Q1
Two sides of an acute-angled triangle are 6 cm and 2 cm respectively. Which one of the following represents the correct range of the third side in cm?
Elementary MathematicsGeometry
UPSC CDS · Solved paper
All 100 questions, textbook-style — tap “Show answer” on any question to reveal the correct option and explanation.
Two sides of an acute-angled triangle are 6 cm and 2 cm respectively. Which one of the following represents the correct range of the third side in cm?
What is the last digit in the expansion of 34798 ?
What is 1/(√9-√8) - 1/(√8-√7) + 1/(√7-√6) - 1/(√6-√5) + 1/(√5-√4) equal to?
What are the factors of x2 + 4y2 + 4y-4xy-2x-8 ?
What is the HCF of 4x3 + 3x2y-9xy2 + 2y3 and x2+xy-2y2?
What is the sum of the digits of the least number which, when divided by 52, leaves 33 as remainder; when divided by 78, leaves 59 and when divided by 117, leaves 98 as remainder?
If f(x) and g(x) are two polynomials with integral coefficients which vanish at x = 1 / 2, then what is the factor of HCF of f(x) and g(x) ?
If the expression (px3 + x2-2x-q) is divisible by (x – 1) and (x + 1), what are the values of p and q respectively?
The ratio between the ages of A and B is 2 : 5. After 8 years, their ages will be in the ratio 1:2. What is the difference between their present ages?
A mixture contains milk and water in the ratio 5:1. On Adding 5 litres of water, the ratio of milk and water becomes 5:2. What is the quantity of milk in the original mixture?
What is the smallest positive integer which when divided by 4, 5, 8, 9 leaves remainders 3, 4, 7,8 respectively?
What is the value of x for which x, x + 1, x+3 are all’ prime numbers?
A person spends 30% of monthly salary on rent, 25% on food, 20% on children’s education and 12% on electricity, and the balance of Rs 1,040 on the remaining items. What is the monthly salary of the person?
If p^x = r^y = m and r^w = p^z = n, then which one of the following is correct?
A father and his son start at a point A with speeds of 12 kmph and 18 kmph respectively, and reach another point B. If his son starts 60 minutes after his father at A and reaches B 60 minutes before his father, what is the distance between A and B?
If the equation (a2 +b2)x2 -2(ac+bd)x + (c2 +d2) = 0 has equal roots, then which one of the following is correct?
Two trains of lengths 100 m and 150 m are traveling in opposite directions! at speeds of 75 kmph and 50 kmph respectively. What is the time taken by them to cross each other?
If (a – b): (a +b) = 1:5, then what is (a2 -b2): (a2 + b2) equal to?
If x varies as, the mth power of y, y varies as the nth power of z and x varies as the pth power of z, then which one of the following is correct?
If x = (b-c)(a-d), y = (c-a)(b-d), z =(a-b)(c-d), then what is x3 + y3 +z3 equal to?
If the 14th term of an arithmetic series is 6 and 6th term is 14, then what is the 95th term?
What is the solution of the equation √(x/(x+3)) - √((x+3)/x) = -3/2 ?
If n is a positive integer, then what is the digit in the unit place of 3^(2n+1) + 2^(2n+1) ?
If k is any even positive integer, then (k2 +2k) is ,
If salary of \X is 20% more than salary of Y, then by how much percentage is salary of Y less than X?
What are the roots of the equation 4^x - 3·2^(x+2) + 32 = 0 ?
If α and β are the roots of the equation x² - x - 1 = 0, then what is the value of (α⁴ + β⁴) ?
Under what condition do the equations kx-y = 2 and 6x-2y =3 have a unique solution?
If sum as well as product of roots of a quadratic equation is 9, then what is the equation?
Which one of the following is a non-terminating and repeating decimal?
If a + b + c = 0, then what is the value , a2/bc+ b2/ca+ c2/ab?
If 1 is added to the denominator of a fraction, it becomes 1/2 and if 1is added to the numerator, the fraction becomes 1. What is the fraction?
A motorboat takes 2 hours to travel a distance of 9 km down the current and it takes 6 hours to travel the same distance against the current. What is the speed of the boat in still water in kmph?
A can do a piece of work in 24 days. If B is 60% more efficient than A, then how many days does B require to do the same work?
A sum of money lent on simple interest trebles itself in 15 years and 6 months. In how many years will it be doubled?
On selling an article for Rs 240, a trader loses 4%. In order to gain 10%, he must sell the article for
In a class, the number of boys is more than the number of girls by 12% of the total strength. What is the ratio of number of boys to that of girls?
Three numbers are in the ratio 3:2:5 and the sum of their squares is 1862. What are the three numbers?
If one root of the equation ax2 +x-3 = 0 is -1, then what is the other root?
What is the value of (1/2)log₁₀ 25 - 2log₁₀ 3 + log₁₀ 18 ?
