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Solved maths problems — page 13

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

  1. Prove that in two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact. [Figure required: t

    In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.

  2. A coin is tossed three times, consider the following events. A: 'No head appears', B: 'Exactly one head appears' and C: 'Atleast two heads appear'. Do they form

    Yes, the events A, B, and C form a set of mutually exclusive and exhaustive events.

  3. Let the system of equations x + 5y - z = 1, 4x + 3y - 3z = 7, and 24x + y + z =, with, R, have infinitely many solutions. Then the number of solutions of this s

    12

  4. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching th

    The time taken by the car to reach the foot of the tower from point C is 3 seconds.

  5. Let the area of a PQR with vertices P(5, 4), Q(-2, 4) and R(a, b) be 35 square units. If its orthocenter and centroid are O(2, (14)/(5)) and C(c, d) respectivel

    3

  6. Let the mean and variance of five observations x_1=1, x_2=3, x_3=a, x_4=7 and x_5=b, with a>b, be 5 and 10 respectively. Then the variance of the observations n

    16

  7. If sin x + sin² x = 1, x (0, (π)/(2)), then (cos^(12) x + tan^(12) x) + 3(cos^(10) x + tan^(10) x + cos^8 x + tan^8 x) + (cos^6 x + tan^6 x) is equal to:

    2

  8. Find a relation between x and y such that the point (x, y) is equidistant from the points (7, 1) and (3, 5).

    The relation between x and y is y = x - 2 or x - y - 2 = 0.

  9. Solve the following pair of equations by substitution method: 7x - 15y = 2 (1) and x + 2y = 3 (2).

    The solution to the system of equations is x = (49)/(29) and y = (19)/(29).

  10. A racing track is build around an elliptical ground whose equation is given by 9x² + 16y² = 144. The width of the track is 3 m as shown below: Based on given in

    (i) y = ±(3)/(4)√(16 - x²) (ii) (3)/(4)[(x)/(2)√(16-x²) + 8sin^(-1)((x)/(4))] + C (iii) (a) 12π m² (iii) (b) P=(7, 0), Q=(0, 6) Area of POQ = 21 square units

  11. The area (in sq. units) of the region (x, y): 0 ≤ y ≤ 2|x| + 1, 0 ≤ y ≤ x² + 1, |x| ≤ 3 is

    The area of the region is (16)/(3) square units.

  12. Discuss the continuity of the function f given by f(x)= x, & if x ≥ 0 x², & if x<0

    The function f(x) is continuous for all real numbers x.

  13. Find the number of 4 letter words, with or without meaning, which can be formed out of the letters of the word ROSE, where the repetition of the letters is not

    24

  14. Let the line passing through the points (-1, 2, 1) and parallel to the line (x-1)/(2) = (y+1)/(3) = (z)/(4) intersect the line (x+2)/(3) = (y-3)/(2) = (z-4)/(3)

    5√5

  15. Given tan A = (4)/(3), find the other trigonometric ratios of the angle A.

    The other trigonometric ratios are sin A = (4)/(5), cos A = (3)/(5), cosec A = (5)/(4), sec A = (5)/(3), and cot A = (3)/(4).

  16. Let z_1 and z_2 be two complex numbers such that z_1+i z_2=0 and (z_1 z_2)=π. Then find (z_1).

    (3π)/(4)

  17. Find the value of x: sin(x) = cos(x) for x in [0, π/2].

    x = (π)/(4)

  18. Determine the convergence of the series Σ(n=1 to ∞) (n!)²/(2n)! using the ratio test.

    The series converges.

  19. Let the point A divide the line segment joining the points P(-1,-1,2) and Q(5,5,10) internally in the ratio r:1 (r>0). If O is the origin and ( OQ · OA - (1)/(5

    7

  20. The sum of all local minimum values of the function f(x) = 1 - 2x, & x < -1 (1)/(3)(7 + 2|x|), & -1 ≤ x ≤ 2 (11)/(18)(x - 4)(x - 5), & x > 2 is

    (157)/(72)

  21. Find the roots of the quadratic equation 2x² - 5x + 3 = 0, by factorisation.

    The roots of the equation 2x² - 5x + 3 = 0 are x = 1 and x = (3)/(2).

