an AP as a-3d, a-d, a+d, a+3d and their sum is 56
Then find a= 14 and d= ±2
(33) 312, [2× {2 × (1 + 2 + 3 + ⋯ + 12)}]
(34) 250,230,210,190,170,150,130,110,90,70
(35) The second term is 1, The 3rd, 10th, and nth terms are −1, −15 and (5 − 2n) respectively
(36) (i) Production in 6th year = 800 9th year = 1130, means 𝑎6 = 800 & 𝑎9 = 1130
Means a+5d=800 & a+8d=1130 we get a=250 and d=110 First year production = 250
(ii) 11th year, [apply nth term= a+(n-1)d, 1350=250+(n-1)110 we get n=11]
(37) (i) There is an AP: 3,5,7, … a=3 & d=2 so apply let there are n rows
𝑛 𝑛
sum of n terms=2 {2𝑎 + (𝑛 − 1)𝑑} , 120 = 2 {2 × 3 + (𝑛 − 1)2}

we get 𝑛2 + 2𝑛 − 120 = 0 𝑡ℎ𝑒𝑛 𝑛 = 10 & − 12 So there are 10 rows

(ii) 7th row=3+(7-1)X2=3+12=15 and 3rd row=3+(3-1)X2=7 their diff.= 8
2
(38) (i) 26 m [s = 2{5+8+11+…up to 2 terms} = 2{2 {2 × 5 + (2 − 1) × 3} = 26 m]
12
(ii) 516 m [s = 2{5+8+11+…up to 12 terms} = 2× {2 × 5 + (12 − 1) × 3} = 516 m]
2

(39) (i) a=51 & d=(-2) so nth term=a+(n-1)Xd, since goal is 31 minutes
so 31=51+(n-1)X(-2) hence n =11th day
(ii) common difference= 𝑎𝑛 − 𝑎𝑛−1 = (2𝑛 + 3) − {2(𝑛 − 1) + 3} = 3 − 1 = 2
(40) (i) AP: 1000, 1100, 1200, … so a=1000, d=100 so 30th instalment = 1000+29X100=3900
(ii) Total Amount paid=118000
𝑛 𝑛 𝑛
𝑆𝑛 = 2 {2𝑎 + (𝑛 − 1)𝑑}, 118000 = 2 {2 × 1000 + (𝑛 − 1) × 100} = 2 {1900 + 100𝑛}

We get 100𝑛2 + 1900𝑛 − 236000 = 0 𝑚𝑒𝑎𝑛𝑠 𝑛2 + 19𝑛 − 2360 = 0

𝑛2 + 59𝑛 − 40𝑛 − 2360 = 0 𝑚𝑒𝑎𝑛𝑠 (𝑛 + 59(𝑛 − 40) = 0 𝑠𝑜 𝑛 = 40

17
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 3. TOPIC – Tangents To A Circle
VERY SHORT ANSWER TYPE QUESTIONS(2marks)

Q1. O is the Centre of a circle of radius 8 cm. The tangent at a point A on the circle cuts a line
through O at B such that AB= 15cm. Find the radius of the circle.

Q2. If PT is a tangent at T to a circle whose center is O and OP= 17cm, OT= 8cm, Find the length of
the tangent segment PT.

Q3. If TP and TQ are two tangents to a circle with center O so that ∠POQ = 110°, then, what is the
value of ∠PTQ?
Q4. From a point Q, the length of the tangent to a circle is 24cm and the distance of Q from the
Centre is 26cm.Find the radius of the circle.

Q5. If from an external point B of a circle with Centre O, two tangents BC and BD are drawn such
that ∠DBC = 120°, prove that BC + BD = BO.

Q6. In figure, AB and CD are common tangents to two circles of unequal radii. Prove that
AB = CD.

Q7. If a chord AB subtends an angle of 60°at the Centre of a circle, then find angle between the
tangents at A and B.

Q8. If angle between two tangents drawn from a point ‘P’ to a circle of radius ‘a’ and Centre O is
90◦, then find OP.

18
Q9. Show that the tangent to the circumcircle of an isosceles triangle ABC at A, in which

AB = AC, is parallel to BC.

Q10. A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC

SHORT ANSWER TYPE QUESTION- (3marks)

Q1. If a number of circles touch a given line segment PQ at a point A, then where will the centers of
all circle lie?

Q2. AB is a diameter of a circle and AC is its chord such that<BAC = 30°. If the tangent at C
intersect AB extended at D, then show that BC = BD.

19
Q3. What is the length of the tangent PQ at a point P of a circle of radius 12cm meets a line through
the Centre O at a point Q so that OQ =20 cm.?

Q4. There are two concentric circle with center O of radii 5cm and 3cm. From an external point P,
tangent PA and PB are drawn to these circles. If AP= 12cm, Find the length of BP.

Q5. If PA and PB are tangents from an external point P to a circle with center O. LN touches the
circle at M. Prove that PL + LM = PN + MN.

Q6. From an external point P, tangents PA = PB are drawn to a circle with Centre O. If ∠PAB = 50°,
then find ∠AOB.

Q7. Out of the two concentric circles, the radius of the outer circle is 10 cm and the chord AC of
length 16 cm is a tangent to the inner circle. Find the radius of the inner circle.

Q8. Two tangents PQ and PR are drawn from an external point to a circle with Centre O. Prove that
QORP is a cyclic quadrilateral.

20
Q9. In figure, O is the Centre of a circle of radius 8 cm, T is a point such that 0T= 17 cm and 0T
intersects the circle at E. If AB is the tangent to the circle at E, find the length of AB.

