Q1
If a² − bc = α, b² − ac = β, c² − ab = γ, then what is (aα + bβ + cγ) / ((a + b + c)(α + β + γ)) equal to?
Elementary MathematicsAlgebra
UPSC CDS · Solved paper
All 100 questions, textbook-style — tap “Show answer” on any question to reveal the correct option and explanation.
If a² − bc = α, b² − ac = β, c² − ab = γ, then what is (aα + bβ + cγ) / ((a + b + c)(α + β + γ)) equal to?
If (x − 1)³ is a factor of x⁴ + αx³ + βx² + γx − 1, then the other factor will be:
A 2-digit number is such that the sum of the number and the number obtained by reversing the order of the digits of the number is 55. Further, the difference of the given number and the number obtained by reversing the order of the digits of the number is 45. What is the product of the digits?
If A and B can finish a work in 10 days, B and C can finish the same work in 12 days, C and A can finish the same work in 15 days; then in how many days can A, B and C together finish half of the work?
A person borrowed Rs. 10,000 at 12% rate of interest per annum compounded quarterly for a period of 9 months. What is the interest paid by him to settle his account after 9 months?
For what relation between a and b is the equation sin θ = (a + b) / (2√(ab)) possible?
Three persons A, B and C together can do a piece of work in 36 days. A and B together can do five times as much work as C alone; B and C together can do as much work as A alone. If A and C together can do n times as much work as B alone, then what is the value of n?
If 2a/3 = 4b/5 = 3c/4, then what is the value of (18/a)√(a² + c² − b²)?
The sum of deviations of n numbers from 10 and 20 are a, b respectively. If b/a = −4, then what is the mean of these n numbers?
If the median of observations 12, 1, 8, 54, 61, 28, 45, 35, 21, 17 is M, then what is the value of 2M + 5?
How many real roots does the equation √(x + 9) = x − 3 have?
If x = 97 + 56√3, then what is the value of x^(1/4) + 1/x^(1/4)?
Let L be the LCM and H be the HCF of two given numbers. L and H are in the ratio 3 : 2. If the sum of the two numbers is 45, then what is the product of the numbers?
A man walks at an average speed of 3 km/h from his home and reaches office 40 minutes early. If he walks at an average speed of 2 km/h, he would reach office 40 minutes late. What is the distance between his home and office?
If 3^(x−1) + 3^(3−x) = 6, then what is 2^(x−1) + 2^(3−x) equal to?
If x[(a − b) + ab/(a − b)] = y[(a + b) − ab/(a + b)] and x + y = 2a³, then what is x − y equal to?
Which one of the following is a factor of 3√3x³ + 2√2y³ − 18xy + 6√6?
What is the number of digits in the expansion of 125^100? (Given log10 2 = 0.301)
What is the HCF of acx³ + bcx² + adx² + acdx + bdx + bcd and adx³ + acx² + bdx² + bcx + acdx + bcd if HCF(c, d) = 1, c ≠ d?
If x^n − py^n + qz^n is divisible by x² + abyz − bzx − axy, then what is p/a^n − q/b^n equal to?
Consider the following statements:
Which of the statements given above is/are correct?
If (3a + 6b + c + 2d) × (3a − 6b − c + 2d) = (3a − 6b + c − 2d) × (3a + 6b − c − 2d), then which one of the following is correct?
If 3sin θ + 5cos θ = 5, then what is the value of 5sin θ − 3cos θ equal to?
The combined age of a man and his wife is 6 times the combined age of their children. Two years ago their combined age was 10 times the combined age of their children; and six years later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 years old?
What is 3(sin x − cos x)⁴ + 6(sin x + cos x)² + 4(sin⁶x) + 4(cos⁶x) equal to?
What is the value of sin θ + cos θ, if θ satisfies the equation cot²θ − (√3 + 1)cot θ + √3 = 0; 0 < θ < π/4?
Which one of the following is a value of θ, if θ satisfies the equation tan 2θ tan 4θ − 1 = 0; 0 < θ < π/2?
If tan x = (sin θ + cos θ)/(sin θ − cos θ), π/4 < θ < π/2, then what is √2 sin x equal to?
How many values of θ will satisfy the equation (sin²θ − 4sin θ + 3)(4 − cos²θ + 4sin θ) = 0, where 0 < θ < π/2?
