Q1
What is the ratio of interior angle to exterior angle of a regular polygon of n sides?
Elementary MathematicsQuadrilaterals and Polygons
UPSC CDS · Solved paper
All 100 questions, textbook-style — tap “Show answer” on any question to reveal the correct option and explanation.
What is the ratio of interior angle to exterior angle of a regular polygon of n sides?
41⁴³ + 43⁴³ is divisible by
If x = 7 + 4√3, then what is the value of √x + (1)/(√x)?
4⁶¹ + 4⁶² + 4⁶³ + 4⁶⁴ is divisible by
Suppose p and q are the LCM and HCF respectively of two positive numbers. If p : q = 14 : 1 and pq = 1134, then what is the difference between the two numbers?
What is the value of the following?((5.4)³ - 0.064)/((5.4)² + 2.16 + 0.16)
A man walks at an average speed of 3 km/hr from his residence and reaches office 40 minutes early. If he walks at an average speed of 2 km/hr, he reaches 40 minutes late. what is the distance between his residence and office?
What is the value of the following?(1)/(5√4 + 4√5) + (1)/(6√5 + 5√6) + (1)/(7√6 + 6√7) + (1)/(8√7 + 7√8) + (1)/(9√8 + 8√9)
If x = 9999, then what is the value of the following?(4x³ - x)/((2x + 1)(6x - 3))
If (x + √(1 + x²))(y + √(1 + y²)) = 1, where x and y are real numbers, then what is the value of (x + y)²?
What is the value of the following (2sin 68∘)/(cos22∘) - (2cot 15∘)/(5tan75∘)-(3tan20∘tan40∘tan45∘tan50∘tan70∘)/(5)
The perpendicular dropped from a vertex of the right-angled triangle upon the hypotenuse divides it into two segments of lengths 9 units and 16 units respectively. What is the length of the perpendicular?
If 43x × 47y = (2021)², x ≠ 0, y ≠ 0, then what is the value of the following?(4xy + x + y)/(2xy - x - y)
Let a, b, c, d be positive integers. If (1)/(a + (1)/(b +(1)/(c + (1)/(d))))= (17)/(60) then what is the product of a, b, c, d?
If x² = 17x + y and y² = x + 17y, x ≠ y, then what is the value of √(x²+y²+1)?
What is the least value of n if 194480 + n = m⁴, where m and n are natural numbers?
If (x-y)/(x√y + y√x) = (1)/(√x); (x > 0,y > 0) then what is the value of (x)/(y)?
What is the area of the region enclosed by three identical circles (each of radius 4 cm) touching each other?
A car travels from A to B at a speed of 40 km/hr, travels back from B to A at a speed of 30 km/hr and again goes from A to B at a speed of 60 km/hr. What is the average speed of the car?
What is the smallest natural number from the following which must be subtracted from 9410 to make the remaining number a perfect square?
27⁵ + 3¹³ is divisible by
Let p = 2²ⁿ ⁺ ² + m and q = 2⁴ⁿ - m (where n is even natural number). What should be the least value of m such that p as well as q is divisible by 5?
If squaring a positive real number x is same as adding 12 in it, then what is x equal to?
What is (1)/(1 + √2) + (1)/(√2 + √3) + (1)/(√3 + √4) +... + (1)/(√2020 + √2021) equal to?
If x + (1)/(x)= (5)/(2) then what is the value of the following?(5x)/(7x² - 3x + 7)
If a + b = 2, (1)/(a) + (1)/(b) = 2 then what is the value of a³ + b³?
8 men or 12 women can do a piece of work in 24 days. In how many days can the work be done by 8 men and 12 women?
A car takes p minutes to travel a distance of 350 km with an average speed of u km/hr. Another car takes q minutes to travel the same distance with an average speed of v km/hr. If u - v = 5 and q - p = 140, then what is the value of u?
How many minutes are there in x weeks and x days?
The arithmetic mean and the geometric mean of two positive numbers p and q (p > q) are A and G respectively. Which one of the following is correct?
If α and β are the roots of the quadratic equation x² + α x + β = 0, where β ≠ 0, then what is the value of α - β?
