Ch 1Orienting Yourself: The Use of Coordinates
2-Mark Questions
Fig. 1.3 shows Reiaan's room with points OABC marking its corners. The x- and y-axes are marked in the figure. Point O is the origin. Referring to Fig. 1.3, answer the following questions: (i) If $D_1R_1$ represents the door to Reiaan's room, how far is the door from the left wall (the y-axis) of the room? How far is the door from the x-axis? (ii) What are the coordinates of $D_1$ ? (iii) If $R_1$ is the point $(11.5, 0)$ , how wide is the door? Do you think this is a comfortable width for the room door? If a person in a wheelchair wants to enter the room, will he/she be able to do so easily? (iv) If $B_1(0, 1.5)$ and $B_2(0, 4)$ represent the ends of the bathroom door, is the bathroom door narrower or wider than the room door?
(i) The door is $8$ units from the left wall (the y-axis) and $0$ units from the x-axis. (ii) $D_1 = (8, 0)$ . (iii) The width is $11.5 - 8 = 3.5$ units. A width of $3.5$ ft is comfortable for a room door and should allow a wheelchair to enter easily. (iv) The bathroom door width is $4 - 1.5 = 2.5$ units, so it is narrower than the room door.
Place Reiaan's rectangular study table with three of its feet at the points $(8, 9)$ , $(11, 9)$ and $(11, 7)$ . (i) Where will the fourth foot of the table be? (ii) Is this a good spot for the table? (iii) What is the width of the table? The length? Can you make out the height of the table?
(i) The fourth foot will be at $(8, 7)$ . (ii) Yes, it is a good spot because the table fits in the open space near the upper-right part of the room without crossing the bed, wardrobe or walls. (iii) The sides are $11 - 8 = 3$ ft and $9 - 7 = 2$ ft. From the top view, we can find only the length and width, not the height.
If the bathroom door has a hinge at $B_1$ and opens into the bedroom, will it hit the wardrobe? Are there any changes you would suggest if the door is made wider?
No, it will not hit the wardrobe in the present plan. The door hinged at $B_1(0,1.5)$ has width $2.5$ ft, so when it opens into the bedroom it reaches only up to about $x=2.5$ along the floor direction. The wardrobe begins at $x=3$ , so there is a small clearance. If the door is made wider than $3$ ft, it may hit the wardrobe; a sliding door or an outward-opening door would be better.
Ch 2Introduction to Linear Polynomials
2-Mark Questions
Find the degrees of the following polynomials: (i) $2x^2 - 5x + 3$ (ii) $y^3 + 2y - 1$ (iii) $-9$ (iv) $4z - 3$
(i) Degree $2$ (ii) Degree $3$ (iii) Degree $0$ (iv) Degree $1$
Write polynomials of degrees 1, 2 and 3.
Examples: degree 1: $x+2$ ; degree 2: $x^2+3x+1$ ; degree 3: $x^3-x+5$ .
What are the coefficients of $x^2$ and $x^3$ in the polynomial $x^4 - 3x^3 + 6x^2 - 2x + 7$ ?
The coefficient of $x^2$ is $6$ and the coefficient of $x^3$ is $-3$ .
Ch 3The World of Numbers
2-Mark Questions
A merchant in the port city of Lothal is exchanging bags of spices for copper ingots. He receives 15 ingots for every 2 bags of spices. If he brings 12 bags of spices to the market, how many copper ingots will he leave with?
He will leave with $90$ copper ingots.
Look at the sequence of numbers on one column of the Ishango bone: 11, 13, 17, 19. What do these numbers have in common? List the next three numbers that fit this pattern.
They are prime numbers between 10 and 20. The next three primes after 19 are $23, 29, 31$ .
We know that Natural Numbers are closed under addition (the sum of any two natural numbers is always a natural number). Are they closed under subtraction? Provide a couple of examples to justify your answer.
No. Natural numbers are not closed under subtraction. For example, $5-2=3$ is natural, but $2-5=-3$ is not a natural number. Also, $4-4=0$ , and 0 is not a natural number in the usual NCERT convention $\mathbb{N}=\{1,2,3,\ldots\}$ .
