Q1
What is the largest number which divides both 2³⁵ - 1 and 2⁹¹ - 1?
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.
What is the largest number which divides both 2³⁵ - 1 and 2⁹¹ - 1?
What is the largest power of 10 that divides the product 29 × 28 × 27 ×...2 × 1?
What is the remainder when 65⁹⁹ is divided by 11?
If the roots of the equation x² - bx + c = 5 differ by 5, then which one of the following is correct?
In a party of 150 persons, 75 persons take tea, 60 persons take coffee and 50 persons take milk. 15 of them take both tea and coffee, but no one taking milk takes tea. If each person in the party takes at least one drink, then what is the number of persons taking milk only?
A, B, C, D and E enter into a business. They invest money in the ratio 2 : 3 : 4 : 5 : 6. However, the time invested by them is in the ratio 6 : 5 : 4 : 3 : 2. If the profit distributed is directly proportional to time and money invested, then who receives the highest amount of profit?
Consider the following numbers :
How many of the above numbers are prime?
A can do certain work at twice the speed of B. Further, B can do the same work at 1.5 times the speed of C. All of them together can finish the work in 12 days. In how many days can C alone finish the work?
The sum of digits of a 2-digit number is 12. When the digits are reversed, the number becomes greater by eighteen. What is the difference between the digits in the number?
The time taken by a train to cross a man travelling in another train is 10 seconds, when the other train is travelling in the opposite direction. However, it takes 20 seconds, if both the trains are travelling in the same direction. The length of the first train is 200 m and that of the second train is 150 m. What is the speed of the first train?
If a, b, c, d, e and f satisfy2a = 3b = 6c = 9d = 12e = 18f, then what is the value of (a + b)/(c + d + e + f)?
If a, b, c are non-zero real numbers such that a + b + c = 0, then what are the roots of the equation ax² + bx + c = 0?
Twelve percent of bananas bought by a fruit vendor got lost during transportation. On selling the remaining bananas, the vendor's overall profit turned out to be 4%. If the vendor had not lost any bananas and had sold them at the price of the remaining bananas, what would have been his profit percentage?
If the positive square root of (5 + 3√2) (5 - 3√2) is α, then what is the positive square root of 8 + 2α?
When every even power of every odd integer (greater than 1) is divided by 8, what is the remainder?
Consider the following statements :
Which of the statements given above is/are correct?
It is given that 5 does not divide n - 1, n and n + 1, where n is a positive integer. Which one of the following is correct?
What is the largest 5-digit number, which leaves the remainder 7, when divided by 18 as well as by 11?
In a business dealing, A owes B Rs. 20,000 payable after 5 years, whereas B owes A Rs. 12,000 payable after 4 years. They want to settle it now at the rate of 5% simple interest. Who gives how much money in this settlement?
Average marks in Mathematics of Section A comprising 30 students is 65 and that of Section B comprising 35 students is 70. What are the average marks (approximately) of both the sections if it was detected later that an entry of 47 marks was wrongly made as 74?
If α and β are the roots of the equation x² - 7x + 1 = 0, then what is the value of α⁴ + β⁴?
Consider the following statements in respect of all factors of 360 :
Which of the above statements is/are correct?
Consider a 6-digit number of the form XYXYXY. The number is divisible by :
What is the HCF of 3²⁹ - 9 and 3³⁸ - 9?
If x = √(4√(4√(4√4…))) then what is the value of x?
Let m and n be natural numbers. What is the minimum value of (m + n) such that 33m + 22n is divisible by 121?
The product of two numbers is 2160 and their HCF is 12. If the sum of the squares of the two numbers is 4896, then what is the mean of the two numbers?
The age of Q exceeds the age of P by 3 years. The age of R is twice the age of P and the age of Q is twice the age of S. Further, the age difference of R and S is 30 years. What is the sum of the ages of P and Q?
If a, b and c are the sides of a triangle ABC, then √a + √b - √c is always :
There are four bells which ring at an interval of 15 minutes, 25 minutes, 35 minutes and 45 minutes respectively. If all of them ring at 9 A.M., how many more times will they ring together in the next 72 hours?
Let a, b, c and d be four positive integers such that a + b + c + d = 200. If S = (-1)a + (-1)b + (-1)c + (-1)d, then what is the number of possible values of S?
The number 97³⁰ - 14³⁰ is divisible by :
Consider the following statements :
Which of the statements given above is/are correct?
If 17 women and 24 men can do a piece of work in 5 days and 12 women and 23 men can do it in 6 days, then which one of the following is correct?
Three taps A, B and C together can fill a tank in 6 hours. Tap C alone can fill the tank in 12 hours. To fill the tank, when it is empty, all the three taps are started together. After working t hours, tap C is closed and the tank is filled in 8 more hours. What is t equal to?
A, B and C can complete a work in x, 1.5x and 2x days respectively. If they complete the work together, in what ratio should they be paid?
