Q1
The number 2 × 3 × 5 × 7 × 11 + 1 is
Elementary MathematicsNumber System
UPSC CDS · Solved paper
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The number 2 × 3 × 5 × 7 × 11 + 1 is
Two unequal pairs of numbers satisfy the following conditions: (i) The product of the two numbers in each pair is 2160 (ii) The HCF of the two numbers in each pair is 12. If x is the mean of the numbers in the first pair and y is the mean of the numbers in the second pair, then what is the mean of x and y?
How many digits are there in (54)^10? (Given that log₁₀2 = 0.301 and log₁₀3 = 0.477)
Which one of the following is a set of solution of the equation x^(√x) = ⁿ√(x^x) if n is a positive integer?
In a competitive examination, 250 students have registered. Out of these, 50 students have registered for Physics, 75 students for Mathematics and 35 students for both Mathematics and Physics. What is the number of students who have registered neither for Physics nor for Mathematics?
If the sum of the digits of a number 10^n − 1, where n is a natural number, is equal to 3798, then what is the value of n?
Which one of the following is the largest number among 2222², 222^22, 22^222, 2^2222?
If m is the number of prime numbers between 0 and 50; and n is the number of prime numbers between 50 and 100, then what is (m – n) equal to?
If a/b = 1/3, b/c = 2, c/d = 1/2, d/e = 3 and e/f = 1/4, then what is the value of abc/def?
Which one of the following is the largest divisor of 3^x + 3^(x+1) + 3^(x+2), if x is any natural number?
A two-digit number is 9 more than four times of the number obtained by interchanging its digits. If the product of digits in the two-digit number is 8, then what is the number?
If α and β are the roots of the quadratic equation x² + kx - 15 = 0 such that α - β = 8, then what is the positive value of k?
What are the values of p and q respectively, if (x-1) and (x+2) divide the polynomial x³+4x²+px+q?
If (x + k) is the HCF of x² + 5x + 6 and x² + 8x + 15, then what is the value of k?
If 5^(x+1) - 5^(x-1) = 600, then what is the value of 10^(2x)?
A number divides 12288, 28200 and 44333 so as to leave the same remainder in each case. What is that number?
If x² + 9y² = 6xy, then what is y : x equal to?
If m and n are positive integers such that m^n = 1331, then what is the value of (m - 1)^(n-1)?
What is 1/(a^(m-n) - 1) + 1/(a^(n-m) - 1) equal to?
If x = √2, y = ∛3 and z = ⁶√6 then which one of the following is correct?
If log x = 1.25 and y = x^(log x), then what is log y equal to?
If f(x) is divided by (x – α)(x – β) where α ≠ β, then what is the remainder?
If the area of a square is 2401x⁴ + 196x² + 4, then what is its side length?
If x varies as yz, then y varies inversely as
If the points P and Q represents real number 0.73̅ and 0.56̅ on the number line, then what is the distance between P and Q?
What is the point on the xy-plane satisfying 5x + 2y = 7xy and 10x + 3y = 8xy?
The price of an article X increases by 20% every year and price of article Y increases by 10% every year. In the year 2010, the price of article X was Rs. 5000 and price of article Y was Rs. 2000. In which year the difference in their prices exceeded Rs. 5000 for the first time?
The ratio of speeds of X and Y is 5 : 6. If Y allows X a start of 70 m in a 1.2 km race, then who will win the race and by what distance?
A train takes two hours less for a journey of 300 km if its speed is increased by 5 km/hr from its usual speed. What is its usual speed?
If 6 men and 8 women can do a piece of work in 10 days; and 13 men and 24 women can do the same work in 4 days, then what is the ratio of daily work done by a man to that of a woman?
Students of a class are made to sit in rows of equal number of chairs. If number of students is increased by 2 in each row, then the number of rows decrease by 3. If number of students is increased by 4 in each row, then the number of rows decreases by 5. What is the number of students in the class?
A sum was put at simple interest at certain rate for 2 years. Had it been put at 1% higher rate of interest, it would have fetched Rs. 24 more. What is the sum?
The populations of two villages are 1525 and 2600 respectively. If the ratio of male to female population in the first village is 27 : 34 and the ratio of male to female population in the second village is 6 : 7, then what is the ratio of male to female population of these two villages taken together?
