Q1
What is the remainder when 4^{96} is divided by 6 ?
Elementary MathematicsNumber System
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 remainder when 4^{96} is divided by 6 ?
If 37/13 = 2 + 1/(x + 1/(y + 1/z)) where x, y, z are natural numbers, then what is z equal to?
For what value of k is (x-5) a factor of x³-3x²+kx-10?
What is [√(5+√10)] / [5√5 − 2√20 − √32 + √50] equal to?
The digit in the units place of the product 81 × 82 × 83 × 84 × × 99 is
The square root of (0.75)³/1-0.75+[0.75+(0.75)²+1] is
What is the remainder obtained when 1421 × 1423 × 1425 is divided by 12 ?
Consider the following statements :
Which of the above statements is/are correct ?
A student was asked to multiply a number by 25. He instead multiplied the number by 52 and got the answer 324 more than the correct answer. The number to be multiplied was
What is the number of possible pairs of (P,Q) if the number 357P25Q is divisible by both 3 and 5?
The difference between the squares of two consecutive odd integers is always divisible by
The expression x³q²-x³pt+4x²pt-4x²q²+3xq²-3xpt is divisible by
The difference between compound interest and simple interest at the same rate of interest R percent per annum on an amount of ₹ 15,000 for 2 years is ₹ 96. What is the value of R?
What is the maximum value of m if the number N = 35 × 45 × 55 × 60 × 124 × 75 is divisible by 5^{m} ?
A milkman claims to sell milk at its cost price only, but he is making a profit of 20% since he has mixed some amount of water in the milk. What is the percentage of milk in the mixture?
There is 60% increase in an amount in 6 years at simple interest. What will be the compound interest on ₹ 12,000 after 3 years at the same rate of interest?
16 litres of a mixture contains milk and water in the ratio 5 : 3. If 4 litres of milk is added to this mixture, the ratio of milk to water in the new mixture would be
A person could save 10% of his income. But 2 years later, when his income increased by 20%, he could save the same amount only as before. By how much percentage has his expenditure increased?
In how much time would the simple interest on a principal amount be 0.125 times the principal amount at 10% per annum?
A man rows downstream 32 km and 14 km upstream, and he takes 6 hours to cover each distance. What is the speed of the current?
A car travels the first one-third of a certain distance with a speed of 10 km/hr, the next one-third distance with a speed of 20 km/hr and the last one-third distance with a speed of 60 km/hr. The average speed of the car for the whole journey is
Two persons A and B start simultaneously from two places c km apart, and walk in the same direction. If A travels at the rate of p km/hr and B travels at the rate of q km/hr, then A has travelled before he overtakes B a distance of
For which value of k does the pair of equations x²-y²=0 and (x-k)²+y²=1 yield a unique positive solution of x?
A person goes to a market between 4 p.m. and 5 p.m. When he comes back, he finds that the hour hand and minute hand have interchanged their positions. For how much time (approximately) was he out of his house?
The efficiency of P is twice that of Q, whereas the efficiency of P and Q together is three times that of R. If P, Q and R work together on a job, in what ratio should they share their earnings?
In a flight of 600 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/hr and the time of flight increased by 30 minutes. The duration of the flight is
With a uniform speed, a car covers a distance in 8 hours. Had the speed been increased by 4 km/hr, the same distance could have been covered in 7 hours and 30 minutes. What is the distance covered?
A run 12/3 times as fast as B. If A gives B a start of 80 m, how far must the winning post from the starting point be so that A and B might reach it at the same time?
A thief is noticed by a policeman from a distance of 200 m. The thief starts running and the policeman chases him. The thief and the policeman run at the speed of 10 km/hr and 11 km/hr respectively. What is the distance between them after 6 minutes?
When a ball bounces, it rises to 2/3 of the height from which it fell. If the ball is dropped from a height of 36 m, how high will it rise at the third bounce?
A light was seen regularly at an interval of 13 seconds. It was seen for the first time at 1 hour 54 minutes 50 seconds (a.m.) and the last time at 3 hours 17 minutes 49 seconds (a.m.). How many times was the light seen?
