SolveForX

Solved maths problems — page 14

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

  1. The integral ∫_-1^((3)/(2)) | π² x sin(π x) | dx is equal to:

    3π + 1

  2. Line L_1 of slope 2 and line L_2 of slope 12 intersect at the origin O. In the first quadrant, P_1,P_2,,P_12 are 12 points on L_1 and Q_1,Q_2,,Q_9 are 9 points

    1134

  3. The relation R = (x, y): x, y Z and x + y is even is:

    The relation R is an equivalence relation.

  4. Let the set of all values of p R, for which both the roots of the equation x² - (p + 2)x + (2p + 9) = 0 are negative real numbers, be the interval (α, β]. Then

    5

  5. If the equation of the parabola with vertex V((3)/(2), 3) and the directrix x + 2y = 0 is α x² + β y² - xy - 30x - 60y + 225 = 0, then α + β + is equal to:

    9

  6. Let f be a function such that f(x) + 3 f (24/x) = 4x, x 0. Then f(3) + f(8) is equal to:

    11

  7. The remainder when (64^(64))^(64) is divided by 7 is equal to:

    1

  8. Given the three orthographic views of a solid as shown, find the volume of the solid.

    The volume of the solid is (4√3)/(3) cubic units.

  9. Find the discriminant of the quadratic equation 2x² - 4x + 3 = 0, and hence find the nature of its roots.

    The discriminant is -8. Since the discriminant is less than 0, the quadratic equation has no real roots.

  10. MODIFIED: Find the equation of the tangent to the ellipse variable_97²/16 + variable_97²/9 = 1 that passes through the point (8, 0).

    The equations of the tangents are y = (√3)/(4)x - 2√3 and y = -(√3)/(4)x + 2√3.

  11. The value of cot^(-1) ( (√(1 + tan²(2)) - 1)/(tan(2))) - cot^(-1) ( (√(1 + tan²((1)/(2))) + 1)/(tan((1)/(2)))) is equal to:

    (π)/(2) - (5)/(4)

  12. A sphere of radius r is inscribed in a cone with base radius R and height h. Find the relationship between r, R, and h.

    The relationship between r, R, and h is r = (Rh)/(R+h).

  13. Show that a one-one function f:1,2,3 →1,2,3 must be onto.

    A one-one function f:1,2,3 →1,2,3 must be onto because the number of distinct images equals the number of elements in the codomain, making the range equal to th

  14. Find the Green's function for the operator L[u] = -u'' with boundary conditions u(0) = u(1) = 0.

    The Green's function is G(x,) = x(1 -) & x < (1 - x) & x >.

  15. 8 Statement 1 | R is a splitting field of some polynomial over Q. Statement 2 | There is a field with 60 elements.

    Both Statement 1 and Statement 2 are false.

  16. Find the equation of the circle which passes through the points (2,-2), and (3,4) and whose centre lies on the line x+y=2.

    The equation of the circle is (x-(7)/(10))² + (y-(13)/(10))² = (629)/(50).

  17. Let a = i + 2 j + 3 k, b = 3 i + j - k and c be three vectors such that c is coplanar with a and b. If the vector c is perpendicular to b and a · c = 5, then |

    (√66)/(6)

  18. The ratio of men to women on a bus is 5:9. If the total number of passengers on the bus is 84, and 20 women alight from the bus at the next stop, how many women

    34 women have remained on the bus.

  19. Find the shape of a flexible cable hanging between two points that minimizes its potential energy (catenary problem).

