Q1
ABC is a triangle. AD, BE, CF are altitudes of
Assertion (A)(AB 2 + BC 2 + CA 2 ) > (AD 2 +BE 2 + CF 2 )
Reason (R)(AE 2 – AF 2 ) + (BE 2 – BD 2 ) + (CD 2 – CE 2 )
Elementary MathematicsStatistics
UPSC CDS · Solved paper
All 100 questions, textbook-style — tap “Show answer” on any question to reveal the correct option and explanation.
ABC is a triangle. AD, BE, CF are altitudes of
ABC is a triangle. Let D, E denote the mid points of BC, CA respectively. Let AD and BE intersect at C. Let O be a point on AD such AO:OD = 2:7.
If x+(1/x)=2 cos a than what is the value of x 2 +(1/x 2 )?
If a is the angle of first quadrant such that cosec 4 a = 17 + cot 4 a, then what is the value of sin a?
Data on percentage distribution of area of land in acres owned by households in two districts of a particular state are as follows: Land holding District A District B 0.01-0.99 5.62 13.53 1.0-2.49 18.35 21.84 2.5-7.49 47.12 39.32 7.5-12.49 19.34 12.15 12.5-19.99 7.21 7.43 20.0-29.99 2.36 5.73 What is the appropriate diagram to represent the above data?
The mean of 25 observations is 36. The mean of first 13 observations is 32 and, that of last 13 observations is 39, what is the value of 13th observation?
Consider the following types of data Marks of students who appeared for a test of 100 marks. Collar sizes of 200 shirts sold in a week. Monthly incomes of 250 employees of a factory. For which of the above data, mode is suitable measure of central tendency?
The arithmetic mean of a set of 10 numbers is 20. If each number is first multiplied by 2 and then increased by 5, then what is the mean of new numbers?
If sin x + sin y = a and cos x + cos y = b, what is sin x . sin y + cos x . cos y equal to?
Which one of the following represents statistical data?
The angle of elevation of the top of an unfinished pillar at a point 150 m from its base is 30°. If the angle of elevation at the same point to be 45°, then the pillar has to be raised to a height of how many metres?
A radio transmitter antenna of height 100 m stands at the top of a tall building. At a point on the ground, the angle of elevation of bottom of the antenna is 45° and that of top of antenna is 60°. What is the height of the building?
What is the value of 5sin 75°sin77°+ 2cosl3°cosl5°/ cos 15°sin 77° -7 sin 81°/ cos 9° ?
If p = tan 2 x + cot 2 x, the which one of the following is correct?
If 0 ≤ x ≤ x /2, then which one of the following is always correct?
If cos x/cos y =n, sin x /sin y =m, then what is(m 2 -n 2 ) sin 2 y equal to?
If x + y = 90° and sin x: sin y √3= :1, then what is x:y equal to?
If cos x/1 + cosec x + cos x/cosec x -1=2 which one of the following is one of the values of a?
If tan 2 y cosec 2 x – 1 – tan 2 y, then which one of the following is correct?
If sin x cos x = 1/2, then what is the value of sin x – cos x?
If cos x + cos 2 x = 1, then what is the value of sin 2 x + sin 4 x?
Consider the following equations cosec 2 x + sec 2 x – cosec 2 x sec 2 x sec 2 x + tan 2 x = sec 2 x tan 2 x cosec 2 x + tan 2 x = cot 2 x + sec 2 x Which of the above statements are correct?
What is log (tan 1°) + log (tan 2°) + log (tan 3°) + … + log (tan 89°) equal to?
Let AB and AC be two rays intersecting at A. Let D, E be the point lying on AB, respectively and P be the point such that P divides the line DE such that PD : PE = AD: AE. What is the locus of the point P?
In the figure given above, O is the centre of a circle circumscribing a quadrilateral ABCD. If AB = BC and ∠BAC = 40°, then what is ∠ADC equal to?
In the figure above, AP = 3 cm, PB = 5 cm, AQ = 2 cm and QC = x. What is the value of x ?
In the figure given above, YAX is a tangent to the circle with centre O. If ƉBAX = 70° and ƉBAQ = 40°, then what is ƉABQ equal to?
In the figure given above, M is the mid point of the side CD of the parallelogram ABCD. What is ON: OB?
ABCD is a concyclic quadrilateral. The tangents at A and C intersect each other at P. If ABC = 100°, then what is APC equal to?
