The factor(s) of 5px − 10qy + 2rpx − 4qry is/are

Option C: Both (5+2r) and (px−2qy)

UPSC CDS · Solved paper

CDS 2013 — Elementary Mathematics (CDS-I)

All 100 questions, textbook-style — tap “Show answer” on any question to reveal the correct option and explanation.

Q1
Consider the following statements:
1There is a finite number of rational numbers between any two rational numbers.
2There is an infinite number of rational numbers between any two rational numbers.
3There is a finite number of irrational numbers between any two rational numbers.
Which of the above statements is/are correct?
Elementary MathematicsNumber System
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Q2
Representation of 0.2341 (with bar over 41) in the form p/q, where p, q are integers and q ≠ 0, is
Elementary MathematicsNumber System
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Q3
In a school there are 30 teachers who teach Mathematics or Physics. Of these, 20 teach Mathematics and 15 teach Physics, 5 teach both Mathematics and Physics. The number of teachers teaching only Mathematics is
Elementary MathematicsSets
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Q4
'X' can complete a job in 12 days. If 'X' and 'Y' work together, they can complete the job in 6 2/3 days. 'Y' alone can complete the job in
Elementary MathematicsArithmetic
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Q5
If a body covers a distance at the rate of x km/hr and another equal distance at the rate of y km/hr, then the average speed in km/hr is
Elementary MathematicsArithmetic
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Q6
A sailor sails a distance of 48 km along the flow of a river in 8 hours. If it takes 12 hours to return the same distance, then the speed of the flow of the river is
Elementary MathematicsArithmetic
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Q7
A mason can build a tank in 12 hours. After working for 6 hours, he took the help of a boy and finished the work in another 5 hours. The time that the boy will take alone to complete the work is
Elementary MathematicsArithmetic
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Q8
The principal on which a simple interest of Rs. 55 will be obtained after 9 months at the rate of 3 2/3% per annum is
Elementary MathematicsArithmetic
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Q9
The expression ((√2)^√2)^√2 is
Elementary MathematicsNumber System
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Q10
The LCM of two numbers is 2376 while their HCF is 33. If one of the numbers is 297, then the other number is
Elementary MathematicsNumber System
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Q11
A person travels a certain distance at 3 km/hr and reaches 15 minutes late. If he travels at 4 km/hr, he reaches 15 minutes earlier. The distance he has to travel is
Elementary MathematicsArithmetic
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Q12
The two-digit number, which when divided by the sum of the digits and the product of the digits respectively the remainder is the same, and the difference of quotients is one, is
Elementary MathematicsNumber System
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Q13
The short and long hands of a clock are 4 cm and 6 cm long respectively. Then the ratio of distances travelled by tips of short hand in 2 days and long hand in 3 days is
Elementary MathematicsMensuration
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Q14
What is the value of (725×725×725 + 371×371×371)/(725×725 − 725×371 + 371×371)?
Elementary MathematicsAlgebra
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Q15
If n, (n+2), (n+4) are prime numbers, then the number of possible solutions for n are
Elementary MathematicsNumber System
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Q16
The sum of two numbers is 7 and the sum of their squares is 25. The product of the two numbers is
Elementary MathematicsAlgebra
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Q17
The system of equations 3x + y − 4 = 0 and 6x + 2y − 8 = 0 has
Elementary MathematicsAlgebra
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Q18
(a+1)^4 − a^4 is divisible by
Elementary MathematicsAlgebra
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Q19
The factor(s) of 5px − 10qy + 2rpx − 4qry is/are
Elementary MathematicsAlgebra
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Q20
If one of the roots of the equation x² − bx + c = 0 is the square of the other, then which one of the following is correct?
Elementary MathematicsAlgebra
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Q21
If x + 1/x = 2, then what is x − 1/x equal to?
Elementary MathematicsAlgebra
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Q22
What is the value of √(6+√(6+√(6+…)))?
Elementary MathematicsAlgebra
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Q23
One of the factors of the polynomial x^4 − 7x³ + 5x² − 6x + 81 is
Elementary MathematicsAlgebra
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Q24
Let A = {x : x is an odd integer} and B = {x : x² − 8x + 15 = 0}. Then which one of the following is correct?