Q10. The tangent at a point C of a circle and a diameter AB when extended intersect at P. If ∠PCA =
120°, Find ∠CBA.

LONG ANSWER TYPE QUESTIONS :(4marks)

Q1. A chord PQ of a circle is parallel to the tangent drawn at a point R of the circle. Prove that R
bisects the arc PRQ.

Q2. If tangent PQ and PR are drawn from an external point P to a circle with Centre O, such that
∠RPQ =30°. A chord RS is drawn parallel to the tangent PQ. Find ∠RQS.

Q3. If tangents PA and PB from a point P to a circle with Centre O are inclined to each other at an
angle of 60°, then find < POA.

Q4. Two tangents TP and TQ are drawn to a circle with Centre O from an external point T. Prove
that ∠ PTQ = 2 ∠ OPQ.

21
Q5Prove that the parallelogram circumscribing a circle is a rhombus.

Q6. If a hexagon ABCDEF circumscribes a circle, prove that

AB + CD + EF = BC + DE + FA.

Q7. Let s denote the semi-perimeter of a triangle ABC in which BC = a, CA = b, AB = c. If a circle

touches the sides BC, CA, AB at D, E, F respectively, prove that BD = s – b.

Q8. From an external point P, two tangents, PA and PB are drawn to a circle with Centre O. At one-
point E on the circle tangent is drawn which intersects PA and PB at C and D, respectively. If PA =
20 cm, find the perimeter of the triangle PCD.

Q9. If AB is a chord of a circle with Centre O, AOC is a diameter and AT is the tangent at A as
shown in figure. Prove that ∠BAT = ∠ACB

22
Q10. Two circles with centers O and O’ of radii 6cm and 8 cm, respectively intersect at two points P
and Q such that OP and O’P are tangents to the two circles. Find the length of the common chord
PQ.

Q11. In a right triangle ABC in which ∠B = 90°, a circle is drawn with AB as diameter intersecting
the hypotenuse AC at P. Prove that the tangent to the circle at P bisects BC.

Q12. Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining
the end points of the arc.

Q13. In figure, the common tangent, AB and CD to two circles with centers O and O’ intersect at E.
Prove that the points O, E, O’ are collinear.

Q14. If an isosceles triangle ABC, in which AB = AC = 6 cm, is inscribed in a circle of radius 9 cm,
find the area of the triangle.

23
Q15 A is a point at a distance 13 cm from the Centre 0 of a circle of radius 5 cm. AP and AQ are the
tangents to the circle at P and Q. If a tangent BC is drawn at a point R lying on the minor arc PQ to
intersect AP at B and AQ at C, find the perimeter of the ∆ABC.

CASE STUDY BASED QUESTIONS

CASE STUDY 1:
A Ferris wheel (or a big wheel in the United Kingdom) is an amusement ride consisting of a rotating
upright wheel with multiple passenger-carrying components (commonly referred to as passenger
cars, cabins, tubs, capsules, gondolas, or pods) attached to the rim in such a way that as the wheel
turns, they are kept upright, usually by gravity.
After taking a ride in Ferris wheel, Aarti came out from the crowd and was observing her friends
who were enjoying the ride. She was curious about the different angles and measures that the wheel
will form. She forms the figure as given below.

1. In
the
given
figure
find ∠ROQ
a) 60°
b) 100 °
c) 150°
d) 90°

2. Find ∠RQP
a) 75°
b) 60°
c) 30°
d) 90°

3. Find ∠RSQ
24
 a) 60°
b) 75°
c) 100 °
d) 30°
4. Find ∠ORP
a) 90°
b) 70°
c) 100°
d) 60°

CASE STUDY 2:

Varun has been selected by his School to design logo for Sports Day T-shirts for students and staff .
The logo design is as given in the figure and he is working on the fonts and different colours
according to the theme. In given figure, a circle with center O is inscribed in a ΔABC, such that it
touches the sides AB, BC and CA at points D, E and F respectively. The
lengths of sides AB, BC and CA are 12 cm, 8 cm and 10 cm respectively.

1. Find the length of AD

a) 7
b) 8
c) 5
d) 9

2. Find the Length of BE

a) 8
b) 5
c) 2
d) 9
3. Find the length of CF
a) 9
b) 5
c) 2
d) 3

25
4. If radius of the circle is 4cm, Find the area of ∆OAB
a) 20
b) 36
c) 24
d) 48
5. Find area of ∆ABC
a) 50
b) 60
c) 100
d) 90

CASE STUDY 3:

There girls Reshma, Salma, Mandeep are playing a game by standing on a circle. Reshma throws a
ball to Salma, Salma to Mandeep, Mandeep to Reshma. The distance between Reshma and Mandeep
is 6m, and between Reshma and Salma is 8m if O is the center of the circle, then

1. Find diameter of the circle

a) 6m b) 8m c) 10m d) 12m

2. Measure of ∠MRS
a)180° b)90° c)100° d) 80°

3. Area of the ∆RMS is

a) 10 m2 b)20cm2 c)24 cm2d)40 cm2

4. length of the longest chord of the circle.

a)6m b) 8m c)10m d)12m

5. The radius of the circle is

a)6m b) 3m c)4m d)5m

………………………………………………………………………………………………….

