SolveForX

Solved maths problems — page 24

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

  1. Let U be the universal set of all the students of Class XI of a coeducational school and A be the set of all girls in Class XI. Find A'.

    The set A' is the set of all boys in Class XI.

  2. A committee of two persons is selected from two men and two women. What is the probability that the committee will have (a) no man? (b) one man? (c) two men?

    (a) The probability that the committee will have no man is (1)/(6). (b) The probability that the committee will have one man is (2)/(3). (c) The probability tha

  3. Ford owns a garden and he grows 40 roses every week. He supplies Roses to the local flower shops. The first flower shop orders 20 roses, the second flower shop

    Ford lacks 100 roses to supply all the flower shops every month.

  4. Let z_1 and z_2 be two complex numbers such that |z_1+z_2|=|z_1|+|z_2|. Then show that (z_1)- (z_2)=0.

    We have shown that (z_1)- (z_2)=0 (modulo 2π).

  5. A box contains 3 blue, 2 white, and 4 red marbles. If a marble is drawn at random from the box, what is the probability that it will be (i) white? (ii) blue? (i

    The probability of drawing a white marble is (2)/(9). The probability of drawing a blue marble is (1)/(3). The probability of drawing a red marble is (4)/(9).

  6. The number of real solution(s) of the equation x² + 3x + 2 = min |x - 3|, |x + 2| is:

    2

  7. Use Stokes' theorem to evaluate ∮_C F⃗ · dr⃗ where F⃗ = (y², xz, z²y) and C is the intersection of x² + y² = 1 and z = x + y.

    -(π)/(2)

  8. Let A, B, C be three points in xy -plane, whose position vectors are given by √3 i+ j, i+√3 j and a i+(1-a) j respectively with respect to the origin O. If the

    1

  9. Let the foci of a hyperbola be (1, 14) and (1, -12). If it passes through the point (1, 6), then the length of its latus-rectum is:

    The length of the latus-rectum is (288)/(5).

  10. आकृति 6.30 में P ज्ञात कीजिए। [आकृति: ABC में AB = 3.8, BC = 6, CA = 3√3, A = 80°, B = 60°; तथा PQR में PQ = 12, QR = 7.6, RP = 6√3।]

    P = 40°

  11. Let the area of the bounded region (x, y): 0 ≤ 9x ≤ y², y ≥ 3x - 6 be A. Then 6A is equal to _____

    81

  12. The angle between two lines is (π)/(4) and slope of one of the lines is (1)/(2) find the slope of the other line.

    The slope of the other line can be either -(1)/(3) or 3.

  13. Find values of x for which | ll3 & x x & 1 |=| ll3 & 2 4 & 1 |.

    x = ± 2√2

  14. Janet’s ducks lay 16 eggs per day. She eats three for breakfast every morning and bakes muffins for her friends every day with four. She sells the remainder at

    Janet makes 18 every day at the farmers' market.

  15. If all the words with or without meaning made using all the letters of the word 'KANPUR' are arranged as in a dictionary, then the word at 440^(th) position in

    PRKANU

  16. Finally, we evaluate the definite integral of the line equation from x = 0 to x = 7. This calculates the area under the line segment PQ, which is the area of th

    The area of POQ is 21 square units.

  17. Let the position vectors of three vertices of a triangle be 4 p + q - 3 r, -5 p + q + 2 r, and 2 p - q + 2 r. If the position vectors of the orthocenter and the

    2

  18. If θ [-(7π)/(6), (4π)/(3)], then the number of solutions of √3csc²θ - 2(√3 - 1)cscθ - 4 = 0, is equal to:

    The number of solutions is 6.

  19. If y=A sin x+B cos x, then prove that (d² y)/(d x²)+y=0.

    The proof is complete, as (d² y)/(d x²)+y=0.