Elementary MathematicsSets
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Q25
Consider the following in respect of the sets A and B:
1(A ∩ B) ⊆ A
2(A ∩ B) ⊆ B
3A ⊆ (A ∪ B)
Which of the above are correct?
Elementary MathematicsSets
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Q26
What is the HCF of the polynomials x³+8, x²+5x+6 and x³+2x²+4x+8?
Elementary MathematicsAlgebra
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Q27
The set of natural numbers is closed under
1addition
2subtraction
3multiplication
4division
Which of the above is/are correct?
Elementary MathematicsNumber System
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Q28
For what value of k is (x+5) a factor of 6x² + kx + 10?
Elementary MathematicsAlgebra
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Q29
If 3 men and 4 boys can do a piece of work in 8 days, then 6 men and 8 boys can do the same work in
Elementary MathematicsArithmetic
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Q30
Y and Z had taken dinner together. The cost of the meal of Z was 20% more than that of Y and the cost of the meal of X was 5/6 as much as the cost of the meal of Z. If X paid Rs. 100, then what was the total amount that all three of them had paid?
Elementary MathematicsArithmetic
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Q31
Which one of the following numbers is divisible by 11?
Elementary MathematicsNumber System
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Q32
Ten chairs and six tables together cost Rs. 6,200; three chairs and two tables together cost Rs. 1,900. The cost of 4 chairs and 5 tables is
Elementary MathematicsArithmetic
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Q33
If a, b, c, d, e are in continued proportion, then a/e is equal to
Elementary MathematicsAlgebra
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Q34
If the expression x³ + 3x² + 4x + p contains (x+6) as a factor, then the value of p is
Elementary MathematicsAlgebra
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Q35
The difference in the roots of the equation 2x² − 11x + 5 = 0 is
Elementary MathematicsAlgebra
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Q36
The HCF of (x^4 − y^4) and (x^6 − y^6) is
Elementary MathematicsAlgebra
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Q37
The LCM of (x³ − x² − 2x) and (x³ + x²) is
Elementary MathematicsAlgebra
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Q38
The smallest positive prime (say p) such that 2^p − 1 is not a prime is
Elementary MathematicsNumber System
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Q39
A number consists of two digits whose sum is 10. If the digits of the number are reversed, then the number decreases by 36. Which of the following is/are correct?
1The number is divisible by a composite number.
2The number is a multiple of a prime number.
Select the correct answer using the code given below:
Elementary MathematicsNumber System
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Q40
If 16 × 8^(n+2) = 2^m, then m equals
Elementary MathematicsAlgebra
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Q41
Consider the following statements for 0 ≤ θ ≤ 90°:
1The value of sinθ + cosθ is always greater than 1. 2. The value of tanθ + cotθ is always greater than 1.
Which of the above statements is/are correct?
Elementary MathematicsTrigonometry
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Q42
What is (tan A − sin A)/sin³A equal to?
Elementary MathematicsTrigonometry
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Q43
If sin A = 3/5 and A is an acute angle, then tan A + sec A is equal to
Elementary MathematicsTrigonometry
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Q44
If sin θ = (x²−y²)/(x²+y²), then which one of the following is correct?
Elementary MathematicsTrigonometry
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Q45
If a² = (1+2sinθcosθ)/(1−2sinθcosθ), then what is (a+1)/(a−1) equal to?
Elementary MathematicsTrigonometry
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Q46
If cos A + cos²A = 1, then what is 2(sin²A + sin^4A) equal to?
Elementary MathematicsTrigonometry
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Q47
What is (1−tan A)² + (1+tan A)² + (1−cot A)² + (1+cot A)² equal to?
Elementary MathematicsTrigonometry
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Q48
As seen from the top and bottom of a building of height h metres, the angles of elevation of the top of a tower are α and β respectively. If β = 30°, then what is tan α equal to? (Tower height = (3+√3)h/2 m)
Elementary MathematicsTrigonometry
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Q49
(Same setup as Q48) If α = 30°, then what is tan β equal to?
Elementary MathematicsTrigonometry
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Q50
(Same setup) If α = 30° and h = 30 m, then what is the distance between the base of the building and the base of the tower?
Elementary MathematicsTrigonometry
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Q51
(Same setup) If β = 30° and θ is the angle of depression of the foot of the tower as seen from the top of the building, then what is tan θ equal to?
