SolveForX

Solved maths problems — page 15

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

  1. Find the missing value of X between 3X +5 equal to 11

    The missing value of X is 2.

  2. If θ [-2π, 2π], then the number of solutions of 2√2cos²θ + (2-√6)cosθ - √3 = 0 is equal to:

    8

  3. Find the area of the segment AYB shown in Fig. 11.6, if radius of the circle is 21 cm and AOB = 120^°. (Use π = (22)/(7))

    The area of the segment AYB is (462 - (441√3)/(4)) cm²

  4. The centroid of a triangle ABC is at the point (1,1,1). If the coordinates of A and B are (3,-5,7) and (-1,7,-6), respectively, find the coordinates of the poin

    The coordinates of point C are (1,1,2).

  5. If the image of the point P(1, 0, 3) in the line joining the points A(4, 7, 1) and B(3, 5, 3) is Q(α, β,), then α + β + is equal to:

    (46)/(3)

  6. If A × B=(p, q),(p, r),(m, q),(m, r), find A and B.

    A = p, m and B = q, r

  7. Express the following in the form of a+b i: (i) (-5 i)((1)/(8) i) (ii) (-i)(2 i)(-(1)/(8) i)³

    (i) (5)/(8) + 0i (ii) 0 + (1)/(256) i

  8. If R is the set of all real numbers, what do the cartesian products R × R and R × R × R represent?

    The Cartesian product R × R represents the two-dimensional Cartesian coordinate plane, and R × R × R represents the three-dimensional Cartesian coordinate space

  9. Let A=1,2,3. The number of relations on A, containing (1,2) and (2,3), which are reflexive and transitive but not symmetric, is:

    5

  10. If Σ_r=1^n T_r = ((2n-1)(2n+1)(2n+3)(2n+5))/(64), then lim_n → ∞ Σ_r=1^n ((1)/(T_r)) is equal to:

    (2)/(3)

  11. Let A=[a_ij] be a 2 × 2 matrix such that a_ij 0,1 for all i and j. Let the random variable X denote the possible values of the determinant of the matrix A. Then

    The variance of X is (3)/(8).

  12. Consider a function f:[0, (π)/(2)] → R given by f(x)=sin x and g:[0, (π)/(2)] → R given by g(x)=cos x. Show that f and g are one-one, but f+g is not one-one.

    The functions f(x) = sin x and g(x) = cos x are one-one on [0, (π)/(2)], but their sum (f+g)(x) = sin x + cos x is not one-one on this interval.

  13. Show that tan 3 x tan 2 x tan x=tan 3 x-tan 2 x-tan x

    The identity tan 3 x tan 2 x tan x=tan 3 x-tan 2 x-tan x is shown to be true by using the tangent addition formula and algebraic manipulation.

  14. The distance of the point (7,10,11) from the line (x-4)/(1)=(y-4)/(0)=(z-2)/(3) along the line (x-9)/(2)=(y-13)/(-3)=(z-17)/(6) is:

    26

  15. If the system of equations ( - 1)x + ( - 4)y + z = 5, x + ( - 1)y + ( - 4)z = 7, ( + 1)x + ( + 2)y - ( + 2)z = 9 has infinitely many solutions, then ² + is equa

    12

  16. Let the focal chord PQ of the parabola y² = 4x make an angle of 60^° with the positive x-axis, where P lies in the first quadrant. If the circle, whose one diam

    15

  17. The sum of first three terms of a G.P. is (13)/(12) and their product is -1. Find the common ratio and the terms.

    The common ratios are r = -3/4 and r = -4/3. The terms of the G.P. are either 4/3, -1, 3/4 or 3/4, -1, 4/3.

  18. Let the function f(x) = (x)/(3) + (3)/(x) + 3, x 0 be strictly increasing in (-∞, _1) ( _2, ∞) and strictly decreasing in ( _1, _2) ( _4, _5). Then Σ_i=1^(5) _i

    18

  19. Prove that (cos 7 x+cos 5 x)/(sin 7 x-sin 5 x)=cot x

    cot x

  20. Solve for x: sin⁻¹(x) + sin⁻¹(√(1-x²)) = π/2 for x ∈ [0,1].

    The solution for x is x [0,1].

