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P = sin θ (sin θ + sin 3θ)
= sin θ (2sin 2θ. cos θ)
[∵ sin C + sin D = 2sin (C+D/2).cos (C−D/2)]
⇒ P = sin 2θ (2sin θ. cos θ)
= sin 2θ.sin 2θ = sin^{2} 2θ
P = sin^{2} 2θ ≥ 0
for all real θ
4^{(n+m1+mn2m)}x5^{(m1+m+n22mn)}x3^{(mn+m+n22m+2}
= 4^{1} x 5^{3} x 30
= 1/4 x 1/125 x 1 = 1/500
Perimeter of an equilateral triangle is equal to the circumference of a circle. The ratio of their areas is :
(Use π= 22/7)
3a = 2πr
⇒ a = 2/3πr
Area of an equilateral triangle:
AB = 8 cm and CD = 6 cm are parallel chords on the same side of the centre of a circle. The distance between them is 1cm. The radius of the circle is:
Let OE= x
OF = x+1, OA=OC= r
AE= 4cm, CF= 3cm
From ∆ OAE
r^{2 }= 16+x^{2}
x^{2 }= r^{2} 16 (1)
From ∆ OCF
(x+1)^{2} = r^{2}  9  (2)
By equation (2)  (1)
x = 3cm
r = 5cm
A tank that would normally be filled in 8 hours is now taking 2 hours more because of a leak. In how much time will the leak empty the full tank?
Time taken to fill the tank due to leak = 8 + 2 = 10 h
Let the leak take x hours to empty the filled tank.
Now, according to the question:
Time taken by leak to empty the tank = X = 40 hours
A sum of Rs. 210 was taken as a loan. This is to be paid back in two equal installments. If the rate of interest be 10% compounded annually, then the value of each installment is
Let x be the value of installment.
Principal = Present value of x for 1 year + Present value of x for 2 years
The difference between simple and compound interest for the fourth year is ₹ 7280 at 20% p.a. What is the sum?
Difference between CI and SI for the n^{th} year
CI for 4^{th} year = 20736  17280 = 3456
SI for 4^{th} year = 2000
Difference between CI and SI = 1456 for Rs10000
So, the difference of Rs7280 is for Rs50,000
At present, the sum of ages of R and K is 63 years. The ratio of their ages after 7 years will be 7:4; what is the present age of R?
⇒ present age of K is 63  x
7 years later
R's age = x+7
K's age = 63x+7 = 70x
⇒ (x+7)/(70x) = 7/4
⇒ 4x+28 = 4907x
⇒ 11x = 462
⇒ x = 42 years
therefore present age of R = 42 years
If cosec θ = x + 1/4x, then the value of cosec θ + cot θ is:
⇒ cosec θ  cot θ = 1/k
On adding, we get
2 cosec θ = k + 1/k = 2(x+1/4x) = k+1/k
⇒ 2x+1/2x = k+1/k ⇒ k = 2x
Base of a right pyramid is a square of area 324 sqm. If the volume of the pyramid is 1296 cu.m., then the area (in m^{2}) of the slant surface is:
In the given figure AB  CD  EF  GH and BH = 100 cm. Find the value of DF.
AG/BH = CE/DF
120/100 = 30/DF ⇒ DF = 100×30/120 = 25
If sin (10° 6’ 32”) = a, then the value of cos (79° 53’ 28”) + tan (10° 6’ 32”) is:
cos 79^{0} 53’ 28’’ + tan 10^{0} + 6’ 32’’
In the figure AB  CD, ∠DPL= 1/2∠NPO, OP is the perpendicular on MN, the value of xº is:
∠ONP = 60°
∠OPN = 90°  ∠ONP = 90° 60°
⇒ OPN = 30°
Find the area between two concentric circles of circumference 88 cm and 132 cm.
A.T.Q.,
A man moving at a speed of 36 kmph covered the shadow of a tower of height 75 m in 3 seconds. If the height of the person is 1.75 m, the length of the shadow cast by the man in meters is:
Distance travelled in 3 seconds with a speed of 36 kmph is =
(5/18)×(3)×(36) = 30m.
From the figure tan θ = BC/AB = 75/30 = 5/2
The height of the person is 1.75 m
Let the length of the shadow be x m.
tan θ = 1.75/x ⇒ 52 = 1.75/x
⇒ x = 1.75/5(2) = 0.7
A dress is 25% off. If both the cost and selling prices are ₹ 100 less, the profit would be 35%. Find its cost price?
