Solved maths problems — page 10

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

1. Find the equation of the circle with centre (-3,2) and radius 8.

   The equation of the circle is (x + 3)² + (y - 2)² = 64

2. Find the equation of the circle with centre (-3,2) and radius 6.

   The equation of the circle is (x+3)² + (y-2)² = 36.

3. Find the equation of the circle with centre (-3,8) and radius 7.

   The equation of the circle is (x+3)² + (y-8)² = 49.

4. Find the equation of the circle with centre (-2,7) and radius 5.

   The equation of the circle is (x+2)² + (y-7)² = 25.

5. Find the equation of the circle with centre (-5,4) and radius 6.

   (x+5)² + (y-4)² = 36

6. x + y = 4 xy = 16 (system of equations, variables, algebra)

   There are no real solutions for x and y.

7. Find the value of tan (π)/(8).

   √2 - 1

8. Find the mean of the following data by the direct method: class intervals 0-10, 10-20, 20-30, 30-40, 40-50 with frequencies 5, 8, 15, 16, 6.

   The mean of the given data is 27.

9. Show that an onto function f:1,2,3 →1,2,3 is always one-one.

   An onto function f:1,2,3 →1,2,3 is always one-one because for the range to cover all elements of the codomain, each distinct element in the domain must map to a

10. Nissa hires 60 seasonal workers to play elves in her department store's Santa village. A third of the elves quit after children vomit on them, then 10 of the re

    30 elves are left.

11. Find (d y)/(d x), if x=a(θ+sin θ), y=a(1-cos θ).

    (d y)/(d x) = tan((θ)/(2))

12. If y=3 e² x+2 e³ x, prove that (d² y)/(d x²)-5 (d y)/(d x)+6 y=0.

    The given equation (d² y)/(d x²)-5 (d y)/(d x)+6 y=0 is proven to be true.

13. Let f(x)=∫_0^x t(t²-9t+20) dt, 1 ≤ x ≤ 5. If the range of f is [α, β], then 4(α+β) equals:

    157

14. Find all values of (1+i)^(1-i) in the form a + bi.

    e^(( (1)/(2)ln 2 + (π)/(4) + 2nπ)) [ cos( (π)/(4) + 2nπ - (1)/(2)ln 2) + i sin( (π)/(4) + 2nπ - (1)/(2)ln 2) ] for n Z

15. Let S= N 0. Define a relation R from S to R by: R= (x, y): log_e y = x log_e ((x)/(5)), x S, y R. Then, the sum of all the elements in the range of R is equal t

    The sum of all the elements in the range of R is infinite.

16. If 1² 151² + 2² 152² + 3² 153² + + 15² 1515² = 2^m 3^n 5^k, where m,n,k are positive integers, then m + n + k is equal to:

    12

17. The x-coordinate of the point which lies on the line represented by 3x - y - 1 = 0 and whose y-coordinate is 5, is:

    The x -coordinate of the point is 2.

18. Find the flux of F⃗ = (x³, y³, z³) across the surface of the sphere x² + y² + z² = a².

    The flux of F across the surface of the sphere is (12π a^5)/(5).

19. cos ( sin^(-1) (3)/(8) + sin^(-1) (5)/(13) + sin^(-1) (33)/(65)) is equal to:

    (3√55 - 12)/(40)

20. In the coordinate plane, triangle ABC is equilateral with B(1,0) and C(3,0). A line through the origin O meets AB and AC at M and N, respectively. If OM=MN, fin

    The coordinates of M are ((5)/(4), (√3)/(4)).

21. Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of the triangles ABC, ACD and ADB be 5, 6 and 7 square uni

    18 + √110 square units

22. Find the inverse of the matrix A = [[1, 2, 5], [2, 3, 1], [-1, 1, 1]] if it exists.

    The inverse of the matrix A is A^(-1) = 2/21 & 3/21 & -13/21 -3/21 & 6/21 & 9/21 5/21 & -3/21 & -1/21.

