Q1
A real number x is such that the sum of the number and four times its square is the least. What is that number?
Elementary MathematicsAlgebra
UPSC CDS · Solved paper
All 100 questions, textbook-style — tap “Show answer” on any question to reveal the correct option and explanation.
A real number x is such that the sum of the number and four times its square is the least. What is that number?
The difference of the square of two natural numbers m and n(m > n) is 72. How many pairs of natural numbers will satisfy?
Let N be a 5 -digit number. When N is divided by 6, 12, 15, 24 it leaves respectively 2, 8, 11, 20 as remainders. What is the greatest value of N?
What is the remainder when 111²²² + 222³³³ + 333⁴⁴⁴ is divided by 5?
What are the last three digits in the multiplication of 4321012345 × 98766789
p varies directly as (x² + y² + z²). When x = 1, y = 2, z = 3, then p = 70. What is the value of p when x = -1, y = 1, z = 5?
Let N be the least positive multiple of 11 that leaves a remainder of 5 when divided by 6, 12, 15, 18. Which one of the following is correct?
What is (1)/(√(10)+√9)+(1)/(√(11)+√(10))+(1)/(√(12)+√(11))+…+(1)/(√(196)+√(195)) equal to?
Train X crosses a man standing on the platform in 24 seconds and train Y crosses a man standing on the platform in 18 seconds. They cross each other while running in opposite directions in 20 seconds. What is the ratio of speed of X to speed of Y?
Let p, q be the roots of the equation x²+mx-n=0 and m, n be the roots of the equation x²+px-q=0 (m, n, p, q are non-zero numbers). Which of the following statements is/are correct?
What is the maximum value of 8sinθ - 4sin2θ?
What is (1 + tanα tanβ)² + (tanα − tanβ)² equal to?
Consider the following statements :
If 0⩽(α -β )⩽(α +β )⩽(π )/(2)tan(α +β )=√3 and tan(α -β )=(1)/(√3), then what is tanα.cot2β equal to?
What is the value of sin2θ cos2θ (sec2θ + cosec2θ) equal to?
If 64^(sin²θ)+64^(cos²θ)=16 where 0 le θ le (π)/(2), then what is the value of tanθ + cotθ?
If cosecθ - cotθ = m and secθ - tanθ = n, then what is cosecθ + secθ equal to?
From a point X on a bridge across a river, the angles of depression of two points P and Q on the banks on opposite side of the river are α and β respectively. If the point X is at a height h above the surface of the river, what is the width of the river if α and β are complementary?
In a triangle ABC, ∠ABC = 60° and AD is the altitude. If AB = 6 cm and BC = 8 cm, then what is the area of the triangle?
If p and q are the roots of the equation x² - sin²θ x - cos²θ = 0, then what is the minimum value of p²+q²?
The arithmetic mean of n numbers is M. If the sum of first (n - 1) terms is k, then what is the n^(th) number?
What is the geometric mean of 3, 9, 27, 81, 243, 729, 2187 ?
A person purchases one kg of tea powder from each of the four places A, B, C, D at the rate of ₹1000 per 1 kg, 2 kg, 4 kg, 5 kg. If on an average he purchased x kg of tea powder per ₹ 1000, then what is the approximate value of x?
What is the sum of the largest and the smallest 4-digit numbers made by using single digit prime numbers (without repetition)?
What is the remainder when 3255 is divided by 28?
What is the value of x, (0 < x < 8) if (100⁹⁷ + 100⁵⁴ + x + 1) leaves a remainder 0 when divided by 9?
In a triangle ABC, D is a point on BC. If AB.DC = AC.BD, ∠BAD = α and ∠CAD = β then which one of the following is correct?
Let N = 12345678AB be a 10-digit number, where A, B are digits. If N is divisible by 9, then which of the following statements is/are correct?
If x³+(1)/(x³)=(65)/(8) and y³+(1)/(y³)=(730)/(27) then which one of the following is a value of xy?
If 11x + 5y is a prime number where x, y are natural numbers then what is the minimum value of (x + y)?
A 4-digit number N has exactly 15 distinct divisors. What is the total number of distinct divisors of N²?
If p, q and r are the lengths (in cm) of the sides of a right-angled triangle, then (p - q - r) (q - r - p) (r - p - q) is always
What is the minimum value of (a⁸+a⁴+1)(b⁸+b⁴+1)a⁴b⁴ where a > 0, b > 0?
In a class containing 200 students, n students prefer both tea and coffee; 2n students prefer coffee, 3n students prefer tea; 4n students prefer neither tea nor coffee. What is the value of n?
Let ABC be a triangle with area 36 square cm. If AB = 9 cm, BC = 12 cm and ∠ABC = θ, then what is cos θ equal to?
Let n be a natural number. The HCF of n, n + 10 is 10. If the LCM is x (a 2 - digit number), then how many values of x are possible?