In a class room the ratio of number of girls to that of boys is 3 : 4. The average height of students in the class is 4.6 feet. If the average height of the boys in the class is 4.8 feet, then what is the average height of the girls in the class?
What is the median of the data 3, 5, 9, 4, 6, 11, 18?
In a pi-diagram there are three sectors. If the ratio of the angles of the sectors is 1 : 2 : 3, then what is the angle of the largest sector?
The maximum marks in a test are converted from 250 to 50 for the purpose of an Internal Assessment. The highest marks scored were 170 and lowest marks were 70. What is the difference between the maximum and minimum marks scored in the Internal Assessment?
Jails with highest and smallest percentage of trained convicts are respectively:
Jail with highest placement rate of trained convicts is:
Jails from which more than half of the trained convicts are offered jobs, are
The number of three digit numbers (all digits are different) which are divisible by 7 and also divisible by 7 on reversing the order of the digits, is
How many integral values of x and y satisfy the equation 5x + 9y = 7, where −500 < x < 500 and −500 < y < 500?
Let XYZ be a 3-digit number. Let S = XYZ + YZX + ZXY. Which of the following statements is/are correct?
Select the correct answer using the code given below:
In covering certain distance, the average speeds of X and Y are in the ratio 4 : 5. If X takes 45 minutes more than Y to reach the destination, then what is the time taken by Y to reach the destination?
For two observations, the sum is S and product is P. What is the harmonic mean of these two observations?
If the annual income of X is 20% more than that of Y, then the income of Y is less than that of X by p%. What is the value of p?
What is the least perfect square which is divisible by 3, 4, 5, 6 and 7?
In a water tank there are two outlets. It takes 20 minutes to empty the tank if both the outlets are opened. If the first outlet is opened, the tank is emptied in 30 minutes. What is the time taken to empty the tank by second outlet?
If (x² − 1) is a factor of ax⁴ + bx³ + cx² + dx + e, then which one of the following is correct?
If (x⁸ + 1/x⁸) = 47, what is the value of (x⁶ + 1/x⁶)?
A wheel makes 360 revolutions in one minute. What is the number of radians it turns in one second?
What is the least value of (25csc²x + sec²x)?
Let 0° < θ < 90° and 100θ = 90°. If α = Π_(n=1)^(99) cot(nθ), then which one of the following is correct?
If tan 6θ = cot 2θ, where 0 < 6θ < π/2, then what is the value of sec 4θ?
A tree of height 15 m is broken by wind in such a way that its top touches the ground and makes an angle 30° with the ground. What is the height from the ground to the point where tree is broken?
On a plane area there are two vertical towers separated by 100 feet apart. The shorter tower is 40 feet tall. A pole of length 6 feet stands on the line joining the base of two towers so that the tip of the towers and tip of the pole are also on the same line. If the distance of the pole from the shorter tower is 75 feet, then what is the height of the taller tower (approximately)?
The angles of elevation of the top of a tower from two points at distances p and q from the base and on the same straight line are 27° and 63° respectively. What is the height of the tower?
What is the value of sin²6° + sin²12° + sin²18° + … + sin²84° + sin²90°?
What is (cosθ)/(1 + sinθ) + 1/cotθ equal to?
What is (sinθ - cosθ + 1)/(sinθ + cosθ - 1) - (sinθ + 1)/(cosθ) equal to?
What is (tan x + tan y)(1 – cot x cot y) + (cot x + cot y)(1 – tan x tan y) equal to?
What is √((secx - tanx)/(secx + tanx)) equal to?
If θ lies in the first quadrant and cotθ = 63/16, then what is the value of (sin θ + cos θ)?
What is the value of (1 - 2sin²θcos²θ)/(sin⁴θ + cos⁴θ) + 4 equal to?
A rectangle is 48 cm long and 14 cm wide. If the diagonal makes an angle θ with the longer side, then what is (sec θ + cosec θ) equal to?
What is the area of the triangle having side lengths y/z + z/x, z/x + x/y, x/y + y/z?
If the angles of a triangle are 30° and 45° and the included side is (√3+1) cm, then what is the area of the triangle?
ABCD is a plate in the shape of a parallelogram. EF is the line parallel to DA and passing through the point of intersection O of the diagonals AC and BD. Further, E lies on DC and F lies on AB. The triangular portion DOE is cut out from the plate ABCD. What is the ratio of area of remaining portion of the plate to the whole?
