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Solved maths problems — page 8

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

  1. The real x for which 2(2x + 3) - 10 < 6 (x - 2) is:

    x > 4

  2. In the following table, the median age of 200 spectators of a football match is 32. Find the missing frequencies p and q. | Age (in years) | Number of Spectator

    The missing frequencies are p = 18 and q = 20.

  3. (a) If Nidhi were 7 years younger than what she actually is, then the square of her age (in years) would be 1 more than 5 times her actual age. What is her pres

    The shopkeeper bought 30 books initially.

  4. (a) The sum of the digits of a 2-digit number is 12. Seven times the number is equal to four times the number obtained by reversing the order of the digits. Fin

    The number is 48.

  5. A school has invited 42 Mathematics teachers, 56 Physics teachers and 70 Chemistry teachers to attend a Science workshop. Find the minimum number of tables requ

    The minimum number of tables required is 12.

  6. If α, β are the zeroes of the polynomial 3x² - 13x - 10, then find the value of (3α + 1)(3β + 1).

    -16

  7. The government rescued 100 people after a train accident. Their ages were recorded in the following table. Find their mean age. | Age (in years) | Number of peo

    The mean age of the rescued people is 44 years.

  8. Prove that: (sin θ - cos θ + 1)/(sin θ + cos θ - 1) = (1)/(sec θ - tan θ).

    (sin θ - cos θ + 1)/(sin θ + cos θ - 1) = (1)/(sec θ - tan θ)

  9. (a) If cos (A + B) = (1)/(2) and tan (A - B) = (1)/(√3), where 0 ≤ A + B ≤ 90^°, then find the value of sec (2A - 3B). OR (b) Find the value of x such that, 3 t

    √2

  10. Check whether there is any natural number 'n' for which (14)^n ends with the digit '0' or '5'.

    No, there is no natural number 'n' for which (14)^n ends with the digit '0' or '5'.

  11. (a) A chord is subtending an angle of 90^° at the centre of a circle of radius 14 cm. Find the area of the corresponding minor segment of the circle. OR (b) Fin

    The area of the corresponding minor segment is 56 cm².

  12. Assertion (A) TA and TB are two tangents drawn from an external point T to a circle with centre 'O'. If TBA = 75^° then ABO = 25^°. Reason (R) The tangent drawn

    Assertion (A) is false, but Reason (R) is true.

  13. Assertion (A) If the graph of a polynomial intersects the x-axis at exactly two points, then the number of zeroes of that polynomial is 2. Reason (R) The number

    Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

  14. A cap is cylindrical in shape, surmounted by a conical top. If the volume of the cylindrical part is equal to that of the conical part, then the ratio of the he

    The ratio of the height of the cylindrical part to the height of the conical part is 1:3.

  15. The value of (1 + tan² θ)/(1 + cot² θ) is:

    tan² θ

  16. If the sector of a circle with diameter 14 cm makes an angle 90^° at the centre, then the perimeter of the sector is:

    The perimeter of the sector is 25 cm.

  17. All queens, jacks and aces are removed from a pack of 52 playing cards. The remaining cards are well-shuffled and one card is picked up at random from it. The p

    The probability of that card to be a king is (1)/(10).

  18. If 5 tan θ = 2, then the value of (10 sin θ - 2 cos θ)/(5 sin θ + 3 cos θ) is:

    (2)/(5)

  19. The probability of getting a chocolate flavoured ice cream at random, in a lot of 600 ice creams is 0.055. The number of chocolate flavoured ice creams in the l

    33

  20. Two A.P.s have the same first term. The common difference of the first A.P. is -3 and of the second A.P. is -5. The difference of the 6^(th) term of the second

    10

  21. If k + 7, 2k - 2 and 2k + 6 are three consecutive terms of an A.P., then the value of k is

    The value of k is 17.

  22. Two positive integers m and n are expressed as m = p^5 q² and n = p³ q^4, where p and q are prime numbers. The LCM of m and n is:

    p^5 q^4

  23. In XYZ, XY = 6 cm. If M and N are two points on XY and XZ respectively such that MN | YZ and XN = (1)/(4) XZ, then the length of XM is:

    The length of XM is 1.5 cm.

