Q1
Let p(x) be a polynomial. When p(x) is divided by (x-1), it leaves 2 as the remainder. When p(x) is divided by (x-2), it leaves 1 as the remainder. What is the remainder when p(x) is divided by (x-1)(x-2)?
Elementary MathematicsAlgebra
UPSC CDS · Solved paper
All 100 questions, textbook-style — tap “Show answer” on any question to reveal the correct option and explanation.
Let p(x) be a polynomial. When p(x) is divided by (x-1), it leaves 2 as the remainder. When p(x) is divided by (x-2), it leaves 1 as the remainder. What is the remainder when p(x) is divided by (x-1)(x-2)?
Consider the following statements for a positive real number x:
Let p and q be natural numbers such that q > p. What is the largest value of p such that q^2 - 5p - 4 is negative?
Let x and y be natural numbers, each less than 20, such that x, y, x+y and x-y are prime numbers. How many such combinations of (x, y, x+y, x-y) are possible?
If (x+1)(x+p)(x^2+p^2) = x^4 - 1, then what is the value of p?
If (2+√3)^x + (2-√3)^x = 2, then what is the value of (2+√3)^x - (2-√3)^x?
If 1/a + 1/b = 5/6 and 1/a^2 + 1/b^2 = 13/36, then what is the value of 1/a^3 + 1/b^3?
What is the remainder when x^6 is divided by x^2 + 1?
Which one of the following polynomials has (x+2) as a factor?
Given log10 2 = 0.301 and log10 3 = 0.477, [the exact quantity asked could not be verified from available sources]
What is the remainder when (17^25 + 19^25) is divided by 18?
Let H be the HCF of x and y (natural numbers). Let p=(x^3+y^3)/(x^2-xy+y^2) and q=(x^3-y^3)/(x^2+xy+y^2). Consider:
If x^4 = x^2 + 1, where x > 0, then what is the value of 2x^4?
If (p+q)/(q+r) = (r+s)/(s+p), where denominators are non-zero, then which one of the following is correct?
If n is a natural number less than 7, then what is the number of values of n for which (12n+2) and (8n+1) are relatively prime?
What is the HCF of x^3+y^3+3xy-1 and [the second polynomial could not be verified from available sources]?
Let x = n(n+1)(n+2), where n is an even natural number. Which of the following statements is/are correct?
What is the LCM of [the exact list of expressions involving x^4+x^2y^2+y^4 could not be verified from available sources]?
Let XYZ be a 3-digit number. Let D be the difference between XYZ and ZYX. What is the remainder when D is divided by 99?
Let p and q be two natural numbers such that (p+q)^(p+q) is divisible by 512. What is the least value of (p+q)?
[The exact algebraic expression could not be verified from available sources]
If a^b = b^a, then what is the value of [the exact expression could not be verified]?
If x = 2 + 2^(1/2) + 2^(3/2), then what is the value of x^2 - 4x - 2?
If (√(p+x)+√(p-x))/(√(p+x)-√(p-x)) = p, then what is the value of x?
If one root of (a-b/2)x^2 - (a+b/2)x + b = 0 is 1, then what is the other root?
If x - 1/x = 2, where x > 0, then what is the value of x^2 - 1/x^2?
If (a-b)^2 + (b-c)^2 + (c-a)^2 = 6 and a^2+b^2+c^2 = 29, then what is the value of a+b+c?
If p = (√5-2)/(√5+2) and q = (√5+2)/(√5-2), then what is the value of p/q + q/p?
What is the hundreds digit of 25^10?
What is the smallest four-digit number N which leaves remainder 3 when divided by 4, 6, 7, or 9?
If √(2+√(2+√(2+...))) = cosecθ, then what is the value of sinθ?
If 8sinθ - cosθ = 4, where 0 < θ < π/2, then what is cosecθ equal to?
[The exact trigonometric equation could not be verified from available sources]
What is the value of (secθ-tanθ) - √((1-sinθ)/(1+sinθ))?
If cotθ = √7, then what is the value of [the exact expression could not be verified]?
The difference between the acute angles of a right-angled triangle is π/12. What is one of the acute angles?
If α and β are the roots of [the exact equation could not be verified], then what is [the exact expression]?
If x^4 + y^4 = 14x^2y^2, consider:
Consider:
[The exact cyclic fractional expression could not be verified from available sources]
If p sin^2α + q cos^2α = m and q sin^2β + p cos^2β = n, then what is (tanα/tanβ)^2 equal to?
[The exact question text could not be verified from available sources]
If cosecθ - sinθ = p and secθ - cosθ = q, then what is p sinθ + q cosθ equal to?
If cosecθ - sinθ = p and secθ - cosθ = q, then what is p^2q^2(p^2+q^2+3) equal to?
If sinα/sinβ = 4√2/3 and cosα/cosβ = 2√3/9, then what is [the exact requested expression]?
If sinα/sinβ = 4√2/3 and cosα/cosβ = 2√3/9, then what is [the exact requested expression]?
If (1+sinθ)/cosθ = p+√(p^2+1), then what is secθ equal to?
If (1+sinθ)/cosθ = p+√(p^2+1), then what is tanθ equal to?
If sinθ+cosθ = p and secθ+cosecθ = q, where p≠1, then which one of the following is correct?
If sinθ+cosθ = p and secθ+cosecθ = q, then what is tanθ+cotθ equal to?
Two trains X and Y are travelling in the same direction. [Exact speeds and times could not be fully verified] What is the length of train Y?
Two persons X and Y leave place P for place Q at 7:00 a.m. and 7:10 a.m. respectively, travelling at 4.8 km/h and 6 km/h. At what distance from P will they meet?
