Q1
What can be said about the expansion of 2^12n – 6^4n, where n is a positive integer ?
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 can be said about the expansion of 2^12n – 6^4n, where n is a positive integer ?
What is the locus of centres of circles which touch a given line at a given point ?
If cos 1° = p and cos 89° = q, then which one of the following is correct ?
Consider those numbers between 300 and 400 such that when each number is divided by 6, 9 and 12, it leaves 4 as remainder in each case. What is the sum of the numbers ?
Consider the following assumption and two statements: Assumption : A number ‘ABCDE’ is divisible by 11.
Which one of the following is correct ?
Which of the following is correct in the light of the above statements ?
(For the next two questions to follow):
The average age of 06 persons living in a house is 23.5 years. Three of them are majors and their average age is 42 years. The difference in ages of the three minor children is same.
What is the mean of the ages of minor children ?
(For the next two questions to follow):
The average age of 06 persons living in a house is 23.5 years. Three of them are majors and their average age is 42 years. The difference in ages of the three minor children is same.
What is the median of the ages of minor children ?
A three digit number is divisible by 11 and has its digit in the unit’s place equal to 1. The number is 297 more than the number obtained by reversing the digits. What is the number?
If u, v, w are real numbers such that u³ – 8v³ – 27w³ = 18uvw, which one of the following is correct ?
Which one of the following is correct ?
Which one of the following is a correct statement ?
What is the least integral value of k for which the equation x² – 2(k-1)x + (2k+1) = 0 has both the roots positive ?
What is the value of (1/3)log₁₀125 – 2log₁₀4 + log₁₀32 + log₁₀1 ?
x⁴ + 4y⁴ is divisible by which one of the following ?
What is the value of [(1 – sin² θ)sec² θ + tan² θ] (cos² θ + 1) when 0 < θ < 90° ?
If the highest common factor of two positive integers is 24 then their least common multiple cannot be
If r and s are any real numbers such that 0 ≤ s ≤ 1 and r + s = 1, then what is the maximum value of the product rs ?
r is a non-zero real number such that r75 > r90 . This is possible only when
The median BD of the triangle ABC meets AC at D. If BD = ½ AC, then which one of the following is correct ?
The remainder on dividing given integers a and b by 7 are respectively 5 and 4. What is the remainder when ab is divided by 7 ?
What is the value of √29.16 + √0.2916 + √0.002916 + √0.00002916 ?
Three circular laminas of the same radius are cut out from a larger circular lamina. When the radius of each lamina cut out is the largest possible, then what is the ratio (approximate) of the area of the residual piece of the original lamina to its original total area ?
The three sides of a triangle are 10, 100 and x. Which one of the following is correct ?
1/48 of a work is completed in half a day by 5 persons. Then, 1/40 of the work can be completed by 6 persons in how many days ?
The shadow of a pole 6 metre high is 15 metre long and at the same time the shadow of a tree is 25 metre long. What is the height of the tree ?
The speed of a boat in still water is 11 km/h. It can go 12 km upstream and return downstream to the engine point in 2 hours 45 minutes. What is the speed of stream?
In the figure given above (a circle with diameter AB, chord CD perpendicular to AB at E; AB = 10 cm, AE = 2 cm), what is the length of ED ?
[Figure: circle with diameter AB horizontal, chord CD vertical intersecting AB at E, C above, D below]
The simple interest on a certain sum of money for 3 years at 8% per annum is half the compound interest on Rs. 4,000 for 2 years at 10% per annum. What is the sum placed on simple interest ?
A man bought a number of oranges at 3 for a rupee and an equal number at 2 for a rupee. At what price per dozen, should he sell them to make a profit of 20% ?
If 0 ≤ θ ≤ π/2 and cos θ + √3 sin θ = 2, then what is the value of θ ?
If 6 men and 8 boys can do a piece of work in 10 days while 26 men and 48 boys can do the same in 2 days, what is the time taken by 15 men and 20 boys in doing the same type of work ?
If 50% of (x – y) = 40% of (x + y) then what per cent of x is y ?
The wages: of labourers in a factory has increased in the ratio 22 : 25 and their number decreased in the ratio 3:2. What was the original wage bill of the factory if the present bill is Rs. 5,000/- ?
If a quantity y varies as the sum of three quantities of which the first varies as x, the second varies as -x + x2, the third varies as x3 – x2 then what is y equal to ?
For any integer n, what is the HCF of integers m = 2n + 1 and k = 9n + 4 ?
If ABC is a triangle right angled at C and having u units, v units, w units as the lengths of its sides opposite to the vertices A, B, C respectively, then what is (tangent of the angle at A) + (tangent of the angle at B) equal to ?
Which one of the following is an infinite set ?
