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The taxi fare in a city is as follows: For the first kilometer, the fare is Rs. 8 and for the subsequent distance it is Rs. 5 per km. Taking the distance covered as x km and total fare as Rs. y, write a linear equation for this information, and draw its graph.

Let us assume; Distance covered as x km and total fare as ₹ y Fare for the first kilometer = 8 per km Fare after the first 1km = ₹ 5 per km... View Article

A group of 360 people were asked to vote for their favorite season from the three seasons rainy, winter and summer. (i) Which season got the most votes? (ii) Find the central angle of each sector. (iii) Draw a pie chart to show this information

Solution (i) According to the table given in the question, Winter season got the most votes. (ii) Central angle of summer season=... View Article

A photograph of a bacteria enlarged 50,000 times attains a length of 5 cm as shown in the diagram. What is the actual length of the bacteria? If the photograph is enlarged 20,000 times only, what would be its enlarged length?

Solution Let the enlarged length of bacteria be x. Actual length of bacteria = 5/50000 = 1/10000 cm = 10-4 cm Since the length of... View Article

Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is: (i) Intersecting lines (ii) Parallel lines (iii) Coincident lines

(i) Given the linear equation 2x + 3y – 8 = 0. To determine another linear equation in two variables such that the geometrical... View Article

Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle whose sides are 7/5 of the corresponding sides of the first triangle. Give the justification of the construction

To construct a triangle from given conditions 1. Let usraw a line segment AB =5 cm. 2. Take A and B as centre, and draw the arcs of radius 6 cm... View Article

Sum of the digits of a two-digit number is 9. When we interchange the digits, it is found that the resulting new number is greater than the original number by 27. What is the two-digit number?

Solution Let the digit at tens place be x The digit at ones place will be (9 – x). Original two-digit number = 10x + (9 – x)... View Article

Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, -1) , (2, 1) and (0, 3) . Find the ratio of this area to the area of the given triangle.

Let the vertices of the triangle be A (0, -1), B (2, 1), C (0, 3). Let D, E, F be the mid-points of the sides of this triangle. Coordinates of D,... View Article

The volume of a right circular cone is 9856cm³. If the diameter of the base is 28cm, find (i) height of the cone (ii) slant height of the cone (iii) curved surface area of the cone (Assume π = 22/7)

Given Radius of cone (r) = 28/2 = 14 cm Volume of cone (V) = 9856 cm3 Find out We have to find the (i) height of the cone (ii)... View Article

Rehman is making a wheel using spokes. He wants to fix equal spokes in such a way that the angles between any pair of consecutive spokes are equal. Help him by completing the following table. (i) Are the number of spokes and the angles formed between the pairs of consecutive spokes in inverse proportion? (ii) Calculate the angle between a pair of consecutive spokes on a wheel with 15 spokes. (iii) How many spokes would be needed, if the angle between a pair of consecutive spokes is 40 degree ?

Solution The Number of spokes and angle between a pair of consecutive spokes are inversely proportional. Therefore, When the number of... View Article

Represent the following situations mathematically: (i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with. (ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was ` 750. We would like to find out the number of toys produced on that day.

(i) We will assume, the number of marbles John have = x. Hence, the number of marble Jivanti have = 45 – x After losing 5 marbles... View Article

Five cards the ten, jack, queen, king and ace of diamonds, are well-shuffled with their face downwards. One card is then picked up at random. (i) What is the probability that the card is the queen? (ii) If the queen is drawn and put aside, what is the probability that the second card picked up is (a) an ace? (b) a queen?

Given that Five cards the ten, jack, queen, king and ace of diamonds, are well-shuffled with their face downwards. One card is then picked up at... View Article

The vertices of a ∆ ABC are A (4, 6) , B (1, 5) and C (7, 2) . A line is drawn to intersect sides AB and AC at D and E respectively, such that . Calculate the area of the ∆ ADE and compare it with area of ∆ ABC. (Recall Theorem 6.2 and Theorem 6.6)

Given: The vertices of an ∆ ABC are A (4, 6), B (1, 5) and C (7, 2) Point D and Point E divide AB and AC respectively in ratio 1 : 3. Now... View Article

State which pairs of triangles in Figure, are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form:

(i) Given that In ΔABC and ΔPQR, ∠A = ∠P = 60° ∠B = ∠Q = 80° ∠C = ∠R = 40° Therefore... View Article

Calculate the force of gravitation between the earth and the Sun, given that the mass of the earth = 6 × 10²⁴ kg and of the Sun = 2 × 10³⁰ kg. The average distance between the two is 1.5 × 10¹¹ m.

Given that Mass of the sun ms = 2 × 1030 kg Mass of the earth me = 6 × 1024 kg Gravitation constant G = 6.67 x 10-11 N m2/ kg2 Average... View Article

Line l is the bisector of an angle A and B is any point on l. BP and BQ are perpendiculars from B to the arms of A (see Fig.) . Show that: (i) ΔAPB ΔAQB (ii) BP = BQ or B is equidistant from the arms of A.

Solution Given Line “l” is the bisector of angle A and the line segments BP and BQ are perpendiculars drawn from l. To Prove (i) ΔAPB ΔAQB... View Article

Form the pair of linear equations for the following problems and find their solution by substitution method (i) A fraction becomes 9/11 , if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the denominator it becomes 5/6. Find the fraction. (ii) Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages?

(i) A fraction becomes 9/11 if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the denominator it... View Article

Classroom activity (Constructing the ‘square root spiral’) : Take a large sheet of paper and construct the ‘square root spiral’ in the following fashion. Start with a point O and draw a line segment OP1 of unit length. Draw a line segment P1P2 perpendicular to OP1 of unit length (see Fig) . Now draw a line segment P2P3 perpendicular to OP2. Then draw a line segment P3P4 perpendicular to OP3. Continuing in Fig.: Constructing this manner, you can get the line segment Pn-1Pn by square root spiral drawing a line segment of unit length perpendicular to OPn-1. In this manner, you will have created the points P2, P3,….,Pn,… ., and joined them to create a beautiful spiral depicting √2, √3, √4, …

Solution Let us follow the following procedure to execute the activity 1: Mark a point O on the paper. Here, O will be the center of the square... View Article

A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope (see Fig.) . Find (i) the area of that part of the field in which the horse can graze. (ii) the increase in the grazing area if the rope were 10 m long instead of 5 m. (Use π = 3.14)

As the horse is tied at one end of a square field, it will graze only a quarter (i.e. sector with θ = 90°) of the field with radius 5... View Article

A piggy bank contains hundred 50p coins, fifty Rs. 1 coins, twenty Rs. 2 coins and ten Rs. 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, what is the probability that the coin (i) will be a 50 p coin? (ii) will not be a Rs. 5 coin?

Given that A piggy bank contains hundred 50p coins, fifty ₹1 coins, twenty ₹2 coins and ten ₹5 coins. Find out The probability that the coin... View Article

What length of tarpaulin 3 m wide will be required to make conical tent of height 8 m and base radius 6m? Assume that the extra length of material that will be required for stitching margins and wastage in cutting is approximately 20 cm. [Use π = 3.14]

Given Height of conical tent, h = 8m Radius of base of tent, r = 6m Find out We have to find the length of the required tarpaulin... View Article