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Solved maths problems — page 17

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

  1. Let P be the foot of the perpendicular from the point (1,2,2) on the line L: (x-1)/(1)=(y+1)/(-1)=(z-2)/(2). Let the line r=(- i+ j-2 k)+ ( i- j+ k), R, interse

    27

  2. Let the angle θ, 0 < θ < (π)/(2), be the angle between two unit vectors a and b, and θ = sin^(-1)((√65)/(9)). If the vector c = 3 a + 6 b + 9( a × b), then the

    29

  3. Let f: R - 0 → R be a function such that f(x) - 6 f((1)/(x)) = (35)/(3x) - (5)/(2). If the lim_x → 0 ((1)/(α x) + f(x)) = β; α, β R, then α + 2β is equal to

    -4

  4. Find the derivative of f(x)=1+x+x²+x³+ +x^(50) at x=1.

    The derivative of f(x) at x=1 is 1275.

  5. Let O be the origin, the point A be z_1 = √3 + 2 √2 i, the point B(z_2) be such that √3 |z_2| = |z_1| and (z_2) = (z_1) + (π)/(2). Then

    z_2 = -(2√2)/(√3) + i

  6. Let f: R → R be a twice differentiable function such that f(x + y) = f(x) f(y) for all x, y R. If f'(0) = 4a and f satisfies f''(x) - 3a f'(x) - f(x) = 0, a > 0

    e² - 1

  7. Find the sum of first 24 terms of the A.P. a_1, a_2, a_3, if it is known that a_1+a_5+a_10+a_15+a_20+a_24=225.

    The sum of the first 24 terms of the A.P. is 900.

  8. If 7 = 5 + (1)/(7) (5 + α) + (1)/(77) (5 + 2α) + (1)/(77) (5 + 3α) + ∞, then the value of α is:

    The value of α is (850)/(847).

  9. Let α, β (α ≠ β) be the values of m, for which the equations x+y+z=1, x+2y+4z=m, and x+4y+10z=m² have infinitely many solutions. Then the value of Σ_n=1^(10) (n

    385 + 1 + (1)/(4) + (1)/(9) + (1)/(16) + (1)/(25) + (1)/(36) + (1)/(49) + (1)/(64) + (1)/(81) + (1)/(100) = 386 + (1)/(4) + (1)/(9) + (1)/(16) + (1)/(25) + (1)/

  10. If z_1 and z_2 both satisfy z+ z=2|z-1| (z_1-z_2)=(π)/(4), then find Im(z_1+z_2).

    Im(z_1+z_2) = 2

  11. The function f: (-∞, ∞) → (-∞, 1), defined by f(x) = (2^x - 2^(-x))/(2^x + 2^(-x)), is:

    The function is injective but not surjective.

  12. Differentiate the following w.r.t. x: (i) e^(-x) (ii) sin (log x), x>0 (iii) cos ^(-1)(e^(x)) (iv) e^(cos x)

    (i) -e^(-x) (ii) (cos (log x))/(x) (iii) (-e^x)/(√(1 - e²x)) (iv) -sin x · e^(cos x)

  13. In how many ways can 5 girls and 3 boys be seated in a row so that no two boys are together?

    14400

  14. A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow

    The length of her shadow after 4 seconds is 1.6 m.

  15. Show that the number of equivalence relation in the set 1,2,3 containing (1,2) and (2,1) is two.

    The number of equivalence relations in the set 1,2,3 containing (1,2) and (2,1) is two.

  16. On Tuesday, Clara bought 20 pomegranates at 20 each. At the till she got 2 off because she had a voucher. The next day, the price shot to 30 per fruit, but the

    The difference between the final prices paid for the pomegranates on the two days is 25.

  17. Among the statements: (S1): The set z C -i: |z| = 1 and (z - i)/(z + i) is purely real contains exactly two elements, and (S2): The set z C -1: |z| = 1 and (z -

    Statement (S1) is incorrect, and statement (S2) is correct.

  18. Let A(x, y, z) be a point in xy -plane, which is equidistant from three points (0, 3, 2), (2, 0, 3) and (0, 0, 1). Let B = (1, 4, -1) and C = (2, 0, -2). Then a

    Statement (S1) is true and statement (S2) is false.

  19. If each of the observation x_1, x_2,..., x_n is increased by ' a ', where a is a negative or positive number, show that the variance remains unchanged.

