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Solved maths problems — page 18

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

  1. Show that the points (1, 7), (4, 2), (-1, -1) and (-4, 4) are the vertices of a square.

    The points (1, 7), (4, 2), (-1, -1) and (-4, 4) are the vertices of a square because all four sides are equal in length and both diagonals are equal in length.

  2. SECTION B This section comprises 5 Very Short Answer (VSA) type questions of 2 marks each. 21. A man in a boat goes 12 km downstream and comes back to the start

    The speed with which the man can row the boat in still water is 9 km/h.

  3. Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball

    14

  4. Are the points A (3,6,9), B (10,20,30) and C(25,-41,5), the vertices of a right angled triangle?

    No, the points A (3,6,9), B (10,20,30) and C (25,-41,5) do not form the vertices of a right-angled triangle.

  5. Find the value of sin (31 π)/(3).

    The value of sin (31 π)/(3) is (√3)/(2).

  6. Find the number of words with or without meaning which can be made using all the letters of the word AGAIN. If these words are written as in a dictionary, what

    The total number of words is 60, and the 50th word is NAAIG.

  7. (b) Find: ∫ (x+3)/(x²+4x+5) dx (integral, quadratic, antiderivative, calculus, integration)

    (1)/(2) ln|x²+4x+5| + tan^(-1)(x+2) + C

  8. EXTENSION: Determine the convergence of the series Σ(n=1 to ∞) (n!)²/(2n)! using the ratio test. Additionally, generalize your solution to n dimensions.

    The series Σ_n=1^(∞) ((n!)²)/((2n)!) converges. The generalized series Σ_n=1^(∞) ((n!)^k)/((kn)!) converges for k ≥ 2.

  9. If for some α, β; α ≤ β, α + β = 8 and sec²(tan^(-1) α) + csc²(cot^(-1) β) = 36, then α² + β is

    14

  10. Find the HCF of 6 and 20.

    The HCF of 6 and 20 is 2.

  11. Use conformal mapping to solve the Laplace equation ∇²φ = 0 in the upper half-plane with boundary conditions φ(x,0) = 1 for -1<x<1 and 0 elsewhere.

    The solution to the Laplace equation is (x,y) = (1)/(π) ((z-1)/(z+1)).

  12. ACB लीजिए जिसका कोण C समकोण है जिसमें AB = 29 इकाई, BC = 21 इकाई और ABC = θ हैं तो निम्नलिखित के मान ज्ञात कीजिए: (i) cos² θ + sin² θ, (ii) cos² θ - sin² θ।

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

  13. State which of the following sets are finite or infinite: (i) x: x N and (x - 1)(x - 2) = 0 (ii) x: x N and x²= 4 (iii) x: x N and 2x -1 = 0 (iv) x: x N and x i

    (i) Finite, (ii) Finite, (iii) Finite, (iv) Infinite, (v) Infinite

  14. Let T_r be the r^(th) term of an A.P. If for some m, T_m = (1)/(25), T_25 = (1)/(20), and 20 Σ_r=1^(25) T_r = 13, then 5m Σ_r=m²m T_r is equal to

    126

  15. If the locus of z C, such that ((z-1)/(2z + i)) + ( z-12 z - i) = 2, is a circle of radius r and center (a,b), then (15ab)/(r²) is equal to:

    9

  16. Monica is wrapping Christmas gifts. She has 6 gifts to wrap for her family, 4 gifts to wrap for her friends and 2 gifts to wrap for her teachers. She has 144 in

    Monica can use 12 inches of ribbon for each gift bow.

  17. If a is a nonzero vector such that its projections on the vectors 2 i- j+2 k, i+2 j-2 k, and k are equal, then a unit vector along a is:

    ± (1)/(√155)(7 i + 9 j + 5 k)

  18. Find the coordinates of the foci and the vertices, the eccentricity, the length of the latus rectum of the hyperbolas: (i) (x²)/(9)-(y²)/(16)=1, (ii) y²-16 x²=1

    (i) Foci: (± 5, 0), Vertices: (± 3, 0), Eccentricity: (5)/(3), Length of latus rectum: (32)/(3). (ii) Foci: (0, ± √17), Vertices: (0, ± 4), Eccentricity: (√17)/

  19. Poppy is solving a 1000-piece jigsaw puzzle. She places a quarter of the pieces on the board, then her mom places a third of the remaining pieces. How many jigs

    500 jigsaw pieces are left to be placed.

