Q1
A number consists of two digits. The sum of the digits is 10. On reversing the digits of the number, the number decreases by 36. What is the product of the two digits?
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.
A number consists of two digits. The sum of the digits is 10. On reversing the digits of the number, the number decreases by 36. What is the product of the two digits?
Which one among the following is not correct?
ABCD is a rhombus with diagonals AC and BD. Then which one among the following is correct?
The sum of a number and its reciprocal is 10/3 then the numbers are:
Which one among the following statements is correct?
The following characteristics relate to the persons participating in athletic events:
Which of the following in respect of the above is correct?
Consider the following statements:
Which of the above statements is/are correct?
Which one among the following statements is not correct?
If the surface area of a sphere is 616 square cm, what is its volume?
What are the dimensions (length, breadth and height respectively) of a cuboid with volume 720 cubic cm, surface area 484 square cm and the area of the base 72 square cm?
Consider the following statements:
Which of the statements given above is/are correct?
Consider the following statements: If two straight lines intersect, then:
Which of the statements given above is/are correct?
A large solid metallic cylinder whose radius and height are equal to each other is to be melted and 48 identical solid balls are to be recast from the liquid metal so formed. What is the ratio of the radius of a ball to the radius of the cylinder?
The geometric mean of 32, 4, 8, X, 2 is 8. What is the value of X?
In the triangle ABC, the base BC is trisected at D and E. The line through D, parallel to AB, meets AC at F and the line through E parallel to AC meets AB at G. Let EG and DF intersect at H. What is the ratio of the sum of the area of parallelogram AGHF and the area of the triangle DHE to the area of the triangle ABC?
If x - 2 is a factor of x^3 + kx^2 - 2x - 24 then what is the value of k?
AB is a straight line. C is a point whose distance from AB is 3 cm. What is the number of points which are at a distance of 1 cm from AB and 5 cm from C?
The angle of elevation of the tip of a tower from a point on the ground is 45 degrees. Moving 21 m directly towards the base of the tower, the angle of elevation changes to 60 degrees. What is the height of the tower, to the nearest meter?
A circular ring with centre O is kept in the vertical position by two weightless thin strings TP and TQ attached to the ring at P and Q. The line OT meets the ring at E whereas a tangential string at E meets TP and TQ at A and B respectively. If the radius of the ring is 5 cm and OT = 13 cm, then what is the length of AB?
What is one of the square roots of 16x^6 - 24x^5 + 25x^4 - 20x^3 + 10x^2 - 4x + 1?
The cost of two articles are in the ratio 3:5. If there is 30% loss on the first article and 20% gain on the second article, what is overall percentage of loss or gain?
The number 58129745812974 is divisible by
What is the HCF of 36(3x^4 + 5x^3 - 2x^2), 9(6x^3 + 4x^2 - 2x), 54(27x^4 - x)?
What is the length of the uniform wire of diameter 0.4 cm that can be drawn from a solid sphere of radius 9 cm?
What will be the cost to plaster the inner surface of a well 14 m deep and 4 m in diameter at the rate of Rs. 25 per square meter?
A wheel of radius 2.1 m of a vehicle makes 75 revolutions in one minute. What is the speed of the vehicle?
If a positive integer leaves remainder 28 when divided by 143, then what is the remainder obtained on dividing the same number by 13?
What is the compound interest on Rs. 1600 at 25% per annum for 2 years compounded annually?
The graphs of ax + by = c, dx + ey = f will be
Which of the above statements are correct?
The medians of a triangle ABC intersect at G. Which one of the following is correct?
What is the mean proportional between (15 + sqrt(200)) and (27 - sqrt(648))?
A stock of food is enough for 240 men for 48 days. How long will the same stock last for 160 men?
If 3/(x+y) + 2/(x-y) = 2 and 9/(x+y) - 4/(x-y) = 1 then what is x/y equal to?
The locus of the mid points of all equal chords in a circle is:
If the outer and inner diameters of a stone parapet around a well are 112 cm and 70 cm respectively, then what is the area of the parapet?
