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Solved maths problems — page 23

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

  1. Diagonalize the matrix A = [[4, -1, 1], [-1, 3, 0], [1, 0, 2]] and compute A^10.

    The given matrix is not diagonalizable because the geometric multiplicity of the eigenvalue = 2 is 1, which is less than its algebraic multiplicity of 2. Theref

  2. Rose went to the store on Monday and bought 4 cakes. Tuesday she went to a different store and bought three times that number of cakes. On Wednesday she went to

    Rose bought a total of 76 cakes after all three days.

  3. Let A = (α, β) R × R: |α - 1| ≤ 4 and |β - 5| ≤ 6 and B = (α, β) R × R: 16(α - 2)² + 9(β - 6)² ≤ 144. Then:

    B is a subset of A.

  4. In a shop the cost of 2 pencils and 3 erasers is ₹9 and the cost of 4 pencils and 6 erasers is ₹18. Find the cost of each pencil and each eraser. [Equations: 2x

    The system of equations has infinitely many solutions. We cannot find a unique cost for each pencil and each eraser with the information provided.

  5. how many word combinations can be made with the names in Vijayawada

    151,200

  6. The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can

    5760

  7. The area of the region enclosed by the curves y = e^x, y = |e^x - 1|, and the y -axis is:

    1 - ln 2

  8. Let r be the radius of the circle, which touches the x-axis at point (a, 0), a < 0 and the parabola y² = 9x at the point (4, 6). Then r is equal to _____

    30

  9. 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.

  10. Write the equation of the lines for which tanθ = (1)/(2), where θ is the inclination of the line and (i) y-intercept is (-3)/(2) (ii) x-intercept is 4.

    (i) y = (1)/(2)x - (3)/(2) (ii) y = (1)/(2)x - 2

  11. Let f: (0, ∞) → R be a twice differentiable function. If for some a ≠ 0, ∫_0^1 f( x) d = a f(x), f(1) = 1 and f(16) = (1)/(8), then 16 - f'((1)/(16)) is equal t

    112

  12. If (π)/(2) ≤ x ≤ (3π)/(4), then cos^(-1) ( (12)/(13) cos x + (5)/(13) sin x) is equal to

    x - tan^(-1)((5)/(12))

  13. Let a straight line L pass through the point P(2,-1,3) and be perpendicular to the lines (x-1)/(4)=(y+1)/(4)=(z-3)/(2) and (x-3)/(4)=(y-2)/(3)=(z+2)/(4). If the

    The distance between the points P and Q is (6√5)/(5).

  14. intersection. Hence, write the equation of a line passing through the point of intersection and perpendicular to both the lines. 35. (a) If P = 1 & -1 & 0 2 & 3

    = xy + yz + zx + xyz. If = 0, then x^(-1) + y^(-1) + z^(-1) = -1.

  15. Let f(x)=√x and g(x)=x be two functions defined over the set of nonnegative real numbers. Find (f+g)(x),(f-g)(x),(f g)(x) and ((f)/(g))(x).

    (f+g)(x) = √x + x (f-g)(x) = √x - x (fg)(x) = x³/2 ((f)/(g))(x) = (1)/(√x)

  16. Given 4 flags of different colours, how many different signals can be generated, if a signal requires the use of 2 flags one below the other?

    12 different signals can be generated.

  17. The glee club ordered 20 pizzas and ate 70% of them. The football team ordered twice as many pizzas and ate 80% of them. How many pizzas are left?

    14 pizzas are left.

  18. For an integer n ≥ 2 if the arithmetic mean of all coefficients in the binomial expansion of (x + y)^(2n-3) is 16, then the distance of the point P(2n-1, n²-4n)

    3√2

  19. Convert 6 radians into degree measure.

    343.77^°

  20. Let ABC be the triangle such that the equations of lines AB and AC be 3y - x = 2 and x + y = 2, respectively, and the points B and C lie on the x-axis. If P is

    6 square units

  21. 33. Solve the following Linear Programming Problem graphically: Maximise Z = 600x + 400y subject to the constraints x + 2y ≤ 12 4x + 5y ≥ 20 2x + y ≤ 12 x, y ≥

    The maximum value of Z is 4000, which occurs at x = 4 and y = 4.

