Q1
Two positive integers x and y are in the ratio 17 : 19. If the LCM of the two numbers is 1615, then what is (x + y) equal to?
Elementary MathematicsHCF and LCM
UPSC CDS · Solved paper
All 100 questions, textbook-style — tap “Show answer” on any question to reveal the correct option and explanation.
Two positive integers x and y are in the ratio 17 : 19. If the LCM of the two numbers is 1615, then what is (x + y) equal to?
If p/q = q/r = r/s = k, then what is (p³+q³+r³)/(q³+r³+s³) equal to?
If x² − 4x + 3 is a factor of x⁴ + px² + q, then what is (p − q) equal to?
Suppose p and q are positive integers where q is not a perfect square. If 1/(p+√q) is a root of a quadratic equation having integer coefficients, then which of the following statements is/are correct?
If 1/(2b+2c) + 1/(2c+2a) = 1/(a+b), then what is a²+b²+c² equal to?
A train completely crossed two persons X and Y moving in the same direction as the train in 8 seconds and 8.1 seconds respectively. X is walking at a uniform speed of 3 km/hr and Y is walking at a uniform speed of 4 km/hr. What is the speed of the train?
A number is formed by three digits, each less by unity than the digit that follows it. If 15 is added to the number, then the sum is 30 times the sum of the digits of the number. What is the product of the digits of the number?
Consider the following statements in respect of common logarithms :
Which of the statements given above is/are correct?
What is the number of zeros immediately after the decimal point in (0.5)¹⁰⁰⁰? (Given that log₁₀2 = 0.30103)
If x = 11 + 2√30, then what is the value of x^(1/2) + x^(−1/2)?
If 1/(x+k) + 1/(x+2k) + 1/(x+5k) = 1/k, then which one of the following is the solution of this equation?
Consider the polynomial p(x) = x⁵+2x⁴+x³−x²−2x−1 :
Which of the statements given above is/are correct?
If 5 is a root of the equation x²+px−55 = 0 and the quadratic equation x²+px+k = 0 has equal roots, then what is the value of k?
Had I walked 0.5 km/hr faster, I would have taken 8 minutes less to walk 2 km. What is my original speed?
If the roots of the equation x² + mx + n = 0 are increased by the same quantity k, then they become the roots of the equation x² + nx + m = 0. What is the value of (m+n)?
If α and β are the roots of the equation 2x² − 2(n+1)x + (n²+n+1) = 0, then what is (2α³ + 2β³) equal to?
The product of two numbers is 1050. The quotient when the larger number is divided by the smaller number is 4 and the remainder is 10. What is the sum of the two numbers?
What is the HCF of x³ + x²y − 10xy² + 8y³ and x³ + x²y − 4xy² − 4y³?
What is (√2 + √3 + √5)(√2 + √3 − √5)(√2 − √3 + √5)(−√2 + √3 + √5) equal to?
The LCM and HCF of two polynomials p(x) and q(x) are (x+k)²(x²−7x+6) and (x+k) respectively. If [x³+(2k−1)x²+k(k−2)x−k²] is one of the polynomials, then which of the following statements is/are correct?
A shopkeeper sold his goods at the cost price. By using false weights, he gained 11(1/9)%. What weight did he use for 1 kg?
If the roots of the equation px²−40x+96 = 0 are even integers, then what is the value of p?
A number N consists of two digits. The digit in the tens place is 3 times the digit in the units place. The digits are reversed and the resulting number is denoted by R. If N × R = 3627, then what is the product of the digits of the number N?
A boatman can row to a place (Y) at a distance of 24 km from the starting point (X) and back in 14 hours. If he can row 4 km with the stream in the same time as he can row 3 km against it, what is the speed of the stream?
An alloy (X) of gold and silver is taken in the ratio 1 : 2 and another alloy (Y) of gold and silver is taken in the ratio 2 : 3. How many parts of X and Y must be taken to obtain a new alloy Z consisting of gold and silver in the ratio 4 : 7?
N is a 3-digit number. The middle digit of N is equal to the sum of the other two digits and the sum of the digits is 10. How many such numbers (N) can be possible?
The ratio of the cost price of two articles X and Y is 1 : 2. A businessman earns a profit of p% by selling X and a loss of 3p% by selling Y. If he loses 20% in this transaction, then what is the value of p?