Ch 4Exploring Algebraic Identities
2-Mark Questions
Using the identity $(a + b)^2 = a^2 + 2ab + b^2$ , expand the following: (i) $(7x + 4y)^2$ (ii) $\left(\dfrac{7}{5}x + \dfrac{3}{2}y\right)^2$ (iii) $(2.5p + 1.5q)^2$ (iv) $\left(\dfrac{3}{4}s + 8t\right)^2$ (v) $\left(x + \dfrac{1}{2y}\right)^2$ (vi) $\left(\dfrac{1}{x} + \dfrac{1}{y}\right)^2$
(i) $49x^2 + 56xy + 16y^2$ (ii) $\dfrac{49}{25}x^2 + \dfrac{21}{5}xy + \dfrac{9}{4}y^2$ (iii) $6.25p^2 + 7.5pq + 2.25q^2$ (iv) $\dfrac{9}{16}s^2 + 12st + 64t^2$ (v) $x^2 + \dfrac{x}{y} + \dfrac{1}{4y^2}$ (vi) $\dfrac{1}{x^2} + \dfrac{2}{xy} + \dfrac{1}{y^2}$
Using the same identity, find the values of the following: (i) $(64)^2$ (ii) $(105)^2$ (iii) $(205)^2$
(i) $4096$ (ii) $11025$ (iii) $42025$
Factor completely: (i) $9x^2 + 24xy + 16y^2$ (ii) $4s^2 + 20st + 25t^2$ (iii) $49x^2 + 28xy + 4y^2$ (iv) $64p^2 + \dfrac{32}{3}pq + \dfrac{4}{9}q^2$ (v) $3a^2 + 4ab + \dfrac{4}{3}b^2$ (vi) $\dfrac{9}{5}s^2 + 6sv + 5v^2$
(i) $(3x+4y)^2$ (ii) $(2s+5t)^2$ (iii) $(7x+2y)^2$ (iv) $\left(8p+\dfrac{2}{3}q\right)^2$ (v) $\dfrac{1}{3}(3a+2b)^2$ (vi) $\dfrac{1}{5}(3s+5v)^2$
Ch 5I
2-Mark Questions
What is the least possible radius of a circle through two points A and B?
The least possible radius is $\dfrac{AB}{2}$ .
Show that the triangle formed by a chord and the centre of the circle is isosceles.
Let AB be a chord of a circle with centre O. Join OA and OB. Since A and B lie on the circle, $OA=OB$ because both are radii. Therefore $\triangle AOB$ has two equal sides and is isosceles.
Show that if two such isosceles triangles (occurring in the previous question) have equal base length, they are congruent to each other.
Let the two chords be AB and DE in the same circle with centre O, and suppose $AB=DE$ . Join OA, OB, OD and OE. Since all four are radii, $OA=OB=OD=OE$ . Also $AB=DE$ . Thus $\triangle AOB$ and $\triangle DOE$ have three corresponding sides equal, so they are congruent by SSS.
Ch 6Measuring Space: Perimeter and Area
2-Mark Questions
The perimeter of a circle is 44 cm. What is its radius?
The radius is $7$ cm.
Calculate, correct to 3 significant figures, the circumference of a circle with: (i) radius 7 cm (ii) radius 10 cm (iii) radius 12 cm.
(i) $44.0$ cm (ii) $62.9$ cm (iii) $75.4$ cm
Calculate the length of the arc of a circle if: (i) the radius is 3.5 cm and the angle at the centre is 60°, and (ii) the radius is 6.3 m and the angle at the centre is 120°.
(i) $\dfrac{11}{3}$ cm, i.e. about $3.67$ cm (ii) $13.2$ m
Ch 7The Mathematics of Maybe: Introduction to Probability
2-Mark Questions
Rank the following events on a scale from 0 (Impossible) to 1 (Certain). Label each event: Impossible, less likely, equally likely (even chance), more likely, certain. Give reasons why you gave each event its ranking. (i) The next Monday will come after Sunday. (ii) It will snow in Mumbai in July. (iii) An elephant will walk through your classroom today. (iv) You will greet at least one friend at school tomorrow.