If x sin³θ + y cos³θ = sin θ cos θ and x sin θ − y cos θ = 0, for every θ ∈ (0, π/2), then what is x² + y² equal to?
A flagstaff stands on the top of a vertical tower. The angle of elevation of the top of the flagstaff from a certain place on the same horizontal level with the base of the tower is found to be α. Advancing a distance d towards the tower in the same horizontal plane, the angle of elevation of the top of the flagstaff is observed to be β and that of the top of the tower is observed to be γ. Let H be the height of the top of the flagstaff from the base of the tower and h be the height of the tower. Which one of the following is correct?
A flagstaff stands on the top of a vertical tower. The angle of elevation of the top of the flagstaff from a certain place on the same horizontal level with the base of the tower is found to be α. Advancing a distance d towards the tower in the same horizontal plane, the angle of elevation of the top of the flagstaff is observed to be β and that of the top of the tower is observed to be γ. Let H be the height of the top of the flagstaff from the base of the tower and h be the height of the tower. Which one of the following is correct?
AB is a straight road leading to the foot P of a tower of height h. Q is at distance x from P and R is at a distance y from Q (R is farther from P than Q; R, Q are on the same side). The angle of elevation of the top of the tower at Q is twice of that at R. Which one of the following is correct?
AB is a straight road leading to the foot P of a tower of height h. Q is at distance x from P and R is at a distance y from Q (R is farther from P than Q; R, Q are on the same side). The angle of elevation of the top of the tower at Q is twice of that at R. Which one of the following is correct?
Two parallel chords AB and CD of a circle are of lengths 60 cm and 80 cm respectively. They are on the same side of the centre O and 10 cm apart. What is the diameter of the circle?
If the chord AB subtends an angle α and chord CD subtends an angle β at the centre O, then what is the value of tan(β/2) − tan(α/2)?
Let p = (y⁴ − z²x²)/(x⁴ − y²z²), q = (z⁴ − x²y²)/(y⁴ − z²x²), r = (x⁴ − y²z²)/(z⁴ − x²y²). What is px² + qy² + rz² equal to?
Let p = (y⁴ − z²x²)/(x⁴ − y²z²), q = (z⁴ − x²y²)/(y⁴ − z²x²), r = (x⁴ − y²z²)/(z⁴ − x²y²). What is x²(px² + qy² + rz²) + qr − p² equal to?
A right conical cap just covers two spheres placed one above the other on a table such that it touches both the spheres. Let r be the radius of the smaller sphere and R be the radius of the bigger sphere. Let 2θ be the vertical angle of the cone. What is the height of the cone?
A right conical cap just covers two spheres placed one above the other on a table such that it touches both the spheres. Let r be the radius of the smaller sphere and R be the radius of the bigger sphere. Let 2θ be the vertical angle of the cone. What is the radius of the base of the cone?
A line segment AB is bisected at C and semi-circles S1, S2 and S3 are drawn respectively on AB, AC and CB as diameters such that they all lie on same side of AB. A circle S is drawn touching internally S1 and externally S2 and S3. If r is the radius of S and R is the radius of S2, then which one of the following is correct?
A line segment AB is bisected at C and semi-circles S1, S2 and S3 are drawn respectively on AB, AC and CB as diameters such that they all lie on same side of AB. A circle S is drawn touching internally S1 and externally S2 and S3. If m is the area of the circle S and n is the area of semi-circle S1, then which one of the following is correct?
Let (x − a)(x − b)/((x − ma)(x − mb)) = (x + a)(x + b)/((x + ma)(x + mb)); m, a, b > 0. What is (x² + ab)/(x² + m²ab) equal to?
Let (x − a)(x − b)/((x − ma)(x − mb)) = (x + a)(x + b)/((x + ma)(x + mb)); m, a, b > 0. What is x equal to?
The total monthly electricity bill for a house consists of the sum of two parts, one part is proportional to number of rooms and the other part is proportional to number of units consumed. Rs. 400 is the monthly electricity bill for a house with 8 rooms and consuming 240 units and Rs. 320 is the monthly electricity bill for a house with 6 rooms and consuming 200 units. What is the monthly electricity bill for a house with m rooms and consuming n units?
The total monthly electricity bill for a house consists of the sum of two parts, one part is proportional to number of rooms and the other part is proportional to number of units consumed. Rs. 400 is the monthly electricity bill for a house with 8 rooms and consuming 240 units and Rs. 320 is the monthly electricity bill for a house with 6 rooms and consuming 200 units. What is the monthly electricity bill for a house with 7 rooms consuming 300 units?