    The variance remains unchanged.

  20. The number of words, which can be formed using all the letters of the word DAUGHTER, so that all the vowels never come together, is

    36000

  21. Mishka bought 3 pairs of shorts, 3 pairs of pants, and 3 pairs of shoes. One pair of shorts costs 16.50. One pair of pants costs 22.50 and one pair of shoes cos

    Mishka spent 243.00 on all the clothing items.

  22. If _x 0 ((tan x)/(x))^1/x²=p, then 96log_e p is equal to:

    32

  23. Let X ~ N(μ, σ²) and Y = e^X (log-normal distribution). Find E[Y] and Var(Y).

    E[Y] = e^( + ( ²)/(2)) Var(Y) = e² + ²(e^ ² - 1)

  24. The variance of the numbers 8, 21, 34, 47,, 320 is:

    The variance of the numbers is 8788.

  25. Let y = y(x) be the solution of the differential equation (dy)/(dx) + 2y sec² x = 2sec² x + 3 tan x · sec² x, such that y(0) = (5)/(4). Then 12 ( y((π)/(4)) - e

    15 + 6 e^(-2)

  26. The table below gives the percentage distribution of female teachers in the primary schools of rural areas of various states and union territories (U.T.) of Ind

    The mean percentage of female teachers is approximately 39.71%.

  27. Let ABCD be a trapezium whose vertices lie on the parabola y² = 4x. Let the sides AD and BC of the trapezium be parallel to y -axis. If the diagonal AC is of le

    (49)/(4)

  28. Express the following in the form a+i b ll (i) (5+√2 i)/(1-√2 i) & (ii) i^(-35)

    (i) 1 + 2√2 i (ii) 0 + 1i

  29. The variance of 20 observations is 5. If each observation is multiplied by 2, find the new variance of the resulting observations.

    The new variance of the resulting observations is 20.

  30. A population grows according to the logistic equation dP/dt = rP(1-P/K). If P(0) = P₀, find P(t) and analyze its behavior.

    P(t) = (K P_0)/(P_0 + (K-P_0)e^(-rt)) The population P(t) starts at P_0 and approaches the carrying capacity K as t → ∞.

  31. Let f: [1, ∞) → [2, ∞) be a differentiable function. If ∫_1^x f(t) dt = 5x f(x) - x^5 - 9 for all x ≥ 1, then the value of f(3) is:

    (274)/(27)

  32. Let f: R→ R be a twice-differentiable function such that (sin xcos y) [f(2x+2y)-f(2x-2y) ]=(cos xsin y) [f(2x+2y)+f(2x-2y) ] for all x,y R. If f'(0)= 12, then t

    -3

  33. The value of ∫_-1^(1) ((1 + √(|x|) - x)e^x + (√(|x|) - x)e^(-x))/(e^x + e^(-x)) dx is equal to:

    (7)/(3)

  34. To prove: n N (n+1)/(2) N m N: n = 2m + 1. (natural numbers, implication, existence, odd number)

    The statement n N (n+1)/(2) N m N: n = 2m + 1 is proven. By setting k = (n+1)/(2), we derived n = 2k-1. Then, by defining m = k-1, we showed that n = 2m+1. Sinc

  35. Prove Wilson's theorem: (p-1)! ≡ -1 (mod p) for prime p.

    The proof shows that (p-1)! -1 p for prime p.

  36. In parallelogram ABCD, BAD=60^°, AB=6, and diagonal AC=6(1)/(3). Find the area of ABCD.

    (-9√3 + √354) square units

  37. Find the distance between the parallel lines 3x - 4y +7 = 0 and 3x - 4y + 5 = 0.

    The distance between the parallel lines is (2)/(5) units.

  38. Solve -5 ≤ (5-3 x)/(2) ≤ 8.

    -(11)/(3) ≤ x ≤ 5

  39. If the line 3x - 2y + 12 = 0 intersects the parabola 4y = 3x² at the points A and B, then at the vertex of the parabola, the line segment AB subtends an angle e

    The angle subtended by the line segment AB at the vertex of the parabola is tan^(-1)((9)/(7)).

  40. 32. Show that f: R → R defined as f(x) = (x)/(√(1+x²)) is one-one but not onto. (function, domain, codomain, one-to-one, onto)

    The function f(x) = (x)/(√(1+x²)) is one-one but not onto.

