ICSE Class 10 Maths Solved Competency Focused Questions — Long Answer Questions 1 (Graph-based)

This section provides detailed solutions for ICSE Class 10 Maths graph long answer questions, a critical component for scoring well in your board exams. We will be working through the problems from the Class 10 – ICSE Maths Solved Competency Focused Questions textbook, specifically focusing on Chapter: Long Answer Questions 1 (Graph-based). These 5-mark questions test your understanding of two important topics: coordinate geometry, involving plotting points and their reflections across axes or lines, and statistics, where you’ll construct graphical representations like ogives from frequency distribution tables. Mastering the precise steps for these questions is essential, as they frequently appear in the exam and carry significant weight.

You are likely here because you need help solving a specific graph-based question or want to verify your method for plotting and interpreting graphical data. This page provides clear, step-by-step solutions for all 4 long answer questions in this section. Each solution is prepared following the exact methodology and presentation style that the ICSE board expects, ensuring you learn the correct techniques for your final exams. Here, you will find reliable, easy-to-follow guides to master these important 5-mark problems and secure full marks.

Long Answer Graph Based (5 Marks Each)

Question 106

Plot the points A(2, 2) and B(6, -2) in the graph and answer the following :

(a) Reflect points A in origin to point D and write the co-ordinates of point D.

(b) Reflect points A in line y = -2 to point C and write the co-ordinates of point C.

(c) Find a point P on CD which is invariant under reflection in x = 0, write its co-ordinates.

(d) Write the geometrical name of the closed figure ABCD.

(e) Write the co-ordinates of the point of intersection of the diagonals of ABCD.

The graph depicting the required points is shown below:

(a) Upon inspecting the graph,

Co-ordinates of D = (-2, -2).

(b) Upon inspecting the graph,

Co-ordinates of C = (2, -6).

(c) Notice that P(0, -4) sits on the y-axis.

∴ It stays invariant when reflected across x = 0.

Co-ordinates of P = (0, -4).

(d) Upon inspecting the graph,

ABCD is a square.

(e) Upon inspecting the graph,

H marks the spot where the two diagonals of ABCD cross each other.

Co-ordinates of H = (2, -2).

Question 107

Plot points A(0, 3), B(4, 0), C(6, 2) and D(5, 0). Reflect the points as given below and write their coordinates :

(a) Reflect A on x-axis to A’.

(b) Reflect B on y-axis to B’.

(c) Reflect C on x-axis to C’.

(d) D remain invariant when reflected on the line whose equation is …………… .

(e) Join the points A, B, C, D, C’, B, A’, B’ and A to form a closed figure. Name the closed figure BCDC’.

The graph depicting the required reflections is shown below:

(a) Coordinates of A’ = (0, -3).

(b) Coordinates of B’ = (-4, 0).

(c) Coordinates of C’ = (6, -2).

(d) From the graph we observe,

D lies on the x-axis.

∴ It remains unchanged under reflection in the x-axis, i.e., the line y = 0.

D remain invariant when reflected on the line whose equation is y = 0.

(e) From the graph we observe,

BCDC’ is a concave quadrilateral.

Question 108

The following data represents the daily wages in rupees of a certain number of employees of a company :

Use a graph to answer the following questions :

(a) Represent the above distribution by an ogive.

(b) Find the following on the graph drawn:

(i) median wage.

(ii) percentage of employees who earn more than ₹ 84 per day.

(iii) number of employees who earn ₹56 and below.

Begin by preparing the cumulative frequency distribution table:

Here, total frequency n = 120, an even number.

Median position = \dfrac{n}{2} = \dfrac{120}{2} = 60th term.

Steps of construction :

1. Take daily wages along the x-axis.

2. Take cumulative frequency along the y-axis.

3. Plot the upper-class-boundary points (40, 8), (50, 22), (60, 34), (70, 51), (80, 71), (90, 97), (100, 110) and (110, 120).

4. Join these plotted points with a smooth free-hand curve, beginning at the lower limit of the first class and ending at the upper limit of the last class.

5. Locate A = 60 on the y-axis; draw a horizontal line through A to meet the ogive at B.

6. From B, drop a perpendicular to the x-axis, meeting it at C. The x-coordinate of C gives the median. ∴ Median wage is ₹74.

7. Mark D = 84 on the x-axis; erect a vertical line to cut the ogive at E.

8. From E, draw a horizontal line to the y-axis, meeting it at F. This y-coordinate tells how many employees earn ₹84 or less per day.

From the ogive,

F = 81.

Employees earning strictly more than ₹84 = 120 − 81 = 39.

Percentage of employees earning more than ₹84 = \dfrac{\text{No. of employees earning more than ₹ 84}}{\text{Total employees}} \times 100

= \dfrac{39}{120} \times 100 = \dfrac{3900}{120} = 32.5 %.

1. Mark G = 56 on the x-axis; erect a vertical line to meet the ogive at H.

2. From H, draw a horizontal line to the y-axis, meeting it at I. This y-coordinate tells how many employees earn ₹56 or less per day.

From the ogive,

I = 30.

No. of employees earning less than or equal to ₹ 56 per day = 30.

Question 109

Study the graph and answer the questions that follow :

(a) Make a frequency table for the information provided in the graph.

(b) The number of students whose height is less than 150 cm.

(c) The total number of students.

(d) The modal height.

(e) The difference in the modal height and the mean height, if the average height of the students is 145.5 cm.

(a)

(b) Referring to the table above,

The number of students whose height is less than 150 cm = 69.

(c) Referring to the table above,

The total number of students = 103.

(d) From the graph,

The modal height = 143 cm.

(e) We are told the mean height equals the average height = 145.5 cm.

Gap between modal height and mean height :

145.5 − 143 = 2.5 cm

Hence, difference between modal height and the mean height = 2.5 cm