  21. If A.M. and G.M. of two positive numbers a and b are 10 and 8, respectively, find the numbers.

    The two numbers are 4 and 16.

  22. 9 out of 10 cheerleaders are 64 tall. The 10th cheerleader is 60 tall. If they build a human pyramid, where 4 girls are on the bottom, 3 stand on top of the 4,

    The human pyramid is 21 feet tall.

  23. Let m and n, (m<n) be two 2-digit numbers. Then the total number of pairs (m,n) such that (m,n)=6 is:

    84

  24. Find the number of permutations of the letters of the word ALLAHABAD.

    The number of permutations of the letters of the word ALLAHABAD is 7560.

  25. Find all continuous functions f: ℝ → ℝ such that f(x+y) = f(x) + f(y) for all x,y ∈ ℝ.

    The continuous functions f: R → R satisfying f(x+y) = f(x) + f(y) are of the form f(x) = cx for some constant c R.

  26. Find the value of a such that the sum of the squares of the roots of the equation x²-(a-2) x-(a+1)=0 is least.

    The value of a for which the sum of the squares of the roots is least is 1.

  27. A wall mural has four different colors of paint in it: red, white, purple, and yellow. There are equal amounts of red, white, and purple paint in the mural. Hal

    2 pints

  28. If a line intersects sides AB and AC of a ABC at D and E respectively and is parallel to BC, prove that (AD)/(AB) = (AE)/(AC) (see Fig. 6.13). [Figure: triangle

    Thus, it is proven that if a line intersects sides AB and AC of a ABC at D and E respectively and is parallel to BC, then (AD)/(AB) = (AE)/(AC).

  29. Find the distance between the points (1, 2) and (4, 6).

    The distance between the points (1, 2) and (4, 6) is 5 units.

  30. Evaluate ∫₀^(π/2) (sin x)^4 (cos x)³ dx using reduction formulas.

    (2)/(35)

  31. Find g of and f ° g, if f: R → R and g: R → R are given by f(x)=cos x and g(x)=3 x². Show that g of ≠ fog.

    g ° f(x) = 3cos² x and f ° g(x) = cos(3x²). Since 3cos² x ≠ cos(3x²), it is shown that g ° f ≠ f ° g.

  32. 35. (a) Represent the equations of lines l_1 and l_2 in vector form and check whether they are intersecting or not. l_1: (x+3)/(-3) = (y-1)/(1) = (z-5)/(5) l_2:

    The vector form of line l_1 is r = (-3 i + j + 5 k) + (-3 i + j + 5 k). The vector form of line l_2 is r = (- i + 2 j + 5 k) + (- i + 2 j + 5 k). The lines inte

  33. Differentiate x^(sin x), x>0 w.r.t. x.

    (dy)/(dx) = x^(sin x) ((sin x)/(x) + cos x log x)

  34. Find the equation of the ellipse, whose length of the major axis is 20 and foci are (0, ± 5).

    The equation of the ellipse is (x²)/(75) + (y²)/(100) = 1.

  35. Solve the system of equations using matrix method: x - y + 2z = 7, 3x + 4y - 5z = -5, 2x - y + 3z = 12.

    x = -10, y = 1, z = 3

  36. Let a random variable X take values 0, 1, 2 and 3 with P(X = 0) = P(X = 1) = p and P(X = 2) = P(X = 3) = (1 - 2p)/(2). If E(X²) = 2E(X), then the value of 8p -

    0

  37. Evaluate the limit: lim_x → ∞ (tan(5x^(1/3)) log_e(1 + 3x²))/((tan^(-1)(3√x))² (e^(5x^(4/3)) - 1)) is equal to:

    The limit does not exist.

  38. Find the Jordan canonical form of the matrix A = [[2, 1, 0], [0, 2, 1], [0, 0, 2]].

    The Jordan canonical form of the matrix A is J = 2 & 1 & 0 0 & 2 & 1 0 & 0 & 2.

  39. (b) Opposite sides of a square are along the lines: r = i + 2 j - 4 k + (2 i + 3 j + 6 k) r = 3 i + 3 j - 5 k + (2 i + 3 j + 6 k) Find the area of the square if

    The area of the square is (293)/(49) square units, and the value of p is -2.

