Q1
What is the number of integral solutions of the equations HCF (a, b) = 5 and a + b = 65?
Elementary MathematicsNumber System
UPSC CDS · Solved paper
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What is the number of integral solutions of the equations HCF (a, b) = 5 and a + b = 65?
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In solving a problem, one student makes a mistake in the coefficient of the first degree term and obtains -9 and -1 for the roots. Another student makes a mistake in the constant term of the equation and obtains 8 and 2 for the roots. The correct equation was
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If the roots of the equation Ax^2 + Bx + C = 0 are -1 and 1, then which one of the following is correct?
If A and B are any two non-empty subsets of a set E, then what is A ∪ (A ∩ B) equal to?
Consider the following in respect of the numbers √2, ∛3, ⁶√6:
Which of the above statements is/are correct?
A water pipe is cut into two pieces. The longer piece is 70% of the length of the pipe. By how much percentage is the longer piece longer than the shorter piece?
The sum of two positive numbers x and y is 2.5 times their difference. If the product of the numbers is 84, then what is the sum of the two numbers?
If n is a whole number greater than 1, then n^2(n^2 - 1) is always divisible by
If the roots of the equation (a^2 - bc)x^2 + 2(b^2 - ac)x + (c^2 - ab) = 0 are equal, where b ≠ 0, then which one of the following is correct?
What is the remainder when 4^1000 is divided by 7?
A, B and C can do a piece of work individually in 8, 12 and 15 days respectively. A and B start working, but A quits after working for 2 days. After this, C joins B till the completion of work. In how many days will the work be completed?
If x + y = 5, y + z = 10 and z + x = 15, then which one of the following is correct?
On a 20% discount sale, an article costs Rs. 596. What was the original price of the article?
What is the HCF of 8(x^5 - x^3 + x) and 28(x^6 + 1)?
Two chairs and one table cost Rs. 700 and 1 chair and 2 tables cost Rs. 800. If the cost of m tables and m chairs is Rs. 30,000, then what is m equal to?
Consider the following statements:
Which of the above statements is/are correct?
For any integer n, what is HCF (22n + 7, 33n + 10) equal to?
In a fire range, 4 shooters are firing at their respective targets. The first, the second, the third and the fourth shooter hit the target once every 5 s, 6 s, 7 s and 8 s respectively. If all of them hit their target at 9:00 am, when will they hit their target together again?
If A is a non-empty subset of a set E, then what is E ∪ ((A∩Φ) - (A - Φ)) equal to?
If A : B = 2 : 3, B : C = 5 : 7 and C : D = 3 : 10, then what is A : D equal to?
If the average of A and B is 30, the average of C and D is 20, then which of the following is/are correct?
1. The average of B and C must be greater than 25.
2. The average of A and D must be less than 25.
Select the correct answer using the code given below:
Out of 105 students taking an examination for English and Mathematics, 80 students pass in English, 75 students pass in Mathematics, and 10 students fail in both the subjects. How many students pass in only one subject?
At what point of time after 3 o'clock, do the hour-hand and the minute-hand of a clock occur at right angles for the first time?
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AB and CD are two diameters of a circle of radius r and they are mutually perpendicular. What is the ratio of the area of the circle to the area of the triangle ACD?
A circle of radius 10 cm has an equilateral triangle inscribed in it. The length of the perpendicular drawn from the centre to any side of the triangle is
The dimensions of a field are 15 m by 12 m. A pit 8 m long, 2.5 m wide and 2 m deep is dug in one corner of the field and the earth removed is evenly spread over the remaining area of the field. The level of the field is raised by
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How many circular plates of diameter d can be taken out of a square plate of side 2d with minimum loss of material?
The volume of a hollow cube is 216x^3. The surface area of the largest sphere which can be enclosed in it is
What is the diameter of the largest circle lying on the surface of a sphere of surface area 616 square cm?
What is the total area of three equilateral triangles inscribed in a semicircle of radius 2 cm?
In a trapezium, the two non-parallel sides are equal in length, each being of 5 units. The parallel sides are at a distance of 3 units apart. If the smaller side of the parallel sides is of length 2 units, then the sum of the diagonals of the trapezium is
The shadow of a tower standing on a level plane is found to be 50 m longer when the Sun's elevation is 30° than when it is 60°. What is the height of the tower?
The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is 60° and the angle of elevation of the top of the second tower from the foot of the first tower is 30°. The distance between the two towers is m times the height of the shorter tower. What is m equal to?
If cos x + sec x = 2, then what is cos^n x + sec^n x equal to, where n is a positive integer?
What is sin 25° sin 35° sec 65° sec 55° equal to?
The value of cos 25° - sin 25° is
In a right-angled triangle ABC, right-angled at B, if cos A = 4/5, then what is sin C equal to?
If α and β are complementary angles, then what is √(cos α cosec β - cos α sin β) equal to?
If 2 cot θ = 3, then what is (2 cos θ - sin θ)/(2 cos θ + sin θ) equal to?
If sin θ cos θ = 1/2, then what is sin^6 θ + cos^6 θ equal to?
If sec θ + tan θ = 2, what is the value of sec θ?