Two circles of radii 20 cm and 16 cm intersect and the length of common chord is 24 cm. If d is the distance between their centres, then which one of the following is correct?
In a circle of radius 5 cm, AB and AC are two chords such that AB = AC = 8 cm. What is the length of chord BC?
Two circles touch internally. The sum of their areas is 136π cm² and distance between their centres is 4 cm. What are the radii of the circles?
If area of a circle and a square are same, then what is the ratio of their perimeters?
A circle of diameter 8 cm is placed in such a manner that it touches two perpendicular lines. Then another smaller circle is placed in the gap such that it touches the lines and the circle. What is the diameter of the smaller circle?
The thickness of a cylinder is 1 foot, the inner radius of the cylinder is 3 feet and height is 7 feet. To paint the inner surface it requires one litre of a particular colour. How much quantity of the same colour is required to paint all the surfaces of the cylinder?
A square and a rectangle have equal areas. If one side of the rectangle is of length numerically equal to the square of the length of the side of the square then the other side of the rectangle is
The length and breadth of a rectangle are in the ratio 4 : 3. Then what is the ratio of the area of the triangle formed by the parts of the diagonals with a long side to the area of the triangle formed by the parts of diagonals with a short side?
Suppose a region is formed by removing a sector of 20° from a circular region of radius 30 feet. What is the area of this new region?
ABCD is a parallelogram where AC and BD are the diagonals. If ∠BAD = 60°, ∠ADB = 90°, then what is BD² equal to?
A line through the vertex A of a parallelogram ABCD meets DC in P and BC produced in Q. If P is the mid-point of DC, then which of the following is/are correct?
Select the correct answer using the code given below:
How many cubic metre of earth is to be dug out to dig a well of radius 1.4 m and depth 5 m?
If the diagonals of a rhombus are x and y, then what is its area?
The lengths of sides of triangle are 3x, 4√y, 5∛z, where 3x < 4√y < 5∛z. If one of the angles is 90°, then what are the minimum integral values of x, y, z respectively?
What is the maximum number of circum-circles that a triangle can have?
If an arc of a circle of radius 6 cm subtends a central angle measuring 30°, then which one of the following is an approximate length of the arc?
A ladder 5 m long is placed in a room so as to reach a point 4.8 m high on a wall and on turning the ladder over to the opposite side of the wall without moving the base, it reaches a point 1.4 m high. What is the breadth of the room?
What is the area of the largest square plate cut from a circular disk of radius one unit?
Out of 4 identical balls of radius r, 3 balls are placed on a plane such that each ball touches the other two balls. The 4th ball is placed on them such that this ball touches all the three balls. What is the distance of centre of 4th ball from the plane?
A right circular cylinder just encloses a sphere. If p is the surface area of the sphere and q is the curved surface area of the cylinder, then which one of the following is correct?
ABCD is a quadrilateral such that AD = DC = CA = 20 units, BC = 12 units and ∠ABC = 90°. What is the approximate area of the quadrilateral ABCD?
Let PQRS be the diameter of a circle of radius 9 cm. The length PQ, QR and RS are equal. Semi-circle is drawn with QS as diameter (as shown in the given figure). What is the ratio of the shaded region to that of the unshaded region?
What is the area of the shaded region in the given figure, if the radius of each of the circles is 2 cm?
In the given figure, what is the area of the shaded region?
In the given figure, if y/x = 6 and z/x = 5, then what is the value of x?
ABCD is a trapezium, where AB is parallel to DC. If AB = 4 cm, BC = 3 cm, CD = 7 cm and DA = 2 cm, then what is the area of the trapezium?
Angles are shown in the given figure. What is value of angle1 + angle2 + angle3 + angle4 + angle5 + angle6 + angle7 + angle8?
In the given figure PQ is parallel to RS, ∠AEF = 95°, ∠BHS = 110°, and ∠ABC = x°. Then what is the value of x?
In the given figure AB is parallel to CD and AC is parallel to BD. If ∠EAC = 40°, ∠FDG = 55°, ∠HAB = x°, then what is the value of x?
In the given figure, there are three semi circles ABC, AEF and CDF. The distance between A and C is 28 units and F is the mid-point of AC. What is the total area of the three semi circles?
What is the approximate area of the shaded region in the figure given?
Consider the following statements with reference to the given figure:
Which of the above statements are correct?