26
 Circles (Answer key)
SHORT ANSWER TYPE QUESTIONS(2marks)

1. 17cm
2. 15cm
3. 70°
4. 7cm
7.120°
8. a√2
SHORT ANSWERTYPE QUESTIONS (3 marks)

1. Perpendicular line of PQ True

3. 16cm
4. 4√10 𝑐𝑚
6. 100°
7. DO=6cm
9. 48/5cm
10. 60°
LONG ANSWER TYPE QUESTION (4 marks)

2. ∠RQS=75°
3. ∠POA =60°
8. 40cm
10. pq= 9.6cm
14. 8√2 𝑐𝑚2
15. 24cm
CASE BASED QUESTIONS

CASE STUDY 1:
1. c) 150°
2. a) 75°
3. b) 75°
4. a) 90°
CASE STUDY 2:

1. a) 7
2. b) 5
3. d) 3
4. c) 24
5. b) 60
CASE STUDY 3:

1: - c) 10m
2: -b) 90°
3: -c) 24 cm2
4: -d) 10m
5: - 5m
27
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 4.TOPIC- CONSTRUCTIONS

SHORT ANSWER TYPE QUESTION (2 MARKS)

Q.1. In the given figure, A1, A2, A3 --- and B1, B2, B3 ,----- are marked at equal distances. Answer
the following questions.

(i) In what ratio point C divides AB ? [Ans: 8:5]

(ii) If AB = 13cm then find the length of AC. [Ans: 8cm]
Q.2, In the given figure, A1, A2, A3, A4, A5 are marked at equal distances. Answer the following
questions.

(i) In what ratio point C divides AB? [Ans: 3:2]

(ii) If AB = 5cm then find the length of AC. [Ans: 3cm]

LONG ANSWER TYPE QUESTION (3 MARKS)

Q.1. Draw a line segment of length 6 cm. Using compasses and ruler, find a point P on it which
divides it in the ratio 3:1.
Solution:
p

Steps of Construction : 1. Draw AB = 6 cm with the help of scale.

2. Draw any ray AX, making an acute angle with AB.
3. Locate 4 (= 3 + 1) points A1, A2, A3 and A4 on AX so that AA1 = A1 A2 = A2 A3 = A3 A4 .
4. Join BA4 .
5. Through the point A3 (m = 3), draw a line parallel to A3 P (by making an angle equal to ∠
AA4B) at
A3 intersecting AB at the point P. Then, AP:PB = 3 : 1
Q.2. Draw a line segment of length 8 cm and divide it in the ratio 3 :5.Measure the two parts.
Q.3. Draw a line segment of length 5 cm and divide it in the ratio 2:3. Measure the two parts.
28
Q.4. Draw a pair of tangents to a circle of radius 3 cm, which are inclined to each other at an angle of
60°.
Q.5. Draw a circle of radius 4 cm. From a point P, 9 cm away from the centre of the circle, draw two
tangents to the circle. Also, measure the angle between two radii through point of contacts of two
tangents.

Solution:

Steps of construction:

1. A circle, with centre O and radius 4 cm is drawn.

2. A point P is taken, outside the circle at a distance of 9 cm from O.
3. Perpendicular bisector of OP is drawn, meeting OP at L.
4. With L as centre and OL as radius a circle is drawn meeting the given circle at A and B.
5. PA and PB are joined.
6. Then PA and PB are the required tangents to the circle and PA = PB = 6.7 cm (approx.)
Q.6. Draw a circle of radius 3 cm. From a point P, 7 cm away from the centre of the circle, draw two
tangents to the circle. Also, measure the lengths of the tangents.
Q.7. Draw two concentric circles of radii 3 cm and 5 cm.Construct a tangent to smaller circle from a
point on the larger circle. Also measure its length.
Q.8. Draw a pair of tangents to a circle of radius 4 cm which are inclined to each other at an
angle of 60°. Measure the length of the two tangents also.

Q.9. Draw a circle of radius 4cm. Mark a point P on it .Draw a tangents passing through it. Measure
the angle between two tangents at P.

Solution:

Now after measuring, PA and PB comes out to be 4 cm.

Steps of construction of tangents:

1. Take point O. Draw 2 concentric circles of radii 3 cm and 5 cm respectively.

2. Locate point P on the circumference of larger circle.
3. Join OP and bisect it. Let M be mid-point of OP.
4. Taking M as centre and MP as radius, draw an arc intersecting smaller circle at A and B.
5. Join PA and PB. Thus, PA, PB are required tangents

29
 LONG ANSWER TYPE QUESTION (4-MARKS)
Q.1. Draw two tangents to a circle of radius 4 cm from a point P at a distance of 6 cm from its centre.
Measure the angle between two tangents.
Q.2. Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of
tangents to the circle and measure their lengths.
Q.3. Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6
cm and measure its length. Also verify the measurement by actual calculation.
Q.4. Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and
taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the
centre of the other circle.
Q.5. Draw a pair of tangents to a circle of radius 6cm which are inclined to each other at an angle of
600. Also find the length of the tangent.
Q.6. Construct two concentric circles of radii 3cm and 7cm. Draw two tangents to the smaller circle
from a point P which lies on the bigger circle.
Q7. Let ABC be a right triangle in which AB = 6 cm, BC = 8 cm and ∠ B = 90°. BD is the
perpendicular from B on AC. The circle through B, C, D is drawn. Construct the tangents from A to
this circle.
Q8. Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an
angle of 45°. Measure the angle between two radii through point of contact at centre of the circle.

