Q1
If the roots of the equation lx²+mx+n=0 are in the ratio p : q,then√(p/q)+√(q/p)+√(m/l) is equal to
Elementary MathematicsAlgebra
UPSC CDS · Solved paper
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If the roots of the equation lx²+mx+n=0 are in the ratio p : q,then√(p/q)+√(q/p)+√(m/l) is equal to
If √(3x²-7x-30) - √(2x²-7x-5) = x-5 has α and β as its roots, then the value of αβ is
If p/x+q/y=m and q/x+p/y=n, then what is x/y equal to ?
If a²-by-cz=0,ax-b²+cz=0 and ax+by-c²=0, then the value of x/(a+x)+y/(b+y)+z/(c+z) will be
If the equations x²-px+q=0 and x²+qx-p=0 have a common root,then which one of the following iscorrect?
If x = 2^(1/3) + 2^(−1/3), then the value of 2x³ − 6x − 5 is equal to
The sum and difference of two expressions are 5x²-x-4 and x²+9x-10 respectively. The HCF of the two expressions will be
If(s - a) + (s - b) + (s - c) = s, then the value of ((s - a)² + (s - b)² + (s - c)²)/(a²+b²+c²) will be
If the polynomial x⁶+px⁵+qx⁴-x²-x-3 is divisible by (x⁴-1), then the value of p²+q² is
Let p and q be non-zero integers. Consider the polynomial A(x)=x²+px+q It is given that (x - m) and (x - km) are simple factors of A(x), where m is a non-zero integer and k is a positive integer,k ≥ 2. Which one of the following is correct?
Let m be a non-zero integer and n be a positive integer. Let R be the remainder obtained on dividing the polynomial xⁿ+mⁿ by (x-m.) Then
If 4ˣ2ʸ=128 and 3³ˣ3²ʸ-9ˣʸ=0,then the value of x + y can be equal to
If the linear factors of ax²-(a² +1 )x + a are p and q, then p + q is equal to
if x = (√(a+2b) + √(a-2b))/(√(a+2b) - √(a-2b)) then bx² - ax + b is equal to (given that b≠ 0)
If a³ = 117 + b³ and a = 3 + b, then the value of a + b is (given a > 0 and b > 0)
If the sum of the roots of ax² + bx + c = 0 is equal to the sum of the squares of their reciprocals, then which one of the following relations is correct?
Consider the following statements in respect of the expression Sₙ = n(n+1)/2 where n is an integer:
Which of the above statements is/are correct?
Consider the following statements in respect of two different non zero integers p and q :
Which of the above statements is/are correct?
If a/b = b/c = c/d, then which of the following is/are correct?1. (b³+c³+d³)/(a³+b³+c³) = d2. (a²+b²+c²)/(b²+c²+d²) = a/d Select the correct answer using the code given below.
7¹⁰-5¹⁰ is divisible by
Let a two-digit number be k times the sum of its digits. If the number formed by interchanging the digits is m times the sum of the digits, then the value of m is
A man walking at 5 km/hour noticed that a 225 m long train coming in the opposite direction crossed him in 9 seconds. The speed of the train is
((sin 35°)/(cos 55°))²-((cos 55°)/(sin 35°))²+2sin 30° is equal to
A cyclist moves non-stop from A to B, a distance of 14 km, at a certain average speed. If his average speed reduces by 1 km per hour, he takes 20 minutes more to cover the same distance. The original average speed of the cyclist is
If a sum of money at a certain rate of simple interest per year doubles in 5 years and at a different rate of simple interest per year becomes three times in 12 years, then the difference in the two rates of simple interest per year is
Let A B C and A' B' C' be two triangles in which A B > A' B', B C > B' C' and C A > C' A'. Let D, E and F be the midpoints of the sides B C, C A and A B respectively. Let D', E' and F' be the midpoints of the sides B' C', C' A' and A' B' respectively. Consider the following statements :
The annual incomes of two persons are in the ratio 9 : 7 and their expenses are in the ratio 4:3. If each of them saves ₹ 2,000 per year, what is the difference in their annual incomes?