C.P = 25+10/10×100
= ₹ 350
The speed of a car during the second hour of its journey is thrice that in the first hour. Also, its thirdhour speed is the average speed of the first two hours. Had the car travelled at the second hour’s speed during all the first three hours, then it would have travelled 150 km more. Find the percentage reduction in time if the original distance covered by the second hour’s speed:
∴ Percentage increase in speed = 3x/6x × 100 = 50%
Since speed is increased by (50%) ½
Therefore, time will reduce by (33.33%) 1/3
If A+B = 45°, then find the value of tan A + tan B + tan A tan B.
A+B = 45°
tan (A+B) = tan 45°
tan A + tanB + tanAtanB = 1
If a + b +c = 13, what is the maximum value of (a 3)(b 2) (c + 1) ?
∵ a + b +c = 13
∴ (a 3) + (b 2) + (c + 1) = 13 – 3 – 2 + 1 = 9
For the maximum value of (a 3)(b 2) (c + 1)
= (a 3)= (b 2)= (c + 1)= 9/3 = 3
So, (a 3)(b 2) (c + 1) = 3 × 3 × 3 = 27
The length of three medians of a triangle are 9 cm, 12 cm and 15 cm. The area (in sq. cm) of the triangle is:
Area of triangle = Area of a triangle formed by taking median as a side of the triangle
A = 9, B = 12, C = 15
The floor of a room is of size 4 m x 3 m, and its height is 3 m. The walls and ceiling of the room require painting. The area to be painted is
= 2 x height (length + breadth)
= 2 x 3(4+3) = 42 sq. metre
Area of ceiling = 4 x 3
= 12 sq. metre
Required area = 42 + 12
= 54 sq. metre
If 5 girls can embroider a dress in 9 days, then the number of days taken by 3 girls will be _____.
Days taken by 1 girl to embroider a dress = 9 x 5 days
Days taken by 3 girls to embroider a dress = 9 × 5/3 days = 15 days
How many metres length of cloth 10 m wide will be required to make a conical tent with base radius of 14 m and height is 48 m?
= √(14^{2} + 48^{2})
= √196 + 2304
= √2500
l = 50 m
Lateral surface area of a cone = πrl
Hence length of cloth =
= 220 m
Two trains are travelling in the same direction at 40 kmph, and 22 kmph completely pass off another in 1 minute. If the length of the first train is 125 metres, what is the length of the second train?
In 1 minute= 18×1000/60 = 300 m
∴ Length of second train = (300125) = 175 m
The ratio between the ages of a father and a son at present is 5: 2, respectively. Four years hence the ratio between the ages of the son and his mother will be 1: 2 respectively. What is the ratio between the present ages of the father and the mother, respectively?
After 4 years, Son’s age = (2x + 4) years
According to question
After 4 years
The ratio between the present age of the father and mother = 5x: (4x + 4)
∴ We cannot find the exact value of the ratio.
Age of mother = 2 x (2x+ 4) = 4x+ 8 years
Present age of Mother = 4x + 8  4
= 4x +4
The ratio between the present age of the father and mother = 5x: (4x + 4)
∴ We cannot find the exact value of the ratio.
If x^{4} + 1/x^{4} =119 and x > 1, then the value of x^{3} − 1/x^{3} is:
The total annual Co_{2} emissions from various sectors are 5 MMT. In the pie chart given below, the percentage distribution to Co_{2} emission from various sectors are indicated.
Q. Emission of Domestic sector is how much % of transport and commercial sector combined?
15/50 × 100 = 30%
The total annual Co_{2} emissions from various sectors are 5 MMT. In the pie chart given below, the percentage distribution to Co_{2} emission from various sectors are indicated.
Q. Which of the following sectors have an emission difference of 1 MMT between them?
1 mmt = 20% = Thermal – Domestic
Twenty percent of Anuj’s annual salary is equal to seventyfive per cent of Raj’s annual salary. Raj’s monthly salary is 60% of Ravi’s monthly salary. If Ravi’s annual salary is Rs. 1.44 lacs, what is Anuj’s monthly salary?
Given:
20% of Anuj's annual salary = 75% of Raj's annual salary
Raj's annual salary = 60% of Ravi's monthly salary
Ravi's annual salary = Rs. 1.44 lakhs = Rs. 144000
Concept:
20% = 1/5, 75% = 3/4, 60% = 3/5
Calculation:
Ravi's monthly salary = 144000/12 = 12000
60% of Ravi's monthly salary = Raj's annual salary
⇒ (3/5) × 12000 = Rs. 7200 = Raj's annual salary
Now, 1/5 of Anuj's annual salary = 3/4 of 7200 = 5400
∴ Anuj's annual salary = 5 × 5400 = Rs. 27000
The product of two successive numbers is 4032. Which is the greater of the two numbers?
63 x 64 = 4032
In the following question, select the related letter/word/number from the given alternatives.
MUSIC: TUNE
Find out similar set of the given set:
(64, 216, 125)
(4, 6, 5). Similarly, (8, 27, 64) is the perfect cube of (2, 3, 4).