  22. Find the sum of first n terms and the sum of first 5 terms of the geometric series 1+(2)/(3)+(4)/(9)+

    The sum of the first n terms is S_n = 3(1-((2)/(3))^n) and the sum of the first 5 terms is S_5 = (211)/(81).

  23. If x+i y=(a+i b)/(a-i b), prove that x²+y²=1.

    x²+y²=1

  24. Use Laplace transforms to solve: y'' + 4y = δ(t-π), y(0)=0, y'(0)=0, where δ is the Dirac delta function.

    y(t) = (1)/(2)u(t-π)sin(2t)

  25. The number of different 5-digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last

    4608

  26. Do the points (3, 2), (-2, -3) and (2, 3) form a triangle? If so, name the type of triangle formed.

    Yes, the points (3, 2), (-2, -3) and (2, 3) form a right-angled triangle.

  27. Let A = [a_ij] be a 3 × 3 matrix such that A 0 1 0 = 0 0 1, A 4 1 3 = 0 1 0 and A 2 1 2 = 1 0 0, then a_23 equals:

    -1

  28. A class of 200 students is split into 3 groups such that 2 of them are equal in number and the last one (which is the smallest) is 10 less than each of the othe

    The number of students in the smallest group is 60.

  29. Let A=1,2,3, B=3,4 and C=4,5,6. Find (i) A ×(B C) (ii) (A × B) (A × C) (iii) A ×(B C) (iv) (A × B) (A × C)

    (i) A × (B C) = (1,4), (2,4), (3,4) (ii) (A × B) (A × C) = (1,4), (2,4), (3,4) (iii) A × (B C) (1,3), (1,4), (1,5), (1,6), (2,3), (2,4), (2,5), (2,6), (3,3)

  30. An observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is 45^°. What is the height of the chimney?

    The height of the chimney is 30 m.

  31. Evaluate the integral of (x² + 1) / (x^4 + 1) dx from 0 to infinity.

    (π)/(√2)

  32. The least value of n for which the number of integral terms in the Binomial expansion of ( [3]7 + [3]11)^n is 183, is:

    546

  33. Let A = 1, 2, 3,, 10 and R be a relation on A such that R = (a, b): a = 2b + 1. Let (a_1, a_2), (a_2, a_3), (a_3, a_4),, (a_k, a_k+1) be a sequence of k element

    2

  34. The 8th term of an A.P., whose first two terms are - 5 and 2 respectively, is:

    The 8th term of the A.P. is 44.

  35. Let f: R→ R be a thrice‐differentiable odd function satisfying f''(x)=f(x), f(0)=0, and f'(0)=3. Then 9f(ln 3) is equal to:

    36

  36. Find the equation of the hyperbola with foci (0, ± 3) and vertices (0, ± (√11)/(2)).

    The equation of the hyperbola is (4y²)/(11) - (4x²)/(25) = 1.

  37. Let f(x) = x - 1 and g(x) = e^x for x R. If (dy)/(dx) = ( e^(-2√x) g(f(f(f(x)))) - (y)/(√x)), y(0) = 0, then y(1) is:

    y(1) = e^(-4) - e^(-5)

  38. Find (d y)/(d x), if x=a t², y=2 a t.

    (d y)/(d x) = (1)/(t)

  39. Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be 2a and 2b, respectively, and one focus and the corresponding directrix o

    (152109)/(850)

  40. A line passing through the point P(a, θ) makes an acute angle α with the positive x-axis. Let this line be rotated about the point P through an angle (α)/(2) in

    The problem is underdetermined. The value of 3a² tan² α - 2√3 simplifies to a² - 2√3. However, the value of a cannot be uniquely determined from the given infor

  41. The sum, of the squares of all the roots of the equation x² + |2x - 3| - 4 = 0, is

    12 - 6√2

  42. A bag contains 19 unbiased coins and one coin with heads on both sides. One coin drawn at random is tossed and a head turns up. If the probability that the draw

    80

  43. lim_x → 0 csc x (√(2 cos² x + 3 cos x) - √(cos² x + sin x + 4)) is:

    -(1)/(2√5)

  44. Solve (3 x-4)/(2) ≥ (x+1)/(4)-1. Show the graph of the solutions on number line.

    The solution to the inequality is x ≥ 1.