  22. In OPQ, right-angled at P, OP = 7 cm and OQ - PQ = 1 cm. Determine the values of sin Q and cos Q.

    The values are sin Q = (7)/(25) and cos Q = (24)/(25).

  23. Find the maximum value of the directional derivative of f(x,y,z) = x²yz³ at point (1,1,1) in any direction.

    The maximum value of the directional derivative is √14.

  24. The distance of ( - 4, 7) from y-axis is:

    The distance of (-4, 7) from the y -axis is 4 units.

  25. If ∫ (2x² + 5x + 9)/(√(x² + x + 1)) dx = x √(x² + x + 1) + α √(x² + x + 1) + β log_e | x + (1)/(2) + √(x² + x + 1) | + C, where C is the constant of integration

    16

  26. If X and Y are independent standard normal random variables, find the probability density function of Z = X/Y.

    The probability density function of Z = X/Y is f_Z(z) = (1)/(π(z²+1)).

  27. Solve the differential equation: (1+x²)y'' + 2xy' - 2y = 0 given that y₁ = x is a known solution.

    y = A x + B (x² - 1)

  28. यदि B और Q ऐसे न्यूनकोण हों जिससे कि sin B = sin Q, तो सिद्ध कीजिए कि B = Q।

    चूंकि sin B = sin Q दिया गया है, और हमने दिखाया कि यह दो समरूप त्रिभुजों को दर्शाता है (जहाँ B और Q संगत कोण हैं), इसलिए B = Q।

  29. Prove that the space C[0,1] of continuous functions with the sup norm is complete.

    The space C[0,1] of continuous functions with the sup norm is complete.

  30. Let ^n C_r-1 = 28, ^n C_r = 56 and ^n C_r+1 = 70. Let A(4 cos t, 4 sin t), B(2 sin t, -2 cos t) and C(3r - n, r² - n - 1) be the vertices of a triangle ABC, whe

    α = 20

  31. Let f(x)+2f ((1)/(x))=x²+5 and 2g(x)-3g ((1)/(x))=x, x>0. If α=∫_1² f(x) dx and β=∫_1² g(x) dx, then the value of 9α+β is:

    19 + (3)/(5)(1 + ln 2)

  32. The product of all the rational roots of the equation (x² - 9x + 11)² - (x - 4)(x - 5) = 3, is equal to

    14

  33. Find the mean deviation about the mean for the following data: 6,7,10,12,13,4,8,12.

    2.75

  34. If 10sin^4θ + 15cos^4θ = 6 then the value of (27csc^6θ + 8sec^6θ)/(16sec^8θ) is:

    (2)/(5)

  35. The sum 1 + (1+3)/(2!) + (1+3+5)/(3!) + (1+3+5+7)/(4!) + up to infinitely many terms is equal to:

    2e

  36. Find the roots of the quadratic equation 3x² - 2√6x + 2 = 0.

    The roots of the equation are x = (√6)/(3) and x = (√6)/(3).

  37. Find the limits: (i) lim _x → 1[(x²+1)/(x+100)] (ii) lim _x → 2[(x³-4 x²+4 x)/(x²-4)] (iii) lim _x → 2[(x²-4)/(x³-4 x²+4 x)] (iv) lim _x → 2[(x³-2 x²)/(x²-5 x+6

    (i) (2)/(101) (ii) 0 (iii) Does not exist (iv) -4 (v) 2

  38. Two numbers k_1 and k_2 are randomly chosen from the set of natural numbers. Then, the probability that the value of i^(k_1) + i^(k_2), (i = √(-1)) is non-zero,

    The probability that the value of i^(k_1) + i^(k_2) is non-zero is (3)/(4).

