SolveForX

Solved maths problems — page 25

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

  1. Show that 5 - √3 is irrational.

    Therefore, 5 - √3 is an irrational number.

  2. Let a = 2 i - 3 j + k, b = 3 i + 2 j + 5 k and a vector c be such that ( a - c) × b = -18 i - 3 j + 12 k and a · c = 3. If b × c = d, then | a · d| is equal to:

    15

  3. The number of solutions of the equation cos 2 (θ)/(2) + cos(5θ)/(2) = 2cos³(5θ)/(2) in [-(π)/(2),(π)/(2)] is:

    7

  4. Prove that the square root of 2 is irrational.

    √2 is irrational.

  5. VARIATION: Find the Taylor series expansion of f(x) = e^(x²) centered at x = 0 and determine its interval of convergence. Find the numerical value if exact solu

    The Taylor series expansion of f(x) = e^x² centered at x = 0 is Σ_n=0^(∞) x²nn! = 1 + x² + (x^4)/(2!) + (x^6)/(3!) +. The interval of convergence is (-∞, ∞).

  6. Let the equation x(x+2)(12-k)=2 have equal roots. Then the distance of the point (k,(k)/(2)) from the line 3x+4y+5=0 is:

    15

  7. Let A= α-1 & -16 & β, α>0, such that det(A)=0 and α+β=1. If I denotes the 2 2 identity matrix, then the matrix (I+A)^8 is:

    I + 8A

  8. Three distinct numbers are selected randomly from the set 1,2,3,…,40. If the probability that the selected numbers are in an increasing G.P. is m/n, gcd(m,n)=1,

    1981

  9. Prove the Cauchy-Schwarz inequality in general inner product spaces.

    | u, v |² ≤ u, u v, v

  10. Let S = p_1, p_2,, p_10 be the set of first ten prime numbers. Let A = S P, where P is the set of all possible products of distinct elements of S. Then the numb

    5120

  11. Find the missing value of X between 5X +5 equal to 11

    The missing value of X is (6)/(5).

  12. Suppose A and B are the coefficients of 30^(th) and 12^(th) terms respectively in the binomial expansion of (1 + x)²n - 1. If 2A = 5B, then n is equal to:

    n = (41)/(2)

  13. Let A = 0, 1, 2, 3, 4, 5. Let R be a relation on A defined by (x, y) R if and only if maxx, y 3, 4. Then among the statements (S₁): The number of elements in R

    Only statement (S 2) is correct.

  14. 4∫_0^1 ((1)/(√(3 + x²) + √(1 + x²))) dx - 3 log_e(√3) is equal to:

    2 - √2 - log_e(1 + √2)

  15. Jessica makes 2,000.00 a month. She sets 25% of her paycheck aside to put towards fancy shoes. Each pair of shoes she buys costs 1,000.00. How many shoes can sh

    Jessica can buy 6 pairs of shoes in a year.

  16. The marks obtained by a student of Class XI in first and second terminal examination are 62 and 48, respectively. Find the minimum marks he should get in the an

    The student must obtain a minimum of 70 marks in the annual examination.

  17. Farmer Brown's farm is 200 acres, and Farmer Smith's farm is 100 acres more than twice that. How many acres do the two farms have, together?

    The two farms have 700 acres together.

  18. Find the value of cos (-1710^(°)).

    The value of cos (-1710^(°)) is 0.

  19. Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively, and the sum and the

    540

  20. Show that A B = A B implies A = B.

    The statement A B = A B implies A = B.

  21. Find the minor of element 6 in the determinant =| lll1 & 2 & 3 4 & 5 & 6 7 & 8 & 9 |

    The minor of element 6 is -6.

  22. Find the derivative of f(x)=10 x.

    The derivative of f(x)=10x is f'(x)=10.

  23. A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how many ways can this be done? How many of these committees would consist o

    There are 10 ways to constitute a committee of 3 persons. Out of these, 6 committees would consist of 1 man and 2 women.

  24. Find the equation of the ellipse whose vertices are ( ± 13,0) and foci are ( ± 5,0).

    The equation of the ellipse is (x²)/(169) + (y²)/(144) = 1.

  25. Rasheed got a playing top (lattu) as his birthday present, which surprisingly had no colour on it. He wanted to colour it with his crayons. The top is shaped li

    The area Rasheed has to colour is approximately 38.58 cm².

  26. Let the circle C touch the line x - y + 1 = 0, have the centre on the positive x -axis, and cut off a chord of length (4)/(√13) along the line -3x + 2y = 1. Let

    16 + 12√2

  27. Insert three numbers between 1 and 256 so that the resulting sequence is a G.P.

    The two possible sets of three numbers to be inserted are 4, 16, 64 or -4, 16, -64.

  28. Consider the region R = (x, y): x ≤ y ≤ 9 - (11)/(3) x², x ≥ 0. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R,

    The area of the largest rectangle is (567)/(121).

  29. Find the zeroes of the quadratic polynomial x² - 2x - 8 and verify the relationship between the zeroes and the coefficients.

    The zeroes of the quadratic polynomial x² - 2x - 8 are 4 and -2. The relationships between the zeroes and coefficients are verified as: Sum of zeroes (4 + (-2)

  30. If a, b, c, d and p are different real numbers such that (a²+b²+c²) p²-2(a b+b c+c d) p+(b²+c²+d²) ≤ 0, then show that a, b, c and d are in G.P.