  20. The number of non-empty equivalence relations on the set 1, 2, 3 is:

    5

  21. The sum of the squares of the roots of |x + 2|² + |x - 2| - 2 = 0 and the squares of the roots of x² - 2|x - 3| - 5 = 0, is:

    26

  22. If x=f(y) is the solution of the differential equation (1+y²) + (x - 2e^(tan^(-1) y)) (dy)/(dx) = 0, y (-(π)/(2), (π)/(2)) with f(0)=1, then f((1)/(√3)) is equa

    e^((π)/(6))

  23. Let a line passing through the point (4,1,0) intersect the line L_1: (x-1)/(2)=(y-2)/(3)=(z-3)/(4) at the point A(α,β,) and the line L_2: x-6 = y - z + 4 at the

    2

  24. A pole has to be erected at a point on the boundary of a circular park of diameter 13 metres in such a way that the differences of its distances from two diamet

    Yes, it is possible to erect the pole. The pole should be erected at a distance of 5 m from gate B and 12 m from gate A.

  25. Find the value of tan(π/7)·tan(2π/7)·tan(3π/7).

    √7

  26. Evaluate the integral ∫₀^∞ ln(x)/(x² + 1) dx using complex analysis techniques.

    0

  27. Let R = (1, 2), (2, 3), (3, 3) be a relation defined on the set 1, 2, 3, 4. Then the minimum number of elements, needed to be added in R so that R becomes an eq

    7

  28. Find the sum of the sequence 7, 77, 777, 7777,... to n terms.

    S_n = (7)/(9)((10(10^n - 1))/(9) - n)

  29. Let f: X → Y be a function. Define a relation R in X given by R=(a, b): f(a)=f(b). Examine whether R is an equivalence relation or not.

    The relation R is an equivalence relation.

  30. EXTENSION: A sphere of radius r is inscribed in a cone with base radius R and height h. Find the relationship between r, R, and h. Additionally, generalize your

    The relationship between r, R, and h is r = (Rh)/(R + √(R² + h²)). The generalization to n is not applicable as the problem describes a specific geometric confi

  31. Evaluate (n!)/(r!(n-r)!), when n = 5, r = 2.

    10

  32. The minimum value of the expression 3^(x)+3^(1-x), x R, is (A) 0 (B) (1)/(3) (C) 3 (D) 2 √3

    2√3

  33. Let f:2,3,4,5 →3,4,5,9 and g:3,4,5,9 →7,11,15 be functions defined as f(2)=3, f(3)=4, f(4)=f(5)=5 and g(3)=g(4)=7 and g(5)=g(9)=11. Find g ° f.

    g ° f = (2,7), (3,7), (4,11), (5,11)

  34. The zeroes of the polynomial x² + 5x + 6 are:

    The zeroes of the polynomial x² + 5x + 6 are -2 and -3.

  35. If the set of all a R 1, for which the roots of the equation (1 - a)x² + 2(a - 3)x + 9 = 0 are positive is (-∞, -α] [β,), then 2α + β + is equal to:

    7

  36. Ava and Emma want to know who is better at the new video game Ava got for her birthday. They are each going to play one level and whoever has the highest score

    The difference between their two scores is -25, meaning Ava's score is 25 points lower than Emma's score.

  37. Find the area of a sector of a circle with radius 6 cm if the angle of the sector is 60 degrees. (Use π ≈ 3.14)

    The area of the sector is 18.84 cm².

  38. Find the coordinates of the foci, the vertices, the lengths of major and minor axes and the eccentricity of the ellipse 9 x²+4 y²=36.

    The coordinates of the foci are (0, ± √5). The coordinates of the vertices are (0, ± 3). The length of the major axis is 6 units. The length of the minor axis i

  39. The distribution below shows the number of wickets taken by bowlers in one-day cricket matches. Find the mean number of wickets by choosing a suitable method. W

    The mean number of wickets is approximately 180.44. This signifies that, on an average, the number of wickets taken by these 45 bowlers in one-day cricket is ap

  40. Let R be the set of real numbers. Define the real function f: R → R by f(x)=x+10 and sketch the graph of this function.

    The graph of the function f(x) = x+10 is a straight line passing through the points (-10, 0) and (0, 10).

  41. Harpreet tosses two different coins simultaneously (say, one is of 1 and other of 2). What is the probability that she gets at least one head?

