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Solved maths problems — page 22

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

  1. Write the set (1)/(2),(2)/(3),(3)/(4),(4)/(5),(5)/(6),(6)/(7) in the set-builder form.

    The set in set-builder form is (n)/(n+1): n N, 1 ≤ n ≤ 6.

  2. Two rails are represented by the equations x + 2y - 4 = 0 and 2x + 4y - 12 = 0. Will the rails cross each other?

    The rails will not cross each other.

  3. Let α and β be the roots of x² + √3 x - 16 = 0, and and be the roots of x² + 3x - 1 = 0. If P_n = α^n + β^n and Q_n = ^n + ^n, then (P_25 + √3 P_24)/(2P_23) + (

    5

  4. Three circles of radius 1 touch each other externally. Find the area of the triangle formed by joining their centers.

    The area of the triangle formed by joining their centers is √3 square units.

  5. Show that the sequence f_n(x) = x^n converges pointwise but not uniformly on [0,1].

    The sequence f_n(x) = x^n converges pointwise to f(x) = 0 & if 0 ≤ x < 1 1 & if x = 1 on [0,1]. It does not converge uniformly on [0,1] because lim_n → ∞ _x [0,

  6. Match each of the set on the left described in the roster form with the same set on the right described in the set-builder form: (i) P, R, I, N, C, A, L (a) x:

    (i) P, R, I, N, C, A, L (d) x: x is a letter of the word PRINCIPAL (ii) 0 (c) x: x is an integer and x + 1= 1 (iii) 1, 2, 3, 6, 9, 18 (a) x: x is a positive int

  7. The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is:

    17280

  8. There are 40 students in Class X of a school of whom 25 are girls and 15 are boys. The class teacher has to select one student as a class representative. She wr

    (i) The probability that the name written on the card is the name of a girl is (5)/(8). (ii) The probability that the name written on the card is the name of a

  9. In parallelogram ABCD, the diagonals intersect at O. Given AC = 10, BD = 18, and AD = 12, find the perimeter of triangle BOC.

    The perimeter of triangle BOC is 26 units.

  10. If lim _x → 0 (cos(2x) + acos(4x) - b)/(x^4) is finite, then (a + b) is equal to:

    (1)/(2)

  11. Let _θ and _θ be the distinct roots of 2x² + (cos θ) x - 1 = 0, θ (0, 2π). If m and M are the minimum and the maximum values of _θ^4 + _θ^4, then 16(M + m) equa

    13

  12. Solve the following question—Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three

    Aftab's present age is 42 years and his daughter's present age is 12 years.

  13. For some a, b, let f(x) = | ccc a + (sin x)/(x) & 1 & b a & 1 + (sin x)/(x) & b a & 1 & b + (sin x)/(x) |, x ≠ 0, lim_x → 0 f(x) = + a + b. Then ( + +)² is equa

    16

  14. Let C be the circle of minimum area enclosing the ellipse (x²)/(a²)+(y²)/(b²)=1 with eccentricity 12 and foci ( 2,0). Let PQR be a variable triangle whose verte

    29(2 + √3)

  15. Find the pairs of equal sets, if any, give reasons: A = 0 B = x: x > 15 and x < 5 C = x: x - 5 = 0 D = x: x² = 25 E = x: x is an integral positive root of the e

    The pairs of equal sets are C = E because both sets contain only the element 5.

  16. Let S_n = (1)/(2) + (1)/(6) + (1)/(12) + (1)/(20) + up to n terms. If the sum of the first six terms of an A.P. with first term -p and common difference p is √(

    25

  17. Let a line pass through two distinct points P(-2,-1,3) and Q, and be parallel to the vector 3 i+2 j+2 k. If the distance of the point Q from the point R(1,3,3)

    34

  18. Solve the following Linear Programming Problem graphically: Maximise Z = 600x + 400y subject to the constraints x + 2y ≤ 12 4x + 5y ≥ 20 2x + y ≤ 12 x, y ≥ 0 (l

    The maximum value of Z is 4000 at x=4, y=4.

  19. If y=sin ^(-1) x, show that (1-x²) (d² y)/(d x²)-x (d y)/(d x)=0.

    The given equation is proven: (1-x²) (d² y)/(d x²)-x (d y)/(d x)=0.

  20. The number of sequences of ten terms, whose terms are either 0, 1, or 2, that contain exactly five 1 ’s and exactly three 2 ’s is equal to:

    2520

  21. In the square grid shown, what is tan AOB?

    (15)/(8)

  22. Let S = m Z: A^(m² + A^m = 3 I - A^(-6), where A = [ cc 2 & -1 1 & 0 ]. Then n(S) is equal to

    n(S) = 2

  23. If the function f(x)=2x³ - 9a x² + 12a² x + 1, where a>0, attains its local maximum and local minimum at p and q respectively, such that p² = q, then f(3) is eq

    37

  24. Let P be the foot of the perpendicular from the point Q(10, -3, -1) on the line (x - 3)/(7) = (y - 2)/(-1) = (z + 2)/(-2). Then the area of the right-angled tri

    The area of the right-angled triangle PQR is (1)/(2)√1346 square units.

  25. Q. 1: In the given figure, PS/SQ = PT/TR and ∠ PST = ∠ PRQ. Prove that PQR is an isosceles triangle. (isosceles triangle, ratio, angles, proportionality, parall

    PQR is an isosceles triangle.

  26. In a G.P. of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the G.P. is (A) (-4)/(5) (B) (1)/(5) (C) 4 (D)

    4

  27. Let A be the set of all functions f: Z → Z and R be a relation on A such that R = (f,g): f(0)=g(1) and f(1)=g(0). Then R is:

    The relation R is symmetric but neither reflexive nor transitive.