A grouped frequency distribution is given below: Weekly wages (Rs.): 2050–2550 (5 workers), 2550–3050 (10), 3050–3550 (k), 3550–4050 (8), 4050–4550 (2), 4550–5050 (10). If average weekly wages earned by a worker is Rs. 3,520, then what is the value of k?
A grouped frequency distribution is given below: Weekly wages (Rs.): 2050–2550 (5 workers), 2550–3050 (10), 3050–3550 (k), 3550–4050 (8), 4050–4550 (2), 4550–5050 (10). What is the median (approximate value) of the distribution?
A quadratic equation is given by (a + b + c)x² − (2a + 2b)x + (a + b − c) = 0, where a, b and c are real and distinct. What are the roots of the equation?
Consider the following statements:
Which of the statements given above is/are correct?
What is the radius of the sphere passing through the corners of the cuboid with edges 8 cm, 12 cm and 24 cm?
A lamp shade is in the shape of a part of a cone and its top and bottom ends are circles whose circumferences are respectively 30 cm and 40 cm. The perpendicular distance between the ends is 6 cm. If the cone were to be completed, then how far would its vertex be from the top end?
A sum of money at 20% rate of compound interest per annum becomes more than 100 times in n years. What is the least value of n? (Use log10 2 = 0.301, log10 3 = 0.477).
The corners of an equilateral triangular plate were cut in such a manner that it forms a regular hexagonal plate. What is the ratio of the area of the triangular plate to the area of the hexagonal plate?
Two equal arcs of different circles C1 and C2 subtend angles of 60° and 75° respectively, at the centres. What is the ratio of the radius of C1 to the radius of C2?
ABC is a triangle with sides AB = 41 cm, BC = 28 cm and CA = 15 cm. If D, E and F are the mid-points of AB, BC and CA respectively, then what is the area of the triangle DEF?
A triangle has sides 13 cm, 14 cm and 15 cm long. What is the length of the smallest altitude of the triangle?
The circumference of a circle exceeds the diameter by 16.8 cm. What is the diameter of the circle? (Take π = 22/7)
The hypotenuse AC of a right-angled triangle ABC is 3√5 cm. If AB is doubled and BC is tripled such that ABC remains a right-angled triangle, the hypotenuse becomes 15 cm. What is AB + BC equal to?
What is the area of the region between two concentric circles if the chord of the outer circle of length 14 cm is a tangent of the inner circle? (Take π = 22/7)
A pendulum swings through an angle of 9° and its end describes an arc of length 14.3 cm. What is the length of the pendulum? (Take π = 22/7)
The arch of a bridge is in the form of an arc of a circle. If the span of the bridge is 40 m and height in the middle is 8 m, then what is the radius of curvature of the bridge?
If a, b and c are the sides of a right-angled triangle, where a > b > c, then what is the value of the expression (a + b + c)(a + b − c)(a − b + c)(a − b − c)?
The cube root of x varies inversely as the square root of y. x = 8 when y = 3. What is the value of x when y = 3√3?
Three solid lead spheres of diameters 6 cm, 8 cm and 10 cm are melted together and recast as a solid sphere. What is the percentage diminution of the surface area as compared to the sum of the surface areas of the three spheres?
A solid sphere of radius 3 cm is melted to form a hollow cylinder of height 4 cm and external diameter 10 cm. What is the thickness of the cylinder?
Three glasses P, Q and R have capacities in the ratio 1 : 2 : 3. All these glasses are completely filled with mixtures of milk and water. The ratio of milk to water in P is 1 : 2, in Q it is 2 : 3 and in R it is 3 : 1. If the content of all three glasses are put into a bigger container, what will be the ratio of milk to water in the container?
What is the LCM of x⁴ + x²y² + y⁴, x³ + y³, x³ − y³?
What is (x² − y² − z² − 2yz)/(x² + y² − z² + 2xy) + (x² − y² − z² − 2yz)/(x² − y² + z² − 2xz) equal to?
If tan A + cot A = 2, where 0 < A < 90°, then what is the value of tan²A + tan³A + tan⁴A + … + tanⁿA?
Which of the following is/are identity/identities?
Select the correct answer using the code given below:
If 7sin⁴θ + 9cos⁴θ + 42sin²θ = 16, 0 < θ < π/2, then what is tan θ equal to?