30
 5. TOPIC: SOME APPLICATIONS OF TRIGONOMETRY

(3 MARKS QUESTIONS)

1. In the figure, AB is a 6 m high pole and CD is a ladder inclined at an angle of 60° to the
horizontal and reaches up to a point D of pole. If AD = 2.54 m. Find the length of the ladder.
(Use √3 = 1.73

2. The tops of two towers of height x and y, standing on level ground, subtend angles of 30° and
60° respectively at the centre of the line joining their feet, then find x : y.

3. The angles of depression of two ships from the top of a light house and on the same side of it
are found to be 45° and 30°. If the ships are 200 m apart, find the height of the light house.

4. The angle of elevation of the top of the tower from two points at the distance of 4m and 9m
from the base of the tower and in the same straight line with it are complementary. Find
height of tower?

5. An observer 1.5m tall is 28.5m away from a chimney . The angle of elevation of the top of
the chimney from her eyes is 450. What is the height of the chimney. ?

6. If a tower 30 m high, casts a shadow 10√3 m long on the ground, then what is the angle of
elevation of the sun?

7. A tree is break due to storm and the broken part bends so that the top of the tree touches the
ground making an angle 300 with it . The distance between the foot of the tree to the point
where the top touches the ground is 8m. Find the height of the tree.
8. As observed from the top of a 60 m high light house from the sea-level, the angles of
depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same
side of the light-house, find the distance between the two ships. (Use √3 = 1.732]

9. The shadow of a tower standing on level ground is found to be 40 m longer when the Sun’s
altitude is 30° than when it is 60°. Find the height of the tower.

31
 10. The angle of elevation of the top of a tower from two points distant a and b from its foot are
complementary. Prove that the height of the tower is √ab

11. The angle of elevation of the top of a hill at the foot of a tower is 60° and the angle of
elevation of the top of the tower from the foot of the hill is 30°. If the tower is 50 m high,
what is the height of the hill?

12. Two men on either side of a 75 m high building and in line with base of building observe the
angles of elevation of the top of the building as 30° and 60°. Find the distance between the
two men

13. A kite is flying at a height of 60 m above the ground. The string attached to the kite is
temporarily tied to a point on the ground. The inclination of the string with the ground is 60°.
Find the length of the string, assuming that there is no slack in the string.

14. From a point on a bridge across a river, the angles of depression of the banks on opposite
sides of the river are 30° and 45° respectively. If the bridge is at a height of 3 m from the
banks, find the width of the river.

15. A man standing on the deck of a ship, which is 10 m above the water level, observes the
angle of elevation of the top of a hill as 60° and the angle of depression of the base of the hill
as 30°. Calculate the height of the hill.

Long answer question (4 marks)

1 . The angles of elevation and depression of the top and bottom of a lighthouse from the top of a
building, 60 m high, are 30° and 60° respectively. Find
(i) the difference between the heights of the lighthouse and the building.
(ii) distance between the lighthouse and the building.

2. A vertical tower stands on a horizontal plane and is surmounted by a flagstaff of height 5 m. From
a point on the ground the angles of elevation of the top and bottom of the flagstaff are 60° and 30°
respectively. Find

(1). The height of the tower .

(2) The distance of the point from the tower. (Take √3 = 1.732)

3. The angles of depression of the top and the bottom of a 8 m tall building from the top of a multi-
storied building are 30° and 45°, respectively. Find

(1)The height of the multi-storied building

( 2) The distance between the two buildings.

4. In Figure , from the top of a building AB, 60 meters high, the angles of depression of the top and
bottom of a vertical lamp post CD height h meter are observed to be 30° and 60°, respectively. Find

32
(i) the horizontal distance between AB and CD.
(ii) the height of the lamp post.

5. The angle of elevation of an aeroplane from a point on the ground is 60°. After a flight of 30
seconds the angle of elevation becomes 30°. If the aeroplane is flying at a constant height of 3000 √3
m, find the speed of the aeroplane.

6. A TV tower stands vertically on bank of a canal. From a point on the other bank directly opposite
the tower, the angle of elevation of the top of tower is 600. From another point 20m away from this
point on the line joining this point to the foot of the tower, the angle of elevation of the tower is 30 0.
Find
1.The height of the tower
2.The width of the canal.

7. A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height h.
At a point on the plane, the angles of elevation of the bottom and top of the flagstaff are 𝛼 and β
ℎ tan 𝛼
respectively. Prove that the height of the tower is tan β−tan 𝛼

8. A spherical balloon of radius r subtends an angle α at the eye of an observer. If the angle of elevation
of its center is β find the height of centre of the balloon.

9. A man on the deck of a ship, which is 10 m above water level, observes the angle of elevation of the
top of a cliff as 600 and the angle of depression of the base of the cliff as 300. Calculate
1. The distance of the cliff from the ship
2. The height of the cliff.

10. At a point, the angle of elevation of a tower is such that its tangent is 5/12 On walking 240 m to
the tower, the tangent of the angle of elevation becomes 3/4. Find the height of the tower.

33
11. A group of students of class X visited India gate on an education trip the teacher and students had
interested in history as well. the narrate the India gate. Official name Delhi Memorial originally called
All- India War Memorial, monumental sand stone arch in new Delhi dedicated to the troops of British
India who died in wars fought between 1914 and 1919. The teacher also said that india gate, which is
located at the eastern end of the Rajpath (formerly called the Kingsway) is about 138 feet (42 metres)
in height.

(i) if the altitude of the sun is at 600. then the height of the vertical tower that will cast a shadow of
length 20 m is?

(ii) The ratio of the length of a Rod and its shadow is 1:1. The angle of elevation of the sun is?