Elementary MathematicsTrigonometry
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Q52
The angles A, B, C, D of a quadrilateral ABCD are in the ratio 1:2:4:5. What is cos(A+B) equal to?
Elementary MathematicsTrigonometry
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Q53
(Same quadrilateral) What is cosec(C−D+B) equal to?
Elementary MathematicsTrigonometry
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Q54
Consider the following statements:
1ABCD is a cyclic quadrilateral.
2sin(B−A) = cos(D−C).
Which of the above statements is/are correct?
Elementary MathematicsGeometry
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Q55
What is sec²D − tan²D equal to?
Elementary MathematicsTrigonometry
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Q56
An isosceles right-angled triangle has area 1 square unit. What is its perimeter?
Elementary MathematicsMensuration
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Q57
If the heights and the areas of the base of a right circular cone and a pyramid with square base are the same, then they have
Elementary MathematicsMensuration
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Q58
A circular water fountain 6.6 m in diameter is surrounded by a path of width 1.5 m. The area of this path in square metres is
Elementary MathematicsMensuration
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Q59
The area of a rectangular field is 4500 sq m. If its length and breadth are in the ratio 9:5, then its perimeter is
Elementary MathematicsMensuration
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Q60
The area of a square inscribed in a circle of radius 8 cm is
Elementary MathematicsMensuration
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Q61
A solid spherical ball of iron of radius 4 cm is melted to form spheres of radius 2 cm each. The number of spheres so formed is
Elementary MathematicsMensuration
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Q62
From a solid wooden right circular cylinder, a right circular cone whose radius and height are same as the radius and height of the cylinder respectively is carved out. What is the ratio of the volume of the utilized wood to that of the wasted wood?
Elementary MathematicsMensuration
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Q63
The arc AB of the circle with centre at O and radius 10 cm has length 16 cm. What is the area of the sector bounded by the radii OA, OB and the arc AB?
Elementary MathematicsMensuration
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Q64
The height of a cylinder is 15 cm. The lateral surface area is 660 sq cm. Its volume is
Elementary MathematicsMensuration
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Q65
A bucket is of height 25 cm. Its top and bottom radii are 20 cm and 10 cm respectively. Its capacity in litres is
Elementary MathematicsMensuration
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Q66
The volume of a right circular cone of height 3 cm and slant height 5 cm is
Elementary MathematicsMensuration
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Q67
A cylindrical tube open at both ends is made of metal. The internal diameter of the tube is 6 cm and length of the tube is 10 cm. If the thickness of the metal used is 1 cm, then the outer curved surface area of the tube is
Elementary MathematicsMensuration
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Q68
The ratio of surface area to diameter of a sphere whose volume is 36π cu cm is
Elementary MathematicsMensuration
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Q69
The volume of the material of a hemispherical shell with outer and inner radii 9 cm and 7 cm respectively is approximately
Elementary MathematicsMensuration
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Q70
The length, breadth and height of a box are respectively 14 m, 12 m and 13 m. The length of the greatest rod that can be put in it is
Elementary MathematicsMensuration
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Q71
Each side of an equilateral triangle is 6 cm. Its altitude is
Elementary MathematicsMensuration
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Q72
What is the quantity of cloth required to roll up to form a right circular tent whose base is of radius 12 m and height 5 m?
Elementary MathematicsMensuration
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Q73
The minute hand of a watch is 2.5 cm long. The distance its extreme end traverses in 40 minutes is
Elementary MathematicsMensuration
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Q74
A swimming pool 70 m long, 44 m wide and 3 m deep is filled by water flowing from a pipe at the rate of 30,800 cm³ per second. The time taken to fill the swimming pool is
Elementary MathematicsMensuration
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Q75
If the area of a regular hexagon is 96√3 sq cm, then its perimeter is
Elementary MathematicsMensuration
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Q76
If the arms of one angle are respectively parallel to the arms of another angle, then the two angles are
Elementary MathematicsGeometry
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Q77
The side BC of a triangle ABC is produced to D; bisectors of the angles ABC and ACD meet at P. If ∠BPC = x° and ∠BAC = y°, then which one of the following is correct?