Pointing to a boy Neha said, ‘He is the son of my grandfather’s only child. How is she related to that boy?
In a code language, ‘S1357’ means ‘SWe are very happy’, ‘S2639’ means ‘Sthey are extremely lucky’, and ‘S794’ means ‘SHappy and lucky’. Which digit in that code language stands for ‘Svery’?
2639 → they are extremely lucky ......(ii)
794 → happy and lucky ......(iii)
from (i) and (ii)
3 → are
from (ii) and (iii)
9 → lucky
from (i) and (iii)
7 → happy
from (iii)
4 → and
from (i)
1, 5 → we, very
Hence, we cannot find the exact code for 'very'.
In a certain code language, the word ‘FUTILE’ is written as ‘HYVMNI’. How will the word ‘PENCIL’ be written in that language?
Laxman started walking from his house and went 15 km to the North, then he turned West and covered 10 km. Then he turned South and covered 5 km. Finally, turning to the East, he covered 10 km. In which direction he is from his house?
From the figure, it's clear that Laxman is in the north of his house.
Given below are two Matrices of Twentyfive Cells, each containing two classes of alphabets. The columns and rows of Matrix I are numbered from 0 to 4 and that of Matrix II from 5 to 9. A letter from these matrices can be represented first by its row number and next by its column number. In each of the following questions, identify one set of number pairs out of (1), (2), (3) and (4), which represents the given word.
RUST
RUST
R– 57, 97, 99
U– 68, 78
S– 56, 58, 98
T– 59, 67
Therefore, the correct answer is option (3) i.e. 97, 68, 56, 59
Arrange the following in a meaningful order:
1. Curd
2. Grass
3. Butter
4. Milk
5. Cow
In the questions given below, some relationship has been expressed through symbols as shown below. Based on the meaning of these symbols and choose the correct answer.
ϕ means ‘less than.’
Δ means ‘not greater than.’
 means ‘equal to.’
+ means ‘not equal to.’
× means ‘not less than.’
= means ‘greater than.’
X ϕ Y  Z implies
X ϕ Y  Z ⇒ X < Y = Z
Using the proper notations/symbols in option (c), we get
X + Y × Z ⇒ X ≠ Y > Z ⇒ X < Y = Z
Therefore, X ϕ Y  Z ⇒ X + Y × Z.
Choose the odd one out from the given alternatives.
Pattern is
Find the odd one out from the given alternatives.
In the question below, there are few statements followed by few conclusions. You have to take the given statements to be true even if they seem to be at variance with commonly known facts and then decide which of the given conclusions logically follow(s) from the given statements.
Statements:
All horses are rivers.
All rivers are jungles.
No Jungle is a flower.
Conclusions:
I. No horse is a flower.
II. No flower is a horse.
If both Conclusions I and II follow.
Choose the missing terms out of the given alternatives.
Ef_h_gfe_hef_ehgefg_
What should come next in the following letter sequence:
A A B A B C A B C D A B C D E A B C D E F A B C D E F G A B C D E F G
In the given figure, the circle represents youths, the triangle represents the persons who are in jobs, and the rectangle represents the illiterate person.
Which of the following numbers depicts illiterate youths who have jobs?
Find the water image of the object given in the question figure.
A Piece of paper is folded and cut as shown below in the question figure. From the given answer figures, indicate how it will appear when opened.
In the following questions, four/five figures are given. Three/four are similar in a certain way, and so form a group. Find out which one of the figures does not belong to that group.
Which of the following words cannot be formed from 'GENERALIZATION'?
If ‘+’ stands for multiplication, ‘x’ stands for division, ‘–’ stands for addition, and ‘÷’ stands for subtraction, what is the answer for the following equation?
20 – 5 ÷ 18 x (3 + 2) = ?
Given expression is
20  5 +18x (3 + 2) = ?
After conversions:
? = 20 + 5 —18 ÷(3 x 2)
or, ? = 20 + 5  18 +6
or, ? 20 + 5  3
or. ? = 25  3 = 22
The day on 18.09.1977 was Sunday. A couple was married on this date. In the next 15 years, how many marriage anniversaries would fall on Sunday?
Therefore,
Number of odd days from 18.09.1977 to 18.09.1983=7, i.e., 0 odd day
It means that in 1983, 18th September would fall on Sunday.
and number of odd days from 18.09.1977 to 18.09.1988 = 14, i.e., 0 odd days.
Now, it is clear that 2 marriage anniversaries would fall on Sunday in the next 15 yr.
In a row of girls, if Rohini who is 12th from the left and Mihika who is 9th from the right, if they interchange their seats, then Rohini becomes 18th from the left. How many girls are there in the row?
Study the diagram given below and answer the question.
Which part represents those clerks who are graduates and have government jobs?