A sum of money at the rate of 5% per annum compounded annually becomes n times in 100 years. What is the value of n? (Given log₁₀2 = 0.301, log₁₀3 = 0.477 and log₁₀7 = 0.845)
A variable y is directly proportional to a variable quantity xⁿ. Given that when x = 2, y = 20.8 and when x = 3, y = 105.3. Which one of the following is the constant of proportionality if it is greater than 1?
If x = 3 − ∛3 − ∛9, then what is x(x−3)(x−6) equal to?
Consider the following statements:
Which of the statements given above is/are correct?
Consider the following sets:
The set of even prime numbers
How many of the above are null sets?
Consider the following statements:
Which of the statements given above is/are correct?
What should be added to (x²−1)/(x²+1) to get (2x³+3x²−1)/((x+2)(x²+1))?
If x+y+z = 0, then what is x(y+z)² + y(z+x)² + z(x+y)² equal to?
If tan 6θ ⋅ tan 3θ = 1, where 0 < θ < 30°, then what is θ equal to?
If (1−tan x)/(1+tan x) = 1 − 2tan x/(1+tan²x), 0 ≤ x < π/2, x ≠ π/4, then what is (sin x + cos x) equal to?
If cos²x + cos⁴x = 1, where 0 < x < π/2, then what is (sin²x + sin³x) + (sin³x + sin⁴x) equal to?
If sin⁴x + cos⁴x = 1, where 0 ≤ x ≤ π/2, then what is sin x ⋅ cos x equal to?
If 3cos θ + 4sin θ = 4, where 0 ≤ θ < π/2, then what is 4cos θ + 3sin θ equal to?
If cot θ + cos θ = m, cot θ − cos θ = n, where 0 < θ < π/2, then what is (m²−n²)/√(mn) equal to?
If 1/(cosec θ + cot θ) − 1/sin θ = p, then what is 1/sin θ − 1/(cosec θ − cot θ) equal to?
If 8sin θ − cos θ = 4, where π/6 < θ < π/3, then what is sin θ equal to?
A person walking along a straight road observes that at two consecutive kilometer-stones the angles of elevation of a hill in front of him are 30° and 60° respectively. What is the height of the hill?
A tower subtends an angle α at a point P on the same level as the foot of the tower. Q is a point vertically above P and PQ = h. If the angle of depression of the foot of the tower measured from Q is β, then what is the height of the tower?
If 5sin θ + 12cos θ = 13, where 0 < θ < π/2, then what is tan θ + cot θ equal to?
What is the ratio of the greatest value of sin²x + 2 (where 0 ≤ x ≤ π/2) to its least value?
If A + B + C = π (A, B, C > 0) and the angle C is obtuse, then which of the following is/are correct? I. sin A ⋅ sin B < 1 II. tan A ⋅ tan B > 1 Select the answer using the code given below:
If (cos 35° + cos 55°) = p, then what is sin 35° ⋅ cos 35° equal to?
If p = sec θ + tan θ and q = cosec θ − cot θ, then what is (p − q − pq) equal to?
In a cyclic quadrilateral ABCD, ∠A = (4x + 3)°, ∠B = (3y + 9)°, ∠C = (4y − 3)°, ∠D = (5x − 10)°. What is ∠A + ∠B equal to?
The area of a rhombus is 720 cm² and the sum of its diagonals is 98 cm. What is the perimeter of the rhombus?
A chord of a circle makes an angle of 90° at the centre of the circle. If the area of the major segment is k times the area of the minor segment, then what is the value of k? (Take π = 22/7)
ABC is a triangle in which (AB + BC) exceeds CA by 10 cm, (BC + CA) exceeds AB by 8 cm and (CA + AB) exceeds BC by 72 cm. What is the area of the triangle?
A reservoir is in the form of a cuboid. Its length is 30 m. If 24 kL of water is removed from the reservoir, the water level goes down by 20 cm. What is the width of the reservoir?
A well is dug with a diameter of 3.5 m and 16 m depth. The earth soil so excavated is spread in the form of a right circular cone of radius 7 m. What is the height of the cone?
The chord of a circle is 20 cm and the height of the minor segment is 5 cm. What is the diameter of the circle?