What is 0.00777... + 17.8383... + 310.020222... equal to? (bars denote recurring decimals: 0.00(7) + 17.(83) + 310.020(2))
A man loses 20% of his money. After spending 25% of the remainder, he has Rs. 480 left. What is the amount of money he originally had?
Assuming that profit of a shopkeeper in a particular commodity is a linear expression of transportation charge (t) and the quantity of commodity (q). He earns a profit of Rs. 10,000 by selling 20 units at the transport charge of Rs. 400. He also earns a profit of Rs. 12,000 by selling 25 units at the transport charge of Rs. 600. What is the linear expression in t and q?
If x : y = 7 : 5, then what is (5x - 2y) : (3x + 2y) equal to?
The sum of two numbers is 232 and their HCF is 29. What is the number of such pairs of numbers satisfying the above condition?
A person bought 8 quintal of rice for certain rupees. After a week, he sold 3 quintal of rice at 10% profit, 3 quintal of rice with neither profit nor loss and 2 quintal at 5% loss. In this transaction, what is the profit?
What is the angle of elevation of the sun when the shadow of a pole is sqrt(3) times the length of the pole?
x^4 + xy^3 + x^3y + xz^3 + y^4 + yz^3 is divisible by:
'A' can do a piece of work in 'x' days and 'B' can do the same work in 3x days. To finish the work together they take 12 days. What is 'x' equal to?
ABCD is a rectangle. Let E be a point on AB and F a point on CD such that DE is parallel to BF. If AE = 3 cm and if the area of triangle BFC = 6 square cm. Consider the following statements:
Which of the above statements is/are correct?
A cardboard sheet in the form of a circular sector of radius 30 cm and central angle 144 degrees is folded to make a cone. What is the radius of the cone?
Two transversals S and T cut a set of distinct parallel lines. S cuts the parallel lines in points A, B, C and D and T cuts the parallel lines in points E, F, G, H respectively. If AB = 4, CD = 3 and EF = 12, then what is length of GH?
If the area of a circle is equal to the area of a square with side 2*sqrt(pi) units, what is the diameter of the circle?
If x + 1/x = a, then what is x^3 + x^2 + 1/x^3 + 1/x^2 equal to?
From the top of a cliff 200 m high, the angles of depression of the top and bottom of a tower are observed to be 30 degrees and 45 degrees respectively. What is the height of the tower?
If x^3 + 5x^2 + 10k leaves remainder -2x when divided by x^2 + 2, then what is the value of k?
If one of the roots of quadratic equation 7x^2 - 50x + k = 0 is 7, then what is the value of k?
If a^x = b, b^y = c and xyz = 1, then what is c^z equal to?
Let ABCD be a parallelogram. Let m, n be positive integers such that n < m < 2n. Let AC = 2mn and BD = m^2 - n^2. Let AB = (m^2 + n^2)/2.
Which one of the following is correct in respect of the above statements?
A square, a circle and an equilateral triangle have same perimeter. Consider the following statements:
Which of the statements is/are correct?
The expression sin^2 x + cos^2 x - 1 = 0 is satisfied by how many values of x?
The area of a rectangle whose length is 5 more than twice its width is 75 square units. What is the perimeter of the rectangle?
Two numbers are in the ratio 2 : 3. If 9 is added to each number, they will be in the ratio 3 : 4. What is the product of the two numbers?
Consider the following statements in respect of an equilateral triangle ABC.
Which of the above statements is/are correct?
What is the value of (1.001^3 + 0.999^3)/(1.001^2 - 1.001*0.999 + 0.999^2)?
If p(x) is a common multiple of degree 6 of the polynomials f(x) = x^3 + x^2 - x - 1 and g(x) = x^3 - x^2 + x - 1 then which one of the following is correct?
If the roots of the equation x^2 - 2ax + a^2 + a - 3 = 0 are real and less than 3, then which one of the following is correct?
