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Solved maths problems — page 21

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

  1. If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then a + b + ab is equal to:

    103

  2. Let A be a square matrix of order 3 such that det(A) = -2 and det(3 adj(-6 adj(3 A))) = 2^(m+n) · 3^(mn), m > n. Then 4m + 2n is equal to:

    38

  3. For n ≥ 2, let S_n denote the set of all subsets of 1, 2,, n with no two consecutive numbers. For example, 1, 3, 5 S_6, but 1, 2, 4 S_6. Then n(S_5) is equal to

    13

  4. Let α, β, and be the coefficients of x^7, x^5, x³ and x respectively in the expansion of (x+√(x³-1))^5+(x-√(x³-1))^5, x>1. If u and v satisfy the equations α u+

    5

  5. If A and B are two events such that P(A) = 0.7, P(B) = 0.4 and P(A B) = 0.5, where B denotes the complement of B, then P(B (A B)) is equal to:

    (0.2)/(0.8) = 0.25

  6. Find the QR decomposition of A = [[1, 1, 0], [1, 0, 1], [0, 1, 1]].

    Q = 1/√2 & 1/√6 & -1/√3 1/√2 & -1/√6 & 1/√3 0 & 2/√6 & 1/√3, R = √2 & 1/√2 & 1/√2 0 & 3/√6 & 1/√6 0 & 0 & 2/√3

  7. If a curve y=y(x) passes through the point (1,(π)/(2)) and satisfies the differential equation (7x^4cot y - e^xcsc y)(dx)/(dy) = x^5, x 1, then at x=2, the valu

    (e(2e - 1))/(128)

  8. Find f^()(x) if f^()(x)=(sin x)^(sin x) for all 0<x<π.

    f^()(x) = (sin x)^(sin x) cos x [ln (sin x) + 1]

  9. Solve the differential equation: (x² - y²)dx + 2xy dy = 0 given that y(1) = 1.

    x² + y² = 2x

  10. Find the roots of the quadratic equation: 3x² - 5x + 2 = 0 using the quadratic formula.

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

  11. If the arcs of the same lengths in two circles subtend angles 65^(°) and 110^(°) at the centre, find the ratio of their radii.

    The ratio of their radii is 22:13.

  12. If the system of linear equations: x + y + 2z &= 6 2x + 3y + az &= a + 1 -x - 3y + bz &= 2b where a, b R, has infinitely many solutions, then 7a + 3b is equal t

    42

  13. If the imaginary part of (2 z+1)/(i z+1) is -2, then show that the locus of the point representing z in the argand plane is a straight line.

    The locus of the point representing z is the straight line given by the equation x+2y-2=0.

  14. There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that ca

    210

  15. (b) Given that P = [[2, -1], [3, 4]], Q = [[5, 2], [7, 4]] and R = [[2, 5], [3, 8]] find a matrix S such that PQ - RS is a null matrix.

    The matrix S is -191 & -110 77 & 44.

  16. Let the line L pass through (1, 1, 1) and intersect the lines (x - 1)/(2) = (y + 1)/(3) = (z - 1)/(4) and (x - 3)/(1) = (y - 4)/(2) = (z)/(1). Then, which of th

    The point (-1, 0, -3) lies on the line L.

  17. So I had ₹100 and in six years became ₹300 assuming there was a equal growth every year equal percentage growth so what's the annual CAGR

    The annual CAGR is approximately 20.09%.

  18. Let A be a 3 × 3 matrix such that X^T A X = 0 for all nonzero 3 × 1 matrices X = x y z. If A 1 1 1 = 1 4 -5, A 1 2 1 = 0 4 -8, and det(adj(2(A + I))) = 2^α 3^β

    80

  19. Which of the following pairs of sets are equal? Justify your answer. (i) X, the set of letters in “ALLOY” and B, the set of letters in “LOYAL”. (ii) A = n: n Z

    (i) The sets X and B are equal. (ii) The sets A and B are not equal.