A shopkeeper marks the price of an article at ₹200. After allowing a discount of 10%, he still gains 20% on the cost price. What is the cost price of the article?
A person borrowed ₹9,000 at 7%, ₹12,000 at 8% and ₹15,000 at 9% simple interest per annum. He had to pay ₹50,700 at the end of n years. What is the value of n?
Consider the following statements :
Which of the above statements is/are correct?
What is the number of divisors of 1000 (excluding 1 and 1000)?
If the sum of the roots of the equation x² - k²x + 30kx - 161x - 64 = 0 is zero, then what is the difference of the roots?
A piece of cloth costs ₹10,000. If a 2 m longer piece of the same cloth is purchased for the same amount, it would cost ₹250 less per metre. What is the original length of the piece of cloth?
What is the condition that the roots of the equation ax² + bx + c = 0 are in the ratio c : 1?
Two sides of a triangle forming a right angle are 6x² and (2x² - 1). If the area of the triangle is 84 square units, then what is the perimeter of the triangle?
A train X takes 2 hours less than a train Y to cover a distance of 192 km between two cities. Their average speeds differ by 16 km/hr. How long does the faster train take to cover the journey?
Consider a question and two statements : Question : Is 3x + 2y positive?Statement - I : x³ = - 29.8Statement - Il : y³ = 3xWhich one of the following is correct in respect of the question and the statements?
Consider a question and two statements : Question : Does the equation ax² + bx + c = 0 have real roots of opposite sign?Statement - I : The discriminant D > 0Statement - II : (c)/(a) > 0Which one of the following is correct in respect of the question and the statements?
Consider a question and two statements : Question : Is a² + b² + c² - ab - bc - ca (a, b, c are distinct real numbers) always positive?Statement - I : a > b > cStatement - II : a + b + c = 0Which one of the following is correct in respect of the question and the statements?
Consider a question and two statement : Question : Is (x⁶ + y⁶)/(x⁴ + y⁴) always greater than (x⁴ + y⁴)/(x² + y²) (x ≠ y ≠ 0)?Statement - I : x > yStatement - II : x² + y² > 2xyWhich one of the following is correct in respect of the question and the statements?
How many quadratic equations have the sum of their roots equal to the product of their roots?
x = (1-cos θ +sinθ )/(1+sinθ ) then what is (sin θ + cosθ -1)/(cosθ ) equal to?
If cos (x + y) = 0 and sin (x - y) = (1)/(2), where x, y ϵ [0, (π )/(2)] then what is the value of cot (2x - y)?
What is the minimum value of sin⁴ θ + cos⁴ θ - 2sin² θ cos²θ?
If cosθ + secθ - 2 = 0, where 0 ≤ θ < (π)/(2), then what is the value of cos⁴θ + sec⁴θ - 2?
If y = cos² x + sec² x, where 0 ≤ x < (π)/(2), then which one of the following is correct?
If A, B, C are acute angles and sin(B + C - A) = cos(C + A - B) = tan(A + B - C) = 1 then what is (A + B + C) equal to?
If A, B, C, D are the angles of a cyclic quadrilateral, then what is the value of the following?sin((A + C)/(2)) + sin((B + D)/(2))
An aeroplane is observed to be approaching the airport. It is at a distance of 10 km from the point of observation on the ground and makes an angle of elevation θ. If the aeroplane is at a height of 8 km above the ground, then which one of the following is correct?
The angle of elevation of the top of a tower of height x metre from a point on the ground is found to be 60°. By going y metre away from that point, it becomes 30°. Which one of the following relations is correct?
What is (sec²α + tanα · tanβ - tan²α)² + (tanα - tanβ)² - sec²α · sec²β equal to?
If tan θ + sec θ = 3, then what is the value of 3 tan θ + 9 sec θ?
Consider the following :
Which of the above is/are an identity/identities?
What is the length of the chord a unit circle which subtends an angle 2θ at the centre, where θ < 45°
If tan² θ + 3 sec θ - 9 = 0, where 0 < θ < 90^°, then what is the value of 12 cot² θ + 3 csc θ?