Consider the following statements :
Which of the statements given above are correct?
What is the last digit of the sum S = 9²⁷ + 27⁹?
If x = (√3+1)/(√3-1) and y = (√3-1)/(√3+1) then what is the value x³ - y³?
The speed of a boat in still water is 15 km/hr. If it can travel 42 km downstream and 28 km upstream in the same time, then what is the speed of the stream?
What is the difference between simple interest and compound interest on Rs. 10,000 for two years at 20% per annum compounded half-yearly?
Consider the following statements in respect of the polynomial a(b - c) (x - b) (x - c) + b(c - a) (x - c) (x - a) + c(a - b) (x - a) (x - b) :
Which of the statements given above is/are correct?
Consider the following statements in respect of the polynomial 1 - x - xⁿ + xⁿ⁺¹ where n is a natural number :
Which of the statements given above is/are correct?
For what values of m, is mx² + mx + 8x + 9 a perfect square?
If x=a+b+((a-b)²)/(4a+4b) and y=(a+b)/(4)+(ab)/(a+b) then what is the value of (x - a)² - (y - b)²?
Consider the following :
Which of the above equations are identities?
If sin θ = (12)/(13) then what is the value of (tan θ + sec θ)² (csc θ - cot θ)⁻² 0 < θ < (π)/(2)
If tan⁸ θ + cot⁸ θ = m, then what is the value of tan θ + cot θ?
What is the minimum value of 6 - 4 sin θ, 0 ≤ θ ≤ (π )/(2)
What is the value of x that satisfies 4 cos² 30° + 2x sin 30° - cot² 30° - 6 tan 15° tan 75° = 0?
What is the value of (cos²32∘+cos²58∘)/(sec²50∘-cot²40∘) + 4 tan 13° tan 37° tan 53° tan 77°?
What is the value of (1+cot²θ)(1+cosθ)(1-cosθ)-(1+tan²θ)(1+sinθ)(1-sinθ)?
If 2 cos² θ + sin θ - 2 = 0, 0 < θ ≤ (π)/(2), then what is the value of θ?
A person on the top of a vertical tower observes a car moving at a uniform speed coming directly towards it. If it takes 6 minutes for the angle of depression to change from 30° to 45°, and further t minutes to reach the tower, which one of the following is correct?
A woman is standing on the deck of a ship, which is h (in metres) above water level. She observes the angle of elevation of the top of a tower as 60° and the angle of depression of the base of the tower as 30°. What is the height of the tower?
Let ABC be a right-angled triangle with sides 5 cm, 12 cm and 13 cm. If p is the length of the perpendicular drawn from vertex A on the hypotenuse BC, then what is the value of 13p?
OABC is a rhombus whose three vertices lie on a circle with centre at O. If the area of the rhombus is 32√3 square cm, then what is the radius of the circle?
The surface area of a cube is increased by 25%. If p is the percentage increase in its length, then which one of the following is correct?
A solid cube is cut into two cuboids of equal volume. What is the ratio of total surface area of the given cube to that of one of the cuboids?
The length of a diagonal of a cuboid is 11 cm. The surface area is 240 square cm. What is the sum of its length, breadth and height?
What is the area of the circle (approximately) inscribed in a triangle with side lengths 12 cm, 16 cm and 20 cm?
Two times the total surface area of a solid right circular cylinder is three times its curved surface area. If h is the height and r is the radius of the base of the cylinder, then which one of the following is correct?
A floor of a big hall has dimensions 30 m 60 cm and 23 m 40 cm. It is to be paved with square tiles of same size. What is the minimum number of tiles required?
How long will a man take to walk around the boundary of a square field of area 25 hectares at the rate of 5 km/hr?
Let x be the area of a square inscribed in a circle of radius r and y be the area of an equilateral triangle inscribed in the same circle. Which one of the following is correct?
If the length of a rectangle is increased by 66(2)/(3)%, then by what percent should the width of the rectangle be decreased in order to maintain the same area?
What is the maximum area that can be covered by three non-intersecting circles drawn inside a rectangle of sides 8 cm and 12 cm?
ABCD is a square field with AB = x. A vertical pole OP of height 2x stands at the centre O of the square field. If ∠APO = θ, then what is cot θ equal to?
A solid iron ball is melted and 64 smaller solid balls of equal size are made using the entire volume of iron. What is the ratio of the surface area of the larger ball to the sum of the surface areas of all the smaller balls?
A triangle ABC has been divided into four smaller triangles P, Q, R, S whose perimeters are 16 cm, 12 cm, 4 cm and 12 cm respectively. P, R and S contain the vertices A, B and C respectively. What is the perimeter of the triangle ABC?
Item contains a Question followed by two Statements. Answer each item using the following instructions : A number 277XY5 (where X, Y are digits) is divisible by 25.Question : What is the value of X?