  20. Let M and m respectively be the maximum and the minimum values of 1 + sin² x & cos² x & 4 sin 4x 1 + sin² x & cos² x & 4 sin 4x sin² x & cos² x & 1 + 4 sin 4x,

    0

  21. If S and S' are the foci of the ellipse (x²)/(18) + (y²)/(9) = 1 and P is a point on the ellipse, then min (SP· S'P) + max (SP· S'P) is equal to:

    27

  22. What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these (i) four cards are of the same suit, (ii) four cards belong

    The number of ways of choosing 4 cards from a pack of 52 playing cards is 270,725. (i) Four cards are of the same suit: 2,860 ways. (ii) Four cards belong to fo

  23. Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then P² R³: S³ is equal to (A) 1: 1 (B) ( common ratio)^(n): 1 (C) (

    1: 1

  24. A wooden toy rocket is in the shape of a cone mounted on a cylinder, as shown in Fig. 12.8. The height of the entire rocket is 26 cm, while the height of the co

    The area painted orange is 63.385 cm² and the area painted yellow is 195.465 cm².

  25. Let f(x)=x² and g(x)=2 x+1 be two real functions. Find (f+g)(x),(f-g)(x),(f g)(x),((f)/(g))(x).

    (f+g)(x) = x² + 2x + 1, (f-g)(x) = x² - 2x - 1, (fg)(x) = 2x³ + x², ((f)/(g))(x) = (x²)/(2x+1) for x ≠ -(1)/(2).

  26. Let f: R → R be a function defined by f(x) = (2 + 3a) x² + ( (a + 2)/(a - 1)) x + b, a ≠ 1. If f(x + y) = f(x) + f(y) + 1 - (2)/(7) xy, then the value of 28 Σ_i

    667

  27. Find the zeroes of the polynomial x² - 3 and verify the relationship between the zeroes and the coefficients.

    The zeroes of the polynomial x² - 3 are √3 and -√3. Verification: Sum of zeroes: √3 + (-√3) = 0. -b/a = -0/1 = 0. (Verified) Product of zeroes: (√3)(-√3) = -3.

  28. Prove that there are infinitely many primes of the form 4k + 3.

    There are infinitely many primes of the form 4k + 3.

  29. Find the equation of the parabola which is symmetric about the y -axis, and passes through the point (2,-3).

    The equation of the parabola is x² = -(4)/(3)y.

  30. Find the equation of the line, which makes intercepts -3 and 2 on the x- and y-axes respectively.

    The equation of the line is 2x - 3y + 6 = 0.

  31. Let the vertices Q and R of the triangle PQR lie on the line (x+3)/(5)=(y-1)/(2)=(z+4)/(3), QR = 5, and the coordinates of the point P be (0,2,3). If the area o

    (5√21)/(2)

  32. The exponent of 3 in the prime factorisation of 243 is:

    5

  33. How many terms of the G.P. 3, (3)/(2), (3)/(4), are needed to give the sum (3069)/(512)?

    10

  34. Let A = a, e, i, o, u and B = a, i, u. Show that A B = A.

    It is shown that A B = A.

  35. The focus of the parabola y² = 4x + 16 is the centre of the circle C of radius 5. If the values of, for which C passes through the point of intersection of the

    15

  36. Find the maximum and minimum values of f(x) = sin(x) + cos(x) on the interval [0, π].

    The maximum value of the function is √2 and the minimum value is -1 on the interval [0, π].

  37. MODIFIED: If variable_94² + variable_94² + variable_94² = variable_94*variable_94 + variable_94*variable_94 + variable_94*variable_94, prove that variable_94 =

    Therefore, a=b=c is proven.

  38. Let (2, 3) be the largest open interval in which the function f(x) = 2 log_e (x - 2) - x² + a x + 1 is strictly increasing and (b, c) be the largest open interv

    360

  39. Let integers a, b [-3, 3] be such that a + b ≠ 0. Then the number of all possible ordered pairs (a, b), for which |(x+1)/(x+b)| = 1 and | ccc x+1 & & ² & z+ ² &

    5

  40. Write the first three terms in each of the following sequences defined by the following: ll (i) a_n=2 n+5, (ii) a_n=(n-3)/(4).

    The first three terms for sequence (i) are 7, 9, 11. The first three terms for sequence (ii) are -(1)/(2), -(1)/(4), 0.