What is cosec (75° + θ) - sec (15° - θ) - tan (55° + θ) + cot (35° - θ) equal to?
If sin θ + 2 cos θ = 1, where 0 < θ < π/2, then what is 2 sin θ - cos θ equal to?
If tan 8θ = cot 2θ, where 0 < 8θ < π/2, then what is the value of tan 5θ?
A square is inscribed in a circle of diameter 2a and another square is circumscribing the circle. The difference between the areas of the outer and inner squares is
The area of a sector of a circle of radius 36 cm is 72π cm^2. The length of the corresponding arc of the sector is
A cone of radius r cm and height h cm is divided into two parts by drawing a plane through the middle point of its height and parallel to the base. What is the ratio of the volume of the original cone to the volume of the smaller cone?
Which one of the following is a Pythagorean triple in which one side differs from the hypotenuse by two units?
A cube has each edge 2 cm and a cuboid is 1 cm long, 2 cm wide and 3 cm high. The paint in a certain container is sufficient to paint an area equal to 54 cm^2. Which one of the following is correct?
ABC is a triangle right-angled at A where AB = 6 cm and AC = 8 cm. Semicircles are drawn (outside the triangle) on AB, AC and BC as diameters which enclose areas x, y and z square units respectively. What is x + y - z equal to?
The three sides of a triangle are 15, 25 and x units. Which one of the following is correct?
Two light rods AB = a + b, CD = a - b are symmetrically lying on a horizontal plane. They are kept intact by two strings AC and BD. The perpendicular distance between the rods is 'a'. The length of AC is given by
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The area of a rectangle lies between 40 cm^2 and 45 cm^2. If one of the sides is 5 cm, then its diagonal lies between
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If sin (A + B) = 1, where 0 < B < 45°, then what is cos (A - B) equal to?
For a plot of land of 100 m x 80 m, the level is to be raised by spreading the earth from a stack of a rectangular base 10 m x 8 m with vertical section being a trapezium of height 2 m. The top of the stack is 8 m x 6 m. By how many centimeters can the level be raised?
AB is a straight line; C and D are points on the same side of AB such that AC is perpendicular to AB and DB is perpendicular to AB. Let AD and BC meet at E. What is AE/AD + BE/BC equal to?
Consider an equilateral triangle of a side of unit length. A new equilateral triangle is formed by joining the mid-points of the first one, then a third equilateral triangle is formed by joining the mid-points of the second. The process is continued. The sum of perimeters of all triangles thus formed is
Let ABCD be a parallelogram. Let P, Q, R, S be the mid-points of sides AB, BC, CD, DA respectively. Consider the following statements:
Select the correct answer using the code given below:
AB and CD are two chords of a circle meeting externally at P. Then which of the following is/are correct?
Select the correct answer using the code given below:
Consider the following statements:
Which of the above statements is/are correct?
ABC is triangle right-angled at B. If AB = 6 cm and BC = 8 cm, then what is the length of the circum-radius of the triangle ABC?
If AD is the internal angular bisector of triangle ABC with AB = 3 cm and AC = 1 cm, then what is BD : BC equal to?
The sides of a triangle are in geometric progression with common ratio r < 1. If the triangle is a right-angled triangle, then r^2 is given by
In a triangle ABC, AD is perpendicular to BC and BE is perpendicular to AC. Which one of the following is correct?
ABCD is a parallelogram. E is a point on BC such that BE : EC = m : n. If AE and DB intersect in F, then what is the ratio of the area of triangle FEB to the area of triangle AFD?
If triangles ABC and DEF are similar such that 2AB = DE and BC = 8 cm, then what is EF equal to?
A rectangle of maximum area is drawn inside a circle of diameter 5 cm. What is the maximum area of such a rectangle?
A drainage tile is a cylindrical shell 21 cm long. The inside and outside diameters are 4.5 cm and 5.1 cm respectively. What is the volume of the clay required for the tile?
The diameter of the base of a cone is 6 cm, altitude is 4 cm. What is the approximate curved surface area of the cone?
What is the area of the larger segment of a circle formed by a chord of length 5 cm subtending an angle of 90° at the centre?
Consider the following in respect of the variate which takes values 2, 2, 2, 7, 7, 2, 7, 7:
Which of the above statements is/are correct?
Consider the following statements pertaining to a frequency polygon of a frequency distribution of a continuous variable with seven class intervals of equal width:
Which of the above statements is/are correct?
The mean of the following distribution is 18.
Class interval: 11-13 (frequency 3), 13-15 (6), 15-17 (9), 17-19 (13), 19-21 (f), 21-23 (5), 23-25 (4).
What is the value of f?
Which of the following items of information is a good example of statistical data?
The following table gives 'less than' type frequency distribution of income per day:
Income less than (Rs.): 1500 (100 persons), 1250 (80), 1000 (70), 750 (55), 500 (32), 250 (12).
What is the modal class?
Consider the following:
Which of the above statements is/are correct?
Consider the following statements in respect of a discrete set of numbers:
Which of the above statements is/are correct?
The geometric mean of (x1, x2, x3, ... xn) is X and the geometric mean of (y1, y2, y3, ... yn) is Y. Which of the following is/are correct?
Select the correct answer using the code given below: