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Solved maths problems — page 19

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

  1. A plague infects ten people. Every day, each infected person infects six others. How many people are infected after three days?

    After three days, 2710 people are infected.

  2. From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include at least 4 batsmen and at least 4 bowlers. One batsman and

    95

  3. Let f(x) = 3x, & x < 0 min 1 + x + [x], x + 2[x], & 0 ≤ x ≤ 2 5, & x > 2 where [.] denotes greatest integer function. If α and β are the number of points, where

    5

  4. Find the singular value decomposition (SVD) of A = [[3, 2, 2], [2, 3, -2]].

    A = U V^T = 1/√2 & 1/√2 1/√2 & -1/√2 5 & 0 & 0 0 & 3 & 0 2/3 & 2/3 & 1/3 1/3 & -2/3 & 2/3 -2/3 & 2/3 & 1/3 ^T

  5. The number of singular matrices of order 2, whose elements are from the set 2, 3, 6, 9, is _____

    36

  6. 38. 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 give

    (i) y = ± (3)/(4) √(16 - x²) (ii) (3)/(4) [ (x)/(2) √(16 - x²) + 8 sin^(-1) ((x)/(4)) ] + C (iii) (a) 12π units² (iii) (b) Coordinates P (7,0), Q (0,6), Area 21

  7. Let P_n=α^n+β^n, n N. If P_10=123, P_9=76, P_8=47 and P_1=1, then the quadratic equation having roots (1)/(α) and (1)/(β) is:

    x² + x - 1 = 0

  8. Find the equation of the line through ( -2, 3) with slope -4.

    The equation of the line is y = -4x - 5.

  9. Express (5-3 i)³ in the form a+i b.

    -10 - 198i

  10. Let circle C be the image of x² + y² - 2x + 4y - 4 = 0 in the line 2x - 3y + 5 = 0 and A be the point on C such that OA is parallel to x -axis and A lies on the

    4

  11. A radioactive substance decays according to the equation dN/dt = -kN. If the initial amount is N₀ and half-life is T, express N(t) in terms of N₀, T, and t.

    N(t) = N_0 2^(-t/T)

  12. MODIFIED: Find the arc length of the curve y = ln(sec x) from x = 0 to x = π/4.

    The arc length of the curve is ln(√2 + 1).

  13. Let f: [0, ∞) → R be a differentiable function such that f(x) = 1 - 2x + ∫_0^x e^(t-x) f(t) dt for all x [0, ∞). Then the area of the region bounded by y = f(x)

    The area of the region bounded by y = f(x) and the coordinate axes is (5√5-1)/(24).

  14. The number of points of discontinuity of the function f(x) = (x²)/(2) - √x, x [0, 4], where · denotes the greatest integer function, is _____

    8

  15. It's a right angle triangle and you know one of the sides is 5 cm the other is 3 cm what is the third side

    The third side can be either √34 cm or 4 cm.

  16. VARIATION: Evaluate ∫₀^(π/2) (sin x)^4 (cos x)³ dx using reduction formulas. Find the numerical value if exact solution is not possible.

    (2)/(35)

  17. Let f(x) be a positive function and I_1 = ∫_-(1)/(2)^(1) 2x f(2x(1 - 2x)) dx and I_2 = ∫_-1^((1)/(2)) f(x(1 - x)) dx. Then the value of (I_2)/(I_1) is equal to:

    The value of (I_2)/(I_1) is equal to 2.

  18. Angle A of triangle is 50. Angle B is 60. What is angle C?

    Angle C is 70^°.

  19. Let C_1 be the circle in the third quadrant of radius 3, that touches both coordinate axes. Let C_2 be the circle with centre (1, 3) that touches C_1 externally

    22

  20. Let y=y(x) be the solution of the differential equation (dy)/(dx) + 3(tan²x) y + 3y = sec²x, with y(0)=(1)/(3)+e³. Then y ((π)/(4)) is equal to:

    (4)/(3)

  21. Solve the system of equations: x + y + z = 6, x² + y² + z² = 14, x³ + y³ + z³ = 36.

    The solutions (x, y, z) are permutations of (1, 2, 3).

  22. Prove that among any 51 integers chosen from 1, 2,..., 100, there exist two that are coprime.

    Among any 51 integers chosen from 1, 2,..., 100, there exist two that are coprime.

  23. Let H_1: (x²)/(a²) - (y²)/(b²) = 1 and H_2: -(x²)/(A²) + (y²)/(B²) = 1 be two hyperbolas having length of latus rectums 15 √2 and 12 √5 respectively. Let their

    55

  24. Check whether 301 is a term of the list of numbers 5, 11, 17, 23,

    No, 301 is not a term of the given list of numbers.

  25. A survey regarding the heights (in cm) of 51 girls of Class X of a school was conducted and the following data was obtained: |l|c| Height (in cm) & Number of gi

    The median height is approximately 149.03 cm.

  26. A Tyrannosaurus rex ate half of a small triceratops it had hunted. When it left, a pack of velociraptors scavenged half of what was left. A group of lazy Allosa

    The triceratops had 1080 kilograms of meat before the T-Rex ate.

  27. Examine whether the function f given by f(x)=x² is continuous at x=0.

    The function f(x)=x² is continuous at x=0.

  28. Evaluate the determinant =| rrr1 & 2 & 4 -1 & 3 & 0 4 & 1 & 0 |.

    The determinant = -52.

  29. Find the general solution of: y'' - 4y' + 4y = e^(2x)/x² using variation of parameters.

    y = c_1e²x + c_2xe²x - e²xln|x| - e²x

  30. If there are (2 n+1) terms in an A.P., then prove that the ratio of the sum of odd terms and the sum of even terms is (n+1): n

    The ratio of the sum of odd terms and the sum of even terms is (n+1): n.

  31. The product of all solutions of the equation e^(5(log_e x)² + 3) = x^8, x > 0, is:

    e^(8/5)

  32. (b) Given that P = 2 & -1 3 & 4, Q = 5 & 2 7 & 4 and R = 2 & 5 3 & 8, find a matrix S such that PQ - RS is a null matrix. (matrix, null matrix, matrix multiplic

    S = -191 & -110 77 & 44

  33. Let the arc AC of a circle subtend a right angle at the centre O. If the point B on the arc AC, divides the arc AC such that (length of arc AB)/(length of arc B

    2 - √3

  34. If the coordinates of the mid-points of the line joining the points (3a, 4) and ( - 2, 2b) are (5, a), then

    The values are a = 4 and b = 2.

  35. Let the product of the focal distances of the point (√3, (1)/(2)) on the ellipse (x²)/(a²) + (y²)/(b²) = 1, (a > b), be (7)/(4). Then the absolute difference of

    (3√3 - 2√6)/(6)

  36. The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must

    1205

  37. If the equation of the line passing through the point (0, -(1)/(2), 0) and perpendicular to the lines r = ( i + a j + b k) and r = ( i - j - 6 k) + (-b i + a j

    14

  38. If lim_x 1 ((x-1)(6 + cos(x-1)) + sin(1-x))/((x-1)³) = -1 where, R then + is equal to:

    18

  39. Prove that the set of rational numbers ℚ is dense in ℝ but has measure zero.

    The set of rational numbers Q is dense in R and has measure zero.

  40. The perpendicular distance, of the line (x-1)/(2)=(y+2)/(-1)=(z+3)/(2) from the point P(2,-10,1), is:

    3√5

  41. Define the function f: R → R by y=f(x)=x², x R. Complete the Table given below by using this definition. What is the domain and range of this function? Draw the

    The completed table is: |l|l|l|l|l|l|l|l|l|l| x & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 y=f(x)=x² & 16 & 9 & 4 & 1 & 0 & 1 & 4 & 9 & 16. The domain of the funct

  42. A kite is flying at a height of 60m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the strin

    The length of the string is 40√3 m.