Item contains a Question followed by two Statements. Answer each item using the following instructions : Question : What are the unique values of a, b and c if 2 is a root of the equation ax² + bx + c = 0?
Item contains a Question followed by two Statements. Answer each item using the following instructions : Question : Is m > n, where m, n are non-zero numbers?Statement I : (m)/(n)>1Statement II : m > 2n
Item contains a Question followed by two Statements. Answer each item using the following instructions : Question : Can a circle be drawn through the points A, B and C?
Item contains a Question followed by two Statements. Answer each item using the following instructions : Question : Can the value of (x + y) be determined uniquely?
Item contains a Question followed by two Statements. Answer each item using the following instructions : Question : Is p² + q² + q odd, where p, q are positive integers?
Item contains a Question followed by two Statements. Answer each item using the following instructions : Question : What is the area of the circle C?
Item contains a Question followed by two Statements. Answer each item using the following instructions : Question : Is triangle right angled?
Item contains a Question followed by two Statements. Answer each item using the following instructions : The lengths of two longer sides of the triangle Δ are 25 cm and 24 cm. Question : What is the length of the shortest side?
Item contains a Question followed by two Statements. Answer each item using the following instructions : A chord PQ of the circle C divides it into two segments such that 3 times the area of the major segment is 4 times the area of the minor segment. Question : What is the radius of C?
Consider the following distribution for the next two (02) items that follow: Class intervals 0-30, 30-60, 60-90, 90-120 have frequencies 4, 5, 7, 4 respectively. What is the mean of the distribution?
Consider the following distribution for the next two (02) items that follow: Class intervals 0-30, 30-60, 60-90, 90-120 have frequencies 4, 5, 7, 4 respectively. What is the standard deviation of the distribution?
Consider the following distribution for the next two (02) items that follow: Class intervals 40-50, 50-60, 60-70, 70-80 have frequencies 4, 3, 1, 2 respectively. What is the mean deviation about the mean of the distribution?
Consider the following distribution for the next two (02) items that follow: Class intervals 40-50, 50-60, 60-70, 70-80 have frequencies 4, 3, 1, 2 respectively. What is the mean deviation about the median of the distribution?
The plinth of a house has an area of 200 square metres. It is rectangular in shape and its length and breadth are in the ratio 2 : 1. The owner of the house extends the terrace by 1 m on each side. What is the percentage of area that has increased in the terrace relative to the plinth?
A square sheet of side length 44 cm is rolled along one of its sides to form a cylinder by making opposite edges just to touch each other. What is the volume of the cylinder? (Take π = 22/7)
The volume of a cuboid is 3600 cubic cm. The areas of two adjacent faces are 225 square cm and 144 square cm. What is the area of the other adjacent face?
The perimeter and the area of a right-angled triangle are 36 cm and 54 square cm respectively. What is the length of the hypotenuse?
Let X = x | x = 2 + 4k, where k = 0, 1, 2, 3,...24. Let S be a subset of X such that the sum of no two elements of S is 100. What is the maximum possible number of elements in S?
The perimeter of a sector of a circle of radius 5.2 cm is 16.4 cm. What is the area of the sector?
Consider the following for the next three (03) items that follow: In triangle ABC, AB = 6 cm, BC = 8 cm and AC = 10 cm. The perpendicular dropped from B meets the side AC at D. A circle of radius BD (with centre B) cuts AB and BC at P and Q respectively. What is the length of QC?
Consider the following for the next three (03) items that follow: In triangle ABC, AB = 6 cm, BC = 8 cm and AC = 10 cm. The perpendicular dropped from B meets the side AC at D. A circle of radius BD (with centre B) cuts AB and BC at P and Q respectively. If ∠ABD = θ, then what is sinθ equal to?
Consider the following for the next three (03) items that follow: In triangle ABC, AB = 6 cm, BC = 8 cm and AC = 10 cm. The perpendicular dropped from B meets the side AC at D. A circle of radius BD (with centre B) cuts AB and BC at P and Q respectively. What is the radius of the circle?
Consider the following for the next three (03) items that follow: A circle of radius 10 cm is inscribed in a square PQRS. A rectangle measuring 4 cm × 2 cm is drawn at the corner P, and a smaller square is drawn at the corner R. What is the area of the circle?
Consider the following for the next three (03) items that follow: A circle of radius 10 cm is inscribed in a square PQRS. A rectangle measuring 4 cm × 2 cm is drawn at the corner P, and a smaller square is drawn at the corner R. What is the area of the smaller square?
Consider the following for the next three (03) items that follow: A circle of radius 10 cm is inscribed in a square PQRS. A rectangle measuring 4 cm × 2 cm is drawn at the corner P, and a smaller square is drawn at the corner R. What is the area of the shaded region?
What is the ratio of the area of the circle to the area of the rectangle?
What is the area of Δ AEC?
What is the relation between x and y?
If y = 15, then what is ∠ACB equal to?