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In any triangle ABC, if the angle bisector of ∠A and perpendicular bisector of BC intersect, prove that they intersect on the circumcircle of the triangle ABC.

Construction: Join BE and CE. Now, since AE is the bisector of ∠BAC, ∠BAE = ∠CAE Also, ∴arc BE = arc EC chord BE =... View Article

Ramkali saved Rs 5 in the first week of a year and then increased her weekly saving by Rs 1.75. If in the nth week, her weekly savings become Rs 20.75, find n.

Given data Ramkali saved Rs.5 in the first week and then started saving each week by Rs.1.75. Find out We need to find out the week,... View Article

Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using a similarity criterion for two triangles, show that AO/OC = OB/OD

Given that Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O To Prove AO/OC = OB/OD Proof In ΔDOC... View Article

Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ΔPQR (see Fig.) . Show that: (i) ΔABM ΔPQN (ii) ΔABC ΔPQR

Solution Given AB=PQ BC=QR AM=PN AM and PN are medians To Prove (i) ΔABM ≅ ΔPQN (ii) ΔABC ≅ PQR Proof In ΔABM and ΔPQN AB=PQ AM=PN... View Article

Diagonals AC and BD of a quadrilateral ABCD intersect each other at P. Show that ar (APB) ×ar (CPD) = ar (APD) ×ar (BPC) . [Hint : From A and C, draw perpendiculars to BD.]

Solution: Given: The diagonal AC and BD of the quadrilateral ABCD, intersect each other at point E. Construction: From A, draw AM... View Article

One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “If you give me ten, I shall be six times as rich as you”. Tell me what is the amount of their (respective) capital? [From the Bijaganita of Bhaskara II] [Hint : x + 100 = 2(y – 100) , y + 10 = 6(x – 10) ].

Let us assume that One friend has capital amount= x Other friend has capital amount =y As per the given conditionsx + 100 = 2(y − 100)x +... View Article

Draw a histogram for the frequency table made for the data in Question 3 and answer the following questions. (i) Which group has the maximum number of workers? (ii) How many workers earn Rs. 850 and more? (iii) How many workers earn less than Rs. 850?

Solution (i) 830-840 is the group having maximum number of workers, 9, compared to other groups. On observing the graph we note... View Article

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Let us assume, The speed of the train be x km/hr. Time taken to cover 360 km = 360/x hr. As per the question given, ⇒... View Article

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

Given Width of canal (w) = 6 m Depth of canal (h) = 1.5 m Speed of canal = 10 km/hr = (10,000 m/hr) Find out We have to find the area... View Article

A hemi spherical tank is made up of an iron sheet 1cm thick. If the inner radius is 1 m, then find the volume of the iron used to make the tank. (Assume π = 22/7)

Given Inner Radius of the tank, (r) = 1m The thickness of hemispherical tank = 0.01m Therefore, the outer Radius (R) = 1 + 0.01 = 1.01m... View Article

The shape of the top surface of a table is a trapezium. Find its area if its parallel sides are 1 m and 1.2 m and perpendicular distance between them is 0.8 m.

Given One parallel side of the trapezium (a) = 1 m The second side (b) = 1.2 m and Height (h) = 0.8 m Find out We have to find out... View Article

A batch of bottles was packed in 25 boxes with 12 bottles in each box. If the same batch is packed using 20 bottles in each box, how many boxes would be filled?

Solution Let the number of boxes be x. the number of bottles and the number of boxes are inversely proportional. Therefore, from inverse... View Article

The slant height of a frustum of a cone is 4 cm and the perimeters (circumference) of its circular ends are 18 cm and 6 cm. Find the surface area of the frustum.

Given Slant height of frustum (l) = 4 cm Find out We have to find the surface area of the frustum Solution Let the radius of smaller... View Article

Hasan buys two kinds of cloth materials for school uniforms, shirt material that costs him Rs. 50 per metre and trouser material that costs him Rs. 90 per metre. For every 3 meters of the shirt material he buys 2 metres of the trouser material. He sells the materials at 12% and 10% profit respectively. His total sale is Rs. 36,600. How much trouser material did he buy?

Solution Let the trouser material bought by him be = 2x Let the shirt material be bought by him be = 3x Selling price of shirt material per... View Article

Three consecutive integers are such that when they are taken in increasing order and multiplied by 2, 3 and 4 respectively, they add up to 74. Find these numbers.

Solution Let the three consecutive integers Ire ‘I’, ‘I+1’ Ind ‘I+2’. According to the given details the... View Article

A joker’s cap is in the form of right circular cone of base radius 7 cm and height 24cm. Find the area of the sheet required to make 10 such caps. (Assume π = 22/7)

Given Radius of conical cap (r) = 7 cm Height of conical cap (h) = 24cm Find out We have to find the area of the sheet required to make 10... View Article

Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer: (i) (- 1, – 2) , (1, 0) , (- 1, 2) , (- 3, 0) (ii) (- 3, 5) , (3, 1) , (0, 3) , (- 1, – 4) (iii) (4, 5) , (7, 6) , (4, 3) , (1, 2)

(i) (- 1, – 2), (1, 0), (- 1, 2), (- 3, 0) Let the points (- 1, – 2), (1, 0), ( – 1, 2), and ( – 3, 0) be representing the vertices A, B, C, and... View Article

Rukshar painted the outside of the cabinet of measure 1 m ×2 m ×1.5 m. How much surface area did she cover if she painted all except the bottom of the cabinet?

Solution: Given Length of the cabinet, l = 2 m The breadth of the cabinet, b = 1 m Height of cabinet, h = 1.5 m Find out We have to... View Article

A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

Solution The chord AB is equal to the radius of the circle. OA and OB are the two radii of the circle. From ΔOAB. AB... View Article

A drinking glass is in the shape of a frustum of a cone of height 14 cm. The diameters of its two circular ends are 4 cm and 2 cm. Find the capacity of the glass.

Given Height of frustum (h) = 14 cm Radius of larger circle end (R) = 4/2 = 2 cm Radius of smaller circle end (r)= 2/2= 1 cm Find out... View Article

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