23. Find the length of the curve defined parametrically by x = a(cos t + t sin t), y = a(sin t - t cos t) for 0 ≤ t ≤ π/2.

    The length of the curve is (aπ²)/(8).

24. Let A=-2,-1,0,1,2,3. Define a relation R on A by xRy if and only if y=max(x,1). If l is the number of elements in R, and m and n are the minimum numbers of orde

    12

25. Which one of the following equations does not have real roots?

    The equation x² - 4x + 3√2 = 0 does not have real roots.

26. Find the last two digits of 7^(7^7).

    43

27. 32. Show that f: R -> R defined as f(x) = x / sqrt(1 + x²) is one-one but not onto. (function, one-one, onto, real numbers, domain, codomain, range)

    The function f(x) = (x)/(√(1 + x²)) is one-one but not onto.

28. Solve the quadratic equation: x³ - 4x + 3 = 0. Find its roots and vertex, and visualize the parabola curve on coordinate axes.

    The roots of the equation x³ - 4x + 3 = 0 are x = 1, x = (-1 + √13)/(2), and x = (-1 - √13)/(2). The concept of a 'vertex' and a 'parabola curve' is applicable

29. One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces ma

    (1)/(2)

30. Prove that =(sin 5 x-2 sin 3 x+sin x)/(cos 5 x-cos x)=tan x

    The proof is complete, as (sin 5 x-2 sin 3 x+sin x)/(cos 5 x-cos x)=tan x.

31. From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30^° and 45^°, respectively. If the bridge is

    The width of the river is 3(√3 + 1) meters.

32. Find all real solutions to the equation: √(x + 3 - 4√(x - 1)) + √(x + 8 - 6√(x - 1)) = 1.

    The real solutions are x [5, 10].

33. Let a R and A be a matrix of order 3 3 such that det(A)=-4 and A+I= 1 & a & 12 & 1 & 0 & 1 & 2, where I is the 3 3 identity. If det ((a+1)adj((a-1)A))=2^m3^n, m

    16

34. Which term of the AP: 21, 18, 15,... is – 81? Also, is any term 0?

    The 35 -th term of the AP is -81. Yes, 0 is the 8 -th term of the AP.

35. Let L_1: (x-1)/(1) = (y-2)/(1) = (z-1)/(2) and L_2: (x+1)/(1) = (y-2)/(2) = (z)/(4) be two lines. Let L_3 be a line passing through the point (α, β,) and be per

    (271)/(5)

36. The roots of the quadratic equation 3x² - px + q = 0 are 10^(th) and 11^(th) terms of an arithmetic progression with common difference (3)/(2). If the sum of th

    474

37. The sum of all values of θ [0, 2π] satisfying 2sin² θ = cos 2θ and 2cos² θ = 3sin θ is:

    π

38. Let f: R → R be a polynomial function of degree four having extreme values at x = 4 and x = 5. If lim_x → 0 (f(x))/(x²) = 5, then f(2) is equal to:

    10

39. Case Study - 1 36. In a park, four poles are standing at positions A, B, C and D around the circular fountain such that the cloth joining the poles AB, BC, CD a

    (i) OSA = 90^° (ii) ABCD is a Kite (iii) (a) AP = 4 cm (iii) (b) QOR = 120^°

40. Let a = i + j + k, b = 2 i + 2 j + k and d = a × b. If c is a vector such that a · c = | c|, | c - 2 a|² = 8 and the angle between d and c is (π)/(4), then |10

    6

41. The area of the region, inside the circle (x - 2√3)² + y² = 12 and outside the parabola y² = 2√3 x is:

    6π - 16

42. Angle A of triangle = 50. B = 30. What is C?

    The measure of angle C is 100^°.

