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Solved maths problems — page 7

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

  1. Radius of a circle is 15cm. How far should the center of another circle be from the midpoint of this circle to ensure that there is no overlap in their circumfe

    The problem is underdetermined. The distance between the center of the first circle and the center of the second circle must be greater than or equal to 15 + R_

  2. A decorative block is made of two solids — a cube and a hemisphere. The base of the block is a cube with edge 5 cm, and the hemisphere fixed on the top has a di

    The total surface area of the block is 163.86 cm².

  3. TRIGONOMETRY (10th Class) Exercise 8.1 Ques 1: In ABC, right angle at B, AB = 24 cm, BC = 7 cm. Determined: (i) sin A, cos A (ii) sin C, cos C Sol: Let us draw

    (i) sin A = (7)/(25), cos A = (24)/(25) (ii) sin C = (24)/(25), cos C = (7)/(25)

  4. Now regarding the shaded region. It definitely looks like a semicircle with PQ as the base. If PQ were a diameter, then O would be the midpoint of PQ. But O is

    The area of the shaded region is 12.5π cm².

  5. Hello! I am your AI Math Tutor. Let's solve this problem step-by-step together. Let's start with Step 1! **Step 1: Visualize the cross-section** To solve this 3

    The cross-section of a cone with an inscribed sphere is a triangle with an inscribed circle.

  6. 8. Solve the following system of linear equations: 3x + y = 10 x - y = 2 (system of linear equations, elimination method, substitution method)

    The solution to the system of equations is x = 3 and y = 1.

  7. Ans: 4/10 8. Solve the following system of linear equations: 3x + y = 10 x - y = 2 Ans: x=3 y=1 (system of equations, linear equations, substitution method, eli

    x = 3, y = 1

  8. Ans 4/10 8. Solve the following system of linear equations: 3x + y = 10 x - y = 2 Ans x = 3, y = 1 (system of linear equations, elimination method, substitution

    x = 3, y = 1

  9. A circle of radius 7 cm is inscribed in a square. Find the area of the shaded region. Use π = 22/7. length of square = 2 × 8 (circle, square, radius, area, pi)

    The area of the shaded region is 42 cm².

  10. Length of square = 2r Area of square = (2r)² = (2 × 7)² = 14² = 196 cm² Area of Circle = π r² = (22)/(7) × 7² = (22 × 7 × 7)/(7) = 22 × 7 = 154 cm² 196 - 154 =

    The area of the shaded region is 42 cm².

  11. A circle of radius 7 cm is inscribed in a square. Find the area of the shaded region. Use π = 22/7. (circle, square, radius, area, inscribed)

    The area of the shaded region is 42 cm².

  12. A Circle of radius 7 cm is inscribed in a Square. Find area of the shaded region. Use π = 22/7. 196 - 154 = 42 cm² (circle, square, radius, area, inscribed)

    The area of the shaded region is 42 cm².

  13. Side of sq = 2r = 14 Area of sq = 14² = 196 cm² Area of circ = π r² = 22 7 × 7 × 7 = 154 cm² Area of shaded region = Area of sq - Area of circle = 196 - 154 = 4

    The area of the shaded region is 42 cm².

  14. A circle of radius 7cm is inscribed in a square. Find the area of the shaded region. Use π = (22)/(7). (circle, radius, square, area, pi)

    The area of the shaded region is 42 cm².

  15. Find the coordinates of the point which divides the line segment joining the points (4, -3) and (8, 5) in the ratio 3:1 internally.

    The coordinates of the point are (7, 3).

  16. Solve the quadratic equation 2x² - 7x + 3 = 0 using the quadratic formula.

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

  17. The present age of Sahil's mother is three times the present age of Sahil. After 5 years their ages will add to 66 years. Find their present ages.

    Sahil's current age is 14 years and his mother's current age is 42 years.

  18. Find the area of a triangular park ABC with sides 50m, 80m, and 120m using Heron's formula.

    The area of the triangular park is approximately 1452.36 m².