On a circular metal plate of uniform thickness, 16 holes, each of diameter 2 cm, are made. If the plate thereby has lost one-ninth of its original weight, then what is the diameter of the plate?
A square of side length 4 cm has its corners cut away in such a manner so as to form a regular octagon. What is the length of the side of the octagon? (Take √2 = 1.41)
A tall cylindrical reservoir kept vertically is 20 m in diameter. Water is poured into it at the rate of 264 m³ per hour. What is the rate at which the water level rises in the reservoir per minute? (Take π = 22/7)
A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option. Question : ABCD is a rhombus of side 10 cm. What is the sum of the squares of the diagonals of the rhombus?
Which one of the following is correct in respect of the above Question and Statements?
A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option. Question : The diagonal of a rectangle ABCD (AB > BC) is 5√2 cm. What is its perimeter?
Which one of the following is correct in respect of the above Question and Statements?
A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option. Question : ABC is a triangle right-angled at A and AD is perpendicular to BC. If AD = 7.2 cm, then what is BD × CD equal to?
Which one of the following is correct in respect of the above Question and Statements?
A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option. Question : ABCD is a quadrilateral with AB = 6 cm, BC = 8 cm, CD = 4√5 cm, DA = 2√5 cm and inscribed in a circle. What is the diameter of the circle?
Which one of the following is correct in respect of the above Question and Statements?
A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option. Question : ABC is a triangle inscribed in a semi-circle such that AC coincides with the diameter of the semi-circle. Let P be any point on the arc of the semi-circle. What is AP² + CP² + AC² equal to?
Which one of the following is correct in respect of the above Question and Statements?
A thin metallic sheet, 1.92 m² in area, is cut into two equal pieces. One piece is used to make a hollow cube of volume P (in m³) and the other is used to make a hollow cuboid of volume Q (in m³) with dimensions in the ratio 4 : 2 : 1. If no sheet is wasted, which one of the following is correct?
ABC is a triangle such that ∠ABC = 120°. If BD is the bisector of ∠B that meets AC at D, then what is the ratio of the area of ΔABD to the area of ΔCBD?
Consider the following statements :
Which of the statements given above is/are correct?
If two similar scalene triangles ABC and PQR have the same area, then which of the following statements is/are correct?
In a triangle ABC, AB = 8 cm and D and E are points on AB and AC respectively such that DE is parallel to BC. If BC = 5DE, then what is AD × BD equal to?
How many square metres of canvas (approximately) will be required to make a conical tent 3 m high so that a man 2 m tall may stand anywhere within a radius of 1 m from its centre without stooping?
PQRS is a square. M is a point on PS such that PM : MS = 2 : 1 and N is a point on SR such that SN : NR = 1 : 2. If the area of triangle MQN is 10 square units, then what is the perimeter of the square?
A cubical wooden block of length 10 cm is sliced to produce a cross-section of maximum area. What is the area of the cross-section?
In a triangle ABC, AD is the bisector of angle A that meets BC at D. If AB = 21 cm, DC = 20 cm and AB : AC = 3 : 4, then what is AC² − BD² equal to?
A square of maximum area is inscribed in an equilateral triangle. If the side of the triangle is equal to (6 + 4√3) cm, then what is the area of the square?
A room is L m long, B m wide and H m high. Further, L > B > H and L, B and H are integers. The length of the longest pole that can be placed in the room is 17 m and the length of the longest pole that can be placed on the floor is 15 m. What is the volume of the room?
A spherical metal ball is molten and made into n smaller identical spheres. In this process, the surface area of the smaller balls increases by 900%. What is the value of n?
ABCD is a parallelogram with ∠ABC = 150° and the sides have integer values. If the area of the parallelogram is 17.5 cm², then consider the following statements :
Which of the statements given above is/are correct?
N number of cubes each of side length equal to 10 cm are joined end to end in a row. If the total surface area of the resulting cuboid is 3800 cm², then what is the value of N?
In a quadrilateral ABCD, ∠ABC = 90° and ∠ACD = 90°. Let AB = p units, BC = q units, AC = r units, CD = s units, AD = t units, where p < q < r < s < t < 15. If p, q, r, s and t are integers, then what is the area of the quadrilateral?