12. Mr. Ram observing from the top of light house finds that Boat A and Boat B are approaching to
light house from opposite direction he finds that the angle of depression of boat A is 450 and angle

34
of depression of Boat B is 300.He also is aware of the height of the light house is 100m

Answer the following question.

1 find length of BC

2 Find length BD

Q 13. A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m
from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60°.
After some time, the angle of elevation reduces to 30° (given fFig. ). Find the distance travelled by
the balloon during the interval.

Q 14. The angle of elevation of the top of a building from the foot of the tower is 30° and the angle
of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high,
find the height of the building.

Q. 15. If the angle of elevation of a cloud from a point h metres above a lake is a and the angle of
depression of its reflection in the lake is B, prove that the height of the cloud is h(tanß−tanα)/tanß–
tanα

35
 Answers
(3 MARKS)
Que (1.) 4m, Que (2.) 1:3, Que (3.) 273m, Que (4) 6m ,
Que (5) 30m , Que (6) 600 Que (8) 8√3 m Que (9) 30√3 ,
Que (10) √ab , Que (11)150m , Que (12) 155.7m Que (13) 40√3m
Que (14) 3(√3+1)m , Que (15) 40m

(4 MARKS)

Ans1. (i) difference between two light house = 20m

(ii) distance between light house and building = 34.64 m

Ans 2. (i) Height of the tower = 2.5 m

(ii) Distance of point of the point of the tower = 4.33 m

Ans 3. (i) The height of the building = 4(3+ √3) m

(ii) Distance between two building 4√3(3+√3)

Ans 4. (i) Horizontal between AB and CD = 20√3 m = 34.64m

(ii) Height of lamppost = 40m

Ans 5. 200m/s OR 720km/h

Ans 6. (i) Height of the tower = 10√3m

(ii) width of the river = 10m

ℎ tan 𝛼
Ans 7. H = tan β−tan 𝛼
Ans 8 height h = r sinβ. cosec α/2

Ans 9. (i) Distance of the cliff from the ship = 17.32 m

Ans 10 Height of the tower = 225 m

Ans 11. (i) 20√3m (ii) 450

Ans 12. (i) 100m (ii) 100√3m

Ans 13. Balloon travel 58√3 m

Ans 14. Height of the building = 50/3 m

Ans 15 Height of the cloud is h(tanß−tanα)/tanß–tanα

36
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 6.TOPIC: SURFACE AREA AND VOLUME
SHORT ANSWER QUESTIONS
2 MARKS
Q1. A toy is in the shape of a right circular cylinder with hemisphere at one end and a cone at the
other. The radius and height of the cylindrical part are 5 cm and 13 cm respectively. The radii of the
hemispherical and conical parts are the same as that of the cylindrical part.If the total height of the
toy is 30 cm, find the total surface area of the toy.

Q2. Three cubes of a metal whose edges are in the ratio 3:4:5 are melted and converted into a single
cube whose diagonal is 12√3 cm.Find the edges of the cubes.

Q3. A cone of maximum size is carved out from a cube of edge 14 cm. Find the surface area of the
cone and the remaining solid left after the cone carved out.

Q4. A vessel is in the form of a hemispherical bowl mounted by a hollow cylinder.The diameter of
the hemisphere is 14 cm and the total height of the vessel is 13cm. Find the capacity of the vessel.

Q5.Two identical cubes each of volume 64 cm3 are joined together end to end.What is the surface
area of the resulting cuboid?

Q6.From a solid cube of side 7cm, a conical cavity of height 7cm and radius 3cm is hollowed out.
Find the volume of the remaining solid.

Q7. Marbles of diameter 1.4 cm are dropped into a cylindrical beaker of diameter 7cm containing
some water. Find the numberof marbles that should be dropped into the beaker so that the water level
rises by 5.6 cm.

Q8.Two cones with same base radius 8 cm and height 15 cm are joined together along their bases.
Find the surface area of the shape so formed.

Q9. A hemispherical bowl of internal diameter 36 cm. contains liquid. This liquid is to be filledin the
cylindrical bottles of radius 3cm and height 6 cm. Find the number of bottles required to empty the
bowl.

Q10. An ice cream cone of radius 5 cm and height 10cm is full of ice cream. Calculate the volume of
ice cream , provided that 1/6 part is left unfilled with ice cream.

3 MARK QUESTIONS

Q 1. A sphere of diameter 18 cm is dropped into a cylindrical vessel of diameter 36 cm, partly filled
with water. If the sphere is completely submerged, then calculate the rise of water level in cm

Q 2. Find the number of solid spheres, each of diameter 6 cm that can be made by melting a solid metal
cylinder of height 45 cm and diameter 4 cm.

Q 3. A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its
base. Find the ratio of the volume of the smaller cone to the whole cone.

37
Q 4. Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of
hemisphere?

Q 5. Two cubes each of side 4 cm are joined end to end. Find the surface area of the resulting cuboid.

Q 6. If the total surface area of a solid hemisphere is 462 𝑐𝑚2 . Find its volume.

LONG ANSWER QUESTIONS

4 MARKS

CASE STUDY 1:
Q1. Adventure camps are the perfect place for the children to practice decision making for
themselves without parents and teachers guiding their every move. Some students of a school
reached for adventure at Sakleshpur. At the camp, the waiters served some students with a welcome
drink in a cylindrical glass and some students in a hemispherical cup whose dimensions are shown
below. After that they went for a jungle trek. The jungle trek was enjoyable but tiring. As dusk fell, it
was time to take shelter. Each group of four students was given a canvas of area 551m2 . Each group
had to make a conical tent to accommodate all the four students. Assuming that all the stitching and
wasting incurred while cutting, would amount to 1m2 , the students put the tents. The radius of the
tent is 7m.