Two students A and B solve an equation of the form x^2 + px + q = 0. A starts with a wrong value of p and obtains the roots as 2 and 6. B starts with a wrong value of q and gets the roots as 2 and -9. What are the correct roots of the equation?
Consider the following:
Which of the above statements is/are correct?
If p = a sin x + b cos x and q = a cos x - b sin x, then what is p^2 + q^2 equal to?
Which one of the following is correct in respect of the above statements?
What is the value of k that (2x - 1) may be a factor of 4x^4 - (k-1)x^3 + kx^2 - 6x + 1?
If 196x^4 = x^6, then x^3 is equal to which one of the following?
The sides of a parallelogram are 12 cm and 8 cm long and one of the diagonals is 10 cm long. If d is the length of other diagonal, then which one of the following is correct?
The base of a triangular wall is 7 times its height. If the cost of painting the wall at Rs. 350 per 100 square meter is Rs. 1225, then what is the base length?
For the next two questions: The areas of the ends of a frustum of a pyramid are P and Q where P < Q and H is its thickness. What is the difference in radii of the ends of the frustum?
What is the volume of the frustum?
What number should be added to 231228 to make it exactly divisible by 33?
Let AB be a line segment such that a point P on the locus satisfies AB^2 - AP^2 - BP^2 = 0. What is the locus?
If 2x + 3y <= 6, x >= 0, y >= 0, then one of the solutions is:
The curved surface area of a right circular cone of radius 14 cm is 440 square cm. What is the slant height of the cone?
If the volume of a cube is 729 cubic cm, what is the length of its diagonal?
The altitude of an equilateral triangle is sqrt(3). What is its perimeter?
The area of a rectangle whose one side is 'a' is 2a^2. What is the area of a square having one of the diagonals of the rectangle as side?
The total surface area of a cube is 150 square cm. What is its volume?
The HCF and LCM of two natural numbers are 12 and 72 respectively. What is the difference between the two numbers if one of the numbers is 24?
A person sold an article for Rs. 136 and got 15% loss. Had he sold it for Rs. x, he would have got a profit of 15%. Which one of the following is correct?
Consider the following statements:
Which of the statements is/are correct?
There are two taps A and B to fill up a water tank. The tank can be filled in 40 minutes if both taps are on. The same tank can be filled in 60 minutes if tap A alone is on. How much time will tap B alone take to fill up the same tank?
The pair of numbers which are relatively prime to each other is:
How many rational numbers are there between 1 and 1000?
The sum of two numbers is 20 and their product is 75. What is the sum of their reciprocals?
The ratio of two complementary angles is 1:5. What is the difference between the two angles?
What is the logarithm of 0.0001 with respect to base 10?
Consider the following statements:
Which of the above is/are correct?
If x lies in the first quadrant and cos x = 5/13, what is tan x - cot x equal to?
In the quadrilateral ABCD shown in the figure, angle DAB = angle DCX = 120 degrees. If angle ABC = 105 degrees, what is angle ADC equal to? (X is a point on the extension of BC beyond C.)
In the figure AB is the diameter of a circle with centre O, and AO = CD where C, D are points on the circle joined to an external point P (A, C, P and B, D, P collinear). What is angle APB equal to?
In the figure angle ABC = angle AED = 90 degrees. Consider the following statements:
Which of the above statements is/are correct?
If 3^(x+y) = 81 and 81^(x-y) = 3, then what is x equal to?
Consider the following:
Which of the above identities is/are correct?
If 3 sin x + 5 cos x = 5, then what is the value of (3 cos x - 5 sin x)?
How many numbers between -11 and 11 are multiples of 2 or 3?
If A = {x : x is an even natural number}, B = {x : x is a natural number and multiple of 5} and C = {x : x is a natural number and multiple of 10}, then what is A intersection (B union C) equal to?
If a set A contains 60 elements and another set B contains 70 elements and there are 50 elements in common, then how many elements does A union B contain?