  19. A circle of radius 7 cm is inscribed in a square. Find the area of the shaded region. Use = (22)/(7). (circle, square, radius, area, pi)

    The area of the shaded region is 42 cm².

  20. If cos(A+B) = 0 and sin(A-B) = (1)/(2), then find the value of A and B, where A and B are acute angles. (trigonometry, acute angles, sum and difference identiti

    A = 60^° and B = 30^°

  21. If 3‾√ cot2θ – 4 cot θ + 3‾√ = 0, then find the value of cot2 θ + tan2θ. (2013) Solution:

    The value of cot² θ + tan²θ is (10)/(3).

  22. Find the area of a triangle with base 6 cm and height 8 cm.

    The area of the triangle is 24 cm².

  23. Partha, a software engineer, lives in Jerusalem for his work. He lives in the most convenient area of the city from where bank, hospital, post office and superm

    (i) The distance between the bank and the hospital is 13 units. (ii) The coordinates of the post office (E) are (7, 0). (iii) (a) The distance Partha needs to c

  24. (a) The largest possible hemisphere is drilled out from a wooden cubical block of side 21 cm such that the base of the hemisphere is on one of the faces of the

    (i) The volume of wood left in the block is 6835.5 cm³. (ii) The total surface area of the remaining solid is 2992.5 cm².

  25. (a) If a hexagon PQRSTU circumscribes a circle, prove that, PQ + RS + TU = QR + ST + UP. OR (b) In the given figure, two concentric circles have radii 3 cm and

    The length of TP is 12 cm.

  26. In the given figure, PA and PB are two tangents drawn to the circle with centre O and radius 5 cm. If APB = 60^°, then the length of PA is:

    The length of PA is 5√3 cm.

  27. Two dice are thrown at the same time and the product of the numbers appearing on them is noted. The probability that the product of the numbers lies between 8 a

    The probability that the product of the numbers lies between 8 and 13 is (7)/(36).

  28. A vertical pole 10 m long casts a shadow of length 5 m on the ground. At the same time, a tower casts a shadow of length 12.5 m on the ground. The height of the

    The height of the tower is 25 m.

  29. A circle is touching the side BC of ABC at P and touching the sides AB and AC produced at Q and R respectively. Prove that AQ = (1)/(2) (AB + BC + AC).

    AQ = (1)/(2) (AB + BC + AC)

  30. The marks obtained by 50 students in a test are tabulated below: | Marks: | 0 - 15 | 15 - 30 | 30 - 45 | 45 - 60 | 60 - 75 | |---|---|---|---|---|---| | Number

    32

  31. The diagonals of a rhombus ABCD intersect at O. Taking 'O' as the centre, an arc of radius 6 cm is drawn intersecting OA and OD at E and F respectively. The are

    9π cm²

  32. Due to short circuit, a fire has broken out in New Home Complex. Two buildings, namely X and Y have mainly been affected. The fire engine has arrived and it has

    (i) Length of the ladder = 15 m (ii) Distance of building Y from point 'O' (OA) = 7.5 m (iii) (a) Horizontal distance between the two buildings = 7.5 + 7.5√2 ≈

  33. In the given figure, x, y and z are the sides of a right triangle, where z is the hypotenuse. Prove that the radius r of the circle which touches the sides of t

    The radius r of the inscribed circle is r = (x + y - z)/(2).

  34. A line segment joining the points P(-5, 11) and Q is divided internally by the point M(2, -3) such that PM: MQ = 7: 2. The coordinates of Q are:

    The coordinates of Q are (4, -7).