Who was recently elected as the Deputy Chairman of the Rajya Sabha in the Parliament?
Earlier in his career, he was a journalist with 'Prabhat Khabar' and then was chosen as the media advisor to former Prime Minister Chandra Shekhar.
Recently, the Ministry of External Affairs organised the 11th World Hindi Conference at which of the following Indian ocean states?
The theme of the Conference: "Hindi Vishwa Aur Bhartiya Sanskriti''. It was organised by the Ministry of External Affairs, Government of India, in association with the Government of Mauritius.
Right to Information, an important legislative act to promote transparency & strengthen the democracy of India, came into being in which year?
Using rtionline.gov.in, citizens of India can seek information to turn India into a participatory democracy.
According to the government guidelines, 'Any person who desires to obtain information shall submit a written or electronic request in English or Hindi or in the official language of the area to the Central Public Information Officer or his/her counterpart at the state level.'
Which Indian state excelled to be the top performer in the recently released 'Ease of Living Index 2018' by the Ministry of Housing & Urban Affairs?
The Index seeks to assist states in undertaking a 360degree assessment of their strengths, weaknesses, opportunities, and threat & innovation to perform in the governance, identity and culture, education, health, safety and security, economy, affordable housing, land use planning etc. for its residents.
Which of the following government entities started 'Seva Bhoj Yojana’  a scheme to reimburse the central share of CGST and IGST on food, prasad, langar etc., offered by religious and charitable institutions?
Institutions like Temples, Gurudwara, Mosque, Church, Dharmik Ashram, Dargah etc., who are established and serving free food for more than 5 years to more than 5000 people every month without any discrimination, are the eligible entities.
Google recently unveiled ____________ which will quickly render any PDF with Indian language content into editable text using Artificial Intelligence.
Under the project, the registered publishers will be provided with a free web hosting service and a branded domain for 3 years and have AdSense support to help monetise their content.
Which of the following Indian states launched a scheme, 'Kanyashree', to incentivise adolescent girl child education to combat the issue of child marriages?
Eligible girls got ₹1,000 a year scholarship from ages 13 to 18 years and a onetime grant of ₹25,000 on remaining unmarried till 18 years. More than 5 million girl children benefited from the scheme.
Recently the annual income limit was relaxed to any girl child within the age limit.
Which of the following government institutions launched "Pitch to MOVE", a business opportunity platform for budding entrepreneurs?
It aims to identify and incentivise the startups, which will help the Government realize its vision of Shared, Connected, Intermodal and EnvironmentFriendly Mobility for India.
The objective is to harness the latest disruption for generating employment and growth in our country.
Ghoomar is a popular traditional folk dance from which region in India?
Attire for women: Ghagras along with colourful chunaris and heavy embroidery. Works garment.
Which of the following International Organisations are headquartered in Washington D.C., USA?
Match the following trophies with their respective sports
Dhauladhar National Park is situated in which of the following Indian states?
Daman & Diu is the Union territory of India comprising 2 different areas separated by which of the following?
With an area of only 112 square km, it is the second smallest UT of India after Lakshadweep.
Recently, BOMBAY NATURAL HISTORY SOCIETY opened its regional centre at which of the following important Wetlands in India?
Here, BNHS would be engaged in identifying the air route of the migratory birds flocking the Chilika lake during winter & sample collection, training related to bird census, among other important conservatory purposes.
With which of the following Financial Institutions would you associate 'Prompt Corrective Action'?
 capital to riskweighted assets ratio (CRAR),
 net nonperforming assets (NPA) and
 Return on Assets (RoA), applicable only to commercial banks and not extended to cooperative banks, nonbanking financial companies (NBFCs) & other financial institutes
Which of the following rays exhibits the property of a light wave?
A ball thrown vertically upwards with an initial velocity of 19.6 m/s returns in 4 seconds. Which one of the following does the ball attain the maximum height?
= 2 Second
Now
Madam Curie invented Radium. She was born in ____.
When a body is immersed in a liquid, the force acting on it is?
Which of the following is the best conductor of Electricity?
Mach number is used in connection with the speed of 
If linear momentum is increased by 50%, the kinetic energy will increase by
Which of the following colours will scatter the most while passing through a prism?
Brass gets discoloured in the air due to the presence of which gas in the air?
At high altitudes, the boiling point of water reduces because 
The point through which the whole weight of the body acts, irrespective of its position, is known as
Avogadro constant, NA is the number of molecules in one mole of a substance, and its value (N_{A}) is
Kaziranga National Park is located in the state of
When CaC_{2} is heated in atmospheric nitrogen in an electric furnace, the compound formed is ______.
In which of the following blocks of the periodic table are the Alkali & Alkaline earth metals included?
Permanent hardness of water may be removed by the addition of ____.
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