PQ is a common chord of two circles. APB is a secant line joining points A and B on the two circles. Two tangents AC and BC are drawn. If ACB = 45°, then what is AQB equal to?
A, B, C, D are four distinct points on a circle whose centre is at O. If OBD – CDB = CBD – ODB, then what is A equal to?
In the figure given above, what is CBA?
In the figure given above, a circle is inscribed in a quadrilateral ABCD. Given that BC = 38 cm, QB = 27 cm, DC = 25 cm and AD is perpendicular to DC. What is the radius of the circles?
The cnetroid and the orthocenter are coincident for which one of the following triangles?
If the medians of two equilateral triangles are in the ratio 3:2, then what is the ratio of their sides?
In the figure given above, P is a point on AB and PQ is parallel to AC. What is the number of pairs of distinct similar triangles in the figure?
If Ca and C, and r1 and r2 are respectively the centroids and radii of incircles of two congruent triangles, then which one of the following is correct?
A ladder 25 m long is leaing against a wall which is perpendicular to the level ground. The bottom of the ladder is 7 m from the base of the wall. If the top of the ladder slips down 4 m, how much will the bottom of the ladder slip?
Which one of the following figures has only one line of symmetry?
ΔPQR is right angled at Q, PR = 5 cm and QR = 4 cm. If the lengths of sides of another triangle ABC are 3 cm, 4 cm and 5 cm, then which one of the following is correct?
If the figure given above ∠ABD = ∠PQD = ∠CDQ = π/2. If AB = x,PQ = z and CD = y, then which one of the following is correct?
What is the angle (in radian) included between the hands of a clock, when the time is 10 min past 5?
A lead pencil is in the shape of a cylinder. The pencil is 21 cm long with radius 0.4 cm and its lead is of radius 0.1 cm. What is the volume of wood in the pencil?
A cylindrical vessel of base radius 14 cm is filled with water to some height. If a rectangular solid of dimensions 22 cm x 7 cm x 5 cm is immersed in it, what is the rise in water level?
A cylinderical rod of length h is melted and cast into a cone of base radius twice that of the cylinder. What is the height of the cone?
A roller of diameter 70 cm and length 2 m is rolloing on the ground. What is the area covered by the roller in 50 revolutions?
If A is the area of a triangle in cm 2 , whose sides are 9 cm, 10 cm and 11 cm, then which one of the following is correct?
What is the area of right angled isosceles triangle whose hypotenuse is 6√2 cm?
The diameter of the moon is approximately one-fourth of that of the earth. What is the (approximately) ratio of the volume of the moon to that of earth?
Three cubes each of side 5 cm are joined end to end. What is the surface area of the resulting cuboid?
From a rectangular metla sheet of sides 25 cm and 20 cm, a circular sheet as large as possible is cut-off. What is the area of the remaining sheet?
Two sides of a parallelogram are 10 cm and 15 cm. If the altitude corresponding to the side of length 15 cm is 5 cm, then what is the altitude to the side of length 10 cm?
A horse is tied to a pole fixed at one corner of a 50 m x 50 m square field of gass by means of a 20 m long rope. What is the area of that part of the field which the horse can graze?
The difference between the area of a square and that of an equilateral triangle on the same base is 1/4 cm 2 . What is length of side of triangle?
A man walking at the rate 3 km/h croses a square field diagonally in 1 min. What is the area of the field?
What is the greatest number which divides 392, 486 and 627 so as to leave the same remainder in each case?
What is the sum of positive integers less than 100 which leave a remainder 1 when divided by 3 and leave a remainder 2 when dividend by 4?
What is the total number of three digit numbers with unit digit 7 and divisible by 11?
There are some coins and rings of either gold or silver in a box, 60% of the objects are coins, 40% of the rings are of gold and 30% of the coins are of silver. What is the percentage of gold articles?
A train of length 150 m takes 10 s to cross another train 100 m long coming from the opposite direction. If the speed of first train is 30 km/h. What is the speed of second train?
Suppose y is equal to the sum of two quantities of which one varies directly as x and the other inversely as x. If y = 6 when x = 4, and y = 10/3 when x = 3, then what is the relation between x and y?
If 1/x +2/y+2+1009/ z +1009 = 1, then what is the value of x/x+1+y/y+2+x/z+1009 ?
If (x/y) = (z/w), then what is (xy + zw) 2 equal to ?
Two taps can fill a tub in 5 min and 7 min A pipe can empty it in 3 min. If all the three are kept open simultaneously, when will the tube be full?