38
(i) The volume of cylindrical cup is
a) 295.75 cm3 b) 7415.5 cm3 c) 384.88 cm3 d) 404.25 cm3

(ii) The volume of hemispherical cup is

a) 179.67 cm3 b) 89.83 cm3 c) 172.25 cm3 d) 210.60 cm3

iii) Which container had more juice and by how much?

a) Hemispherical cup, 195 cm3 b) Cylindrical glass, 207 cm3
3
c) Hemispherical cup, 280.85 cm d) Cylindrical glass, 314.42 cm3

iv) The height of the conical tent prepared to accommodate four students is
a) 18m b) 10m c) 24m d) 14m

v) How much space on the ground is occupied by each student in the conical tent
a) 54 m2 b) 38.5 m2 c) 86 m2 d) 24 m2

CASE STUDY 2:

Q2.The Great Stupa at Sanchi is one of the oldest stone structures in India, and an important
monument of Indian Architecture. It was originally commissioned by the emperor Ashoka in the 3rd
century BCE. Its nucleus was a simple hemispherical brick structure built over the relics of the
Buddha. It is a perfect example of combination of solid figures.

A big hemispherical dome with a cuboidal structure mounted on it. (Take π = 22/7)

(i) Calculate the volume of the hemispherical dome if the height of the dome is 21 m –
a) 19404 cu. m b) 2000 cu .m c) 15000 cu. m d) 19000 cu. m

(ii) The formula to find the Volume of Sphere is -

a) 2/3 πr3 b) 4/3 πr3 c) 4 πr2 d) 2 πr2

(iii) The cloth require to cover the hemispherical dome if the radius of its base is 14m is
a) 1222 sq.m b) 1232 sq.m c) 1200 sq.m d) 1400 sq.

39
(iv) The total surface area of the combined figure i.e. hemispherical dome with radius 14m and
cuboidal shaped top with dimensions 8m, 6m and 4m is

a)1200 sq.m b) 1232 sq.m c) 1392 sq.m d) 1932 sq.m

(v) The volume of the cuboidal shaped top is with dimensions mentioned in question (iv)

a) 182.45 m3 b) 282.45 m3 c) 292 m3 d) 192 m3

CASE STUDY 3:
Q3.On a Sunday, your parents took you to a fair. You could see lot of toys displayed, and you
wanted them to buy a RUBIC’s cube and strawberry ice-cream for you. Observe the figures and
answer the questions: -
(i) The length of the diagonal if each edge measures 6cm is
a) 3√3 b) 3√6 c) √12 d) 6√3

(ii) Volume of the solid figure if the length of the edge is 7cm is

a)256 cm3 b) 196 cm3 c) 343 cm3 d) 434 cm3

3. What is the curved surface area of hemisphere (ice cream) if the base radius is 7cm?
a) 309 cm2 b) 308 cm2 c) 803 cm2 d) 903 cm2

4. Slant height of a cone if the radius is 7cm and the height is 24 cm___
a) 26cm b) 25 cm c) 52 cm d) 62cm

5. The total surface area of cone with hemispherical ice cream is

a) 858 cm2 b) 885 cm2 c) 588 cm2 d) 855 cm2

Q4.The surface area of a solid metallic sphere is 616 cm2 . It is melted and recast into a cone of
height 28 cm. Find the diameter of the base of the cone so formed

Q5. Water in a canal 6m wide and 1.5 m deep , is flowing with a speed of 10 km/hr. How much area
will it irrigate in 30 minutes if 8cm of standing water is needed?

Q6.A building is in the form of a cylinder surmounted by a hemispherical dome.The base of the
dome is equal to 2/3 of the total height of the building .Find the height of the building if it contains
1
67 21 𝑚3 of air.

40
Q7.A toy is in the form of a hemisphere surrmounted by a right circular cone of the same base radius
as that of the hemisphere. If the radius of the base of the cone is 21cm and its volume is 2/3 of the
volume of the hemisphere , calculate the height of the cone and the surface area of the toy.

Q8. A cylindrical vessel with internal diameter 10 cm and height 10.5 cm is full of water. A solid
cone of base diameter 7cm and height 6cm is completely immersed in water .Find the volume(in
litres)of

(i) water displaced out of the cylindrical vessel

(ii) water left in the cylindrical vessel.
Q9. A solid is in the form of a hemisphere surmounted by a right circular cone. The height of the
cone is 4cm and the diameter of the base is 8cm. Determine the volume of the toy. If a cube
circumscribes the toy, then the difference of the volumes of cube and toy. Also, find the total surface
area of the toy.