  41. Fireman Frank has 200 shoes. If he gets 5 pairs of shoes on Monday and gains 15 new pairs on Wednesday and 30 pairs on Friday, how many shoes will he have on Su

    Fireman Frank will have 120 shoes on Sunday.

  42. Find the number of ways to color the vertices of a regular hexagon with 3 colors such that no two adjacent vertices have the same color.

    The number of ways to color the vertices of a regular hexagon with 3 colors such that no two adjacent vertices have the same color is 66.

  43. Show that the points A (1, 2, 3), B (-1, -2, -1), C (2, 3, 2) and D(4,7,6) are the vertices of a parallelogram ABCD, but it is not a rectangle.

    The points A, B, C, D form a parallelogram because AB = - CD and BC = - DA. It is not a rectangle because the dot product of adjacent sides AB · BC = -38 ≠ 0, i

  44. If (x+1, y-2)=(3,1), find the values of x and y.

    The values are x=2 and y=3.

  45. Let a_1, a_2, a_3, be a G.P. of increasing positive terms. If a_1 a_5 = 28 and a_2 + a_4 = 29, then a_6 is equal to:

    784

  46. A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let X denote the number of defective pens. Then the variance of X is:

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

  47. If the orthocentre of the triangle formed by the lines y = x + 1, y = 4x - 8 and y = m x + c is at (3, -1), then m - c is:

    0

  48. Find the derivative at x=2 of the function f(x)=3 x.

    The derivative of the function f(x)=3x at x=2 is 3.

  49. Let [t] be the greatest integer less than or equal to t. Then the least value of p N for which lim_x → 0^+ ( x ( [(1)/(x)] + [(2)/(x)] + + [(p)/(x)]) - x² ( [(1

    No such natural number p exists.

  50. Which term of the AP: 21, 18, 15, is -81? Also, is any term 0? Give reason for your answer.

    The 35th term of the AP is -81. Yes, 0 is a term of the AP, specifically the 8th term, because the calculated value of 'n' for a_n = 0 is a positive integer (n=

  51. Let A=1,2,3,4,5,6. Define a relation R from A to A by R=(x, y): y=x+1 (i) Depict this relation using an arrow diagram. (ii) Write down the domain, codomain and

    The relation R = (1,2), (2,3), (3,4), (4,5), (5,6). Domain (R) = 1,2,3,4,5. Codomain (R) = 1,2,3,4,5,6. Range (R) = 2,3,4,5,6.

  52. If the range of the function f(x) = (5 - x)/(x² - 3x + 2), x ≠ 1,2, is ( -∞, α) [β, ∞), then α² + β² is equal to:

    194

  53. Let A = x (0, π) - (π)/(2): log_(2 / π) |sin x| + log_(2 / π) |cos x| = 2 and B = x 0: √x (√x - 4) - 3 |√x - 2| + 6 = 0. Then n(A B) is equal to:

    8

  54. If the area of the region (x, y): -1 ≤ x ≤ 1, 0 ≤ y ≤ a + e^(|x|) - e^(-x), a > 0 is (π² + 8e + 1)/(e), then the value of a is:

    a = (π² - e² + 10e)/(2e)

  55. Let X=1,2,3,4,5,6,7,8,9. Let R_1 be a relation in X given by R_1=(x, y): x-y is divisible by 3 and R_2 be another relation on X given by R_2=(x, y):x, y 1,4,7 o

    The relations R_1 and R_2 are equal.

  56. Let f(x)=∫_0^x² (t²-8t+15)/(e^t) dt, x R. Then the numbers of local maximum and local minimum points of f, respectively, are:

    The number of local maximum points is 2 and the number of local minimum points is 3.

  57. Given three identical bags each containing 10 balls, whose colours are as follows: Bag I: Red = 3, Blue = 2, Green = 5 Bag II: Red = 4, Blue = 3, Green = 3 Bag

    7

  58. Discuss the continuity of the function f given by f(x)=|x| at x=0.

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

  59. If lim_x → ∞ ( ( (e)/(1-e)) ( (1)/(e) - (x)/(1+e)))^x = α, then the value of (log_e α)/(1 + log_e α) equals:

    (1)/(2-e²)

  60. Differentiate a^(x) w.r.t. x, where a is a positive constant.

    (d)/(dx)(a^x) = a^x ln a

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