  35. The point on x-axis which is equidistant from the points (5, -3) and (4, 2) is

    (7, 0)

  36. A school has decided to plant some endangered trees on 51^(st) World Environment Day in the nearest park. They have decided to plant those trees in few concentr

    (i) 230 trees (ii) 60 trees (iii) (a) 16 rows OR (b) 1550 trees

  37. Using flat rate method, the EMI to repay a loan of ₹ 20,000 in 2 (1)/(2) years at an interest rate of 8% per annum is:

    ₹ 800

  38. The value of 1 & 2 & 3 4 & 5 & 6 7 & 8 & 9 is:

    0

  39. General solution of differential equation y log y dx - x dy = 0 is:

    x = A log y

  40. If x = t² and y = t³, then (d² y)/(dx²) is equal to:

    (3)/(4t)

  41. If ∫_0^4 (dx)/(2x + 1) = log k, then the value of k is:

    3

  42. For the purpose of t-test of significance, if a random sample of size 34 is drawn from a normal population, then the degree of freedom () is:

    33

  43. The total area under a standard normal curve is:

    1

  44. The rate of change of the area of a circle with respect to its radius r (in cm²/s), when r = 6 cm is:

    12π cm²/cm

  45. If AB = A and BA = B, then (B² + B) is equal to:

    2B

  46. The range of variable 't' of the t-distribution is:

    The range of variable ' t ' of the t -distribution is (-∞, ∞).

  47. Let X be a discrete random variable, then the variance of X is:

    The variance of a discrete random variable X is given by the formula Var(X) = E[X²] - (E[X])².

  48. The present value of a perpetuity of ₹ R payable at the end of each period, when the money is worth i per period is:

    The present value of a perpetuity of ₹ R payable at the end of each period, when the money is worth i per period is (R)/(i).

  49. If 'm' is the mean of a Poisson distribution, then its variance is given by:

    m

  50. The number of all possible matrices of order 3 × 2 with each entry 1 or 2 is:

    64

  51. (a) A tent is in the shape of a right circular cylinder up to a height of 3 m and then a right circular cone, with a maximum height of 13.5 m above the ground.

    The volume of the wooden toy is 266.112 cm³ and the total surface area of the toy is 207.996 cm².

  52. (a) Find the area of the minor and the major sectors of a circle with radius 6 cm, if the angle subtended by the minor arc at the centre is 60^°. (Use π = 3.14)

    The area of the corresponding minor segment of the circle is 9.08 cm².

  53. As observed from the top of a lighthouse, 100 m above sea level, the angle of depression of a ship, sailing directly towards it, changes from 30^° to 45^°. Dete

    The distance travelled by the ship is 73.2 m.

  54. State and prove Basic Proportionality theorem.

    If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same rati

  55. Prove that the lengths of tangents drawn from an external point to a circle are equal.

    The lengths of tangents drawn from an external point to a circle are equal, i.e., PQ = PR.

  56. A survey regarding the heights (in cm) of 50 girls of class X of a school was conducted and the following data was obtained: | Height (in cm) | Number of girls

    The mean height is 149.8 cm and the mode height is 154 cm.

  57. In the given figure, DE || BC and all measurements are given in centimetres. The length of AE is:

    The length of AE is 3 cm.

  58. While playing in a garden, Samaira saw a honeycomb and asked her mother about it. Her mother replied that it is a honeycomb. Also, she told her that the shape o

    (i) There are 3 zeroes. (ii) The zeroes are -2, 1, and 3. (iii) (a) a = -6, b = 6. (iii) (b) p = ± 18.

  59. February 14 is celebrated as International Book Giving Day and many countries in the world celebrate this day. Some people in India also started celebrating thi

    (i) 48 books are arranged in each stack. (ii) 7 stacks are used to arrange all the Mathematics books. (iii) (a) The total number of stacks used is 14. (iii) (b)

  60. (a) (i) Prove that: (sin θ - 2sin³ θ)/(2cos³ θ - cos θ) = tan θ (ii) Prove that: (sin A + cos A)/(sin A - cos A) + (sin A - cos A)/(sin A + cos A) = (2)/(2sin²

    The identities are proven as follows: (i) (sin θ - 2sin³ θ)/(2cos³ θ - cos θ) = tan θ (ii) (sin A + cos A)/(sin A - cos A) + (sin A - cos A)/(sin A + cos A) = (

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