  43. Let a_n be a sequence such that a_0 = 0, a_1 = (1)/(2) and 2 a_n+2 = 5 a_n+1 - 3 a_n, n = 0, 1, 2, 3,. Then Σ_k=1^(100) a_k is equal to

    3 ((3)/(2))^(100) - 103

  44. A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8, and B wins if he throws a sum of 8 before A throws a sum of

    The probability that A wins is (9)/(19).

  45. Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.

    The HCF of 96 and 404 is 4, and their LCM is 9696.

  46. Let a=2 - +3 k, b=3 -5 + k and c be a vector such that a × c= c × b and ( a+ c) · ( b+ c)=168. Then the maximum value of | c|² is:

    308

  47. Lorraine and Colleen are trading stickers for buttons. Each large sticker is worth a large button or three small buttons. A small sticker is worth one small but

    Lorraine has a total of 69 buttons by the end.

  48. In a right-angled triangle ABC, right-angled at B, if AB = 3 cm and BC = 4 cm, find sin A and cos A.

    sin A = (4)/(5) and cos A = (3)/(5)

  49. In trapezoid ABCD, AD BC, B=60^°, C=45^°, AD=4, BC=10. Find AB and CD.

    AB = 6√3-6 and CD = 9√2-3√6

  50. Prove that for any square matrix A, the eigenvalues of AAᵀ and AᵀA are the same (excluding zero).

    The non-zero eigenvalues of AA^T and A^TA are the same.

  51. स्मोकिंग से फेफड़ों की परेशानियों का खतरा बढ़ जाता है। Your Lungs / Smoke LUNG CANCER STOP SMOKING Secondhand Smoke Cough Coughing up blood Hoarseness Wheezing

    (i) चुने गए व्यक्ति के महिला होने की प्रायिकता 0.4 है। (ii) यदि एक पुरुष चुना गया है तो उसे फेफड़ों की समस्या न होने की प्रायिकता 0.83 है। (iii) (a) यदि यादृच्छ

  52. Let z_1, z_2, and z_3 be three complex numbers on the circle |z| = 1 with (z_1) = (π)/(4), (z_2) = 0, and (z_3) = (π)/(4). If |z_1 z_2 + z_2 z_3 + z_3 z_1|² = α

    13

  53. Uriah's book bag is getting too heavy for him. He needs to remove 15 pounds from it. His comic books weigh 1/4 pound each and his toys weigh 1/2 pound each. If

    Uriah needs to remove 15 toys.

  54. Let f: R → R be a continuous function satisfying f(0) = 1 and f(2x) - f(x) = x for all x R. If lim_n→∞ [f(x) - f(x/2^n)] = G(x) then Σ_r=1^(10) G(r²) is equal t

    385

  55. Alain's mom bought 5 packs of red pens and also bought twice the amount of black pens than the red. If each pack has 5 pens, how many pens does Alain have?

    Alain has 75 pens.

  56. Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that

    The eccentricity of the ellipse is (√2)/(2).

  57. The length of the latus-rectum of the ellipse whose foci are (2,5) and (2,-3) and eccentricity is (4)/(5) is:

    (18)/(5)

  58. If A, B, C are three events associated with a random experiment, prove that P(A B C) & =P(A)+P(B)+P(C)-P(A B)-P(A C) & -P(B C)+P(A B C)

    P(A B C) = P(A)+P(B)+P(C)-P(A B)-P(A C)-P(B C)+P(A B C)

  59. Find the Laurent series expansion of f(z) = 1/(z² - 3z + 2) valid in the annulus 1 < |z| < 2.

    f(z) = -Σ_n=0^(∞) (1)/(z^(n+1)) - Σ_n=0^(∞) (z^n)/(2^(n+1))

  60. If the product of the zeroes of the quadratic polynomial 5x² + 7x + k is - (2)/(5), then the value of k is:

    The value of k is -2.

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