  39. Find the minimal polynomial of α = √2 + √3 over ℚ and determine [ℚ(α):ℚ].

    The minimal polynomial of α = √2 + √3 over Q is x^4 - 10x² + 1. The degree of the field extension [ Q(α): Q] is 4.

  40. Let for some function y = f(x), ∫_0^x t f(t) dt = x² f(x), x > 0 and f(2) = 3. Then f(6) is equal to

    1

  41. A line passing through the point P(√5,√5) intersects the ellipse (x²)/(36)+(y²)/(25)=1 at points A and B such that (PA)·(PB) is maximum. Then 5 (PA²+PB²) is equ

    (119)/(18)

  42. Let the vectors a and b be of the same magnitude such that ( a + b + a - b)/( a + b - a - b) = √2 + 1. Then ( a + b ²)/( a ²) is:

    2 + √2

  43. Let one focus of the hyperbola H:(x²)/(a²)-(y²)/(b²)=1 be at (√10,0) and the corresponding directrix be x=(9)/(√10). If e and l respectively are the eccentricit

    16

  44. In a group of 3 girls and 4 boys, there are two boys B_1 and B_2. The number of ways, in which these girls and boys can stand in a queue such that all the girls

    144

  45. Let L be the set of all lines in a plane and R be the relation in L defined as R= (L_1, ~L_2): L_1 is perpendicular to L_2. Show that R is symmetric but neither

    The relation R is symmetric but neither reflexive nor transitive.

  46. Find the equations of the lines parallel to axes and passing through ( -2, 3).

    The equations of the lines are y = 3 and x = -2.

  47. Let the line x + y = 1 meet the circle x² + y² = 4 at the points A and B. If the line perpendicular to AB and passing through the midpoint of the chord AB inter

    2√14

  48. Let α be a solution of x² + x + 1 = 0, and for some a and b in R, [4 a b] 1 & 16 & 13 -1 & -1 & 2 -2 & -14 & -8 = [0 0 0]. If (4)/(α^4) + (m)/(α) + (n)/(α²) = 3

    -10

  49. The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers.

    The three numbers are 2, 8, and 14.

  50. The 10th common term between the series 3+7+11+ and 1+6+11+ is (A) 191 (B) 193 (C) 211 (D) None of these

    191

  51. Let f: (0, ∞) → R be a function which is differentiable at all points of its domain and satisfies the condition x² f'(x) = 2x f(x) + 3, with f(1) = 4. Then 2 f(

    39

  52. Let X=R × R. Define a relation R on X as: (a_1, b_1) R (a_2, b_2) b_1=b_2. Statement I: R is an equivalence relation. Statement II: For some (a, b) X, the set S

    Statement I is true but Statement II is false.

  53. From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:

    5148

  54. In Fig. 6.16, (PS)/(SQ) = (PT)/(TR) and PST = PRQ. Prove that PQR is an isosceles triangle. [Figure: triangle PQR with vertex P at top; points S on PQ and T on

    Triangle PQR is an isosceles triangle.

  55. How many two-digit numbers are divisible by 3?

    There are 30 two-digit numbers divisible by 3.

  56. Discuss the continuity of the function f defined by f(x)=(1)/(x), x ≠ 0.

    The function f(x) = (1)/(x) is continuous for all x R, x ≠ 0. It is discontinuous at x=0.

  57. Find a quadratic polynomial, the sum and product of whose zeroes are -3 and 2, respectively.

    The quadratic polynomial is x² + 3x + 2.

  58. The absolute difference between the squares of the radii of the two circles passing through the point (-9,4) and touching the lines x+y=3 and x-y=3 is equal to:

    768

  59. Find the zeroes of the quadratic polynomial x² + 7x + 10, and verify the relationship between the zeroes and the coefficients.

    The zeroes of the quadratic polynomial x² + 7x + 10 are -2 and -5. The relationship between the zeroes and coefficients is verified as the sum of zeroes (-2 + (

  60. A survey conducted on 20 households in a locality by a group of students resulted in the following frequency table for the number of family members in a househo

    The mode of the given data is approximately 3.286.

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