What is HCF of a4 + 2a3 + 3a2 + 2a + 1 and a6 - 2a3 + 1?
If the roots of the equation x² - (k - 2)x + (k + 1) = 0 are equal, then what are the values of k?
What is ((cosθ -sinθ +1)/(cosθ +sinθ -1))(cotθ -cosecθ ) equal to?
What is (sinθ -2sin³θ )/(2cos³θ -cosθ ) equal to?
The mean weight of 150 students in a class is 60 kg. The mean weight of boys in the class is 70 kg and that of girls is 55 kg. What is the ratio of number of boys to number of girls?
Two towers A and B of height 23 m and 11 m respectively, stand 9 m apart. A straight rod is joined to the two tops of the towers. A monkey sitting on the top of A, climbs the rod to reach the top of B. If the monkey takes 5 minutes to reach the other end, what is the average speed of the monkey?
A spherical wooden ball of radius r is to be divided into eight identical parts by cutting by planes passing through the same diameter. What is the surface area of each final piece?
A trolley with two wheels one metre apart is moved clockwise on the circular track around a ground with radius 50 m (described by right wheel). If the size of each wheel is of 1 foot radius and the right wheel turns 1000 times, how many times will the other wheel turn?
What is the remainder when 70 × 71 × 72 × 73 × 74 × 75 × 76 × 77 × 78 × 79 is divided by 1000?
A vertical pole of length 80 m is situated on the horizontal plane. The base of the pole is at P. There are two points A and B such that P, A, B are on the same straight line. Let the angles of elevation of top of the pole from A and B be α and β (α > β) respectively. If PA = 64 m and AB = 36 m, then what is (α + β) equal to?
Let k be a positive integer. What is the quotient when x⁸ᵏ⁺³+x⁸ᵏ⁺⁶+x⁸ᵏ⁺⁹+x^(8k+12) is divided by (1 + x³) (1 + x⁶)?
A square is drawn inside a square of side 14 cm in such a way that the corners of the inner square coincide with the mid points of the sides of the outer square. What is the area lying between the two squares?
The base of a right-angled triangle is 4/3 times the height of triangle. If the area of the triangle is 54 square cm, then what is the perimeter of the triangle?
What is the area of a triangle having sides 4, 4 and 6 units?
Consider the following for the next three (03) items that follow: Let ABC be a triangle right-angled at B. Let P be the point on BC such that BP = PC. If AB = 10 cm, ∠BAP = 45° and ∠CAP = θ. What is tanθ equal to?
Consider the following for the next three (03) items that follow: Let ABC be a triangle right-angled at B. Let P be the point on BC such that BP = PC. If AB = 10 cm, ∠BAP = 45° and ∠CAP = θ. If ∠ACP = γ, then what is tanγ equal to?
Consider the following for the next three (03) items that follow: Let ABC be a triangle right-angled at B. Let P be the point on BC such that BP = PC. If AB = 10 cm, ∠BAP = 45° and ∠CAP = θ. Consider the following statements :
Consider the following for the next two items that follow: A frequency distribution is as follows: Marks: 18–26, 27–35, 36–44, 45–53, 54–62, 63–71, 72–80; Number of Students: 5, 7, 10, 15, 8, 3, 2. What is the median of the distribution?
Consider the following for the next two items that follow: A frequency distribution is as follows: Marks: 18–26, 27–35, 36–44, 45–53, 54–62, 63–71, 72–80; Number of Students: 5, 7, 10, 15, 8, 3, 2. What is the mode of the distribution?
Consider the following for the next two items that follow: ABC is a triangle right-angled at B. Given that AC − AB = 2 cm and BC = 16 cm. If ∠BAC = θ then what is sinθ + cosθ equal to?
Consider the following for the next two items that follow: ABC is a triangle right-angled at B. Given that AC − AB = 2 cm and BC = 16 cm. If BD is the perpendicular on the side AC, then what is the length of BD?
Consider the following for the next three (03) items that follow: Let MN be a chord of length 16 cm of a circle with centre at O and radius 10 cm. The tangents at M and N intersect at a point P. Further, OP intersects MN perpendicularly at Q. What is OQ equal to?
Consider the following for the next three (03) items that follow: Let MN be a chord of length 16 cm of a circle with centre at O and radius 10 cm. The tangents at M and N intersect at a point P. Further, OP intersects MN perpendicularly at Q. What is PM equal to?
Consider the following for the next three (03) items that follow: Let MN be a chord of length 16 cm of a circle with centre at O and radius 10 cm. The tangents at M and N intersect at a point P. Further, OP intersects MN perpendicularly at Q. What is the area of triangle OMN?
A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option. Question : What is the integral value of k for which the expression 4x² - kx + 1 is positive?
A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option. Question : In how many days can A, B and C together finish the work?