There are two employees X and Y. [Exact salary change details could not be fully verified] What is the ratio of their salaries?
In a village consisting of p persons, [exact literacy details could not be fully verified]
Two alloys X and Y of copper and zinc are mixed. [Exact quantities could not be verified]
Two students' test scores are given. [Exact details could not be verified] What is the ratio?
One root of 2x^2 - 5px + 2p^2 = 0 is [exact]. What is [the requested value]?
An amount of ₹10,000 is borrowed at 10% per annum compound interest for 3 years, compounded annually, and paid back in 3 equal annual installments. What is each installment (approximately)?
What is the solution of the inequalities 5x+3 < 8x-9 and 2x+20 > 5x+2?
A shopkeeper gives three consecutive discounts of 10%, 20% and 25%, after which he sells at 8% profit on cost price. If he had sold after the first discount only, what profit would he have got?
In triangle ABC, AD is the altitude. [The exact definitions of p, q, r could not be verified]
The sum of the height and radius of a solid [the exact solid and values could not be verified] What is the volume?
In right triangle ABC, right-angled at B, point P [the exact construction and statements could not be verified]
The length, breadth and height of a cuboid are consecutive integers. If the volume is 336 cubic units, what is the total surface area?
In a circle of radius 14 cm, APB is the shorter arc and P is the midpoint of the arc. Let C be the midpoint of chord AB and PC=7 cm. What is the length of chord AP?
Two poles of heights 10 m and 15 m are 25 m apart. What is the height of the point of intersection of the lines joining the tip of each pole to the foot of the other?
ABC is a triangle right-angled at B. (AB+BC) exceeds AC by 10 units. If the perimeter is 60 units, what is the area?
Two poles are 24 m apart and their heights differ by 10 m. What is the distance between their tips?
X, Y, Z are the midpoints of the sides of a triangle. [The exact statements could not be verified]
Which of the following can be exterior angles of regular polygons: 4°, 5°, 6°, 8°?
In triangle ABC, angle A=30°, AB=7 cm, AC=12 cm. What is the area?
In right triangle ABC, right-angled at B, D is [the exact construction could not be verified]. What is [the requested length]?
The interior angle of a regular polygon exceeds its exterior angle by 120°. How many sides does it have?
An angle is one-fourth of its complement. What is the angle?
A triangle has sides 11 cm, 60 cm and [the third side could not be verified]. What is the area?
A pendulum swings through 30° and the arc length is 55 cm. Taking π=22/7, what is the length of the pendulum?
A conical tent has a vertex angle of 60°. [Further measurements could not be verified] What is the volume?
A cone and a hemisphere have equal base and equal volume. What is the ratio of the cone's height to the hemisphere's radius?
A wire is in the form of an equilateral [shape]. It is reshaped into [another shape]. [Exact details could not be verified]
Let x and y denote [the exact areas could not be verified]. Which relation holds?
A Question is given followed by two Statements I and II.
Question: What is the remainder when x^{2n} - y^{2n} + 1 is divided by x^n + y^n, where n is a natural number?
Which one of the following is correct?
A Question is given followed by two Statements I and II.
Question: The product of a natural number N and the number M written by the same digits of N in reverse order is 252. What is N?
Which one of the following is correct?
A Question is given followed by two Statements I and II.
Question: The last digit in the expansion of (54D)^100 is 1. What is D?
Which one of the following is correct?
A Question is given followed by two Statements I and II.
Question: In triangle ABC, ∠A = ∠B - ∠C. Is angle A acute?
Which one of the following is correct?
A Question is given followed by two Statements I and II.
Question: In triangle ABC right-angled at B, AC=20 cm. What is the circumradius?
Which one of the following is correct?
A Question is given followed by two Statements I and II.
Question: ABCD is a parallelogram with ∠ABC=60°. Area is 7√3 sq units. What is the perimeter?
Which one of the following is correct?
A Question is given followed by two Statements I and II.
Question: AB and CD are chords intersecting at P. If AP×PB=48, what is CP×PD?
Which one of the following is correct?
A Question is given followed by two Statements I and II.
Question: In quadrilateral ABCD, AB=6, BC=18, CD=6, DA=9. What is diagonal BD?
Which one of the following is correct?
A Question is given followed by two Statements I and II.
Question: ABC is isosceles with AB=AC=10. Area is 48. What is base BC?
Which one of the following is correct?
A Question is given followed by two Statements I and II.
Question: Diagonals of rhombus ABCD are in ratio 5:12. Is one diagonal equal to the side?
Which one of the following is correct?
Which measure of central tendency is least affected by the presence of extreme observations?
To find the average ratio like price/unit, work done/hour, km/hour under certain conditions, the suitable measure of central tendency is:
The frequency distribution of marks of 100 students is given. [The exact distribution could not be verified] What is [the required statistic]?
The arithmetic mean of 200 observations is 60. If 5 is multiplied to each observation, what is the new mean?
A distribution has 3 components with frequencies 45, 40, 55 and means 2, 2.5, 2. What is the combined mean?
Which one of the following is a positional average?
Consider the grouped data: Distance classes 0-5, 5-10, 10-15, 15-20, 20-25, 25-30 with first group frequencies u, v, 11, 52, 75, 22 and second group 3u, 2v, 40, 50, 30, 28. Median of first group is 20.8 m, mean of second is 17.3 m. What is u?
[Same data as Q97] What is v?
Consider the distribution with median 24: Marks less than 10, 20, 30, 40, 50 have frequencies 5, 30, 30+k, 48+k, 55+k. What is k? [The options could not be verified]
[Same data as Q99] What is the mean of the distribution?