    The shape of a flexible cable hanging between two points that minimizes its potential energy is a catenary curve, described by the equation y(x) = c ((x-b)/(c))

  20. Evaluate | ccx & x+1 x-1 & x |

    1

  21. If the area of the larger portion bounded between the curves x² + y² = 25 and y = |x - 1| is (1)/(4)(b π + c), b, c N, then b + c is equal to

    77

  22. Consider the lines L_1: x - 1 = y - 2 = z and L_2: x - 2 = y - 2 = z - 1. Let the feet of the perpendiculars from the point P(5,1,-3) on the lines L_1 and L_2 b

    67

  23. Consider the equation x² + 4x - n = 0 where n [20,100] is a natural number. Then the number of all distinct values of n, for which the given equation has integr

    6

  24. Using binomial theorem, prove that 6^n-5 n always leaves remainder 1 when divided by 25.

    The expression 6^n - 5n can be written as 1 + 25K, where K is an integer. Therefore, 6^n - 5n always leaves a remainder of 1 when divided by 25.

  25. (i) If a, b, c, d are four distinct positive quantities in A.P., then show that b c>a d (ii) If a, b, c, d are four distinct positive quantities in G.P., then s

    (i) bc > ad (ii) a+d > b+c

  26. If B and Q are acute angles such that sin B = sin Q, then prove that B = Q.

    B = Q

  27. Differentiate w.r.t. x, the following function: (i) √(3 x+2)+(1)/(√(2 x²+4)) (ii) log _7(log x)

    (i) (3)/(2√(3x+2)) - (2x)/((2x²+4)³/2) (ii) (1)/(x log x log 7)

  28. Evaluate the infinite product Π(n=1 to ∞) (1 - 1/n²) and prove its convergence.

    The infinite product _n=2^(∞) (1 - (1)/(n²)) converges to (1)/(2).

  29. यदि कोई रेखा एक ABC की भुजाओं AB और AC को क्रमश: D और E पर प्रतिच्छेद करे तथा भुजा BC के समांतर हो, तो सिद्ध कीजिए कि (AD)/(AB) = (AE)/(AC) होगा (देखिए आकृति 6.

    हमने सिद्ध किया है कि यदि कोई रेखा ABC की भुजाओं AB और AC को D और E पर प्रतिच्छेद करती है और भुजा BC के समांतर है, तो (AD)/(AB) = (AE)/(AC)।

  30. If (x+i y)^((1)/(3))=a+i b, where x, y, a, b R, show that (x)/(a)-(y)/(b)=-2(a²+b²)

    The identity (x)/(a)-(y)/(b)=-2(a²+b²) is shown to be true.

  31. A die is thrown once. Find the probability of getting a prime number.

    The probability of getting a prime number when a die is thrown once is (1)/(2).

  32. Let E_1: (x²)/(5) + (y²)/(4) = 1 be an ellipse. Ellipses E_i are constructed such that their centres and eccentricities are same as that of E_1, and the length

    50√5

  33. Three people invested 1200 in a joint savings account. After Dylan's investment of 2/5 of the total amount, Frances invested 2/3 of the remaining amount. Skyler

    Skyler's investment was 160.

  34. Let A be the point of intersection of the lines L_1: (x-7)/(1)=(y-5)/(0)=(z-3)/(-1), and L_2: (x-1)/(3)=(y+3)/(4)=(z+7)/(5). Let B and C be points on L_1 and L_

    54

  35. Find (d y)/(d x), if y^(x)+x^(y)+x^(x)=a^(b).

    (dy)/(dx) = - (y^x ln y + x^y (y)/(x) + x^x (ln x + 1))/(y^x (x)/(y) + x^y ln x)

  36. Prove that √3 is irrational.

    √3 is an irrational number.

  37. If the sum of the first 20 terms of the series (4 1)/(4+3 1²+1^4)+(4 2)/(4+3 2²+2^4)+(4 3)/(4+3 3²+3^4)+ is (m)/(n), where m and n are coprime, then m+n is equa

    421

  38. At the end of each year the value of a certain machine has depreciated by 20 % of its value at the beginning of that year. If its initial value was Rs 1250, fin

    The value of the machine at the end of 5 years is Rs 409.60.

  39. Angle A of triangle is 50, B is 30. What is C?

    The measure of angle C is 100^°.