A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option. Question : Can we have a common solution which is prime?
A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option. Question : Is 327n + 173n divisible by 500?
A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option. Question : If the price of petrol goes up by 20%, by what percentage should the consumption be reduced so that the expenditure remains the same?
A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option.
Question: The ratio of P's salary to Q's salary is 6 : 5. How much is P's expenditure?
A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option.
Question: The largest of five different integers is 8 and least is 2. What is the average of these integers?
A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option.
Question: There are three different weights. All the weights are integers and their sum is a prime number. What are the weights?
A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option.
Question: What is the amount at the end of 10 years?
A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option.
Question: Is p2 + pq + q2 odd where p, q are integers?
The total population of an area is 10,000 out of which males and females are equal in number. Out of the total population 30% are Newspaper readers. Out of the total newspaper readers, one-third read English Newspaper. Out of the total English Newspaper readers, 20% are females. What is the number of males who do not read English Newspaper?
What is the maximum area of a rectangle, in square cm, whose perimeter is 400 cm?
What is the remainder if we divide 3¹⁰ by 7?
What is the square root of 64%?
The difference of 10³¹ − 5 and 10³⁰ + p is divisible by 3 where p is a digit. How many values of p are possible?
Consider the following statements :
The average of the temperatures recorded at noontime from Monday to Sunday is 31°C. If the lowest temperature recorded is 30°C, then what is the maximum of temperature that is possible to record at noontime on any one of the days
If (x+(1)/(yz))-(y+(1)/(zx))=(y+(1)/(zx))-(z+(1)/(xy))and x + z ≠ 2y, then what is xyz equal to?
Consider the following statements in respect of p = n(n +
What is the difference between the average of first 50 even natural numbers and the average of first 50 odd natural numbers?
Three amounts x, y, z are such that y is the compound interest on x; and z is the compound interest on y. The rate of interest per annum and the time period in years are same. Which one of the following is correct?
There are n concentric squares. The area of the innermost square is 1 unit and the distance between corresponding corners of any two consecutive squares is 1 unit. Consider the following statements :
In a rectangle ABCD, AC is one of the diagonals. If AC + AB = 3AD and AC - AD = 4 units, then what is the area of the triangle?
The area of the circle circumscribing three identical circles touching each other is π(2 + √3)²/3 square cm. What is the radius of one of the smaller circles?
In a triangle ABC, AB = 21 cm, BC = 20 cm and CA = 13 cm. A perpendicular CD is drawn upon the longest side. What is the area of the triangle BCD?
There are two containers A and B. In container A, the ratio of milk and water is 1 : 3 and in container B, the ratio of milk and water is m : n. If the mixture in the containers A and B are mixed in the ratio 2 : 3 to get 20 litres of a mixture having milk and water in the ratio 3 : 7, then what is the value of (m)/(n)?
A cone, a hemisphere and a cylinder stand on equal base of radius r and have the same height. If the sum of volumes of cone, the hemisphere and the cylinder is equal to volume of a sphere of radius R, then what is R³r³ equal to?
If x3 + px2 + qx + r is an integer for all integral values of x, then consider the following statements :
XYZ is a 3-digit number, where X, Y, Z are distinct non-zero digits. The difference between the two 3-digit numbers XYZ and YXZ is 90. How many possible values exist for the sum (X + Y)?
How many times does the minute hand of a clock coincide with the second hand between 2.01 pm and 4.01 pm on the same day?
What is the HCF of 2³⁶ − 1 and 2⁴⁵ - 1?
The section of a solid right circular cone by a plane containing vertex and perpendicular to base is an equilateral triangle of side 14 cm . What is the volume of the cone? ≤ft(π=22/7)
Three identical cones each with base radius 3 cm are placed on their bases so that each is touching the other two. There will be one and only circle that would pass through each of the vertices of the cones. What is the area of the circle?
A circle is inscribed in a triangle ABC right-angled at B. If AB = 5 cm and BC = 12 cm, then what is the radius of the circle?
The ratio of sum of interior angles to sum of exterior angles of a regular polygon of n sides is 7/2. What is the measure of an interior angle of polygon?
The number 199 can be written as m² − n², where m, n are natural numbers (m > n). What is the value of mn?
How many numbers of the form 2ⁿ - 1 and less than 2000 are prime?
In a class of 160 students, each of them opt at least one language from among English, Hindi and Sanskrit. It is found that 130 students opt English, 120 students Hindi and 110 Sanskrit. If the students opt either only one language or all three languages, then what is the number of students who study all three languages?
Let S = 5ᵃ+7ᵇ+11ᶜ+13ᵈ, where a, b c and d are natural numbers. What is the number of distinct remainders of S when it is divided by 10?
In a right triangle, ABC ∠A = 90° and AD is perpendicular to BC. If ∠CAD = 60° and BC = 6 cm, then what is AB equal to?