  28. The zookeeper feeds all the apes in the zoo. He orders all the bananas from a local farm every 2 months. If the monkeys need 200 bananas, the gorillas need 400

    The zookeeper needs to order 1400 bananas to last for 2 months.

  29. If the area of the region (x,y): |x-5| ≤ y ≤ 4√x is A, then 3A is equal to _____

    200

  30. Let [x] denote the greatest integer function, and let m and n respectively be the numbers of the points where the function f(x) = [x] + |x - 2|, -2 < x < 3, is

    8

  31. Prove that the equation x⁴ + y⁴ = z⁴ has no non-trivial integer solutions (special case of Fermat's Last Theorem).

    The equation x^4 + y^4 = z^4 has no non-trivial integer solutions.

  32. A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at th

    256π cm²

  33. In an arithmetic progression, if S_40 = 1030 and S_12 = 57, then S_30 - S_10 is equal to:

    515

  34. How many words, with or without meaning, each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE?

    2880

  35. Let f(x) be a real differentiable function such that f(0) = 1 and f(x + y) = f(x) f'(y) + f'(x) f(y) for all x, y R. Then Σ_n=1^(100) log_e f(n) is equal to:

    2525

  36. The equation of the chord of the ellipse (x²)/(25) + (y²)/(16) = 1, whose mid-point is (3, 1), is:

    The equation of the chord is 48x + 25y - 169 = 0.

  37. Let x_1, x_2, x_3, x_4 be in a geometric progression. If 2, 7, 9, 5 are subtracted respectively from x_1, x_2, x_3, x_4, then the resulting numbers are in an ar

    216

  38. If y = cos((π)/(3) + cos^(-1)(x)/(2)), then (x - y)² + 3y² is equal to:

    3

  39. In a relay race there are five teams A, B, C, D and E. (a) What is the probability that A, B and C finish first, second and third, respectively. (b) What is the

    (a) The probability that A, B and C finish first, second and third, respectively is (1)/(60). (b) The probability that A, B and C are first three to finish (in

  40. Let L_1: (x-1)/(3) = (y-1)/(4) = (z+1)/(0) and L_2: (x-2)/(2) = (y)/(0) = (z+4)/(0), α R, be two lines, which intersect at the point B. If P is the foot of perp

    175.5

  41. Solve the equation z²= z, where z=x+i y

    z = 0, 1, -(1)/(2) + i(√3)/(2), -(1)/(2) - i(√3)/(2)

  42. How many 2 digit even numbers can be formed from the digits 1, 2, 3, 4, 5 if the digits can be repeated?

    10

  43. If α is a root of the equation x² + x + 1 = 0 and Σ_k=1^n (α^k + α^(-k))² = 20, then n is equal to:

    n = 11

  44. Find the limits: (i) lim _x → 1[x³-x²+1] (ii) lim _x → 3[x(x+1)] (iii) lim _x →-1[1+x+x²+ +x^(10)].

    (i) 1, (ii) 12, (iii) 1

  45. For a 3 × 3 matrix M, let trace(M) denote the sum of all the diagonal elements of M. Let A be a 3 × 3 matrix such that |A|=(1)/(2) and trace(A)=3. If B=adj(adj(

    280

  46. Find the maximum value of 3cos θ + 4sin θ + 5cos(θ + π/6).

    The maximum value of the expression is √(30 + 15√3).

  47. Martha is knitting winter wear for her 3 grandchildren. They're triplets, so they're all the same size. She wants to make a hat, scarf, sweater, mittens, and so

    Martha will need to buy 63 skeins of wool.

  48. Prove that in any group of 6 people, there are either 3 mutual friends or 3 mutual strangers.

    In any group of 6 people, there are either 3 mutual friends or 3 mutual strangers.

  49. Find the 10^ th and n^ th terms of the G.P. 5, 25,125,...

    The 10^ th term is 9,765,625 and the n^ th term is 5^n.

  50. Evaluate ∮_C (z² + 1)/(z² - 1) dz where C is the circle |z| = 3 using residue theorem.

    0

  51. If tan A = 1/7 and tan B = 1/3 find tan(2A + B) without using calculator approximations.

    tan(2A + B) = (9)/(13)

  52. Find the mean deviation about the median for the following data: 3,9,5,3,12,10,18,4,7,19,21.

    The mean deviation about the median is approximately 5.27.

  53. A rectangular sheet is folded as shown and then unfolded. If 1=58^°, find 2.

    2 = 64^°

  54. MODIFIED: Find the shortest distance between the skew lines: r⃗₁ = (i + j + k) + λ(2i + 3j + 4k) and r⃗₂ = (2i + 3j + 5k) + μ(i + 2j + 3k).

    The shortest distance between the given skew lines is (√6)/(6) units.

  55. VARIATION: Three circles of radius 1 touch each other externally. Find the area of the triangle formed by joining their centers. Find the numerical value if exa

    The area of the triangle formed by joining their centers is √3 square units.

  56. Let A be the set of all 50 students of Class X in a school. Let f: A → N be function defined by f(x)= roll number of the student x. Show that f is one-one but n

    The function f is one-one but not onto.

  57. Find the center of mass of the solid bounded by z = 4 - x² - y² and z = 0 with density ρ(x,y,z) = z.

    The center of mass is (0, 0, 1).

  58. Let y² = 12x be the parabola and S be its focus. Let PQ be a focal chord of the parabola such that (SP)(SQ) = (147)/(4). Let C be the circle described taking PQ

    1328

  59. Let A=1,2,3. Then show that the number of relations containing (1,2) and (2,3) which are reflexive and transitive but not symmetric is three.

    The number of relations containing (1,2) and (2,3) which are reflexive and transitive but not symmetric is three.

  60. The remainder, when 7^(103) is divided by 23, is equal to:

    14

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