  24. If x = 5 is a solution of the quadratic equation 2x² + (k - 1)x + 10 = 0, then the value of k is

    k = -11

  25. If (k, 3) is the point of intersection of the lines represented by x + py = 6 and x = 15 then (k, p) will be:

    (15, -3)

  26. If the length of the shadow on the ground of a pole is √3 times the height of the pole, then the angle of elevation of the Sun is:

    The angle of elevation of the Sun is 30^°.

  27. A man has ₹ 15,000 for purchasing rice and wheat. A bag of rice and a bag of wheat cost ₹ 1,800 and ₹ 1,200 respectively. He has a storage capacity of 10 bags.

    (i) Objective function: Z = 100x + 90y. (ii) Constraints: 1800x + 1200y ≤ 15000 (or 3x + 2y ≤ 25), x + y ≤ 10, x ≥ 0, y ≥ 0. (iii) (a) The man should buy 5 bags

  28. When observed over a long period of time, a time series data can predict trends that can forecast increase or decrease or stagnation of a variable under conside

    (i) The trend line is Y = 90 + 2X. (ii) The average change in sales is ₹2 thousand per year. (iii) (b) The expected sales for the year 2025 are ₹98 thousand.

  29. Three schools A, B and C organised a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand-made fans, mats and plates from r

    (i) Price Matrix: P = 25 & 50 & 10 (ii) Sales Matrix: S = 50 & 30 & 35 60 & 35 & 40 40 & 50 & 25 (iii) (a) Funds collected by School B: ₹ 3000 (iii) (b) Total f

  30. Mr. Arya wants to know the amount he should pay for a gold mine expected to yield an annual return of ₹ 4 lakh for the next 10 years, after which it will be wor

    Mr. Arya should pay approximately ₹ 16,47,800.7 for the gold mine.

  31. (a) Two cards are drawn at random and one by one with replacement from a well-shuffled pack of 52 playing cards. Find the probability distribution of the number

    (i) The probability of no defective screws is approximately 0.1353. (ii) The probability of one defective screw is approximately 0.2706.

  32. (a) Find the consumer's surplus for the demand function p = 25 - x - x², where the prevailing market price p_0 = 19. OR (b) Solve the following initial value di

    The consumer's surplus is (22)/(3).

  33. Solve the following inequation: (x - 2)/(x + 5) > 2.

    x (-12, -5)

  34. Solve the following Linear Programming Problem (LPP) graphically: Maximize z = 3x + 5y subject to the constraints: x + 2y ≤ 2000, x + y ≤ 1500, y ≤ 600, x ≥ 0,

    The maximum value of z is 5500 at x=1000 and y=500.

  35. A company XYZ Ltd. has issued a bond having a face value of ₹ 10,000 paying annual dividend at 8.5% p.a. The bond will be redeemed at par at the end of 10 years

    ₹ 10,335.06

  36. A soap manufacturing company was distributing a particular brand of a soap through a large number of retail shops. Before a heavy advertisement campaign, the me

    Yes, the advertisement campaign can be considered effective.

  37. (a) Find the intervals in R for which the function f(x) = x^4 - 2x² is increasing or decreasing. OR (b) Find: ∫ (2x + 1)/(√(18 - 4x - x²)) dx.

    The function f(x) = x^4 - 2x² is increasing on the intervals (-1, 0) (1, ∞) and decreasing on the intervals (-∞, -1) (0, 1).

  38. Using Cramer's rule, show that the following system of linear equations is consistent and hence solve it: x + y + z = 1, 2x + 3y + 2z = 2, x + y + 2z = 4.

    x = -2, y = 0, z = 3

  39. (a) (i) Apply addition modulo to positive integers 17 and 13 for modulo 30. (ii) Find subtraction modulo 8 for numbers 11 and 3. OR (b) Three pipes A, B and C c

    Pipe B alone can fill the tank in 24 hours.

  40. A man takes a personal loan of ₹ 2,00,000 at an interest rate of 15% p.a. compounded monthly, to be repaid by equal monthly instalments in 4 years. Calculate th

    The EMI is approximately ₹ 5555.55.

  41. (a) For a Poisson distribution, if mean (m) = 1, then find P(r = 1). OR (b) Find the mean and standard deviation of the Binomial distribution B(4, (1)/(3)).