If (cosθ)/(cscθ+1)+(cosθ)/(cscθ-1)=2 where 0 < θ<90°, then what is the value of sin⁴θ+cos⁴θ?
Consider the following statements :
Which of the above statements is/are correct?
A circle of radius 25 cm has a chord of length 48 cm. What is the length of perpendicular drawn from the centre of the circle to the chord?
The surface area of a cube of length x is equal to the surface area of a sphere of radius y. Consider the following statements :
Which of the above statements is/are correct?
The radius and height of a right circular cone are in the ratio 3 : 7. If the volume of the cone is 528 cm³, then what is the height of the cone? (Take π=(22)/(7))
A cylindrical pipe has inner diameter of 14 cm. Water flows through it at a rate of 154 litres per minute. What is the speed of water in km/hr?(Take π =(22)/(7))
A spherical ball of lead 6 cm in diameter is melted and recast into three spherical balls. The diameters of the balls are 2 cm, 4 cm and x cm. Which one of the following is correct?
A rectangle of length 10 units and breadth 8 units is split into two squares each of area x square units and two rectangles each of area y square units. Consider the following statements :
Which of the above statements is/are correct?
A square sheet is formed by joining n identical square sheets of same size. If the length of the diagonal of the bigger square sheet so formed is m, then what is the side length of a smaller square sheet?
A rectangular sheet is of length x and breadth y. If p is the volume of the cylinder formed by rolling the sheet along its breadth and q is the volume of the cylinder formed by rolling the sheet along its length, and q = 2p, then which one of the following is correct?
A tall cylindrical container with circular base of radius 18 cm contains a good quantity of water. Metal balls each of radius 0.9 cm are immersed in it. How many balls are required to raise the water lever by 3 cm?
An equilateral triangular sheet is formed by joining 9 equilateral triangular sheets each of area 9√3 cm². What is the height of the bigger triangular sheet?
A farmland is in the shape of a rhombus. The perimeter of the land is 100 m and the length of one of the diagonals is 40 m. The land is divided into four equal parts. What is the area of each part?
ABCD is a trapezium in which AB is parallel to DC. Let E and F be the midpoints on AD and BC respectively. If EF = 10 cm and AB - DC = 4 cm, then what is the value of AB × DC?
ABCD is a parallelogram with AB = 15 cm and AD = 8 cm. If θ is the acute angle between AB and AD, then what is the area of the parallelogram in square cm?
If the perimeter of an isosceles right triangle is 4(2+√2) cm, then what is its area in triangle cm?
The diagonal of a square is 12√2 cm and the area of an equilateral triangle is 64√3 square cm. Which of the following statements is/are correct?1. The square and the triangle have the same perimeter.2. Four times the area of the square is equal to 3√3 times the area of the triangle.Select the correct answer using the code given below.
Consider the following statements :
Which of the above statements is/are correct?
Let D, E and F be the midpoints of the sides BC, CA and AB respectively of a triangle ABC. Triangle DEF is congruent to which of the following triangles?
Select the correct answer using the code given below.
In a triangle ABC, AB = AC and BC is produced to D such that ∠ACD = x, then what is ∠BAC equal to?
ABC is a triangle right angled at B with AC = 2BC. If ∠A = x, then what is ∠C equal to?
In a triangle ABC, AB = 16 cm, AC = 12 cm and AD is the bisector of ∠A. If BD = 4 cm, then what is CD equal to?
An equilateral triangle of side x is inscribed in a circle of radius y. which one of the following is correct?
ABC is a triangle right angled at B. Let D be the midpoint on AC. If BD = 6.5 cm, then what is AB² + BC² equal to?
Water is trickling out of a completely filled cylindrical tank of height 1 m and diameter 2 m. Every second a spherical droplet of 1 cm radius trickles down from the bottom of the tank. The tank will be emptied in approximately.
The length, breadth and height of a cuboid are in the ratio 27 : 8 : 1. The cuboid is melted and recast into a cube. If p is the surface area of the cuboid and q is the surface area of the cube, then what is p/q equal to?
In a right triangle ABC, BD is perpendicular on hypotenuse AC. If AC = 9 cm and AD = 4 cm, then what is AB + BC approximately equal to?