Let S be a set of first fourteen natural numbers. The possible number of pairs (a, b), where a, b ∈ S and a≠ b such that ab leaves remainder 1 when divided by 15, is
A clock strikes once at 1 o’clock, twice at 2 o’clock and thrice at 3 o’clock, and so on. If it takes 8 seconds to strike at 5 o’clock, the time taken by it to strike at 10 o’clock is
In a circle of radius 2 units, a diameter AB intersects a chord of length 2 units perpendicularly at P. If AP > BP, then AP is equal to
Under what condition on p and q, one of the roots of the equation x²+x+q=0 is the square of the other?
What is the maximum value of m, if the number N = 90 x 42 x 324 x 55 is divisible by 3ᵐ?
A bike consumes 20 mL of petrol per kilometre, if it is driven at a speed in the range of 25-50 km/hour and consumes 40 mL of petrol per kilometre at any other speed. How much petrol is consumed by the bike in travelling a distance of 50 km, if the bike is driven at a speed of 40 km/hour for the first 10 km, at a speed of 60 km/hour for the next 30 km and at a speed of 30 km/hour for the last 10 km?
Consider the following statements :
Which of the above statements are correct?
For what value of k is (x+1) a factor of x³+k x²-x+2 ?
Consider the following statements :
Which of the above statements is/are correct?
The solution of the inequation 1+1/x-1/x² ≥ 0 is (given that x ≠ 0)
Which of the points P(5, -1), Q(3, -2) and R(1, 1) lie in the solution of the system of inequations x + y ≤ 4 and x - y ≥ 2
A passenger train takes 1 hour less for a journey of 120 km, if its speed is increased by 10 km/hour from its usual speed. What is its usual speed?
The value of k, for which the system of equations 3x-ky-20=0 and 6x - 10y + 40 = 0 has no solution, is
Consider the following in respect of natural numbers a, b and c:
Which of the above are correct?
There are three brothers. The sums of ages of two of them at a time are 4 years, 6 years and 8 years. The age difference between the eldest and the youngest is
A person goes to a market between 4 p.m. and 5 p.m. When he comes back, he finds that the hour hand and the minute hand of the clock have interchanged their positions. For how much time (approximately) was he out of his house?
In a gathering of 100 people, 70 of them can speak Hindi, 60 can speak English and 30 can speak French. Further, 30 of them can speak both Hindi and English, 20 can speak both Hindi and French. If x is the number of people who can speak both English and French, then which one of the following is correct? (Assume that everyone can speak at least one of the three languages.)
A cloth merchant buys cloth from a weaver and cheats him by using a scale which is 10 cm longer than a normal metre scale. He claims to sell cloth at the cost price to his customers, but while selling uses a scale which is 10 cm shorter than a normal metre scale. What is his gain?
Two pipes A and B can fill a tank in 60 minutes and 75 minutes respectively. There is also an outlet C. If A, B and C are opened together, the tank is full in 50 minutes. How much time will be taken by C to empty the full tank?
In a race of 1000 m, A beats B by 100 m or 10 seconds. If they start a race of 1000 m simultaneously from the same point and if B gets injured after running 50 m less than half the race length and due to which his speed gets halved, then by how much time will A beat B?
The salary of a person is increased by 10% of his original salary. But he received the same amount even after increment. What is the percentage of his salary he did not receive?