  20. नीचे दी हुई सारणी भारत के विभिन्न राज्यों एवं संघीय क्षेत्रों (union territories) के ग्रामीण क्षेत्रों के प्राथमिक विद्यालयों में, महिला शिक्षकों के प्रतिशत बंट

    महिला शिक्षकों का माध्य प्रतिशत 39.71% है।

  21. Write the equation of the line through the points (1, -1) and (3, 5).

    The equation of the line is y = 3x - 4.

  22. A card from a pack of 52 cards is lost. From the remaining 51 cards, n cards are drawn and are found to be spades. If the probability of the lost card being a s

    n = 2

  23. VARIATION: In triangle ABC, points D, E, F are on sides BC, CA, AB respectively such that AD, BE, CF are concurrent. If BD:DC = 2:3, CE:EA = 3:4, find AF:FB usi

    The ratio AF:FB is 2:1.

  24. If ∫ e^x ( (x sin^(-1) x)/(√(1-x²)) + (x ln^(-1) x)/((1-x²)³/2) + (x)/(1-x²)) dx = g(x) + C, where C is the constant of integration, then g((1)/(2)) equals:

    (π e^(1/2))/(6√3)

  25. In Fig. 6.31, OA · OB = OC · OD. Show that A = C and B = D. [Figure: segments AB and CD intersect at point O; A and D on one side, C and B on the other, forming

    It has been shown that A = C and B = D using the SAS similarity criterion for triangles AOD and COB.

  26. A juice seller was serving his customers using glasses as shown in Fig. 12.13. The inner diameter of the cylindrical glass was 5 cm, but the bottom of the glass

    The apparent capacity of the glass is 196.25 cm³ and its actual capacity is 163.542 cm³.

  27. Prove that (sin (x+y))/(sin (x-y))=(tan x+tan y)/(tan x-tan y).

    The identity is proven: (sin (x+y))/(sin (x-y))=(tan x+tan y)/(tan x-tan y)

  28. Let I be the identity matrix of order 3 3 and let A= & 2 & 34 & 5 & 67 & -1 & 2 with |A|=-1. Let B be the inverse of the matrix adj(A) adj(A²). Then | B + I| is

    The problem cannot be solved to a specific numerical value without further information about or additional properties of the matrices involved. The expression s

  29. Let A be a matrix of order 3 3 with A =5. If 2 adj (3A adj(2A)) =2^(α)3^(β)5^(), where α,β, N, then α+β+ is equal to:

    27

  30. If the domain of the function f(x)=log_7 (1-log_4(x²-9x+18)) is (α,β) (,), then α+β+ + is equal to:

    18

  31. If the sum of the first 10 terms of the series (4 · 1)/(1 + 4 · 1^4) + (4 · 2)/(1 + 4 · 2^4) + (4 · 3)/(1 + 4 · 3^4) + is (m)/(n), where gcd(m, n) = 1, then m +

    441

  32. Find the equation of the ellipse, with major axis along the x -axis and passing through the points (4,3) and (-1,4).

    7x² + 15y² = 247

  33. Suppose we throw a die once. (i) What is the probability of getting a number greater than 4? (ii) What is the probability of getting a number less than or equal

    (i) The probability of getting a number greater than 4 is (1)/(3). (ii) The probability of getting a number less than or equal to 4 is (2)/(3).

  34. Let z be a complex number such that |z|=1. If 2 + k² zk + z = k z, k R, then the maximum distance of k + i k² from the circle |z - (1 + 2i)| = 1 is:

    The maximum distance is √5 + 1.

  35. If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 term

    -1080

  36. If in the expansion of (1 + x)^p (1 - x)^q, the coefficients of x and x² are 1 and -2, respectively, then p² + q² is equal to:

    13

  37. If Σ_r=0^(10) ((10^(r+1) - 1)/(10^r)) · ^(11)C_r+1 = (α^(11) - 11^(11))/(10^(10)), then α is equal to:

    α = 20

  38. The probability of forming a 12-person committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor is:

    The probability of forming a 12-person committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor is (1)/(11)

  39. Vincent can buy flowers in packages of 3 for 2.50 or in packages of 2 for 1. How much money does he save by buying 18 flowers at the better price?

    Vincent saves 6.00 by buying 18 flowers at the better price.

