Solved maths problems — page 20

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

1. If the system of equations x + 2y - 3z = 2, 2x + y + 5z = 5, 14x + 3y + z = 33 has infinitely many solutions, then + is equal to:

   12

2. Prove that the set of discontinuities of any monotonic function is at most countable.

   The set of discontinuities of any monotonic function is at most countable.

3. The minute hand of a watch is 1.5 ~cm long. How far does its tip move in 40 minutes? (Use π=3.14).

   The tip of the minute hand moves 6.28 cm.

4. 5X+14Y=1679. X=20. What is value of Y?

   Y = (1579)/(14) or Y ≈ 112.79

5. Let A(4, -2), B(1, 1) and C(9, -3) be the vertices of a triangle ABC. Then the maximum area of the parallelogram AFDE, formed with vertices D, E, and F on the s

   The maximum area of the parallelogram AFDE is 3 square units.

6. Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.

   The HCF of 6, 72 and 120 is 6, and the LCM is 360.

7. If f(x) = x²2² + √2, x R, then Σ_k=1^(81) f((x)/(82)) is equal to

   81x²6724(4 + √2)

8. Find the equation of the parabola with vertex at (0,0) and focus at (0,2).

   The equation of the parabola is x² = 8y.

9. The area of the region (x, y): x² + 4x + 2 ≤ y ≤ |x + 2| is equal to

   (20)/(3)

10. Case Study - 1 36. In a park, four poles are standing at positions A, B, C and D around the circular fountain such that the lines joining the poles AB, BC, CD a

    (i) OSA = 90^° (ii) The figure ABCD is a kite. (iii) (a) AP = 4 cm OR (iii) (b) QOR = 120^°

11. एक समकोण त्रिभुज ABC में, जिसका कोण B समकोण है, यदि tan A = 1 तो सत्यापित कीजिए कि 2 sin A cos A = 1।

    सत्यापित हो गया कि 2 sin A cos A = 1।

12. Let R be a relation on the set A of ordered pairs of positive integers defined by (x, y) R(u, v) if and only if x v=y u. Show that R is an equivalence relation.

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

13. In a G.P., the 3^ rd term is 24 and the 6^ th term is 192.Find the 10^ th term.

    The 10^(th) term is 3072.

14. In the coordinate plane, triangle ABC is equilateral with B(1, 0) and C(3, 0). A line through the origin O meets AB and AC at M and N respectively. If OM = MN,

    The coordinates of M are (5/4, √3/4).

15. Prove that cos ((π)/(4)+x)+cos ((π)/(4)-x)=√2 cos x

    The identity is proven: cos ((π)/(4)+x)+cos ((π)/(4)-x)=√2 cos x.

16. Let f(x) = 7 tan^8 x + 7 tan^8 x - 3 tan^4 x - 3 tan² x, I_1 = ∫_0^(π/4) f(x) dx and I_2 = ∫_0^(π/4) x f(x) dx. Then 7 I_1 + 12 I_2 is equal to:

    12 I_2

17. Let X = Ram, Geeta, Akbar be the set of students of Class XI, who are in school hockey team. Let Y = Geeta, David, Ashok be the set of students from Class XI wh

    X Y = Geeta

18. Find the ratio in which P(4, m) divides the line segment joining the points A(2, 3) and B(6, -3). Hence, find m.

    The ratio is 1:1 and m=0.

19. If z_1, z_2, z_3 C are the vertices of an equilateral triangle whose centroid is z_0, then Σ_k=1³ (z_k - z_0)² is equal to:

    0

20. Let the area enclosed between the curves |y|=1-x² and x²+y²=1 be α. If 9α=βπ+, β, are integers, then the value of |β- | equals:

    33

21. 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 using Ceva's t

    The ratio AF:FB is 2:1.

22. आकृति 6.16 में (PS)/(SQ) = (PT)/(TR) है तथा PST = PRQ है। सिद्ध कीजिए कि PQR एक समद्विबाहु त्रिभुज है।

    PQR एक समद्विबाहु त्रिभुज है।

23. Solve the Diophantine equation x² + y² = z² in positive integers (Pythagorean triples).

    The solutions to the Diophantine equation x² + y² = z² in positive integers are given by x = k(m² - n²), y = k(2mn), and z = k(m² + n²), where m and n are copri

24. The decorative block shown in Fig. 12.7 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 fixe

    The total surface area of the block is 163.86 cm².

25. If z_1, z_2, z_3 are complex numbers such that |z_1|=|z_2|=|z_3|=|(1)/(z_1)+(1)/(z_2)+(1)/(z_3)|=1, then find the value of |z_1+z_2+z_3|.

    |z_1+z_2+z_3| = 1

26. Let A = -3, -2, -1, 0, 1, 2, 3. Let R be a relation on A defined by xRy if and only if 0 ≤ x² + 2y ≤ 4. Let be the number of elements in R and m be the minimum

    18

27. Let the shortest distance from (a, 0), a > 0, to the parabola y² = 4x be 4. Then the equation of the circle passing through the point (a, 0) and the focus of th

    (x-3)² + y² = 4

28. Given that α, β, γ are roots of x³ - 3x² + 4 = 0, find the value of α⁴ + β⁴ + γ⁴.

    33

29. Find the value of sin 15^(°).

    The value of sin 15^(°) is (√6 - √2)/(4).