An isosceles triangle has its base length 2a and its height is h. On each side of the triangle a square is drawn external to the triangle. What is the area of the figure thus formed?
If p = a²/((b − a)(c − a)), q = b²/((c − b)(a − b)), r = c²/((a − c)(b − c)), then what is (p + q + r)² equal to?
Which one of the following is a factor of a² − b² − c² + 2bc + a + b − c?
Let α and β be the roots of the equation 1/x = 1/a + 1/b + 1/(x + a + b); a ≠ 0, b ≠ 0, x ≠ 0. Which one of the following is a quadratic equation whose roots are α² and β²?
If x = 6/(7 − 6/(7 − 6/(7 − ...))); x > 1, then what is the value of x² − 3x + 2 equal to?
A train completely overtakes two persons, walking in the same direction with speeds 3 km/h and 4 km/h in 9 seconds and 75/8 seconds respectively. What is the length of the train?
A person bought an article and sold it at a profit of 20%. Had he bought it at 20% less, what would have been the profit percentage if the selling price had been the same?
If 2s = a + b + c, then what is s² + (s − a)(s − b) + (s − b)(s − c) + (s − c)(s − a) equal to?
A sphere of radius 5 cm is dropped in a right circular cylindrical vessel partly filled with water. The radius of the cylindrical vessel is 10 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel?
Consider the following statements:
Which of the statements given above is/are correct?
If a, b, c, x, y, z are real numbers such that (a + b + c)² − 3(ab + bc + ca) + 3(x² + y² + z²) = 0, then which one of the following is correct?
In a triangle ABC, angle B = 90° and p is the length of the perpendicular from B to AC. If BC = 10 cm and AC = 12 cm, then what is the value of p?
The mean of p, q, r, s and t is 280. If the mean of p, r and t is 240, what is the mean of q and s?
A, B, C, D, E, F and G are cousins. D is thrice as old as A. Further, C is as many years younger to B, as G to E and E to D. The average age of D and G is 16 years; the average age of A and E is 11 years; the average age of B and C is also 11 years. B and C have equal weight. A's weight is 10 kg less than that of B; D is 4 kg heavier than E; E is 4 kg heavier than F; F is 4 kg heavier than G. Further, D has age-weight ratio of 9 : 20, where age is in years and weight in kg; A has age-weight ratio of 2 : 5. Moreover, none of them is more than 40 kg. What is D's age (in years)?
A, B, C, D, E, F and G are cousins. D is thrice as old as A. Further, C is as many years younger to B, as G to E and E to D. The average age of D and G is 16 years; the average age of A and E is 11 years; the average age of B and C is also 11 years. B and C have equal weight. A's weight is 10 kg less than that of B; D is 4 kg heavier than E; E is 4 kg heavier than F; F is 4 kg heavier than G. Further, D has age-weight ratio of 9 : 20, where age is in years and weight in kg; A has age-weight ratio of 2 : 5. Moreover, none of them is more than 40 kg. What is the average age (in years) of B, C, D, E and G?
A, B, C, D, E, F and G are cousins. D is thrice as old as A. Further, C is as many years younger to B, as G to E and E to D. The average age of D and G is 16 years; the average age of A and E is 11 years; the average age of B and C is also 11 years. B and C have equal weight. A's weight is 10 kg less than that of B; D is 4 kg heavier than E; E is 4 kg heavier than F; F is 4 kg heavier than G. Further, D has age-weight ratio of 9 : 20, where age is in years and weight in kg; A has age-weight ratio of 2 : 5. Moreover, none of them is more than 40 kg. What is the difference between the weights (in kg) of G and C?
A, B, C, D, E, F and G are cousins. D is thrice as old as A. Further, C is as many years younger to B, as G to E and E to D. The average age of D and G is 16 years; the average age of A and E is 11 years; the average age of B and C is also 11 years. B and C have equal weight. A's weight is 10 kg less than that of B; D is 4 kg heavier than E; E is 4 kg heavier than F; F is 4 kg heavier than G. Further, D has age-weight ratio of 9 : 20, where age is in years and weight in kg; A has age-weight ratio of 2 : 5. Moreover, none of them is more than 40 kg. What is the average weight (in kg) of A, B, C, D, E, F and G?
Consider the following statements:
Which of the statements given above are correct?
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