A bag contains Rs. 114 in the form of 1 rupees, 50 paise and 10 paise coins in the ratio 3:4:10. What is the number of 50 paisa coins?
If a : b = 1 ½ : 2 ¼ and b : c = 2 :3 ½ , then what is a: b: c equal to?
A person A sells a table costing Rs. 2000 to a person B and earns a profit of 6%. The person B sells it to another the person C at a loss of 5%. At what price did B sell the table?
What the seller marked the printed price of a watch purchased at Rs. 380, so that after giving 5% discount, there is 25% profit?
A person invested part of Rs. 45000 at 4% and the rest at 6%. If his annual income from both is equal, then what is the average rate of interest?
The compound interest on a sum for 2 yr is Rs. 832 and the simple interest on the same sum at the same rate for the same period is Rs. 800. What is the rate of interest?
A and B can do a piece of work in 8 days, B and C can do the same work in 12 days. If A, B and C can complete the same work in 6 days, in how many days can A and C complete the same work?
A boy walks from his house to school at 2.5 km/h and arrives 12 min late. The next day he walks at 4 km/h and reaches the school 15 min earlier. What is the distance from his house to school?
Two person P and Q start at the same time form city A for city B, 60 km away. P travels 4km/h slower than Q. Q reaches city B and at once turns back meeting P , 12 km from city B. What is the speed of P?
What is the value
Nine numbers are written in ascending order. The middle number is the average of the nine numbers. The average of the first five larger numbers is 68 and that of five smaller numbers is 44. What is the sum of all numbers?
If we divivde a positive integer by another positive interger, what is the resulting number?
A dishonest dealer professes to sell his goods at cost price, but use a false weight and thus gains 20%. For a kilogram he uses a weight of
If the unit of weight is 15/4 kg. What number will 3/2 quintal represent?
If x cos 60° + cos 0° = 3 and 4x sin 30° -y cot 45° = 2, then what is the value of x?
What is the value of 2 log (5/8) + log (128/ 125) +log (5/2)?
A ball is dropped from a height 64 m above the ground and every time if hits the ground it rises to a height equal to half of the previous. What is the height attained after it hits the ground for the 16th time?
If a x = c q = b and c y = a z = d, then which one of the following is correct?
The product of two alternate odd integers exceeds three times the smaller by 12. What is the larger number?
A person bought 5 tickets from a station P to a station Q and 10 tickets from the station P to a station R. He paid Rs. 350. If the sum of a ticket from P to Q and a ticket from P to R is Rs. 42, then what is the fare from P to Q?
Pooja started her job with certain monthly salary and gets a fixed increment every year. If her salary was Rs. 4200 after 3 yr and Rs. 6800 after 8 yr of service, then what are her initial salary and the annual increment respectively?
What is the solution of the equations x-y = 0.9 and 11 (x+y) -1 =2?
What is the magnitude of difference of the roots of x 2 – ax + b = 0?
If x = 1 + √2, then what is the value of x 3 -4x 3 + 4x 2 ?
If 3 x + 27 (3 -x ) = 12, then what is the value of x?
What is the LCM of (x 2 – y 2 – z 2 – 2yz), (x 2 – y 2 – x 2 + xz) and (x 2 – y 2 – z 2 – 2xy)?
If (x + 2) is the HCF of x2 + ax + b and x2 + cx + d (a ≠ c and b ≠ d), then which one of the following is correct?
What are the values of c when the HCF of x 3 + cx 2 – x + 2c and x 2 + cx – 2 over the rationals is a linear polynomial?
When (x 3 – 2x 2 + px – q) is divided by (x 2 – 2x 3), the remainder is (x – 6). What are the values of p, q respectively?
If the remainder of the polynomial a 0 + a 1 x + a 2 x 2 +,………+ a n x n when divivded by (x – 1) is 1, then which one of the following is correct?
What is x (y – z) (y + z) + y (z – x) (z + x) + z (x – y) (x + y) equal to?
Consider the following statements: A number a l a 2 a 3 a 4 a 5 is divisible by 9 if a 1 + a 2 + a 3 + a 4 + a 5 is divisible by 9. a 1 – a 2 + a 3 – a 4 + a 5 is divisible by 9. Which of the above statements is/are correct?
If log r 6 = m and log r 3 = n, then what is log r (r/2) equal to?
What is the last digit in the expansion of (2457) 754 ?
When a natural number n is divided by 4, the remainder is 3. What is the remainder when 2n is divided by 4?