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Solved maths problems — page 16

1518 problems, newest first. Browse by chapter at NCERT solutions by class.

  1. Two players take turns rolling a fair die. Player A wins if he rolls a 6, Player B wins if he rolls a 5 or 6. If A starts, find the probability that A wins.

    The probability that A wins is (3)/(8).

  2. The number of real roots of the equation x x - 2 + 3 x - 3 + 1 = 0 is:

    The equation has 1 real root.

  3. If the equation of the hyperbola with foci (4,2) and (8,2) is 3x² - y² - ax + by + = 0, then a + b + is equal to:

    132

  4. Solve the system: dx/dt = 3x - y, dy/dt = x + y with initial conditions x(0)=2, y(0)=1.

    x(t) = e²t(t+2), y(t) = e²t(t+1)

  5. Let A=1,2,3,4 and B=1,4,9,16. Then the number of many-one functions f: A → B such that 1 f(A) is equal to:

    175

  6. Find the derivative of sin x at x=0.

    The derivative of sin x at x=0 is 1.

  7. Find the dimensions of the prayer hall discussed in Section 4.1. (The breadth x satisfies 2x² + x - 300 = 0.)

    The breadth of the prayer hall is 12 m and the length is 25 m.

  8. Number of functions f: 1, 2,, 100 → 0, 1, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to:

    392

  9. Kelly has 5 quarters and 2 dimes. If she buys a can of pop for 55 cents, how many cents will she have left?

    Kelly will have 125 cents left.

  10. If the area of the region bounded by the curves y = 4 - (x²)/(4) and y = (x - 4)/(2) is equal to α, then 6α equals:

    250

  11. Differentiate sin ² x w.r.t. e^(cos x).

    -(2 cos x)/(e^(cos x))

  12. Find the multiplicative inverse of 2-3 i.

    The multiplicative inverse of 2-3i is (2)/(13) + (3)/(13)i.

  13. On Monday the post office delivered 425 letters. On Tuesday they delivered 17 more than one-fifth as many as Monday. On Wednesday they delivered 5 more than twi

    The post office delivered a total of 736 letters from Monday to Wednesday.

  14. Evaluate | rr2 & 4 -1 & 2 |.

    8

  15. Let α, β be the roots of the equation x² - ax - b = 0 with Im(α) < Im(β). Let P_n = α^n - β^n. If P_3 = -5√7i, P_4 = -3√7i, P_5 = 11√7i and P_6 = 45√7i, then |α

    158

  16. Find the ratio in which the line x – 3y = 0 divides the line segment joining the points (-2, -5) and (6, 3). Find the coordinates of the point of intersection.

    The line x - 3y = 0 divides the line segment in the ratio 13:3. The coordinates of the point of intersection are ( (9)/(2), (3)/(2)).

  17. If 4 x+i(3 x-y)=3+i(-6), where x and y are real numbers, then find the values of x and y.

    The values are x = (3)/(4) and y = (33)/(4).

  18. The sum of all rational terms in the expansion of (2+√3)^8 is:

    18817

  19. Let c be the projection vector of b = i + 4 k, > 0, on the vector a = i + 2 j + 2 k. If | a + c| = 7, then the area of the parallelogram formed by the vectors b

    16

  20. In triangle ABC, BAC=75^°, ACB=60^°, and AB=8√2. Find BC.

    BC = 8 + (8√3)/(3)

  21. Evaluate lim(n→∞) Σ(k=1 to n) [√(n² + k²) - n].

    (1)/(6)

  22. how many words combinations can you make with the name Rakesh Ramesh

    The total number of word combinations that can be made with the name Rakesh Ramesh is 29,937,600.

  23. If 24 ∫_0^((π)/(4)) (sin |4x - (π)/(12)| + |2 sin x|) dx = 2π + α, where [·] denotes the greatest integer function, then α is equal to:

    60 - 24√2 - 2π

  24. A committee of 5 is to be formed from 6 men and 4 women with at least 3 women. How many ways?

    The total number of ways to form the committee is 66.