ABC is a right triangle with right angle at A. If the value of tan B = 1/√3 then, for any real k the length of the hypotenuse is of the form
For what value of k, will the roots of the equation kx2–5x+6=0 be in the ratio of 2: 3 ?
What is the value of sin²15° + sin²20° + sin²25° + ……… + sin²75° ?
Consider the following data:
X: 1 (f=3), 2 (f=5), 3 (f=9), 4 (f=_), 5 (f=2)
The arithmetic mean of the above distribution is 2.96. What is the missing frequency ?
If α is an acute angle and sin α = √((x-1)/2x), then what is tan α equal to ?
Suppose the angle of elevation of the top of a tree at a point E due east of the tree is 60° and that at a point F due west of the tree is 30°. If the distance between the points E and F is 160 feet, then what is the height of the tree ?
Let WXYZ be a square. Let P,Q,R be the midpoints of WX, XY and ZW respectively and K, L be the midpoints of PQ and PR respectively. What is the value of area of triangle PKL/ area of square WXYZ ?
One saree was purchased for Rs. 564 after getting a discount of 6% and another saree was purchased for Rs. 396 after getting a discount of 1%. Taking both the items as a single transaction, what is the percentage of discount ?
What is the value of 1.3̄85̄ + 4.12̄ ? (bars denote recurring digits)
Three lines intersect each other in pairs. What is the number of angles so formed ?
ABCD is a square. P, Q, R, S are points on the sides AB. BC, CD, DA respectively such that AP = BQ = CR = DS. What is ∠SPQ equal to ?
A new frequency distribution is constructed by doubling each frequency of the original distribution keeping the other entries intact. The following measures are computed for both the tables:
A ladder of 17 ft. length reaches a window which is 15 ft. above the ground on one side of the street. Keeping its foot at the same point the ladder is turned to the other side of the street and now it reaches a window 8 ft. high. What is the width of the street ?
ABC is a triangle. The internal bisector of ∠ABC and the external bisector of ∠ACB meet at D. Which one of the following is correct ?
The following sets of conditions relate to two triangles ABC and DEF. Which set of conditions does not guarantee the congruence of ABC and DEF ?
Consider the following data:
Year: 1911-21, 1921-31, 1931-41, 1941-51, 1951-61, 1961-71, 1971-81
Birth rate: 48.1, 46.4, 45.2, 39.9, 41.7, 41.1, 37.1
Death rate: 38.5, 36.3, 31.2, 27.4, 22.8, 35.9, 14.8
For which period is the natural growth rate minimum ?
The middle points of the parallel sides AB and CD of a parallelogram ABCD are P and Q respectively. If AQ and CP divide the diagonal BD into three parts BX, XY and YD, then which one of the following is correct ? :
Consider the following statements in respect of any triangle :
Which of the statements given above is/are correct ?
The volume of a cone is equal to that of sphere. If the diameter of base of cone is equal to the diameter of the sphere, what is the ratio of height of cone to the diameter of the sphere ?
A parallelogram and a rectangle stand on the same base and on the same side of the base with the same height. If l1,l2 be the perimeters of the parallelogram and the rectangle respectively, then which one of the following is correct ?
A person travelled by train for 1 hour at a speed of 50 kmph. He then travelled by a taxi for 30 minutes at a speed of 32 kmph to complete his journey. What is the average speed at which he travelled during the journey ?
Two similar parallelograms have corresponding sides in the ratio 1 : k. What is the ratio of their areas ?
What is the number of circles passing through a given pair of points ?
Two circles touch each other externally at P. Two secants APB and CPD are drawn through P to meet the circle at A, C and B, D respectively. Then, which one of the following is correct ?
In the above figure (a geometric figure with segment labeled x and side a), what is x equal to ?
[Figure: geometric diagram with length x to be found in terms of a]
In the figure given above (AB parallel to LM, with angles b, c and reflex angle a), what is angle a equal to ?
[Figure: two horizontal parallel lines AB (top) and LM (bottom), with a polygonal line forming angles b, c and reflex angle a]
If the expressions px³ + 3x² – 3 and 2x³ – 5x + p when divided by x – 4 leave the same remainder, then what is the value of p ?
If x(x + y + z) = 9, y(x + y + z) = 16 and z(x + y + z) = 144, then what is x equal to?
A person invested some amount at the rate of 12% simple interest and the remaining at 10%. He received yearly an interest of Rs. 130. Had he interchanged the amounts invested, he would have received an interest of Rs. 134. How much money did he invest at different rates ?
A man is watching from the top of a tower a boat speeding away from the tower. The boat makes an angle of depression of 45° with the man’s eye when at a distance of 60 m from the bottom of tower. After 5 seconds, the angle of depression becomes 30°. What is the approximate speed of the boat assuming that it is running in still water ?