30. For t > -1, let _t and _t be the roots of the equation ((t + 2)^((1)/(7)) - 1)x² + ((t + 2)^((1)/(6)) - 1)x + ((t + 2)^((1)/(21)) - 1) = 0. If lim_t → -1^+ _t =

    98

31. The mean and standard deviation of 100 observations are 40 and 5.1, respectively. By mistake, one observation is taken as 50 instead of 40. If the correct mean

    430

32. If I = ∫_0^((π)/(2)) (sin (1)/(2) x)/(sin² x + cos² x) dx, then ∫_0²I (x sin x cos x)/(sin² x + cos^4 x) dx equals:

    The problem statement is mathematically underdetermined due to an inconsistency in the upper limit of the second integral. If 2I is assumed to be π, the integra

33. Find the area of a sector of a circle with radius 6 cm and sector angle 60 degrees. Use pi = 3.14.

    The area of the sector is 18.84 cm².

34. If the lines 2x + y - 3 = 0, 5x + ky - 3 = 0 and 3x - y - 2 = 0 are concurrent, find the value of k.

    The value of k is -2.

35. The value of ( sin 70^°) ( cot 10^° cot 70^° - 1) is

    1

36. Let y = y(x) be the solution of the differential equation cos x (log_e (cos x))² dy + (sin x - 3y sin x log_e (cos x)) dx = 0, x (0, (π)/(2)). If y((π)/(4)) = (

    12 (log_e ((√3)/(2)))²

37. The coordinates of a point on the x-axis, which is equidistant from ( - 2, 5) and (2, - 3) are:

    (-2, 0)

38. Two students Anil and Ashima appeared in an examination. The probability that Anil will qualify the examination is 0.05 and that Ashima will qualify the examina

    (a) The probability that both Anil and Ashima will not qualify the examination is 0.87. (b) The probability that at least one of them will not qualify the exami

39. The number of relations on the set A = 1, 2, 3 containing at most 6 elements including (1, 2), which are reflexive and transitive but not symmetric, is _____

    5

40. Find the conjugate of ((3-2 i)(2+3 i))/((1+2 i)(2-i)).

    The conjugate of the given expression is (63)/(25) + (16)/(25)i.

41. Let the range of the function f(x) = 6 + 16 cos x · cos ((π)/(3) - x) · cos ((π)/(3) + x) · sin 3x · cos 6x, x R be [α, β]. Then the distance of the point (α, β

    11

42. Write the set x: x is a positive integer and x²< 40 in the roster form.

    The set in roster form is 1, 2, 3, 4, 5, 6.

43. Locate the points for which 3<|z|<4

    The points for which 3<|z|<4 are all points in the complex plane that lie strictly between the circle of radius 3 and the circle of radius 4, both centered at t

44. The sum of all rational terms in the expansion of (1 + 2^(1/2) + 3^(1/2))^6 is equal to

    1296

45. Check the points where the constant function f(x)=k is continuous.

    The constant function f(x)=k is continuous at all points x R.

46. The p^ th term of an A.P. is a and q^ th term is b. Prove that the sum of its (p+q) terms is (p+q)/(2)[a+b+(a-b)/(p-q)].

    The sum of its (p+q) terms is (p+q)/(2)[a+b+(a-b)/(p-q)].

47. Find the roots of 2x² - 5x + 3 = 0 by factorisation.

    The roots of the equation 2x² - 5x + 3 = 0 are x = (3)/(2) and x = 1.

48. Find the distance of the line 4x - y = 0 from the point P (4, 1) measured along the line making an angle of 135^° with the positive x-axis.

    The distance is 3√2 units.

49. Let the distance between two parallel lines be 5 units and a point P lie between the lines at a unit distance from one of them. An equilateral triangle PQR is f

    28

50. Consider the sequence defined by a₁ = 1, aₙ₊₁ = 1 + 1/aₙ. Prove that this sequence converges and find its limit.

    The sequence converges to the limit (1 + √5)/(2).

51. Let f be a continuous function on [a,b] such that ∫ₐᵇ xⁿf(x)dx = 0 for all n ≥ 0. Prove that f(x) = 0 for all x ∈ [a,b].

    Since ∫_a^(b) f(x)² dx = 0 and f(x) is continuous, it must be that f(x) = 0 for all x [a,b].

52. Let y = y(x) be the solution curve of the differential equation x(x² + e^x)dy + (e^x(x - 2)y - x³)dx = 0, x > 0, passing through the point (1, 0). Then y(2) is

    (4)/(4 + e²)

53. Let N be the set of natural numbers. Define a real valued function f: N → N by f(x)=2 x+1. Using this definition, complete the table given below. |r|c|c|c|c|c|c

    The completed table is: |r|c|c|c|c|c|c|c| x & 1 & 2 & 3 & 4 & 5 & 6 & 7 y & f(1)=3 & f(2)=5 & f(3)=7 & f(4)=9 & f(5)=11 & f(6)=13 & f(7)=15

54. If P=1,2, form the set P × P × P.

    P × P × P = (1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1), (2,2,2)

55. If the system of equations 2x - y + z = 4, 5x + y + 3z = 12, 100x - 47y + z = 212 has infinitely many solutions, then - 2 is equal to

    57

56. From a point P on the ground the angle of elevation of the top of a 10 m tall building is 30^°. A flag is hoisted at the top of the building and the angle of el

    The distance of the building from point P is 17.32 m and the length of the flagstaff is 7.32 m.

57. Let the function, f(x) = -3 a x² - 2, & x < 1 a² + b x, & x 1 be differentiable for all x R, where a > 1, b R. If the area of the region enclosed by y = f(x) an

    18

58. Let P(4,4√3) be a point on the parabola y²=4ax and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectivel

    The area of the quadrilateral PQMN is (343√3)/(8) square units.

59. Find the equation of the set of the points P such that its distances from the points A(3,4,-5) and B(-2,1,4) are equal.

    The equation of the set of points P is 10x + 6y - 18z - 29 = 0.

60. Let a and b be two unit vectors such that the angle between them is (π)/(3). If a + 2 b and 3 a - b are perpendicular to each other, then the number of values o

    0

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