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In Fig.ABC and BDE are two equilateral triangles such that D is the mid-point of BC. If AE intersects BC at F, show that: (i) ar (BDE) = 1/4 ar (ABC) (ii) ar (BDE) = 1/2 ar (BAE) (iii) ar (ABC) = 2 ar (BEC) (iv) ar (BFE) = ar (AFD) (v) ar (BFE) = 2 ar (FED) (vi) ar (FED) = 1/8 ar (AFC)

Solution i) Given: ABC and BDE are equilateral triangles D is mid point of BC i.e. BD=DC To prove: Proof: Let AB=x=AC=BC Then... View Article

As observed from the top of a 75 m high lighthouse 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 lighthouse, find the distance between the two ships.

Let AB be the lighthouse of height 75 m. Let C and D be the positions of the ships. 30° and 45° are the angles of depression from the... View Article

The Life insurance agent found the following data for the distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to the persons whose age is 18 years onwards but less than the 60 years.

Age (in years) Number of policy holder Below 20 2 Below 25 6 Below 30 24 Below 35 45 Below 40 78 Below 45 89 Below 50 92 Below... View Article

A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30°. (see fig.)

Given: AC = 20 m and angle C = 30° To Find: Height of the pole Let AB be the vertical pole Solution sin30° = {AB}/{AC} AB = 1/2 *... View Article

There is a pentagonal shaped park as shown in the figure. For finding its area Jyoti and Kavita divided it in two different ways. Find the area of this park using both ways. Can you suggest some other way of finding its area?T

Solution: First way: By Jyoti’s diagram, Area of pentagon = Area of trapezium ABCP + Area of trapezium AEDP = (½)(AP + BC)... View Article

The students of Class VIII of a school donated Rs 2401 in all, for Prime Minister’s National Relief Fund. Each student donated as many rupees as the number of students in the class. Find the number of students in the class.

Solution Let us assume that the number of students be = n So, Each Student has donated Rs n Hence, Total amount donated = n x n That... View Article

A sum of Rs 700 is to be used to give seven 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.

Given that A sum of Rs 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. Each prize is... View Article

The following table shows the ages of the patients admitted in a hospital during a year: Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.

Age (in years) 5-15 15-25 25-35 35-45 45-55 55-65 Number of patients 6 11 21 23 14 5 Solution The greatest frequency in the given table... View Article

From each corner of a square of side 4 cm a quadrant of a circle of radius 1 cm is cut and also a circle of diameter 2 cm is cut as shown in Fig.. Find the area of the remaining portion of the square.

Solution Given Side of the square = 4 cm The radius of the circle = 1 cm Four quadrants of a circle are cut from the corner and one circle... View Article

From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are 45° and 60° respectively. Find the height of the tower.

Let BC be the 20 m high building. D is the point on the ground from where the elevation is taken. Height of transmission tower = AB = AC - BC To... View Article

The heights of 50 students, measured to the nearest centimeters, have been found to be as follows: 161 150 154 165 168 161 154 162 150 151 162 164 171 165 158 154 156 172 160 170 153 159 161 170 162 165 166 168 165 164 154 152 153 156 158 162 160 161 173 166 161 159 162 167 168 159 158 153 154 159 (i) Represent the data given above by a grouped frequency distribution table, taking the class intervals as 160 – 165, 165 – 170, etc. (ii) What can you conclude about their heights from the table?

Solution Grouped frequency distribution table with classes 160 – 165, 165 – 170, etc… The data is represented in the grouped... View Article

A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point.

Let AB be the tower. D is the initial and C is the final position of the car respectively. As man is standing at the top of the tower so,... View Article

A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is 60°. From another point 20 m away from this point on the line joing this point to the foot of the tower, the angle of elevation of the top of the tower is 30° (see Fig.) . Find the height of the tower and the width of the canal.

From diagaram AB = tower = ? CB = canal = ? From right ΔABD, tan 30° = AB/BD 1/√3 = AB/(20+BC) AB = (20+BC)/√3 …... View Article

A farmer was having a field in the form of a parallelogram PQRS. She took any point A on RS and joined it to points P and Q. In how many parts the fields is divided? What are the shapes of these parts? The farmer wants to sow wheat and pulses in equal portions of the field separately. How should she do it?

Solution The field is divided into three parts each in triangular shape. Let us assume, ΔPSA, ΔPAQ and ΔQAR be the... View Article

A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8 (see Fig.) , and these are equally likely outcomes. What is the probability that it will point at (i) 8? (ii) an odd number? (iii) a number greater than 2? (iv) a number less than 9?

Given that A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8 Find... View Article

A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.

The diameter of the hemisphere = D = 14 cm The radius of the hemisphere = r = 7 cm The height of the cylinder = h = (13-7) = 6 cm The radius of... View Article

A stone is thrown vertically upward with an initial velocity of 40 m/s. Taking g = 10 m/s2, find the maximum height reached by the stone. What is the net displacement and the total distance covered by the stone?

Given that Initial velocity u = 40m/s g = 10 m/s2 Max height final velocity = 0 Find out The maximum height reached by the stone and the net... View Article

There is a narrow rectangular plot, reserved for a school, in Mahuli village. The length and breadth of the plot are in the ratio 11:4. At the rate Rs. 100 per metre it will cost the village panchayat Rs. 75000 to fence the plot. What are the dimensions of the plot?

Solution Let the length of the rectangular plot be 11x Let the breadth of the rectangular plot be 4x. Rate of fencing per metre = ₹100... View Article

An ant is moving around a few food pieces of different shapes scattered on the floor. For which food-piece would the ant have to take a longer round? Remember, circumference of a circle can be obtained by using the expression C = 2πr ,where r is the radius of the circle

(a) Radius = Diameter/2 = 2.8/2 cm = 1.4 cm Circumference of semi-circle = ∏.r = (22/7)×1.4 = 4.4 Circumference of semi-circle is... View Article

For each of the following numbers, find the smallest whole number by which it should be divided so as to get a perfect square. Also find the square root of the square number so obtained. i. 252 ii. 2925 iii. 396 iv. 2645 v. 2800 vi. 1620

Solution (i) 252 252 = 2 × 2 × 3 × 3 × 7 = (2 × 2) × (3 × 3) × 7 7 cannot be paired.... View Article