(i) Certain; probability $1$ . (ii) Impossible or almost impossible in ordinary experience; probability close to $0$ . (iii) Less likely, practically close to $0$ . (iv) More likely; probability close to $1$ , though not certain.
A teacher mixes a large bag of sweets of different colours and randomly selects a sample of 30 sweets. She counts the number of sweets of each colour: 10 red sweets | 8 green sweets | 7 yellow sweets | 5 blue sweets. (i) Calculate the probability that a randomly picked sweet from the sample is green. (ii) If there are 600 sweets in total in the large bag, estimate how many are likely to be yellow, based on the sample results.
(i) $\dfrac{8}{30}=\dfrac{4}{15}$ (ii) About $140$ yellow sweets.
A survey is conducted at a school where a random sample of 40 students is asked about their favourite club. The responses are: 14 students: Science Club | 11 students: Arts Club | 9 students: Sports Club | 6 students: Debate Club. Assume there are 800 students in the whole school. (i) What is the probability that a randomly chosen student from the sample prefers the Arts Club? (ii) Using the sample results, estimate how many students in the whole school are likely to prefer the Sports Club.
(i) $\dfrac{11}{40}$ (ii) About $180$ students.
Ch 8Predicting What Comes Next: Exploring Sequences and Progressions
2-Mark Questions
Find the first five terms of the sequence in which the nth term is given by (i) $t_n = 3n - 4$ , (ii) $t_n = 2 - 5n$ , and (iii) $t_n = n^2 - 2n + 3$ for $n \geq 1$ .
(i) $-1, 2, 5, 8, 11$ (ii) $-3, -8, -13, -18, -23$ (iii) $2, 3, 6, 11, 18$
Find the 10th and 15th terms of the sequence $t_n = 5n - 3$ for $n \geq 1$ .
$t_{10}=47$ and $t_{15}=72$ .
Determine whether 97 and 172 are terms of the sequence $t_n = 5n - 3$ for $n \geq 1$ .
Yes. $97$ is the 20th term and $172$ is the 35th term.
Ch 9Circles
2-Mark Questions
Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.
Equal chords of congruent circles subtend equal angles at their centres.
Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.
Chords of congruent circles that subtend equal central angles are equal.
Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.
$6$ cm.
Ch 10Heron
2-Mark Questions
A traffic signal board, indicating 'SCHOOL AHEAD', is an equilateral triangle with side 'a'. Find the area of the signal board, using Heron's formula. If its perimeter is 180 cm, what will be the area of the signal board?
The area is $\dfrac{\sqrt3}{4}a^2$ . If the perimeter is $180$ cm, the area is $900\sqrt3\text{ cm}^2$ .
The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 122 m, 22 m and 120 m. The advertisements yield an earning of ₹ 5000 per m² per year. A company hired one of its walls for 3 months. How much rent did it pay?
The company paid $₹16,50,000$ .
There is a slide in a park. One of its side walls has been painted in some colour with a message "KEEP THE PARK GREEN AND CLEAN". If the sides of the wall are 15 m, 11 m and 6 m, find the area painted in colour.
The painted area is $20\sqrt2\text{ m}^2$ , approximately $28.3\text{ m}^2$ .
Ch 11Surface Areas and Volumes
2-Mark Questions
A plastic box 1.5 m long, 1.25 m wide and 65 cm deep, is to be made. It is opened at the top. Ignoring the thickness of the plastic sheet, determine: (i) The area of the sheet required for making the box. (ii) The cost of sheet for it, if a sheet measuring 1 m² costs ₹20.
(i) $5.375\text{ m}^2$ (ii) $₹107.50$
The length, breadth and height of a room are 5 m, 4 m and 3 m respectively. Find the cost of white washing the walls of the room and the ceiling at the rate of ₹7.50 per m².
$₹555$ .
