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A factory requires 42 machines to produce a given number of articles in 63 days. How many machines would be required to produce the same number of articles in 54 days?

Solution Let the number of machines required be x. The number of machines and the number of days are inversely proportional. Therefore, from... View Article

Parallelogram ABCD and rectangle ABEF are on the same base AB and have equal areas. Show that the perimeter of the parallelogram is greater than that of the rectangle.

Given: ||gm ABCD and rectangle ABEF lie on the same base AB. ar(ABCD)=a(ABEF) To prove: Perimeter of ||gm ABCD> perimeter of rectangle... View Article

In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig.) . Show that: (i) ΔAPD ≅ ΔCQB (ii) AP = CQ (iii) ΔAQB ≅ ΔCPD (iv) AQ = CP (v) APCQ is a parallelogram

ABCD is a parallelogram DP = BQ (i) ∆APD ≅ ∆CQB As ABCD is a parallelogram ∠ADB = ∠CBD [Alternate interior angles are equal]……………….(1)... View Article

The following tables gives production yield per hectare of wheat of 100 farms of a village. Change the distribution to a more than type distribution and draw its ogive.

Production Yield 50-55 55-60 60-65 65-70 70-75 75-80 Number of farms 2 8 12 24 38 16 Solution: Production Yield (kg/ha) Number... View Article

Amina thinks of a number and subtracts 5/2 from it. She multiplies the result by 8. The result now obtained is 3 times the same number she thought of. What is the number?

Solution Let the number be x, According to the given details, the equation becomes (x – 5/2) × 8 = 3x Solving for x we get... View Article

A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the cost of painting the curved surface of the pillar at the rate of Rs. 12.50 per m2. (Assume π = 22/7)

Given Diameter of cylindrical pillar=50 cm Height of cylindrical pillar (h)=3.5 m Find out We have to find the cost of painting the curved... View Article

In Fig. XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that ∠ AOB = 90°.

AP and AC are tangent at same external point A. and, QB and BC are tangent at same external point B. Considering, ∆OAP and ∆OAC in consideration... View Article

Write first four terms of the A.P. when the first term a and the common difference are given as follows: (i) a = 10, d = 10 (ii) a = -2, d = 0 (iii) a = 4, d = – 3 (iv) a = -1 d = 1/2 (v) a = – 1.25, d = – 0.25

(i) a = 10, d = 10 Let us assume the Arithmetic Progression series... View Article

A bus stop is barricaded from the remaining part of the road, by using 50 hollow cones made of recycled cardboard. Each cone has a base diameter of 40 cm and height 1 m. If the outer side of each of the cones is to be painted and the cost of painting is Rs. 12 per m2, what will be the cost of painting all these cones? (Use π = 3.14 and take √(1.04) = 1.02)

Given Radius of cone, r = diameter/2 = 40/2 cm = 20cm = 0.2 m Height of cone, h = 1m Find out We have to find the cost of painting all these... View Article

Two customers Shyam and Ekta are visiting a particular shop in the same week (Tuesday to Saturday) . Each is equally likely to visit the shop on any day as on another day. What is the probability that both will visit the shop on (i) the same day? (ii) consecutive days? (iii) different days?

Given that Two customers Shyam and Ekta visit shop in the same week (Tuesday to Saturday). Each is equally likely to visit the shop on any day... View Article

The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is travelling at a speed of 66 km per hour?

Given The wheels of a car are of diameter 80 cm each. Find out We have to find out the revolutions each wheel makes in 10 minutes when the... View Article

Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.

Given ABCD is a square whose one diagonal is AC. ΔAPC and ΔBQC are two equilateral triangles described on the diagonals AC and... View Article

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 ceiling at the rate of Rs 7.50 per m²

Given, Length of room (l) = 5 m Breadth of room (b) = 4 m Height of room (h) = 3 m Area of walls + Area of ceiling = lb + 2lh + 2bh = (5 *... View Article

A study was conducted to find out the concentration of sulphur dioxide in the air in parts per million (ppm) of a certain city. The data obtained for 30 days is as follows: 0.03 0.08 0.08 0.09 0.04 0.17 0.16 0.05 0.02 0.06 0.18 0.20 0.11 0.08 0.12 0.13 0.22 0.07 0.08 0.01 0.10 0.06 0.09 0.18 0.11 0.07 0.05 0.07 0.01 0.04 (i) Make a grouped frequency distribution table for this data with class intervals as 0.00 – 0.04, 0.04 – 0.08, and so on. (ii) For how many days, was the concentration of Sulphur dioxide more than 0.11 parts per million?

Solution Grouped frequency distribution table with classes 0.00 – 0.04, 0.04 – 0.08, etc.. The data is represented in the grouped... View Article

The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret the two measures

No of Students per teacher Number of states / U.T 15-20 3 20-25 8 25-30 9 30-35 10 35-40 3 40-45 0 45-50 0 50-55 2 Solution... View Article

A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an angle of 115°. Find the total area cleaned at each sweep of the blades.

Given, Radius (r) = 25 cm Sector angle (θ) = 115° Formula The total area of the sector made by wiper = 2×(θ/360°)×π r2 =... View Article

Survey of India [UPSC Notes]

Survey of India is the national survey and mapping organization of India. It is featured regularly in the news as part of various surveying and... View Article

07 July 2021: UPSC Exam Comprehensive News Analysis

CNA 7th July 2021:- Download PDF Here TABLE OF CONTENTS A. GS 1 Related B. GS 2 Related POLITY AND GOVERNANCE 1. Bengal passes resolution to set... View Article

From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is (A) 7 cm (B) 12 cm (C) 15 cm (D) 24.5 cm

Option (A) 7cm Solution So, OP is perpendicular to PQ i.e. OP ⊥ PQ Given that OQ = 25 cm and PQ = 24 cm Find out We have to find out the... View Article

You have studied in Class IX that a median of a triangle divides it into two triangles of equal areas. Verify this result for ΔABC whose vertices are A (4, – 6) , B (3, – 2) and C (5, 2) .

Let us assume the vertices of the triangle be A (4, -6), B (3, -2), and C (5, 2). Let D be the mid-point of side BC of ΔABC. Therefore, AD... View Article