Elementary MathematicsGeometry
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Q78
There are 45 male and 15 female employees in an office. If the mean salary of the 60 employees is Rs. 4,800 and the mean salary of the male employees is Rs. 5,000, then the mean salary of the female employees is
Elementary MathematicsStatistics
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Q79
Pipe A can fill a tank in 10 minutes and pipe B can empty it in 15 minutes. If both the pipes are opened in an empty tank, the time taken to make it full is
Elementary MathematicsArithmetic
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Q80
ABCD is a trapezium with parallel sides AB=2 cm and DC=3 cm. E and F are the mid-points of the non-parallel sides. The ratio of area of ABFE to area of EFCD is
Elementary MathematicsMensuration
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Q81
In the figure given, PQRS is a parallelogram. If AP, AQ, BR, CS are bisectors of ∠P, ∠Q, ∠R and ∠S respectively, then ABCD (formed by the bisectors) is a
Elementary MathematicsGeometry
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Q82
In the figure, the area of rectangle ABCD is 100 sq cm, O is any point on AB and CD = 20 cm. Then the area of triangle ODC is
Elementary MathematicsMensuration
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Q83
ABC is a right-angled triangle such that AB = a−b, BC = a and CA = a+b. D is a point on BC such that BD = AB. The ratio BD:DC for any value of a and b is
Elementary MathematicsGeometry
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Q84
Let ABC be an equilateral triangle. If the side BC is produced to D so that BC = 2CD, then AD² is equal to
Elementary MathematicsGeometry
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Q85
ABC is a triangle where BC = 2AB, ∠C = 30° and ∠A = 90°. The magnitude of side AC is
Elementary MathematicsTrigonometry
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Q86
The bisectors BI and CI of the angles B and C of a triangle ABC meet at I. What is ∠BIC equal to?
Elementary MathematicsGeometry
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Q87
In a triangle ABC, ∠BCA = 90° and CD is perpendicular to AB. If AD = 4 cm and BD = 9 cm, then the value of DC will be
Elementary MathematicsGeometry
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Q88
Consider a circle with centre at O and radius r. Points A and B lie on its circumference and a point M lies outside of it such that M, A and B lie on the same straight line. Then the ratio of MA to MB is
Elementary MathematicsGeometry
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Q89
In the figure, ABCD is a trapezium. EF is parallel to AD and BC. Angle y is equal to
Elementary MathematicsGeometry
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Q90
In the figure, ∠PQR = 90° and QL is a median, PQ = 5 cm and QR = 12 cm, then QL is equal to
Elementary MathematicsGeometry
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Q91
Consider the following statements:
1The tangent of a circle is a line that meets the circle in one and only one point.
2The tangent of a circle at the end point of the diameter is perpendicular to the diameter.
Which of the above statements is/are correct?
Elementary MathematicsGeometry
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Q92
In a right-angled triangle ABC, ∠C = 90° and CD is perpendicular to AB. If AB×CD = CA×CB, then 1/CD² is equal to
Elementary MathematicsGeometry
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Q93
The mean of 7 observations is 7. If each observation is increased by 2, then the new mean is
Elementary MathematicsStatistics
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Q94
Which of the following are examples of discrete variables?
1Number of errors per page in a book.
2Height of individuals measured in cm.
3Waiting time to failure of electric bulbs.
4Number of leaves on branches of a tree.
Select the correct answer using the code given below:
Elementary MathematicsStatistics
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Q95
The average weight of a class of 15 boys and 10 girls is 38.4 kg. If the average weight of the boys is 40 kg, then what is the average weight of the girls?
Elementary MathematicsStatistics
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Q96
In a frequency distribution having class intervals 0–10, 10–20, 20–30 and 30–40, the respective frequencies are x, x+8, x−2 and x−4 and the arithmetic mean is 17.8. The value of x is
Elementary MathematicsStatistics
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Q97
(Same distribution) The median lies in which one of the following class intervals?
Elementary MathematicsStatistics
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Q98
(Same distribution) The modal class is
Elementary MathematicsStatistics
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Q99
Consider the following statements:
1A frequency distribution condenses the data and reveals its important features.
2A frequency distribution is an equivalent representation of original data.
Which of the above statements is/are correct?
Elementary MathematicsStatistics
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Q100
Which one of the following statements is correct?
Elementary MathematicsStatistics
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