  40. If the function f(x) = (tan(tan x) - sin(sin x))/(tan x - sin x) is continuous at x = 0, then f(0) is equal to:

    2

  41. Find all values of k for which the equation (k-2)x² - 2(k-1)x + k = 0 has equal roots.

    The only value of k for which the equation has equal roots is k=2.

  42. If sin A = 3/5 and A is an acute angle, find the values of cos A and tan A.

    cos A = (4)/(5), tan A = (3)/(4)

  43. Let x = -1 and x = 2 be the critical points of the function f(x) = x³ + ax² + b log_2|x| + 1, x 0. Let m and M respectively be the absolute minimum and the abso

    23.68

  44. Consider the experiment of rolling a die. Let A be the event 'getting a prime number', B be the event 'getting an odd number'. Write the sets representing the e

    (i) A B = 1, 2, 3, 5 (ii) A B = 3, 5 (iii) A - B = 2 (iv) A' = 1, 4, 6

  45. Evaluate: (i) lim _x → 1 (x^(15)-1)/(x^(10)-1) (ii) lim _x → 0 (√(1+x)-1)/(x)

    (i) (3)/(2) (ii) (1)/(2)

  46. x और y में एक संबंध ज्ञात कीजिए, ताकि बिंदु (x, y) बिंदुओं (7, 1) और (3, 5) से समदूरस्थ (equidistant) हो।

    निकला हुआ संबंध है x - y = 2।

  47. Find the equation of set of points P such that PA²+PB²=2 k², where A and B are the points (3,4,5) and (-1,3,-7), respectively.

    x²-2x + y²-7y + z²+2z + (109)/(2) - k² = 0

  48. Prove that the square root of 5 is irrational.

    The square root of 5 is irrational.

  49. Find (d y)/(d x), if x=a cos θ, y=a sin θ.

    (d y)/(d x) = -cot θ

  50. Suppose that the number of terms in an A.P. is 2k, k N. If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the

    The value of k is 5.

  51. The value of lim_n → ∞ ( Σ_k=1^n (k³ + 6k² + 11k + 6)/((k+3)!)) is:

    e-1

  52. If A, B, and (adj(A^(-1)) + adj(B^(-1))) are non-singular matrices of same order, then the inverse of A (adj(A^(-1)) + adj(B^(-1)))^(-1) B, is equal to

    B^(-1) + A^(-1)

  53. The number of solutions of the equation (4-√3)sin x - 2√3cos²x = -(4)/(1+√3), x [-2π,(5π)/(2)] is equal to:

    5

  54. If lim_t → 0 (∫_0^1 (3x + 5)^t dx)^((1)/(t)) = (2)/(56) ((8)/(5))^((2)/(α)), then α is equal to:

    The problem statement contains an inconsistency. The calculated limit involves the constant e, which is not present in the given expression (2)/(56) ((8)/(5))^(

  55. Let the three sides of a triangle ABC be given by the vectors 2 i- j+ k, i-3 j-5 k and 3 i-4 j-4 k. Let G be the centroid of the triangle ABC. Then 6 (| AG|²+|

    164

  56. Examine each of the following relations given below and state in each case, giving reasons whether it is a function or not? (i) R=(2,1),(3,1),(4,2), (ii) R=(2,2

    (i) is a function. (ii) is not a function. (iii) is a function.

  57. If P=a, b, c and Q=r, form the sets P × Q and Q × P. Are these two products equal?

    The sets are P × Q = (a, r), (b, r), (c, r) and Q × P = (r, a), (r, b), (r, c). These two products are not equal.

  58. Three defective oranges are accidentally mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are dr

    The variance of x is (28)/(75).

  59. Find a point on the y -axis which is equidistant from the points A(6, 5) and B(-4, 3).

    The point on the y -axis equidistant from A(6, 5) and B(-4, 3) is (0, 9).

  60. As observed from the top of a 100m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the

    The distance between the two ships is 100(√3 - 1) m.

Solve your own maths question free →