The floor of a rectangular hall has a perimeter 250 m. If the cost of painting the four walls at the rate of ₹10 per m² is ₹15000, find the height of the hall.
$6$ m.
Ch 12Statistics
2-Mark Questions
A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house. Number of plants: 0-2, 2-4, 4-6, 6-8, 8-10, 10-12, 12-14. Number of houses: 1, 2, 1, 5, 6, 2, 3. Which method did you use for finding the mean, and why?
The mean number of plants per house is $8.1$ plants. The direct method is convenient because the class marks and frequencies are small.
Consider the following distribution of daily wages of 50 workers of a factory. Daily wages (in ₹): 500-520, 520-540, 540-560, 560-580, 580-600, 600-620, 620-640. Number of workers: 12, 14, 8, 6, 4, 3, 3. Find the mean daily wages of the workers of the factory by using an appropriate method.
The mean daily wage is $₹545.20$ .
The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is ₹18. Find the missing frequency f. Daily pocket allowance (in ₹): 11-13, 13-15, 15-17, 17-19, 19-21, 21-23, 23-25. Number of children: 7, 6, 9, 13, f, 5, 4.
$f=20$ .
Frequently asked questions
- Fig. 1.3 shows Reiaan's room with points OABC marking its corners. The x- and y-axes are marked in the figure. Point O is the origin. Referring to Fig. 1.3, answer the following questions: (i) If $D_1R_1$ represents the door to Reiaan's room, how far is the door from the left wall (the y-axis) of the room? How far is the door from the x-axis? (ii) What are the coordinates of $D_1$ ? (iii) If $R_1$ is the point $(11.5, 0)$ , how wide is the door? Do you think this is a comfortable width for the room door? If a person in a wheelchair wants to enter the room, will he/she be able to do so easily? (iv) If $B_1(0, 1.5)$ and $B_2(0, 4)$ represent the ends of the bathroom door, is the bathroom door narrower or wider than the room door?
- (i) The door is $8$ units from the left wall (the y-axis) and $0$ units from the x-axis. (ii) $D_1 = (8, 0)$ . (iii) The width is $11.5 - 8 = 3.5$ units. A width of $3.5$ ft is comfortable for a room door and should allow a wheelchair to enter easily. (iv) The bathroom door width is $4 - 1.5 = 2.5$ units, so it is narrower than the room door.
- Place Reiaan's rectangular study table with three of its feet at the points $(8, 9)$ , $(11, 9)$ and $(11, 7)$ . (i) Where will the fourth foot of the table be? (ii) Is this a good spot for the table? (iii) What is the width of the table? The length? Can you make out the height of the table?
- (i) The fourth foot will be at $(8, 7)$ . (ii) Yes, it is a good spot because the table fits in the open space near the upper-right part of the room without crossing the bed, wardrobe or walls. (iii) The sides are $11 - 8 = 3$ ft and $9 - 7 = 2$ ft. From the top view, we can find only the length and width, not the height.
- If the bathroom door has a hinge at $B_1$ and opens into the bedroom, will it hit the wardrobe? Are there any changes you would suggest if the door is made wider?
- No, it will not hit the wardrobe in the present plan. The door hinged at $B_1(0,1.5)$ has width $2.5$ ft, so when it opens into the bedroom it reaches only up to about $x=2.5$ along the floor direction. The wardrobe begins at $x=3$ , so there is a small clearance. If the door is made wider than $3$ ft, it may hit the wardrobe; a sliding door or an outward-opening door would be better.
- Find the degrees of the following polynomials: (i) $2x^2 - 5x + 3$ (ii) $y^3 + 2y - 1$ (iii) $-9$ (iv) $4z - 3$
- (i) Degree $2$ (ii) Degree $3$ (iii) Degree $0$ (iv) Degree $1$
These important questions are selected from the NCERT Class 9 Maths textbook book-back exercises to help you revise the most useful questions. Mark weightage (5/2/1) follows the usual exam pattern and may vary by exam — always check your latest syllabus and question pattern. Open each chapter for the complete set of questions and answers.