  40. Let x = x(y) be the solution of the differential equation y = (x - y (dx)/(dy)) sin ((x)/(y)), y > 0 and x(1) = (π)/(2). Then cos (x(2)) is equal to:

    2(ln 2)² - 1

  41. The system of equations x + y + z = 6, x + 2y + 5z = 9, x + 5y + z = has no solution if:

    The system has no solution if = 17 and ≠ 18.

  42. In a flight of 600 km, an aircraft was slowed due to bad weather. Its average speed for the trip was reduced to 200 km/hr and time of flight increased by 30 min

    The original duration of flight was 1 hour.

  43. Evaluate ∫₋∞^∞ sin(x)/x dx using contour integration in the complex plane.

    π

  44. Let T be the set of all triangles in a plane with R a relation in T given by R= (T_1, ~T_2): T_1 is congruent to T_2. Show that R is an equivalence relation.

    The relation R is an equivalence relation because it is reflexive, symmetric, and transitive.

  45. Find the 10th term of the AP: 2, 7, 12,

    The 10th term of the AP is 47.

  46. Find the derivative of the constant function f(x)=a for a fixed real number a.

    The derivative of the constant function f(x)=a is f'(x)=0.

  47. Let the lines 3x - 4y - α = 0, 8x - 11y - 33 = 0, and 2x - 3y + = 0 be concurrent. If the image of the point (1, 2) in the line 2x - 3y + = 0 is ((87)/(13), (-6

    (1073)/(8)

  48. Given below are two statements: **Statement I:** lim_x → 0 ( (tan^(-1)x + log_e ( (√(1 + x) - √(1 - x))/(x)) - 2x)/(x^5)) = (2)/(5) **Statement II:** lim_x → 1

    Both Statement I and Statement II are false.

  49. Let a = 3 i - j + 2 k, b = a × ( i - 2 k), and c = b × k. Then the projection of c - 2 j on a is:

    -(24)/(√14)

  50. Prove that the function f(x) = x sin(1/x) for x ≠ 0 and f(0) = 0 is continuous everywhere but not differentiable at x = 0.

    The function f(x) = x sin(1/x) for x ≠ 0 and f(0) = 0 is continuous everywhere but not differentiable at x = 0.

  51. Let the ellipse E_1: (x²)/(a²) + (y²)/(b²) = 1, a > b and E_2: (x²)/(A²) + (y²)/(B²) = 1, A < B have the same eccentricity (1)/(√3). Let the product of their le

    The area of the quadrilateral ABCD is (24√6)/(5).

  52. Evaluate ∬∫∫_E (x² + y²) dV where E is the region bounded by z = 1 - x² - y² and the xy-plane.

    The value of the integral is (π)/(6).

  53. EXTENSION: Find all values of k for which the equation (k-2)variable_95² - 2(k-1)variable_95 + k = 0 has equal roots. Additionally, generalize your solution to

    There are no values of k for which the given quadratic equation has equal roots in the standard sense. However, if k=2, the equation becomes linear with a singl

  54. Solve the pair of linear equations: x + y = 14 and x - y = 4.

    The solution to the pair of linear equations is x = 9 and y = 5.

  55. If A and B are two events such that P(A B)=0.1, and P(A B) and P(B A) are the roots of the equation 12x²-7x+1=0, then the value of P( A B)P( A B) is:

    2.25

  56. Solve the quadratic equation: 2x² - 7x + 3 = 0

    The solutions are x = 3 and x = (1)/(2).

  57. Let | z - z2z + z | = (1)/(3), z C, be the equation of a circle with center at C. If the area of the triangle, whose vertices are at the points (0, 0), C and (α

    The problem statement contains an inconsistency as the given equation does not represent a circle. Therefore, α² cannot be determined.

  58. Find the mean deviation about the median for the following data: x_i: 3, 6, 9, 12, 13, 15, 21, 22 f_i: 3, 4, 5, 2, 4, 5, 4, 3.

    The mean deviation about the median is 4.967 (approximately).

  59. If α x+β y=109 is the equation of the chord of the ellipse (x²)/(9)+(y²)/(4)=1, whose midpoint is ((5)/(2), (1)/(2)), then α+β is equal to:

    58

  60. Let R be the relation defined in the set A=1,2,3,4,5,6,7 by R=(a, b): both a and b are either odd or even. Show that R is an equivalence relation. Further, show

    The relation R is an equivalence relation because it is reflexive, symmetric, and transitive. All elements within the subset 1,3,5,7 are related to each other (

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