For the next three (03) items that follow : The centre O of a circle inside a triangle ABC is at a distance of 13 cm from each of the vertices of the triangle. The diameter of the circle is 10 cm and the circle touches only two sides of the triangle, AB and AC. If x is the perimeter, in cm, of the triangle, then which one of the following is correct?
For the next three (03) items that follow : The centre O of a circle inside a triangle ABC is at a distance of 13 cm from each of the vertices of the triangle. The diameter of the circle is 10 cm and the circle touches only two sides of the triangle, AB and AC. If y is the area in cm² of the triangle, then which one of the following is correct?
For the next three (03) items that follow : The centre O of a circle inside a triangle ABC is at a distance of 13 cm from each of the vertices of the triangle. The diameter of the circle is 10 cm and the circle touches only two sides of the triangle, AB and AC. Consider the following statements :
Which of the statements given above is/are correct?
For the next three (03) items that follow : A circle with centre O passes through the vertex A of an equilateral triangle ABC and touches BC at its midpoint M. The circle cuts AB at D and AC at E. What is AD : DB equal to?
For the next three (03) items that follow : A circle with centre O passes through the vertex A of an equilateral triangle ABC and touches BC at its midpoint M. The circle cuts AB at D and AC at E. If x is the area of the circle and y is the area of the triangle such that z = (x/y)², then which one of the following is correct?
For the next three (03) items that follow : A circle with centre O passes through the vertex A of an equilateral triangle ABC and touches BC at its midpoint M. The circle cuts AB at D and AC at E. What is the ratio of the area of the triangle ADO to the area of the quadrilateral BDOM?
For the next two (02) items that follow : ABC is an equilateral triangle. BP is perpendicular to AC, PR is perpendicular to AB, and PQ is perpendicular to BC. What is the ratio of AB² : BP² : PR²?
For the next two (02) items that follow : ABC is an equilateral triangle. BP is perpendicular to AC, PR is perpendicular to AB, and PQ is perpendicular to BC. What is the ratio of the area of ΔPRB to the area of ΔPQC?
For the next two (02) items that follow : PQR is a triangle such that QP = QR = 15 cm and PR = 18 cm. PN, QM and RT are the altitudes of the triangle which intersect at O. What is QT : QO equal to?
For the next two (02) items that follow : PQR is a triangle such that QP = QR = 15 cm and PR = 18 cm. PN, QM and RT are the altitudes of the triangle which intersect at O. What is the ratio of QO to OM?
If the mean of a set of 7 observations is 10 and the mean of a set of 3 observations is 5, then what is the combined mean?
The average weekly wages of male employees in a company is ₹4,200 and that of females is ₹3,200. If the average weekly wage of all employees is ₹4,000, then what is the ratio of male to female employees?
Consider the observations x, x + 4, 30, 33, 74, 78, 49, 52, 98, 85. If the median of the data is 67, then what is the value of x?
A cyclist pedals from her house to her office at a speed of 3 kmph and back from the office to her house at 6 kmph. What is the average speed?
If the heights (in cm) of 9 students of a class are 150, 165, 145, 149, 150, 147, 152, 144 and 148, then what is the algebraic sum of the heights measured from their arithmetic mean?
Consider the following grouped data : Class – Frequency : 0–10 – 4; 10–20 – 8; 20–30 – 15; 30–40 – 10; 40–50 – 3. What is the mode of the above distribution?
If the mean of 15 observations, namely x₁, x₂, x₃, ..., x₁₅ is 2.5, then what is the value of Σ(i=1 to 15)[5(4xᵢ + 1)]?
Consider the following discrete grouped data : x – f : 12 – 6; 14 – 8; 10 – 10; 16 – 9; 18 – 3. What is the arithmetic mean of the above distribution?
Consider the following incomplete frequency distribution with a total frequency of 100 : Class – Frequency : 0–10 – 10; 10–20 – x; 20–30 – 20; 30–40 – y. If the mean of the frequency distribution is 25, then what are the missing frequencies?
The following data gives the investment in different categories by a governmental organization : Sr. No. – Item – Investment (₹ in crores) : 1 – Science, Technology and Environment – 7,200; 2 – Transport and Communication – 3,000; 3 – Rural Development – 1,500; 4 – Industry and Minerals – 3,800; 5 – Agricultural Services – 2,500. Which of the following statements is/are correct?