    (a) P(r=1) = e^(-1) (approximately 0.3679) (b) Mean = (4)/(3), Standard Deviation = (2√2)/(3)

  42. (a) Solve the following differential equation: (dy)/(dx) = e^(x - y) + x² e^(-y). OR (b) Solve the following differential equation: (x² - y²) dx + 2xy dy = 0.

    e^y = e^x + (x³)/(3) + C

  43. A man in a boat goes 12 km downstream and comes back to the starting point by rowing non-stop in a total time of 3 hours. If the speed of the stream is 3 km/h,

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

  44. Assertion (A): In sinking fund, a fixed amount at regular intervals is deposited. Reason (R): In Savings Bank Account, any amount, any time can be deposited.

    Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).

  45. Assertion (A): ∫ (1)/(√(9 - x²)) dx = sin^(-1) (x)/(3). Reason (R): ∫ (1)/(√(a² - x²)) dx = sin^(-1) (x)/(a) + C.

    Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

  46. The maximum value of the function z = 7x + 5y, subject to the constraints x ≤ 3, y ≤ 2, x ≥ 0, y ≥ 0 is:

    31

  47. A machine costs ₹ 45,000 with an estimated useful life of 5 years and a scrap value of ₹ 10,000. The annual depreciation of the machine is:

    The annual depreciation of the machine is ₹ 7,000.

  48. If an investment of ₹ 10,000 becomes ₹ 60,000 in 4 years, then the Compound Annual Growth Rate (CAGR) is:

    The Compound Annual Growth Rate (CAGR) is approximately 56.5%.

  49. Draw the graph of the following equations: x + y = 5, x - y = 5, and (i) find the solution of the equations from the graph. (ii) shade the triangular region for

    (i) The solution to the equations is (5, 0). (ii) The triangular region formed by the lines and the y -axis has vertices (0, 5), (0, -5), and (5, 0).

  50. (a) Find the sum of all integers between 50 and 500, which are divisible by 7. OR (b) How many numbers lie between 10 and 300, which when divided by 4 leave a r

    The sum of all integers between 50 and 500, which are divisible by 7, is 17696.

  51. Two water taps together can fill a tank in 3(3)/(13) hours. The tap of larger diameter takes 5 hours less than the smaller one to fill the tank separately. Find

    The smaller tap takes approximately 9.81 hours and the larger tap takes approximately 4.81 hours to fill the tank separately.

  52. The age of the father is twice the sum of the ages of his two children. After 20 years, his age will be equal to the sum of the ages of his children. Find the p

    The present age of the father is 40 years.

  53. Find the ratio in which the point (-1, k) divides the line segment joining the points (-3, 10) and (6, -8). Hence, find the value of k.

    The point (-1, k) divides the line segment in the ratio 2:7, and the value of k is 6.

  54. A circle is touching the side BC of a ABC at the point P and touching AB and AC produced at points Q and R respectively. Prove that AQ = (1)/(2) (Perimeter of A

    AQ = (1)/(2) (Perimeter of ABC)

  55. (a) Show that the points ( -3, 3), (3, -3) and (3√3, 3√3) are the vertices of an equilateral triangle. OR (b) Prove that A(4, 3), B(6, 4), C(5, 6), D(3, 5) are

    The points A(4, 3), B(6, 4), C(5, 6), D(3, 5) are the vertices of a square ABCD.

  56. In the figure, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD BC and EF AC, prove that ABD ECF.

    ABD ECF

  57. (a) If √2 is given as an irrational number, then prove that (5 - 2√2) is an irrational number. OR (b) Check whether 6^n can end with the digit 0 for any natural

    (a) 5 - 2√2 is an irrational number. (b) 6^n cannot end with the digit 0 for any natural number n.

  58. Assertion (A): A fair die is thrown once. The probability of getting a prime number is (1)/(2). Reason (R): A natural number is a prime number if it has only tw

    Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

  59. Assertion (A): Two players, Sania and Ashnam play a tennis match. The probability of Sania winning the match is 0.79 and that of Ashnam winning the match is 0.2

    Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

  60. A number is chosen from the numbers 1, 2, 3 and denoted as x, and a number is chosen from the numbers 1, 4, 9 and denoted as y. Then P(xy < 9) is:

    P(xy < 9) = (5)/(9)

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