  45. Consider the numbers 4^n, where n is a natural number. Check whether there is any value of n for which 4^n ends with the digit zero.

    No, there is no value of n for which the number 4^n ends with the digit zero.

  46. VARIATION: Find all real solutions to the equation: √(variable_93 + 3 - 4√(variable_93 - 1)) + √(variable_93 + 8 - 6√(variable_93 - 1)) = 1. Find the numerical

    The real solutions are x [5, 10].

  47. In the expansion of ((√2)/(3) + (1)/(√3))^n, n N if the ratio of the 15th term from the beginning to the 15th term from the end is (1)/(6) then the value of n3

    2600

  48. Let E: (x²)/(a²) + (y²)/(b²) = 1, a > b and H: (x²)/(A²) - (y²)/(B²) = 1. Let the distance between the foci of E and the foci of H be 2√3. If a - A = 2, and the

    8

  49. The mean of 5 observations is 4.4 and their variance is 8.24. If three of the observations are 1, 2 and 6, find the other two observations.

    The other two observations are 4 and 9.

  50. Let the shortest distance between the lines (x-3)/(3) = (y-α)/(-1) = (z-3)/(1) and (x+3)/(-3) = (y+7)/(2) = (z-β)/(4) be 3√30. Then the positive value of 5α + β

    46

  51. For a positive constant a find (d y)/(d x), where y=a^(i+(1)/(t)), and x=(t+(1)/(t))^a

    (d y)/(d x) = (a^(t+(1)/(t)) ln a)/(a(t+(1)/(t))^(a-1))

  52. Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains n white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag

    6

  53. Find the derivative of f(x) = x³ - 3x² + 2x - 5 with respect to x.

    f'(x) = 3x² - 6x + 2

  54. Consider ACB, right-angled at C, in which AB = 29 units, BC = 21 units and ABC = θ. Determine the values of (i) cos² θ + sin² θ, (ii) cos² θ - sin² θ.

    (i) cos² θ + sin² θ = 1 (ii) cos² θ - sin² θ = (41)/(841)

  55. The sum of the series 2 × 1 × 204 - 3 × 2 × 205 + 4 × 3 × 206 - 5 × 4 × 207 + + 18 × 17 × 2020 is equal to:

    6460

  56. Let ∫ x³ sin x dx = g(x) + C, where C is the constant of integration. If 8(g((π)/(2)) + g'((π)/(2))) = α π³ + β π² +, α, β, Z, then α + β - equals:

    55

  57. Let f: N → Y be a function defined as f(x)=4 x+3, where, Y=y N: y=4 x+3 for some x N. Show that f is invertible. Find the inverse.

    The inverse function is f^(-1)(y) = (y-3)/(4).

  58. Let a be a unit vector perpendicular to the vectors b= i-2 j+3 k and c=2 i+3 j- k, and makes an angle of cos^(-1)(-(1)/(3)) with the vector i+ j+ k. If a makes

    The value of α is -√6.

  59. If α = 1 + Σ_r=1^6 (-3)^(r-1) ^(12)C_2r-1, then the distance of the point (12, √3) from the line α x - √3 y + 1 = 0 is

    5

  60. Let f be a differentiable function such that 2(x + 2)² f(x) - 3(x + 2)² = 10 ∫_0^x (t + 2) f(t) dt, x ≥ 0. Then f(2) is equal to

    19

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