Q1
If a : b : c : d = √4 : √3 : √2 : √1, then what is the value of ((-a²+b²+c²+d²))/((a²-b²+c²-d²)) ?
Elementary MathematicsRatio and Proportion
UPSC CDS · Solved paper
All 100 questions, textbook-style — tap “Show answer” on any question to reveal the correct option and explanation.
If a : b : c : d = √4 : √3 : √2 : √1, then what is the value of ((-a²+b²+c²+d²))/((a²-b²+c²-d²)) ?
The speeds of four cars are 2u, 3u, 4u and xu and the time taken by them to cover the same distance is xt, 4t, 3t and 2t respectively, where x, u, t are real numbers. What is the value of x?
If m : n = 1 : 2 and p : q = 3 : 4, then what is (2m + 4p) : (n + 3q) equal to ?
If the rate of interest is 5%, then what would be the difference between compound interest and simple interest received on ₹ 10,000 (each) after 3 years from now?
A person bought a book at 3/4th of its listed price and sold it at 50% more than its listed price. What is the percentage of gain in the transaction?
If the difference between the interior and exterior angles of a regular polygon is 144°, then what is the number of sides of the polygon ?
If the sum and product of the roots of a quadratic equation are 2 and - 100 respectively, then which one of the following is correct?
If 2 is a zero of the polynomial p(x) = x³ + 3x² - 6x - a, then what is the sum of the squares of the other zeros of the polynomial?
If t = cos 79°, then what is cosec 79° (1 - cos 79°) equal to ?
Suppose p(x) = x⁴ + a₃x³ + a₂x² + a₁x + a₀ and q(x) = x⁴ + b₃x³ + b₂x² + b₁x + b₀ polynomials. If α, β, γ, δ are zeros of p(x) and α, β, γ, λ are zeros of q(x), then what is (p(x)-q(x))/((x-α)(x-β)(x-γ)) equal to ?
If the equation x cos θ = x² + p has a real solution for every θ where 0 ≤ θ ≤ (π)/(4), then which one of the following is correct?
What is the difference between the greatest value and the least value of cos²θ + 3sin²θ + 2 ?
ABC is a right-angled triangle, right-angled at B such that AB = 6 cm and BC = 8 cm. What is the perimeter of the square inscribed in the triangle ABC with maximum area?
What is the greatest value of k for which 2x² - 4x + k = 0 has real roots?
Consider the following data: 110, 41, 43, 95, 127, 99, 61, 92, 71, 93, 110, 36. If 93 is replaced by 94, then consider the following statements: 1. The difference between new median and old median is 1. 2. The difference between new mean and old mean is less than 0.1. 3. The difference between new mode and old mode is zero. Which of the statements given above are correct?
What is the digit at the 100th place of number (225)⁴⁰?
If a, b, c, d are natural numbers, then how many possible remainders are there when 1ᵃ+2ᵇ+3ᶜ+4ᵈ is divided by 10?
If n is a natural number, then what is the sum of all distinct remainders of 4ⁿ + 6ⁿ + 9ⁿ + 11ⁿ when divided by 10 for various values of n?
When the number (12345678910111213 ... 99100) is divided by 16, what will be the remainder?
A, B, C, D can complete a work in 3, 6, 9, 12 hours respectively. Further, only one person can work at a time in each hour and nobody can work for two consecutive hours. It is not necessary to engage all. What is the minimum number of hours that they will take to finish the work?
If p = √((a+√(a²+b³))) + √((a-√(a²+b³))), then what is p³ + 3bp equal to ?
A plank of wood 4.25 m long and 3.4 m wide is to be cut into square pieces of equal size. How many square pieces of largest size can be cut from the plank, if no wastage is allowed?
What is the HCF of x⁴ - 13x²y² - 300y⁴, x³ - 4x²y - 4xy² - 5y³ and x³ - 125y³ ?
If HCF of 768 and x⁶y² is 32xy for natural numbers x ≥ 2, y ≥ 2, then what is the value of (x + y)?
What is the smallest natural number n such that (n + 1) × n × (n - 1) × (n - 2) × ... 3 × 2 × 1 is divisible by 910?
The expression 555⁷⁷⁷ + 777⁵⁵⁵ is divisible by which of the following? 1. 2 2. 3 3. 37 Select the correct answer using the code given below:
Consider the following statements:
Which of the statements given above is/are correct ?
The sum of two positive numbers is 40. If the GM of these two numbers is lower than their AM by 20%, then what is the difference between the two numbers ?
50 men can complete a work in 40 days. They begin the work together but a batch of 5 men left after each period of 10 days. What is the time to complete the work?