A truck moves along a circular path and describes 100 m when it has traced out 36° at the centre. The radius of the circle is equal to
Two poles are placed at P and Q on either side of a road such that the line joining P and Q is perpendicular to the length of the road. A person moves x metre away from P parallel to the road and places another pole at R. Then the person moves further x metre in the same direction and turns and moves a distance y metre away from the road perpendicularly, where he finds himself, Q and R on the same line. The distance between P and Q (i.e., the width of the road) in metre is
Suppose chords AB and CD of a circle intersect at a point P inside the circle. Two right-angled triangles A′P′B′ and C′Q′D′ are formed as shown in the figures below such that A′P′ = AP, B′P′ = BP, C′Q′ = CP, D′Q′ = DP and ∠A′P′B′ = 90° = ∠C′Q′D′. Which of the following statements are not correct?
Suppose ABC is a triangle with AB of unit length. D and E are the points lying on AB and AC respectively such that BC and DE are parallel. If the area of triangle ABC is twice the area of triangle ABE, then the length of AD is
A rhombus is formed by joining midpoints of the sides of a rectangle in the suitable order. If the area of the rhombus is 2 square units, then the area of the rectangle is
If each interior angle of a regular polygon is 140°, then the number of vertices of the polygon is equal to
Let the triangles ABC and DEF be such that ∠ABC = ∠DEF, ∠ACB = ∠DFE and ∠BAC = ∠EDF. Let L be the midpoint of BC and M be the midpoint of EF. Consider the following statements :
Which one of the following is correct in respect of the above statements?
The number of rounds that a wheel of diameter 7/11 metre will make in traversing 4 km will be
The base of an isosceles triangle is 300 units and each of its equal sides is 170 units. Then the area of the triangle is
Four equal discs are placed such that each one touches two others. If the area of empty space enclosed by them is 150/847 square centimetre, then the radius of each disc is equal to
ABC and DEF are similar triangles. If the ratio of side AB to side DE is (√(2)+1) : √(3) then the ratio of area of triangle ABC to that of triangle DEF is
A tangent is drawn from an external point O to a circle of radius 3 units at P such that OP = 4 units. If C is the centre of the circle, then the sine of the angle COP is
A square is inscribed in a right-angled triangle with legs p and q, and has a common right angle with the triangle. The diagonal of the square is given by
If x/a - y/b tanθ = 1 and x/a tanθ + y/b = 1, then the value of x²/a² + y²/b² is
Consider the following :1. √((1-cosθ)/(1+cosθ)) = cscθ - cotθ2. √((1+cosθ)/(1-cosθ)) = cscθ + cotθ Which of the above is/are identity/ identities?
If p=cotθ+tanθ and q=secθ-cosθ, then (p²q)^(2/3)-(q²p)^(2/3) is equal to
Two observers are stationed due north of a tower (of height x metre) at a distance y metre from each other. The angles of elevation of the tower observed by them are 30° and 45° respectively. Then x/y is equal to
If (cos²θ − 3cosθ + 2)/sin²θ = 1, where 0 < θ < 90°:
Select the correct answer using the code given below.
In a triangle ABC if A-B=π/2, then C+2B is equal to
Which of the following is correct in respect of the equation 3-tan²θ =α(1-3tan²θ)? (Given that α is a real number.)
There are five lines in a plane, no two of which are parallel. The maximum number of points in which they can intersect is
Consider a circle with centre at O and radius 7 cm. Let QR be a chord of length 2 cm and let P be the midpoint of QR. Let CD be another chord of this circle passing through P such that ∠CPQ is acute. If M is the midpoint of CD and MP = √(24) cm, then which of the following statements are correct?1. ∠QPD = 135° 2. If CP = m cm and PD = n cm, then m and n are the roots of the quadratic equation x² - 10x + 1 = 0. 3. The ratio of the area of triangle OPR to the area of triangle OMP is 1: 2√(2). Select the correct answer using the code given below.
Let ABC be a triangle in which AB = AC. Let L be the locus of points X inside or on the triangle such that BX = CX. Which of the following statements are correct?
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Consider a circle with centre at C. Let OP, OQ denote respectively the tangents to the circle drawn from a point O outside the circle. Let R be a point on OP and S be a point on OQ such that OR x SQ = OS x RP. Which of the following statements is/are correct?