  25. Show that the relation R in the set 1,2,3 given by R=(1,1),(2,2), (3,3),(1,2),(2,3) is reflexive but neither symmetric nor transitive.

    The relation R is reflexive but neither symmetric nor transitive.

  26. Elise is learning to write and decides to keep re-writing the alphabet until she knows it. She writes it in full twice, writes half of it once, then re-writes e

    Elise has written 130 letters in total.

  27. Steve decides to start eating more tomatoes and decides to grows his own cherry tomatoes. He eats twice as much as his girlfriend. He eats 6 per day. If a vine

    Steve needs 21 vines.

  28. Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of 16((sec^(-1) x)² + (csc^(-1) x)²) is:

    22π²

  29. What is the 20^ th term of the sequence defined by a_n=(n-1)(2-n)(3+n)?

    The 20^ th term of the sequence is -7866.

  30. If sin A + sin B + sin C = cos A + cos B + cos C = 0 prove that sin²A + sin²B + sin²C = cos²A + cos²B + cos²C = 3/2.

    sin²A + sin²B + sin²C = cos²A + cos²B + cos²C = (3)/(2)

  31. In a right triangle the two legs are 3 cm and 4 cm. Find the hypotenuse and sin of the angle opposite the 3 cm side.

    The hypotenuse is 5 cm and the sine of the angle opposite the 3 cm side is (3)/(5).

  32. Product of two consecutive positive integers is 306. Find them.

    The two consecutive positive integers are 17 and 18.

  33. Let the parabola y = x² + p x - 3 meet the coordinate axes at the points P, Q, and R. If the circle C with centre at (-1, -1) passes through the points P, Q, an

    6 square units

  34. Let f(x) = (2^(x+2) + 16)/(2²x+1) + 2^(x+4) + 32. Then the value of 8(f((1)/(15)) + f((2)/(15)) + + f((59)/(15))) is equal to

    118

  35. Let y=f(x) be the solution of the differential equation (dy)/(dx) + (xy)/(x²-1) = (x^4 + 4x)/(√(1-x²)), -1 < x < 1 such that f(0)=0. If 6 ∫_-1/2^(1/2) f(x) dx =

    27

  36. In an experiment, a solution of hydrochloric acid is to be kept between 30^(°) and 35^(°) Celsius. What is the range of temperature in degree Fahrenheit if conv

    The range of temperature in degree Fahrenheit is between 86^(°)F and 95^(°)F.

  37. If a_1, a_2,, a_n are in A.P. with common difference d (where d ≠ 0); then the sum of the series sin d(cosec a_1 cosec a_2+cosec a_2 cosec a_3+ +cosec a_n-1 cos

    The sum of the series is cot a_1 - cot a_n.

  38. Find the area of the sector of a circle with radius 4 cm and of angle 30^°. Also, find the area of the corresponding major sector (Use π = 3.14).

    The area of the minor sector is 4.19 cm² and the area of the major sector is 46.05 cm².

  39. (a) If P = 1 -1 0 2 3 4 0 1 2 and Q = 2 2 -4 -4 2 -4 2 -1 5, find (QP) and hence solve the following system of equations using matrices: x - y = 3, 2x + 3y + 4z

    The product QP = 6 & 0 & 0 0 & 6 & 0 0 & 0 & 6. The solution to the system of equations is x = 2, y = -1, z = 4.

  40. Find the distance of the point (3, -5) from the line 3x - 4y - 26 = 0.

    The distance of the point (3, -5) from the line 3x - 4y - 26 = 0 is (3)/(5).

  41. When Marcus wakes up, his house is 40 degrees. He spends 3 hours baking, and every hour the oven is on it raises the house's temperature by 5 degrees. Then Marc

    The house's final temperature is 49^(°).

  42. Let P be the image of the point Q(7, -2, 5) in the line L: (x - 1)/(2) = (y + 1)/(3) = (z)/(4) and R(5, p, q) be a point on L. Then the square of the area of PQ

    155

  43. Let C be the circle x² + (y-1)² = 2, E_1 and E_2 be two ellipses whose centres lie at the origin and major axes lie on the x-axis and y-axis respectively. Let t

    46

  44. Find the shortest distance between the lines r = (i + 2j + k) + lambda*(i - j + k) and r = (2i - j - k) + mu*(2i + j + 2k).