    The probability of getting at least one head is (3)/(4).

  42. Let the area of the triangle formed by the lines x + 2 = y - 1 = z, (x - 3)/(5) = (y)/(-1) = (z - 1)/(1), and (x)/(-3) = (y - 3)/(3) = (z - 2)/(1) be A. Then A²

    56

  43. If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is:

    The eccentricity of the ellipse is (4√17)/(17).

  44. Let (a, b) be the point of intersection of the curve x² = 2y and the straight line y - 2x - 6 = 0 in the second quadrant. Then the integral I = ∫_a^b (9x²)/(1 +

    24

  45. The centre of a circle C is at the centre of the ellipse E: (x²)/(a²)+(y²)/(b²)=1, a>b. Let C pass through the foci F_1 and F_2 of E such that the circle C and

    13

  46. How many numbers greater than 1000000 can be formed by using the digits 1, 2, 0, 2, 4, 2, 4?

    360

  47. Find the volume of a right circular cone with base radius 6 cm and height 7 cm. Use pi = 22/7.

    The volume of the right circular cone is 264 cm³.

  48. Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that PTQ = 2 OPQ. [Figure required: circle with centre O, external po

    Hence, it is proved that PTQ = 2 OPQ.

  49. To prove: n N (n+1)/(2) N ⇒ m N: n = 2m + 1. (natural numbers, proof, implication, existence, odd number)

    Given n N and (n+1)/(2) N. Let k = (n+1)/(2). Since k N, k is a natural number. Multiplying by 2, we get n+1 = 2k. Subtracting 1, we get n = 2k - 1. We want to

  50. In the Cartesian coordinate plane shown, plot the points A(2,3), B(-2,-1), and C(2,-3), and connect them in order A→ B→ C. Then find the area of ABC.

    The area of ABC is 12 square units.

  51. Let the mean and the standard deviation of the observations 2, 3, 3, 4, 5, 7, a, b be 4 and √2 respectively. Then the mean deviation about the mode of these obs

    1

  52. If |z²-1|=|z|²+1, then show that z lies on imaginary axis.

    The complex number z lies on the imaginary axis.

  53. Let in a ABC, the length of the side AC be 6, the vertex B be (1,2,3) and the vertices A, C lie on the line (x-6)/(3) = (y-7)/(2) = (z-7)/(2). Then the area (in

    The area of ABC is (3√561)/(17) square units.

  54. Let a= i+2 j+ k, b=3 i-3 j+3 k, c=2 i- j+2 k and d be a vector such that b× d= c× d and a· d=4. Then | a× d|² is equal to:

    128

  55. The shortest distance between the curves y² = 8x and x² + y² + 12y + 35 = 0 is:

    2√2 - 1

  56. Solve the system of inequalities: & 3 x-7<5+x & 11-5 x ≤ 1 and represent the solutions on the number line.

    The solution to the system of inequalities is 2 ≤ x < 6.

  57. What is the determinant of the matrix 0 & 7 & 3 1 & 0 & 7 3 & 0 & 0? Simplify your answer. (determinant, matrix, linear algebra, cofactor expansion, square matr

    The determinant of the matrix is 147.

  58. The angle of elevation of the top of a tower from a point on the ground 30 m away from the foot of the tower is 30 degrees. Find the height of the tower.

    The height of the tower is 10√3 meters.

  59. Find the number of arrangements of the letters of the word INDEPENDENCE. In how many of these arrangements, (i) do the words start with P (ii) do all the vowels

    The number of arrangements of the letters of the word INDEPENDENCE is 1663200. (i) The number of arrangements that start with P is 138880. (ii) The number of ar

  60. केंद्र O वाले वृत्त पर बाह्य बिंदु T से दो स्पर्श रेखाएँ TP तथा TQ खींची गई हैं। सिद्ध कीजिए कि PTQ = 2 OPQ है। [आकृति आवश्यक: केंद्र O वाला वृत्त, बाह्य बिंदु

    यह सिद्ध हो गया है कि PTQ = 2 OPQ है।

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