The product of a rational number and an irrational number is
The angle of elevation of the top of a tower from the bottom of a building is twice that from its top. What is the height of the building if the height of the tower is 75 m and the angle of elevation of the top of the tower from the bottom of the building is 60° ?
If cos θ ≥ 1/2 in the first quadrant, then which one of the following is correct?
What is the value of cos 1°cos 2° cos 3°……. cos 90°?
Two poles of height 6m and 11m stand vertically upright on a plane ground. If the distance between their feet is 12 m, what is the distance between their tops?
If sin θ+cos θ= 1, then what is the value of sinθ cos θ?
What is √((1 + sinθ)/(1 - sinθ)) equal to?
The shadow of a tower is 15 m when the Sun’s altitude is 30°. What is the length of the shadow when the Sun’s altitude is 60° ?
The arithmetic mean of 100 numbers was computed as 89.05. It was later found that two numbers 92 and 83 have been misread as 192 and 33 respectively. What is the correct arithmetic mean of the numbers?
In a triangle, if sum of two angles is equal to the third angle (considering the interior angles only), then the triangle is
The side BC of the triangle ABC is extended to D. If ∠ACD = 120°, ∠ABC = 2/3 ∠CAB, then what is ∠BAC ?
Let PAB be a secant to a circle intersecting at points A and B, and PC is a tangent. Which one of the following is correct?
[Figure: secant PAB to a circle with tangent PC from external point P]
The line segments AB and CD intersect at O. OF is the internal bisector of obtuse angle BOC and OE is the internal bisector of acute angle AOC. If ∠BOC = 130°, what is the measure of ∠FOE ?
In the figure given above (cyclic quadrilateral ABCD with secants P-D-A and P-C-B meeting at external point P), if ∠BAD = 60°, ∠ADC = 105°, then what is ∠DPC equal to ?
In the figure given above, ABCD is a quadrilateral with AB parallel to DC and AD parallel to BC. ADC is a right angle. If the perimeter of the triangle ABE is 6 units, what is the area of the quadrilateral?
In the figure given above (PQ parallel to RS; transversal QR making 55° with PQ; at R, RL makes 30° with RQ and 25° with RS; LM drawn from L on RL with 155° above), PQ is parallel to RS. What is the angle between the lines PQ and LM ?
In the figure given above (acute angle ABC with bisector BL; ML parallel to BC), L is any point on the bisector of the acute angle ABC and the line ML is parallel to BC. Which one of the following is correct ?
In the figure given above (circle with diameter PQ, centre O; points R, S on circle; T outside at intersection of extended PR and extended QS), PQ is a diameter of the circle whose centre is at O. If ∠ROS = 44°, then what is the value of ∠RTS ?
In the figure given above (circle with centre O; chords AC and BD intersect at P inside), O is the centre of the circle. AC and BD intersect at P. If ∠AOB = 100°, what is ∠APB ?
In the figure given above (parallelogram ABCD; P on BC with PB:PC = 1:2; DP produced meets AB produced at Q), ABCD is a parallelogram. P is a point on BC such that PB:PC = 1:2. DP produced meets AB produced at Q. If the area of the triangle BPQ is 20 square units, what is the area of the triangle DCP ?
In the figure given above (AB parallel to CD; transversals XO, YO with ∠BXO = 125°, ∠DYO = 35°), AB is parallel to CD. What is ∠XOY ?
What is the radius of the circle inscribed in a triangle having side lengths 35 cm, 44 cm and 75 cm?
What is the maximum length of a rod that can be placed inside a box having the shape of a cuboid of length 30 cm, breadth 24 cm and height 18 cm?
The volume of a sphere is 8 times that of another sphere, What is the ratio of their surface areas?
A rectangular area of 6 square metres is to be painted on a 3 m x 4 m board leaving a border of uniform width on all sides. What should be the width of the border?
A solid cylinder of height 9 m has its curved surface area equal to one-third of the total surface area. What is the radius of the base?
A figure is formed by revolving a rectangular sheet of dimensions 7 cm x 4 cm about its length. What is the volume of the figure thus formed ?
The diagonals of the three faces of a cuboid are x, y, z respectively. What is the volume of the cuboid?
Half of a large cylindrical tank open at the top is filled with water and identical heavy spherical balls are to be dropped into the tank without spilling water out:. If the radius and the height of the tank are equal and each is four times the radius of a ball, what is the maximum number of balls that can be dropped?
The diameters of two circles are 18 cm and 8 cm The distance between their centres is 13 cm. What is the number of common tangents?
If the height of a cone is increased by 50%, then what is the percentage increase in the volume of the cone?
If three cubic biscuits having edges 0-3 m, 0-4 m and 0-5 m respectively are melted and formed into a single cubic biscuit, then what is the total surface area of the cubic biscuit?
A wheel of a bicycle has inner diameter 50 cm and thickness 10 cm. What is the speed of the bicycle if it makes 10 revolutions in 5 seconds?