If x = (1)/(2+(3)/(4+(5)/(6+(7)/(8+(9)/(10))))), then which one of the following is correct?
A bottle contains spirit and water in the ratio 1 : 4 and another identical bottle contains spirit and water in the ratio 4 : 1. In what ratio should the mixtures in the two bottles be mixed to get a new mixture in which the ratio of spirit to water is 1 : 3?
If 3 sin θ + 5 cos θ = 5, then what is the value of 5 sin θ - 3 cos θ ?
Consider the following in respect of the polynomial x⁴ᵏ + x⁴ᵏ⁺² + x⁴ᵏ⁺⁴ + x⁴ᵏ⁺⁶.
Which of the statements given above is / are correct ?
What is the minimum value of (sin²A+5sinA+1)/(sinA) for 0 < A ≤ (π )/(2) ?
What is (3)/(1²× 2²)+(5)/(2²× 3²)+(7)/(3²× 4²) + ... equal to ?
If (1)/(a+(1)/(b+(1)/(c+(1)/(d+(1)/(e)))))=(421)/(972), then what is the value of a × b × c × d × e?
A cube whose edge is 14 cm long has on each of its faces a circle of 7 cm radius painted yellow. What is the total area of unpainted surface? (Takе π = 22/7)
From a circular metal plate of radius 7 cm and thickness 0.16 mm, a sector is cut off containing an angle 150°. The remaining piece is moulded into a spherical bead of radius r. What is the value of r in cm?
The chord AB of a circle with centre at O is 2√3 times the height of the minor segment. If P is the area of the sector OAB and Q is the area of the minor segment of the circle, then what is the approximate value of P/Q ? (Take √3 = 1.7 and π = 3.14)
What is the area of the region between two concentric circles, if the length of a chord of the outer circle touching the inner circle at a particular point of its circumference is 14 cm ? (Takе π = 22/7)
In a right - angled triangle ABC, AB = 15 cm, BC 20 cm and AC = 25 cm. Further, BP is the perpendicular on AC. What is the difference in the area of triangles PAB and PCB ?
Let the positive numbers a₁, a₂, a₃, …, a₃ₙ be in GP. If P is the GM of a₁, a₂, a₃, ...., aₙ and Q is the GM of aₙ₊₁, aₙ₊₂, aₙ₊₃, …, a₃ₙ, then what is the GM of 3n numbers?
The cost price of y articles is equal to selling price of z articles. If y : z = 5 : 4, what is the profit percentage?
A sum of money invested at simple interest triples itself in 8 years and becomes n times in 20 years. What is the value of n?
If the work done by x men in (x + 1) days is equal to the work done by (x + 5) men in (x - 2) days, then what is the value of x?
If (a + b) : (b + c) : (c + a) = 5 : 7 : 6, then what is the value of (a - b + c) : (a + b - c)?
Let x be the compound interest at the end of 3 years on a sum of ₹ 1000 at the rate of 10% compounded annually and y be the simple interest at the end of 3 years on a sum of ₹ 1000 at the annual rate of 11%. What is the difference between x and y ?
In a quadrilateral ABCD, AB = 6 cm, BC = 18 cm, CD = 6 cm and DA = 10 cm. If the diagonal BD = x, then which one of the following is correct?
In a quarter circle of radius R, a circle of radius r is inscribed. What is the ratio of R to r ?
In a quadrilateral ABCD, AB = BC and CD = DA; AC and BD are diagonals such that AC = 6 cm and BD = 12 cm. What is the area of the quadrilateral?
If tan (3A) = cot (A - 22°), where 3A is an acute angle, then what is the value of A?
If (sinθ -cosθ +1)/(sinθ +cosθ -1) = p sec θ + q tan θ, where 0 < θ < (π )/(2), then what is p + q equal to?
The angles of elevation of the top of a tower from two points A and B at a distance of x m and (x + 5) m from the base of the tower of height 6 m and in the same straight line with it are complementary. What is the value of x?
Consider the following statements :
Which of the statements given above is/are correct ?
Two trains A and B leave Delhi for Hyderabad at 7 : 00 a.m. and 7 : 50 a.m. on the same day and travel at 80 kmph and 100 kmph respectively. After how many kilometers from Delhi will the two trains be together?
The length, breadth and height of a cuboid are increased by 10%, 20% and 50% respectively. What is the percentage increase in volume of the cuboid?
₹ 9400 is distributed among P, Q, R in such a way that if ₹ 93, ₹ 24, ₹ 55 are deducted from their respective shares, then they have money in the ratio 3 : 4 : 5. What is the share of P?
If P² varies as R and Q² varies as R, (P ≠ Q), then which of the following are correct?