  41. Find the number of derangements of 1, 2,..., n using inclusion-exclusion principle

    D_n = n! Σ_k=0^(n) ((-1)^k)/(k!) = n! ( 1 - (1)/(1!) + (1)/(2!) - (1)/(3!) + + (-1)^n (1)/(n!))

  42. Let A = [a_ij] be a square matrix of order 2 with entries either 0 or 1. Let E be the event that A is an invertible matrix. Then the probability P(E) is:

    (3)/(8)

  43. Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of

    8925

  44. Find the area of the triangle whose vertices are (3,8),(-4,2) and (5,1).

    The area of the triangle is 30.5 square units.

  45. Two cups of flour are needed to make a dozen cookies. Carla is making 36 cookies today and 30 cookies tomorrow. How many cups of flour will Carla need to bake t

    11 cups of flour

  46. Let L_1: (x-1)/(2) = (y-2)/(3) = (z-3)/(4) and L_2: (x-2)/(3) = (y-4)/(4) = (z-5)/(6) be two lines. Then which of the following points lies on the line of the s

    The point (5, 8, 11) lies on the line of the shortest distance between L_1 and L_2.

  47. Calculate mean, variance and standard deviation for the following distribution: Classes: 30-40, 40-50, 50-60, 60-70, 70-80, 80-90, 90-100; Frequencies: 3, 7, 12

    Mean ( x) = 63.8, Variance ( ²) = 164.56, Standard Deviation () = 12.83

  48. If the shortest distance between the lines (x - 1)/(2) = (y - 2)/(3) = (z - 3)/(4) and (x)/(1) = (y)/(α) = (z - 5)/(1) is (5)/(√6), then the sum of all possible

    -3

  49. In a square grid, the position of ABC is shown. Find tan B.

    tan B = 2

  50. A rectangle has length 40 cm and width 16 cm. Point M is the midpoint of one side. The paper is folded along a line through M. If a vertex of the side containin

    16

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

    The area of the triangle is 24 square units.

  52. Let A=(x, y) R × R: |x+y| 3 and B=(x, y) R × R: |x|+|y| ≤ 3. If C=(x, y) A B: x=0 or y=0, then Σ_(x, y) C|x+y| is:

    12

  53. Let the curve z(1+i) + z(1-i) = 4, z C, divide the region |z-3| ≤ 1 into two parts of areas α and β. Then |α - β| equals:

    (π)/(2) + 1

  54. A bag contains 9 discs of which 4 are red, 3 are blue and 2 are yellow. The discs are similar in shape and size. A disc is drawn at random from the bag. Calcula

    (i) P(Red) = (4)/(9), (ii) P(Yellow) = (2)/(9), (iii) P(Blue) = (1)/(3), (iv) P(Not Blue) = (2)/(3), (v) P(Either Red or Blue) = (7)/(9)

  55. Each of the angles β and that a given line makes with the positive y - and z -axes, respectively, is half of the angle that this line makes with the positive x

    The sum of all possible values of the angle β is (3π)/(4).

  56. A coin is tossed three times. Let X denote the number of times a tail follows a head. If and ² denote the mean and variance of X, then the value of 64( + ²) is:

    48

  57. Look at the graphs in Fig. 2.9 given below. Each is the graph of y = p(x), where p(x) is a polynomial. For each of the graphs, find the number of zeroes of p(x)

    (i) 1, (ii) 2, (iii) 3, (iv) 1, (v) 1, (vi) 4

  58. Show that (x²+x y+y²),(z²+x z+x²) and (y²+y z+z²) are consecutive terms of an A.P., if x, y and z are in A.P.

    The given terms (x²+x y+y²),(z²+x z+x²) and (y²+y z+z²) are consecutive terms of an A.P.

  59. Let the equation of the circle, which touches x -axis at the point (a, 0), a > 0 and cuts off an intercept of length b on y -axis be x² + y² - α x + β y + = 0.

    (2a, 4r² - 4a²)

  60. x का मान ज्ञात कीजिए: sin(x) = cos(x) जहाँ x [0, π/2] के बीच है।

    x = (π)/(4)

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