An automobile financer claims to be lending money at simple interest, but he includes the interest every six months for calculating the principal. If he is charging an interest at the rate of 10%, the effective rate of interest becomes
What are the possible solutions for x of the equation x^{{x}} = [n]{x^x} where x and n are positive integers?
Let A and B be finite non-empty sets with the number of elements in A = m and number of elements in B = n. Let m > n. If for some integer k > 1, the number of non-empty subsets of A = 2k + the number of non-empty subsets of B, then which one of the following is correct?
If n is a natural number and n=p^{x₁₁p^{x₂₂p^{x₃₃ where p₁,p₂,p₃ are distinct prime factors, then the number of prime factors for n is
Consider the following statements for the sequence of numbers given below : 11, 111, 1111, 11111,.....
Which of the above statements is/are correct?
The HCF and LCM of two polynomials are (x + y) and (3x^5+5x^4y+2x³y²-3x²y³-5xy^4-2y^5) respectively. If one of the polynomials is (x² - y²), then the other polynomial is
Let p be a prime number other than 2 or 5. One would like to express the vulgar fraction 1/p in the form of a recurring decimal. Then the decimal will be
Consider the following statements :
Which of the above statements is/are correct?
The sign of the quadratic polynomial ax²+bx+c is always positive if
If A, G and H are the arithmetic, geometric and harmonic means between a and b respectively, then which one of the following relations is correct?
The geometric mean of three positive numbers a, b, c is 3 and the geometric mean of another three positive numbers d, e, f is 4. Also, at least three elements in the set {a, b, c, d, e, f} are distinct. Which one of the following inequalities gives the best information about M, the arithmetic mean of the six numbers?
There are five parties A, B, C, D and E in an election. Out of total 100000 votes cast, 36000 were cast to party A, 24000 to party B, 18000 to party C, 7000 to party D and rest to party E. What angle will be allocated for party E in the pie chart?
Out of 532 savings accounts held in a post office, 218 accounts have deposits over ₹ 10,000 each. Further, in 302 accounts, the first or sole depositors are men, of which the deposits exceed f 10,000 in 102 accounts. In how many accounts the first or sole depositors are women and the deposits are up to ₹ 10,000 only?
If x+y+z=0 then x³+y³+z³+3xyz is equal to
If three metallic spheres of radii 6 cm, 8 cm and 10 cm are melted to form a single sphere, then the diameter of the new sphere will be
If each interior angle of a regular polygon is 135°, then the number of diagonals of the polygon is equal to
The sides of a triangular field are 41 m, 40 m and 9 m. The number of rose beds that can be prepared in the field if each rose bed, on an average, needs 900 square cm space, is
The ratio of the outer and inner perimeters of a circular path is 23 : 22. If the path is 5 m wide, the diameter of the inner circle is
A cylindrical vessel of radius 4 cm contains water. A solid sphere of radius 3 cm is lowered into the water until it is completely immersed. The water level in the vessel will rise by
If the height of a right circular cone is increased by 200% and the radius of the base is reduced by 50%, then the volume of the cone
The angles of a triangle are in the ratio 4:1:1. Then the ratio of the largest side to the perimeter is
Let a, b, c be the sides of a right triangle, where c is the hypotenuse. The radius of the circle which touches the sides of the triangle is
The area of the largest triangle that can be inscribed in a semicircle of radius r is
AD is the diameter of a circle and AB is a chord. If AD = 34 cm, AB = 30 cm, the distance of AB from the centre of the circle is
If a star figure is formed by elongating the sides of a regular pentagon, then the measure of each angle at the angular points of the star figure is
Four equal-sized maximum circular plates are cut off from a square paper sheet of area 784 square cm. The circumference of each plate is
If tan(A + B) = √3 and tan A = 1, then tan(A − B) is equal to
If the radius of a sphere is increased by 10%, then the volume will be increased by
ABCD is a parallelogram, where AB : AD = 2:1. One of the angles of the parallelogram is 60°. The two diagonals are in the ratio
Consider the following statements :
Which of the above statements is/are correct?