In the figure given above (square ABCD of side 7 cm, with arcs BD centred at C and A forming a shaded lens), what is the area of the shaded portion, formed by the arcs BD of the circles with centre at C and A ?
[Figure: square ABCD, shaded lens-shaped region formed by two circular arcs]
The marks of the students of a class who appeared for a test in English are represented in the following frequency table:
Class interval: 1-10 (f=9), 11-20 (f=22), 21-30 (f=_), 31-40 (f=20), 41-50 (f=12), 51-60 (f=8); Total frequency = 100
What is/are the modal class(es) ?
The length, breadth and height of a rectangular parallelopiped are in ratio 6: 3: 1. If the surface area of a cube is equal to the surface area of this parallelopiped; then what is the ratio of the volume of the cube to the volume of the parallelopiped ?
A rectangular tank whose length and breadth are 2.5 m and 1.5 m respectively is half full of water. If 750 litre more water is poured into the tank, what is the height through which water level further goes up ?
What is the weighted mean of first 10 natural numbers whose weights are equal to the corresponding number ?
What is volume of the frustum of a cone with height h and radii r1, r2 ?
A cylinder having base of circumference 60 cm is rolling without sliding at a rate of 5 rounds per second. How much distance will the cylinder roll in 5 seconds ?
A semi circular thin sheet of a metal of diameter 28 cm is bent and an open conical cup is made. What is the capacity of the cup ?
Smaller lead shots are to be prepared by using the material of a spherical lead shot of radius 1 cm. Some possibilities are listed in the statements given below :
Which of the above statements is/are correct ?
A hemispherical bowl of internal radius 20 cm contains sauce. The sauce is to be filled in conical shaped bottles of radius 5 cm and height 8 cm. What is the number of bottles required ?
A hemisphere is made of a sheet of a metal 1 cm thick. If the outer radius is 5 cm, what is the weight of the hemi- sphere.(1 cm3 of the metal weighs 9 g) ?
A sphere and a cube have same surface area. What is the ratio of the square of volume of the sphere to the square of volume of the cube ?
If 7 cos² θ + 3 sin² θ = 4 and 0 < θ < π/2, then what is the value of tan θ ?
With the help of histogram one can prepare
Three straight lines X, Y and Z are parallel and the angles are as shown in the figure above (X top, Y middle, Z bottom; transversals forming 125°, 80° and 30°). What is ∠AFB equal to ?
[Figure: three parallel horizontal lines X, Y, Z with two transversals and marked angles 125°, 80°, 30°]
Consider the following in respect of the above figure (triangle ABC with altitudes AD and BE, ∠ADB = ∠AEB = 90°) :
Which of the above are correct ? [Figure: triangle with two altitudes forming right angles]
The table below gives the number of members of a club classified by sex and nativity :
Locals Migrants Total
Male 85 45 130
Female 35 35 70
Total 120 80 200
The above data are represented by a pie diagram. What is the sectorial angle of the area representing male-migrant category ?
The inner and outer radii of a 7 m long hollow iron right circular cylindrical pipe are 2 cm and 4 cm respectively. If 1000 cm3 of iron weighs 5 kg, what is the weight of the pipe ?
Consider the following statements in respect of a quadrilateral :
Which of the statements given above is/are correct ?
Which one of the following numbers is not a square of any natural number ?
What is the maximum area of a rectangle, the perimeter of which is 18 cm?
If one of the roots of the equation ax2 + x – 3 = 0 is -1.5, then what is the value of a ?
Consider the following statements :
Which of the statements given above are correct ?
If 0 ≤ θ < π/2 and p = sec² θ, then which one of the following is correct ?
In a triangle ABC, ∠ABC = 90°, ∠ACB = 30°, AB = 5 cm. What is the length of AC ?
When the roots of the quadratic equation ax2 + bx + c = 0 are negative of reciprocals of each other, then which one of the following is correct ?
What are the roots of the equation log₁₀(x²-6x + 45) = 2 ?
The sum of the roots of the equation 1/(x+a) + 1/(x+b) = 1/c is zero. What is the product of the roots of the equation ?
Let there be three simultaneous linear equations in two unknowns. What can be the number of solutions (if they do exist) ?
(For the next three questions to follow):
The arithmetic mean, geometric mean and median of 06 positive numbers a, a, b, b, c, c where a < b < c are 7/3, 2, 2 respectively.
What is the sum of the squares of all the six numbers ?
(For the next three questions to follow):
The arithmetic mean, geometric mean and median of 06 positive numbers a, a, b, b, c, c where a < b < c are 7/3, 2, 2 respectively.
What is the value of c?
(For the next three questions to follow):
The arithmetic mean, geometric mean and median of 06 positive numbers a, a, b, b, c, c where a < b < c are 7/3, 2, 2 respectively.
What is the mode?