Q10. A juice seller serves his customers using a glass with bottom(base) as hemispherical portion
raised which reduces the capacity of the glass. If the inner diameter of cylindrical glass is 5cm amd
height is 10cm, find the apparent capacity of the glass and its actual capacity.(Use π =3.14)

ANSWER KEY

SHORT ANSWER QUESTIONS

Q1.770 cm2 Q2. 6cm,8cm,10cm Q3. 154(1 + √5)𝑐𝑚2 , (1022 + 154√5)𝑐𝑚2

Q4. 1642.66cm3 Q5. 160 cm2 Q6. 277.60 cm3 Q7. 150

Q8. 855 cm2 Q9. 72 bottles Q10. 327.375 cm3

LONG ANSWER QUESTIONS

CASE STUDY 1:
Q1. (i)d) 404.25 cm3 (ii) b) 89.83 cm3 (iii)d) Cylindrical glass, 314.42 cm3
(iv) c) 24m (v) b) 38.5 m2

CASE STUDY 2:

Q2. (i) a) 19404 cu.m (ii) b) 4/3 πr3 (iii) b) 1232 sq.m (iv) c) 1392 sq.m (v) d)
192 m3

CASE STUDY 3:
Q3.(i) d) 6√3 (ii) c) 343 cm3 (iii) b) 308 cm2 (iv) b) 25 cm
(v) a) 858 cm2

Q4. 14 cm Q5. 562500 m2 Q6. Height = 6 m Q7. 5082 cm2

Q8. 0.77 litre , 0.748 litre Q9. 1408/7 cm3, 310.86 cm3, 171.68 cm2

Q10. 32.71cm3, 163.54 cm3

41
 7.TOPIC: STATISTICS

IMPORTANT FORMULAS AND CONCEPTS

We will learn the three measures of central tendency namely, mean, median and mode of grouped
data.

1. Mean or Average: - It is the sum of the values of all the observations divided by the total number
of observations.

(a) Direct Method: - Mean of grouped data

(b) Assumed Mean Method: - Mean of grouped data

2. Mode of grouped data: - Mode is that values among the observations which occurs most often or
the value of the observation having the maximum frequency.

Where l = lower limit of the modal class

h = size of the class interval
f1 = frequency of the modal class
f0 = frequency of the class preceding the modal class
f2 = frequency of the class succeeding the modal class
3. Median of grouped data: - median is the measure of central tendency which gives the value of the
middle-most observation in the data.

Where l = lower limit of median class

n = number of observations
cf = cumulative frequency of class preceding the median class
f = frequency of median class
h = class size
The empirical relationship between the three measures of central tendency is: -

3 Median = Mode + 2 Mean

42
I. Case Study and Situation Based Questions: -

1. Under the physical and health education a medical checkup program was conducted in a
Vidyalaya to improve the health and fitness conditions of the students. Reading of the heights of 50
students was obtained as given in the table below:

Hight ( in cm ) Number of students

135 – 140 2

140 – 145 8

145 – 150 10

150 – 155 15

155 – 160 6

160 – 165 5

165 – 170 4

(i) Identify the lower-class limit of the modal class and find the mode of the given data.
(ii) Calculate the mean and median of the above data.
2. In a Vidyalaya there are two sections A and B. 39 students are there in section A and in section B
there are 41 students. A periodic test was conducted to assess the performance of students thereafter
analyze and plan the teaching learning process accordingly. The marks obtained out of 40 are given
below in the table.

43
 Marks obtained by the students Number of students

Less than 5 3

Less than 10 12

Less than 15 22

Less than 20 35

Less than 25 42

Less than 30 60

Less than 35 71

Less than or equal to 40 80

(i) How many students have obtained more than or equal to 35 marks?

(iii) Arrange the given data in class interval and find the median of the marks obtained.

3. An international cricket tournament was organized. Ten teams participated in the tournament. All
the players got opportunity to bat in their first match. The lowest and highest runs scored by an
individual player in their first match are 0 and 99 respectively. Runs scored by the players in their
first match are given below in the table: -

44
 Runs scored in their first match Number of players

More than or equal to 0 110

More than or equal to 10 105

More than or equal to 20 95

More than or equal to 30 81

More than or equal to 40 69

More than or equal to 50 51

More than or equal to 60 45

More than or equal to 70 30

More than or equal to 80 20

More than or equal to 90 8

(i) How many players scored more than or equal to 50 runs and how many players scored less than
10 runs?

(iii) Find the range of the runs scored by individual players.

II. Very Short Answer Type Questions: -

1. Find the mean of first ten whole numbers.

2. If the mode of a distribution is 9 and its mean is 6, then find its median.

3. Write the modal class for the following frequency distribution.

Class Interval 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50

Frequency 5 7 4 10 4

4. Write the empirical relationship between the three measures of central tendency.

5. A data has 9 observations arranged in descending order. Which observation represents the

median of the data?

6. Find the class size of the given class intervals.

45
 Class Interval 0–6 6 – 12 12 – 18 18 – 24 24 – 30 30 – 46

Frequency 3 5 7 4 9 2

7. Find the cumulative frequency of the class interval 20 – 25 in the given frequency

distribution.

Class Interval 0–5 5 – 10 10 – 15 15 – 20 20 – 25 25 – 30

Frequency 4 12 8 3 3 2

8. Find the class mark of the class interval 30 – 40 in a frequency distribution.

III. Short Answer Type Questions: -

1. Find the mean of 20 numbers, such that if the mean of 8 of them is 10 and the mean of 10

of them is 12. The last two numbers are 8 and 12.

2. Find the mean of first 15 natural numbers.

3. The number of pages read by a student during a week are as under: -

Monday Tuesday Wednesday Thursday Friday Saturday Sunday

14 14 12 18 13 15 12

Find the mean number of pages.

4. The observation 15, 24, 32, a + 5, b, 46, 50 is arranged in ascending order. The median is

36. Find the value of a.

5. From the following frequency distribution, find the median class.

Monthly wages Number of workers

18000 – 24000 18

24000 – 30000 25

30000 – 36000 30

36000 – 42000 28

42000 – 48000 35

48000 – 54000 32

54000 – 60000 32

46
6. Find the mode of the following frequency distribution.