In a triangle ABC, AD is the bisector of ∠BAC. If AB = 12 cm, BD = 10 cm and DC = 5 cm, then what is the perimeter of the triangle?
What is the radius of the circle inscribed in a triangle whose sides are 4 cm, 7.5 cm and 8.5 cm?
In a shower, 5 cm of rain falls. What is the volume of water that falls on 2 hectare area of land?
A bicycle wheel of radius 35 cm makes n revolutions in moving 11 km. what is the value of n? (Take π =(22)/(7))
Consider the following for the next two (02) items that follow: The age distribution of a city population is: Below 30 years: 14.00%, 30-34.99 years: 29.75%, 35-39.99 years: 26.25%, 40-44.99 years: 0%, 45-49.99 years: 18.50%, 50 years and above: 11.50%. The number of persons below 40 years is 10.5 lakh. What is the total population of the city (in lakhs)?
Consider the following for the next two (02) items that follow: The age distribution of a city population is: Below 30 years: 14.00%, 30-34.99 years: 29.75%, 35-39.99 years: 26.25%, 40-44.99 years: 0%, 45-49.99 years: 18.50%, 50 years and above: 11.50%. The number of persons below 40 years is 10.5 lakh. If the ratio of taxpayers to other persons in the age group below 30 years is 1 : 2, then what is the number of taxpayers (in lakhs) in that age group?
Consider the following for the next two (02) items that follow: The expenditure (in suitable units) from 2011 to 2017 is: 2011: 13.8, 2012: 15.4, 2013: 10.4, 2014: 13.1, 2015: 15.8, 2016: 17.2, 2017: 19.4. How many times did the increase in expenditure in a year exceed 15% as compared to the previous year?
Consider the following for the next two (02) items that follow: The expenditure (in suitable units) from 2011 to 2017 is: 2011: 13.8, 2012: 15.4, 2013: 10.4, 2014: 13.1, 2015: 15.8, 2016: 17.2, 2017: 19.4. In which year is the percentage increase in expenditure maximum as compared to its previous year?
Consider the following for the next two (02) items that follow: A total budget of ₹300.4 lakh is allocated among five heads A, B, C, D, E with shares 40%, 18%, 9%, 25% and 8% respectively. How much less amount is allocated to A and C together as compared to B, D and E together?
Consider the following for the next two (02) items that follow: A total budget of ₹300.4 lakh is allocated among five heads A, B, C, D, E with shares 40%, 18%, 9%, 25% and 8% respectively. How much amount will be increased on A if the total budget is increased to three times?
Consider the following for the next four (04) items that follow: In a group of 500 candidates, 30 failed in English only, 75 failed in Hindi only, 50 failed in Mathematics only, 15 failed in both English and Hindi (inclusive), 17 failed in both English and Mathematics (inclusive), 17 failed in both Hindi and Mathematics (inclusive), and 5 failed in all three subjects. What is the percentage of candidates who failed in at least two subjects?
Consider the following for the next four (04) items that follow: In a group of 500 candidates, 30 failed in English only, 75 failed in Hindi only, 50 failed in Mathematics only, 15 failed in both English and Hindi (inclusive), 17 failed in both English and Mathematics (inclusive), 17 failed in both Hindi and Mathematics (inclusive), and 5 failed in all three subjects. What is the percentage of candidates who failed in only one subject?
Consider the following for the next four (04) items that follow: In a group of 500 candidates, 30 failed in English only, 75 failed in Hindi only, 50 failed in Mathematics only, 15 failed in both English and Hindi (inclusive), 17 failed in both English and Mathematics (inclusive), 17 failed in both Hindi and Mathematics (inclusive), and 5 failed in all three subjects. What is the percentage of candidates who failed in at least one subject?
Consider the following for the next four (04) items that follow: In a group of 500 candidates, 30 failed in English only, 75 failed in Hindi only, 50 failed in Mathematics only, 15 failed in both English and Hindi (inclusive), 17 failed in both English and Mathematics (inclusive), 17 failed in both Hindi and Mathematics (inclusive), and 5 failed in all three subjects. How many candidates passed in two or more subjects?