    The numbers a, b, c, and d are in Geometric Progression.

  31. Let f=(1,1),(2,3),(0,-1),(-1,-3) be a linear function from Z into Z. Find f(x).

    f(x) = 2x - 1

  32. Consider the lines x(3 +1)+y(7 +2)=17 +5, being a parameter, all passing through a point P. One of these lines (say L) is farthest from the origin. If the dista

    20

  33. Let M denote the set of all real matrices of order 3 × 3 and let S = -3, -2, -1, 1, 2. Let S_1 = A = [a_ij] M: A = A^T and a_ij S, i, j, S_2 = A = [a_ij] M: A =

    1737

  34. Find the equation of the tangent line to the curve y = x² - 2x + 7 which is parallel to the line 2x - y + 9 = 0.

    The equation of the tangent line is y = 2x + 3.

  35. Let A = 1, 2, 3,, 10 and B = (m)/(n): m, n A, m < n and (m, n) = 1. Then n(B) is equal to:

    31

  36. PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP. [Figure: circle centre O radius 5 c

    The length TP is (20)/(3) cm.

  37. Write the set A = 1, 4, 9, 16, 25,... in set-builder form.

    A = x x = n², n N

  38. In a G.P. of positive terms, if any term is equal to the sum of the next two terms. Then the common ratio of the G.P. is (A) sin 18^(°) (B) 2 cos 18^(°) (C) cos

    The common ratio of the G.P. is 2 sin 18^(°).

  39. Find the value of: tan^(-1)(2 * cos(2 * sin^(-1)(1/2)))

    The value of tan^(-1)(2 · cos(2 · sin^(-1)(1/2))) is (π)/(4).

  40. Is it true that x=e^(log x) for all real x?

    No, it is not true that x=e^(log x) for all real x. It is only true for x > 0.

  41. Argo has 200 toys. He gives 40 toys to Alyssa, 80 to Bonnie, and 30 to Nicky. How many toys does Argo have now?

    Argo has 50 toys now.

  42. A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.

    The number of ancestors during the ten generations preceding his own is 2046.

  43. The number of integral terms in the expansion of ( (1)/(5²) + (1)/(7^8))^(1016) is:

    0

  44. Show that the function f: R → R, defined as f(x)=x², is neither one-one nor onto.

    The function f(x) = x² is neither one-one nor onto.

  45. Show that the function f: N → N, given by f(1)=f(2)=1 and f(x)=x-1, for every x>2, is onto but not one-one.

    The function f is onto but not one-one.

  46. One card is drawn from a well shuffled deck of 52 cards. If each outcome is equally likely, calculate the probability that the card will be (i) a diamond (ii) n

    (i) The probability of drawing a diamond is (1)/(4). (ii) The probability of not drawing an ace is (12)/(13). (iii) The probability of drawing a black card is (

  47. Let A be a 3 × 3 real matrix such that A²(A - 2I) - 4(A - I) = O, where I and O are the identity and null matrices, respectively. If A^5 = α A² + β A + I, where

    12

  48. Show that the points P(-2,3,5), Q(1,2,3) and R(7,0,-1) are collinear.

    The points P, Q, and R are collinear because QR = 2 PQ.

  49. Let A=1,2 and B=3,4. Find the number of relations from A to B.

    The number of relations from A to B is 16.

  50. Let a_n be the n th term of an A.P. If S_n = a_1 + a_2 + + a_n = 700 for some n, a_6 = 7 and S_7 = 7, then a_n is equal to:

    64

  51. The number of complex numbers z, satisfying |z| = 1 and |(z)/(2) + (2)/(z)| = 1, is:

    0

  52. If the system of equations 2x + y + 3z = 5 3x + 2y - z = 7 4x + 5y + z = 9 has infinitely many solutions, then ( ² + ²) is equal to:

    29

  53. Let the values of for which the shortest distance between the lines (x-1)/(2) = (y-2)/(3) = (z-3)/(4) and (x -)/(3) = (y - 4)/(4) = (z - 5)/(5) is (1)/(√6) be _

    (5√2)/(3)

  54. Find all positive integers n such that φ(n) = n/3, where φ is Euler's totient function.

    All positive integers n such that (n) = n/3 are of the form n = 2^a · 3^b, where a and b are positive integers (a ≥ 1, b ≥ 1).

  55. If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of

    757

  56. Let the length of a latus rectum of an ellipse (x²)/(a²) + (y²)/(b²) = 1 be 10. If its eccentricity is the minimum value of the function f(t) = t² + t + (11)/(1

    126

  57. A manufacturer has 600 litres of a 12 % solution of acid. How many litres of a 30 % acid solution must be added to it so that acid content in the resulting mixt

    The manufacturer must add more than 120 litres but less than 300 litres of the 30% acid solution.

  58. On her vacations Veena visits four cities (A, B, C and D) in a random order. What is the probability that she visits (i) A before B? (iii) A first and B last? (

    (i) 1/2 (ii) 1/6 (iii) 1/12 (iv) 1/2 (v) 1/4

  59. Let the system of equations: 2x + 3y + 5z &= 9, 7x + 3y - 2z &= 8, 12x + 3y - (4 +)z &= 16 - have infinitely many solutions. Then the radius of the circle centr

    The radius of the circle is (7)/(5).

  60. Find the derivative of f(x)=x².

    The derivative of f(x)=x² is f'(x)=2x.

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