Class Interval Frequency

0 – 10 10

10 – 20 14

20 – 30 12

30 – 40 8

40 – 50 9

7. While finding the mean of 18 observations, an observation 43 was wrongly noted as 34 and

then the mean was 30. Find the correct mean.

8. In the following frequency distribution, find the lower limit of the median class.

Age group ( in years) Number of Students

5–8 45

8 – 11 50

11 – 14 35

14 – 17 60

17 – 20 110

9. The mean of the following frequency distribution is 4.84. Find the value of f.

Class Interval Frequency

0–2 5

2–4 f

4–6 25

6–8 4

8 – 10 6

47
10. Find the missing frequency from the following data, when mode is 27.

Class 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50

Frequency 5 x 15 12 7

IV.Long Answer Type Questions: -

1. A survey was conducted to find the monthly earnings of 500 people in a city. Their

monthly earnings are given by the following frequency distribution table. Find the mean

earnings and modal monthly earnings.

Monthly earnings ( in Rs.) Number of people

20000 – 30000 58

30000 – 40000 56

40000 – 50000 60

50000 – 60000 85

60000 – 70000 37

70000 – 80000 70

80000 – 90000 77

90000 – 100000 57

2. In a Vidyalaya, students were asked to find out the number of people of different ages

those who have been recovered from Covid-19 pandemic. The students asked 100 people

and represented the data as given:

Age ( in years ) Number of people

Below 10 2
Below 20 6
Below 30 22
Below 40 40
Below 50 75
Below 60 84
Below 70 90
Below 80 96
Below 90 99
Below 100 100

48
Calculate the median age and mean age of the people.

3. The percentage of marks scored by 40 students of class X in their board examination is given
below in the table. Find the mean and modal percentage of their marks.

Percentage 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80 80 – 90 90 – 100
of marks

Number of 6 14 20 25 15 12 8
students

4. Apples are supplied to a retail market from a garden. Different number of apples are

packed in the boxes as per their size. The following are the distribution of apples according

to the number of boxes:

Number of 100 - 105 105 – 110 110 – 115 115 – 120 120 – 125
apples

Number of 25 100 130 125 20

boxes

Find the average number of apples kept in a packing box. Also find the mode of the given

distribution.

5. The class teacher of class X A has the following attendance record of 40 students for 200

days. The minimum number of days any student present is 80. Find the mean, median and

modal attendance of the students.

Number of days Number of students present

More than or equal to 80 40

More than or equal to 100 36

More than or equal to 120 30

More than or equal to 140 20

More than or equal to 160 15

More than or equal to 180 8

6. It is good news that the number of covid-19 cases are decreasing day by day. 30 cities of

our country is surveyed and the number of positive cases in a day are recorded as under.

49
 Number of cases Number of cities

Less than 15 3

Less than 30 4

Less than 45 5

Less than 60 7

Less than 75 10

Less than 90 12

Less than 105 15

Less than 120 23

Less than 135 26

Less than 150 30

Find the median and modal number of cases.

7. The distribution below gives the weight of 50 students of class X. Find the median and

modal weight of the students.

Weight ( in kg ) 35 – 45 45 –55 55 – 65 65 – 75 75 – 85

Number of 5 10 20 12 3
students

8. Data of average annual rainfall ( incm )of different states and union territories of our

country is recorded by the students of class X B. It is represented by the following table.

Average annual rainfall ( incm) Number of states or union territories

50 – 74 3
75 – 99 5
100 –124 4
125 – 149 10
150 – 174 8
175 – 199 4
200 – 225 1

Find the median and mean rainfall of the cities.

50
 Answer Key

I. Case Study and Situation Based Questions:

1. (i) Lower class limit of modal is (150 – 155) is 150. Mode of the observation is 151.79

(ii) Mean of the data is 190.13 and median is 150

2. Hint: Convert the less than type data into class intervals.

(i) 80 – 71 = 9

(ii) Median of the marks obtained is 21.92

3. Hint: Convert the more than type data into class intervals and find cumulative frequency.

(i) more than or equal to 50 is 51 and less 10 is 5.

(ii) Range of the runs scored by individual player = 99 – 0 = 99.
II. VSA

1. 2 2. 7 3. 30 – 40 4. 3 Median = Mode + 2 Mean 5. 5 th 6. 6

7. 30 8. 35

III. SA

1. 11 2. 8 3. 14 4. 31(Hint: When data is arranged in ascending or

descending order, the middle value is its median.)

5. 36000 – 42000 6. 16.67 7. 30.5 8. 14 9. f = 10 10. x = 8

IV. Long Answer Type Questions

1. The mean earnings is Rs. 58190 and modal earnings is Rs.53203.

2.(Hint: Convert the ‘below type’ into class intervals as 0 – 10, 10 – 20, 20 – 30 etc.

Median age is 43 years and mean age is 41.05 years.

3.Mean percentage of marks is 64. 70 and modal percentage of marks is 63.33.

4.The average number of apples is 113 (112.69) and mode of the given distribution is 114.29

5.Mean = 144.5, Median = 140 and mode = 128.89

6.Median = 105 and mode = 112.5

7.Median = 60 and mode = 60.56

8. (Hint: convert the data to continuous classes, as 49.5 – 74.5, 74.5 – 99.5, 99.5 – 124.5 and

so on) Median rainfall is138.25 cm and mean rainfall is 134.14 cm.

51
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