Select the correct answer using the code given below:
p number of men can finish a piece of work in q days. If there are 50% more men, then the work will be finished 12 days earlier. What is the value of q?
What is the minimum value of ((a²+3a+1)/(a))((b²+3b+1)/(b)) for a, b > 0?
Consider the following for the next ten (10) items that follow: Each item contains a Question followed by two Statements. Answer each item using the following instructions: Let a, b, c and d be positive integers. Question : Which one of a, b, c, d is closest to the product abcd?
Consider the following for the next ten (10) items that follow: Each item contains a Question followed by two Statements. Answer each item using the following instructions: Let mn = k, where m and n are prime numbers and k is an even number. Question : What is the value of mn - n + 1?
Consider the following for the next ten (10) items that follow: Each item contains a Question followed by two Statements. Answer each item using the following instructions: Question : If p is a positive integer, then what is the remainder when pn is divided by p + 1?
Consider the following for the next ten (10) items that follow: Each item contains a Question followed by two Statements. Answer each item using the following instructions: Question : Is xy positive?
Consider the following for the next ten (10) items that follow: Each item contains a Question followed by two Statements. Answer each item using the following instructions: Let a, b and c be the sides of a triangle ABC. Question : Is the triangle equilateral?
Consider the following for the next ten (10) items that follow: Each item contains a Question followed by two Statements. Answer each item using the following instructions: Area of a rectangle with length x and breadth y is P and area of a parallelogram (which is strictly not a rectangle) with adjacent sides of length x and y is Q. Question : Is P > Q?
Consider the following for the next ten (10) items that follow: Each item contains a Question followed by two Statements. Answer each item using the following instructions: Area of a rectangle with length x and breadth y is P and area of a parallelogram (which is strictly not a rectangle) with adjacent sides of length x and y is Q. A circle touches all the four sides AB, BC, CD, DA of a quadrilateral ABCD. Question : What is the perimeter of the quadrilateral ?
Consider the following for the next ten (10) items that follow: Each item contains a Question followed by two Statements. Answer each item using the following instructions: Area of a rectangle with length x and breadth y is P and area of a parallelogram (which is strictly not a rectangle) with adjacent sides of length x and y is Q. Question : What is the ratio of the lengths of diagonals of a rhombus ?
Consider the following for the next ten (10) items that follow: Each item contains a Question followed by two Statements. Answer each item using the following instructions: Area of a rectangle with length x and breadth y is P and area of a parallelogram (which is strictly not a rectangle) with adjacent sides of length x and y is Q. The chord of a circle of radius R touches at a point on the circumference of a concentric circle of radius r. The length of the chord is 24 units. Question : What are the values of r and R?
Consider the following for the next ten (10) items that follow: Each item contains a Question followed by two Statements. Answer each item using the following instructions: Area of a rectangle with length x and breadth y is P and area of a parallelogram (which is strictly not a rectangle) with adjacent sides of length x and y is Q. P, Q, R, S are the mid - points of sides AB, BC, CD, DA respectively of a quadrilateral ABCD. Question : What is the difference in the area of the quadrilateral ABCD and the area of the quadrilateral PQRS ?
If P is the area of the smallest sector and Q is the area of the largest sector, then what is P + Q equal to ?
If p is the perimeter of the smallest sector, then what is the value of 9p?
How far from station Q will the two trains meet?
At what time will the two trains meet?
If the lengths of the two trains A and B are 400 m and 500 m respectively, then what is the time taken by them to cross each other?
What is the area of the square?
What is tan α + tan β equal to ?
What is the area of triangle CDE?
What is the area of the shaded region?
What is the ratio of the area of the shaded region to the area of the non - shaded region?
What is the length of MN ?
What is the ratio of the area of the quadrilateral PQNM to the area of the quadrilateral RSNM?
If the people coming from a particular State belonging to S are 15% and 24,000 in number, then what is the total number of migrating people belonging to the age group B ?
What is the maximum of differences between the number of people coming from different groups P, Q, R and S?
What is the difference between number of people coming from R having age group A and those coming from Q having age group D ?
What is the area of the overlapping region?
What is the area of the non - overlapping region?
What is AB + BC equal to ?
What is the radius of the circle with centre at O₁?
What is the radius of the circle with centre at O₂?
What is the area of triangle EOF?
What is the area of trapezium AEFB?
What is the area of the shaded region ?
What is the length of the side of the triangle ABC?
What is the area of the shaded region?
What is the sum of the areas of the two circles?
Which one of the following is correct in respect of angle θ ?
What is the area of the shaded region?
What is the ratio of the area of the shaded region to that of the non - shaded region?
What is the perimeter of the shaded region?