If cos A = tan B, cos B = tan C and cos C = tan A, then sin A is equal to
If 3-²A/1-3²A=K where k is a real number, then A(3 A-4³A) is equal to
The angles of elevation of the top of a tower from two points P and Q at distances m² and n² respectively, from the base and in the same straight line with it are complementary. The height of the tower is
The angle of elevation of a cloud from a point 200 m above a lake is 30° and the angle of depression of its reflection in the lake is 60°. The height of the cloud is
If tan A + cot A = 4 then tan⁴A + cot⁴A is equal to
If p={1- x/1+ x}, q=1- x/ x, r=/1+ x then which of the following is/are correct?
Select the correct answer using the code given below
Consider the following :
Which of the above is /are identity/identities?
A B C is a triangle right angled at B and A B : B C = 3 : 4. What is A+ B+ C equal to?
The value of ²67^{}+²57^{}-²33^{}-²23^{} is
Consider the following statements :
Which of the above statements is/are correct?
From the top of a tower, the angles of depression of two objects P and Q (situated on the ground on the same side of the tower) separated at a distance of 100(3 -{3}) m are 45° and 60° respectively. The height of the tower is
If sin x + cos x = c then sin⁶x + cos⁶x is equal to
Consider the following statements :
Which of the above statements is/are correct?
A triangle ABC is inscribed in a circle. If sum of the squares of sides of the triangle is equal to twice the square of the diameter, then ²A+²B+²C is equal to
The area of a rhombus with side 13 cm and one diagonal 10 cm will be
Two circles, each of radius r, with centres P and Q, are such that each circle passes through the centre of the other circle. Then the area common to the circles is less than one-third of the sum of the areas of the two circles by
A sphere and a cube have same surface area. The ratio of square of their volumes is
The radius of a sphere is equal to the radius of the base of a right circular cone, and the volume of the sphere is double the volume of the cone. The ratio of the height of the cone to the radius of its base is
Water flows through a cylindrical pipe of internal diameter 7 cm at the rate of 5 m/s. The time, in minutes, the pipe would take tofill an empty rectangular tank 4 m x 3 m x 2'31 m i s
The total outer surface area of a right circular cone of height 24 cm with a hemisphere of radius 7 cm upon its base is
The last digit in the expansion of 17²56} is
A rectangular block of wood having dimensions 3m x 2m x 175m has to be painted on all its faces. The layer of paint must be 0.1 mm thick. Paint comes in cubical boxes having their edges equal to 10 cm. The minimum number of boxes of paint to be purchased is
The diagonals of three faces of a cuboid are 13, {281} and 20 linear units. Then the total surface area of the cuboid is
A rectangular paper of 44 cm long and 6 cm wide is rolled to form a cylinder of height equal to width of the paper. The radius of the base of the cylinder so roiled is
If a-b =4 and a²+b²=40 where a and b are positive integers, then a³+b^6 is equal to
The sides of a triangle are 25 cm, 39 cm and 56 cm. The perpendicular from the opposite vertex on the side of 56 cm is
The diagonals of a trapezium are at right angles, and the slant sides, if produced, form an equilateral triangle with the greater of the two parallel sides. If the area of the trapezium is 16 square cm, then the distance between the parallel sides is
From a circular piece of cardboard of radius 3 cm, two sectors of 40° each have been cut off. The area of the remaining portion is
Three equal circles each of diameter d are drawn on a plane in such a way that each circle touches the other two. What is the area of the region enclosed by the three circles?
If the angle between the radii of a circle is 130°, then the angle between the tangents at the ends of the radii is
Let OA, OB, OC and OD are rays in the anticlockwise direction such that ∠AOB=∠COD=100°,∠BOC=82° and ∠AOD=78° Consider the following statements :
Which of the above statements is /are correct?
At 8:30, the hour hand and the minute hand of a clock form an angle of
The complement angle of 80° is
Consider the following statements :
Which of the above statements is/are correct?
Out of two concentric circles, the diameter of the outer circle is 26 cm and the chord MN of length 24 cm is tangent to the inner circle. The radius of the inner circle is
What is the mean of the distribution?
What is the median class?
What is the median of the distribution?
What is the mode of the distribution?