    The shortest distance between the given lines is (3√2)/(2) units.

  45. Consider the sets, A = 1, 3, B = 1, 5, 9, C = 1, 3, 5, 7, 9. Insert the symbol or ot between each of the following pair of sets: (i)... B (ii) A... B (iii) A...

    (i) B (ii) A B (iii) A C (iv) B C

  46. Find the domain of the function f(x)=(x²+3 x+5)/(x²-5 x+4)

    The domain of the function is all real numbers except 1 and 4, which can be written as R 1, 4.

  47. Evaluate =| ccc0 & sin α & -cos α -sin α & 0 & sin β cos α & -sin β & 0 | Solution Expanding along R_1, we get & =0| cc 0 & sin β -sin β & 0 |-sin α| cc -sin α

    The value of the determinant is 0.

  48. Let the sequence a_n be defined as follows: a_1=1, a_n=a_n-1+2 for n ≥ 2. Find first five terms and write corresponding series.

    The first five terms are 1, 3, 5, 7, 9. The corresponding series is 1 + 3 + 5 + 7 + 9.

  49. Let a_1, a_2,, a_2024 be an Arithmetic Progression such that a_1 + (a_5 + a_10 + a_15 + + a_2020) + a_2024 = 2233. Then a_1 + a_2 + a_3 + + a_2024 is equal to:

    11132

  50. If cot x=-(5)/(12), x lies in second quadrant, find the values of other five trigonometric functions.

    The values of the other five trigonometric functions are: tan x = -(12)/(5), csc x = (13)/(12), sin x = (12)/(13), cos x = -(5)/(13), and sec x = -(13)/(5).

  51. Find the number of different 8-letter arrangements that can be made from the letters of the word DAUGHTER so that (i) all vowels occur together (ii) all vowels

    (i) 4320, (ii) 36000

  52. If x² + y² + z² = xy + yz + zx, prove that x = y = z and find the value of (x+y+z)²/(x²+y²+z²).

    The value of ((x+y+z)²)/(x²+y²+z²) is 3.

  53. (b) Find a particular solution of the differential equation (x+1) (dy)/(dx) = 2e^(-y) - 1, given that y=0 when x=0. (differential equation, particular solution,

    y = ln|(2x + 1)/(x+1)|

  54. A herd consists of camels and dromedaries. There are 180 heads and 304 bumps. How many dromedaries are there if camels have two humps each and dromedaries have

    There are 56 dromedaries.

  55. Let y = y(x) be the solution of the differential equation (x² + 1)y' - 2xy = (x^4 + 2x² + 1)cos x, with y(0) = 1. Then ∫_-3³ y(x) dx is:

    24

  56. If the four distinct points (4,6), (-1,5), (0,0) and (k,3k) lie on a circle of radius r, then 10k + r² is equal to:

    35

  57. Let y = y(x) be the solution of the differential equation 2 cos x (dy)/(dx) = sin 2x - 4y sin x, x (0, (π)/(2)). If y((π)/(3)) = 0, then y'((π)/(4)) + y((π)/(4)

    (5√2)/(4)

  58. EXTENSION: Find the volume of the solid generated by revolving the region bounded by y = x³, y = 0 and x = 2 about the y-axis using the washer method. Additiona

    The volume of the solid is (64π)/(5) cubic units.

  59. Let a_1,a_2,a_3, be in an A.P. such that Σ_k=1^(12)a_2k-1=-(72)/(5)a_1, a_1 0. If Σ_k=1^n a_k=0, then n is:

    n=11

  60. A line makes angles alpha, beta, gamma and delta with the four diagonals of a cube, prove that cos²(alpha) + cos²(beta) + cos²(gamma) + cos²(delta) = 4/3.

    cos²(α) + cos²(β) + cos²() + cos²() = 4/3 is proven.

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