A wire of length 36 cm is bent in the form of a semicircle. What is the radius of the semicircle?
In order to fix an electric pole along a roadside, a pit with dimensions 50 cm x 50 cm is dug with the help of a spade. The pit is prepared by removing earth by 250 strokes of spade. If one stroke of spade removes 500 cm³ of earth, then what is the depth of the pit ?
In the figure given above (triangle ABC with circles centred at A, B, C), what is the sum of the angles formed around A, B, C except the angles of the triangle ABC ?
In the figure given above (circle with diameter AB, centre P; C on circumference; PC drawn), A and B are end points of diameter of a circle with centre at P, and C is a point on the circumference of the circle such that ∠ABC = 35°, then what is ∠PCA ?
In the figure given above, ABC is an equilateral triangle of side length 30 cm. XY is parallel to BC, XP is parallel to AC and YQ is parallel to AB. If (XY + XP + YQ) is 40 cm, then what is PQ equal to?
[Figure: see official question paper]
In the figure given above, AS is a line of length 2a, with M as mid- point. Semicircles are drawn on one side with AM, MB and AB as diameters. A circle with centre O and radius r is drawn such that this circle touches all the three semicircles. What is the value of r ?
[Figure: see official question paper]
The mean weight of 150 students in a certain class is 60 kg. The mean weight of the boys from the class is 70 kg, while that of girls is 55 kg. What is the number of girls in the class?
In an examination, 40% of the candidates wrote their answers in Hindi and the others in English. The average marks of the candidates written in Hindi is 74 and the average marks of the candidates written in English is 77. What is the average marks of all the candidates?
Consider the following frequency distribution :
Class: 0-10, 0-20, 0-30, 0-40, 0-50
Frequency: 3, 8, 14, 20, 25
What is the above frequency distribution known as ?
Consider the following statements in respect of histogram :
Which of the statements given above is/are correct?
What is the median of the values 11, 7, 6, 9, 12, 15, 19 ?
Let (x₁, y₁), (x₂, y₂), …, (xₙ, yₙ) be n pairs of positive numbers. The arithmetic mean and geometric mean of any set of positive numbers (c₁, c₂, …, cₙ) are denoted by M(cᵢ), G(cᵢ) respectively.
Consider the following :
Which of the above is/are correct ?
A circle and a square have the same perimeter. Which one of the following is correct?
If (x4 + x-4) = 322, what is one of the values of (x-x-1 )?
Consider the following statements :
Which of the statements given above is/are correct?
If two corresponding sides of two similar triangles; are in the ratio 9 : 4, then what is the ratio of their areas?
ABCD is a quadrilateral, the sides of which touch a circle. Which one of the following is correct?
By selling 8 dozen pencils, a shopkeeper gains the selling price of 1 dozen pencils. What is the gain?
The ratio of A to B is x: 8 and the ratio of B to C is 12: z. If the ratio of A to C is 2 : 1, then what is the ratio of x:z?
If α and β are the roots of the equation x² + px + q = 0, then -α⁻¹, -β⁻¹ are the roots of which one of the following equations ?
For a set of positive numbers, consider the following statements :
Which of the above statements is/are correct ?
(For the next three questions to follow):
Let C be a right-circular cone. It is given that the two ends of a frustum of C are of radii 3 cm and 6 cm, and the height of the frustum is 9 cm.
What is the slant height of the given frustum ?
(For the next three questions to follow):
Let C be a right-circular cone. It is given that the two ends of a frustum of C are of radii 3 cm and 6 cm, and the height of the frustum is 9 cm.
What is the height of the cone ?
(For the next three questions to follow):
Let C be a right-circular cone. It is given that the two ends of a frustum of C are of radii 3 cm and 6 cm, and the height of the frustum is 9 cm.
What is the total surface area of the given frustum ?
(For the next three questions to follow):
Let ABCD be a quadrilateral. Let the diagonals AC and BD meet at O. Let the perpendicular drawn from A to CD meet CD at E. Further, AO:OC = BO:OD, AB = 30 cm, CD = 40 cm and the area of the quadrilateral ABCD is 1050 cm².
What is BE equal to ?
(For the next three questions to follow):
Let ABCD be a quadrilateral. Let the diagonals AC and BD meet at O. Let the perpendicular drawn from A to CD meet CD at E. Further, AO:OC = BO:OD, AB = 30 cm, CD = 40 cm and the area of the quadrilateral ABCD is 1050 cm².
What is the area of the triangle ADC equal to ?
(For the next three questions to follow):
Let ABCD be a quadrilateral. Let the diagonals AC and BD meet at O. Let the perpendicular drawn from A to CD meet CD at E. Further, AO:OC = BO:OD, AB = 30 cm, CD = 40 cm and the area of the quadrilateral ABCD is 1050 cm².
What is ∠AEB equal to ?