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If m and n are distinct natural numbers, then which of the following is/are integer / integers?1. m/n+n/m2. mn(m/n+n/m)(m²+n²)⁻¹3. mn/(m²+n²)Select the correct answer using the code given below.
Outside a meeting room, Madhukar was told by a person that each meeting takes place after 13/4 hours. The last meeting has been over just 45 minutes ago and the next meeting will take place at 2 p.m. At what time did Madhukar receive this information?
The expenditure of a household for a certain month is ₹ 20,000, out of which ₹ 8,000 is spent on education, ₹ 5,900 on food, ₹ 2,800 on shopping and the rest on personal care. What percentage of expenditure is spent on personal care?
If 61/19=3+1/(x+1/(y+1/z)) where x, y and z are natural numbers, then what is z equal to?
A circular path is made from two concentric circular rings in such a way that the smaller ring when allowed to roll over the circumference of the bigger ring, it takes three full revolutions. If the area of the pathway is equal to n times the area of the smaller ring, then n is equal to
Consider the following statements :
Consider the following statements :
Which of the above statements is/are correct?
In a triangle P Q R, point X is on P Q and point Y is on P R such that X P = 1.5 units, X Q = 6 units, P Y = 2 units and Y R = 8 units. Which of the following are correct?
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If a transversal intersects four parallel straight lines, then the number of distinct values of the angles formed will be
A person travels 7 km eastwards and then turns right and travels 3 km and further turns right and travels 13 km. What is the shortest distance of the present position of the person from his starting point?
ABC is a triangle in which D is the midpoint of BC and E is the midpoint of AD. Which of the following statements is/are correct?
Select the correct answer using the code given below.
A circle of 3 m radius is divided into three areas by semicircles of radii 1 m and 2 m as shown in the figure above. The ratio of the three areas A, B and C will be
If √(x/y) = 10/3 - √(y/x) and x-y=8, then the value of xy is equal to
ABC is a triangle right angled at C as shown in the figure above. Which one of the following is correct?
In the figure given above, AD = CD = BC. What is the value of ∠ CDB?
ABC is an equilateral triangle and X, Y and Z are the points on BC, CA and AB respectively such that BX = CY = AZ. Which of the following is/are correct?
Select the correct answer using the code given below.
In the figure given above, p and q are parallel lines. What are the values of the angles x, y and z?
The linear inequations, for which the shaded area in the figure given above is the solution set, are
The election result in which six parties contested was depicted by a pie chart. Party A had an angle 135° on this pie chart. If it secured 21960 votes, how many valid votes in total were cast?
The mean and median of 5 observations are 9 and 8 respectively. If 1 is subtracted from each observation, then the new mean and the new median will respectively be
The age distribution of 40 children is as follows: | Age (in years) | 5–6 | 6–7 | 7–8 | 8–9 | 9–10 | 10–11 | | No. of children | 4 | 7 | 9 | 12 | 6 | 2 |. Consider the following statements in respect of the above frequency distribution:
Which of the above statements are correct?
Supposexᵢ = λⁱ for 0 ≤ i ≤ 10, where λ > 1. Which one of the following is correct?
Suppose xᵢ = 1/i for i = 1, 2, 3, …, 11. Which one of the following is not correct?
The value of (1/5)log₁₀3125 − 4log₁₀2 + log₁₀32 is
If tanθ + cotθ = 4/√3, where 0 < θ < π/2, then sinθ + cosθ is equal to
The value of the expression ((243+647)²+(243-647)²)/(243 × 243 + 647 × 647) is equal to
If a and b are negative real numbers and c is a positive real number, then which of the following is/are correct?
Select the correct answer using the code given below.
AD is the diameter of a circle with area 707 m² and AB=BC=CD as shown in the figure above.All curves inside the circle are semicircles with their diameters on AD.What is the cost of levelling the shaded region at the rate of ₹ 63 per square metre?