  40. Solve the differential equation dy/dx + y*cot(x) = 2*x + x²*cot(x) given y(pi/2) = 0.

    y = x² - (π²)/(4)csc(x)

  41. If the image of the point (4, 4, 3) in the line (x - 1)/(2) = (y - 2)/(1) = (z - 1)/(3) is (α, β,), then α + β + is equal to

    9

  42. Let |z_1 - 8 - 2i| ≤ 1 and |z_2 - 2 + 6i| ≤ 2, z_1, z_2 C. Then the minimum value of |z_1 - z_2| is:

    7

  43. Find the derivative of the function f(x)=2 x²+3 x-5 at x=-1. Also prove that f^()(0)+3 f^()(-1)=0.

    The derivative of the function f(x)=2 x²+3 x-5 at x=-1 is -1. The relation f^()(0)+3 f^()(-1)=0 is proven.

  44. Let the area of the triangle formed by a straight line L: x + by + c = 0 with coordinate axes be 48 square units. If the perpendicular drawn from the origin to

    97

  45. A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. Determine t

    The volume of the toy is 25.12 cm³. The difference between the volumes of the cylinder and the toy is 25.12 cm³.

  46. Find (d y)/(d x) if x-y=π.

    (dy)/(dx) = 1

  47. Compute the derivative of tan x.

    The derivative of tan x is sec² x.

  48. Find the geodesics on the surface of a sphere using the Euler-Lagrange equation.

    The geodesics on the surface of a sphere are great circles, which can be expressed by the equation cotθ = A cos( - _0), where A and _0 are constants determined

  49. Let P be the parabola, whose focus is (-2, 1) and directrix is 2x + y + 2 = 0. Then the sum of the ordinates of the points on P, whose abscissa is -2, is:

    The sum of the ordinates of the points on P, whose abscissa is -2, is (3)/(2).

  50. Find the Fourier series of f(x) = x² on [-π, π] and use it to evaluate Σ(n=1 to ∞) 1/n⁴.

    The Fourier series of f(x) = x² on [-π, π] is x² = (π²)/(3) + Σ_n=1^(∞) (4(-1)^n)/(n²) cos(nx). Using this, Σ_n=1^(∞) (1)/(n^4) = (π^4)/(90).

  51. Prove that every polynomial of odd degree with real coefficients has at least one real root.

    Every polynomial of odd degree with real coefficients has at least one real root.

  52. Show that the function f given by f(x)= x³+3, & if x ≠ 0 1, & if x=0 is not continuous at x=0.

    The function f(x) is not continuous at x=0 because lim_x → 0 f(x) = 3 while f(0) = 1, and 3 ≠ 1.

  53. Let a curve y = f(x) pass through the points (0, 5) and (log_e 2, k). If the curve satisfies the differential equation 2(3 + y) e²x dx - (7 + e²x) dy = 0, then

    k=8

  54. In triangle ABC, C=90^°. Point D is the midpoint of BC, and AD=BC. Find sin BAD.

    The value of sin BAD is (√21)/(14).

  55. Find the Taylor series expansion of f(x) = e^(x²) centered at x = 0 and determine its interval of convergence.

    The Taylor series expansion of f(x) = e^x² centered at x = 0 is Σ_n=0^(∞) x²nn!. The interval of convergence is (-∞, ∞).

  56. 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.

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

  57. The integral ∫_0^(π)(8x)/(4cos²x + sin²x) dx is equal to:

    2π²

  58. If the square of the shortest distance between the lines (x-2)/(1)=(y-1)/(2)=(z+3)/(3) and (x+1)/(2)=(y+3)/(4)=(z+5)/(5) is (m)/(n), where m, n are coprime numb

    9

  59. Let the area of the region (x, y): 2y ≤ x² + 3, y + |x| ≤ 3, y ≥ |x-1| be A. Then 6A is equal to:

    24

  60. Colby wants to buy some gumballs that cost a nickel each. If he has 8 quarters, 6 dimes, 14 nickels, and 15 pennies, how many can he buy?

    Colby can buy 69 gumballs.

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