43. Differentiate the following w.r.t. x. (i) cos ^(-1)(sin x) (ii) tan ^(-1)((sin x)/(1+cos x)) (iii) sin ^(-1)((2^(x+1))/(1+4^x))

    (i) -1 (ii) (1)/(2) (iii) (2^(x+1) log 2)/(1+4^x)

44. A particle moves in the plane with position vector r⃗(t) = (t², e^t). Find its velocity, acceleration, and the tangential and normal components of acceleration

    At t=1: Velocity v(1) = (2, e), Acceleration a(1) = (2, e), Tangential component of acceleration a_T = √(4 + e²), Normal component of acceleration a_N = 0.

45. In how many ways can 4 red, 3 yellow and 2 green discs be arranged in a row if the discs of the same colour are indistinguishable?

    The discs can be arranged in 1260 ways.

46. Consider two vectors u=3i-j and v=2i+j-, > 0. The angle between them is given by cos^(-1) ((√5)/(2√7)). Let v=v_1+v_2 where v_1 is parallel to u and v_2 is perp

    14

47. Let the points ((11)/(2), α) lie on or inside the triangle with sides x + y = 11, x + 2y = 16, and 2x + 3y = 29. Then the product of the smallest and the larges

    33

48. The line L_1 is parallel to the vector a = -3 i + 2 j + 4 k and passes through the point (7, 6, 2) and the line L_2 is parallel to the vector b = 2 i + j + 3 k

    The shortest distance between the lines L_1 and L_2 is (69)/(√342).

49. The integral 80 ∫_0^((π)/(2)) ((sin θ + cos θ)/(9 + 16 sin 2 θ)) dθ is equal to:

    8 ln 3

50. How many numbers lying between 100 and 1000 can be formed with the digits 0, 1, 2, 3, 4, 5, if the repetition of the digits is not allowed?

    100

51. Find the number of all one-one functions from set A=1,2,3 to itself.

    The number of all one-one functions from set A=1,2,3 to itself is 6.

52. Find the centre and the radius of the circle x²+y²+8 x+10 y-8=0.

    The center of the circle is (-4, -5) and the radius is 7.

53. किसी स्कूल की कक्षा X की 51 लड़कियों की ऊँचाइयों का एक सर्वेक्षण किया गया और निम्नलिखित आँकड़े प्राप्त किए गए: |l|c| ऊँचाई (cm में) & लड़कियों की संख्या 140 से

    माध्यक ऊँचाई लगभग 149.03 cm है।

54. Let A be the set of all students of a boys school. Show that the relation R in A given by R=(a, b): a is sister of b is the empty relation and R^()=(a, b): the

    The relation R is an empty relation because no student in a boys' school can be a sister. The relation R^() is a universal relation because the height differenc

55. If the components of a=α i+β j+ k along and perpendicular to b= i+ j- k respectively, are (16)/(11)(3 i+ j- k) and (1)/(11)(-4 i-5 j-17 k), then α²+β²+ ² is equ

    26

56. Prove that if f(z) is analytic in |z| < 1, continuous on |z| ≤ 1, and |f(z)| ≤ M on |z| = 1, then |f^(n)(0)| ≤ n!M for all n ≥ 0.

    The proof shows that if f(z) is analytic in |z| < 1, continuous on |z| ≤ 1, and |f(z)| ≤ M on |z| = 1, then |f^((n))(0)| ≤ n!M for all n ≥ 0.

57. Let A be a 3 × 3 matrix such that |adj (adj(adj A))| = 81. If S = n Z: |adj(adj A)|^(((n-1)²)/(2)) = |A|³n² - 5n - 4, then Σ_n S | A^(n² + n) | is equal to:

    732

58. The length of the chord of the ellipse (x²)/(4) + (y²)/(2) = 1, whose mid-point is (1, (1)/(2)), is:

    The length of the chord is √((11)/(3)).

59. How many ways can a 2×n rectangle be tiled using 1×2 dominoes?

    The number of ways to tile a 2 × n rectangle using 1 × 2 dominoes is F_n+1, where F_n is the n -th Fibonacci number with F_1=1 and F_2=1.

60. Find the area of the region bounded by the curves y = x² and y = |x|.

    The area of the region bounded by the curves y = x² and y = |x| is (1)/(3) square units.

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