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NDA 2018 (II), Paper I, Mathematics question paper

120 questions · 300 marks · 150 minutes · +2.5 per correct answer, -0.83 per wrong one

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  1. 1
    What is the value of log⁡7log⁡7777\log_7 \log_7 \sqrt{7\sqrt{7\sqrt{7}}} equal to?

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  2. 2
    If an infinite GP has the first term x and the sum 5, then which one of the following is correct?

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  3. 3
    Consider the following expressions: 1. x+x2−1xx + x^2 - \frac{1}{x} 2. ax2+bx+x−c+dx−ex2\sqrt{ax^2 + bx + x - c + \frac{d}{x} - \frac{e}{x^2}} 3. 3x2−5x+ab3x^2 - 5x + ab 4. 2x2−ax+b3\frac{2}{x^2 - ax + b^3} 5. 1x−2x+5\frac{1}{x} - \frac{2}{x+5} Which of the above are rational expressions?

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  4. 4
    A square matrix A is called orthogonal if (where A' is the transpose of A)

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  5. 5
    If A, B and C are subsets of a Universal set, then which one of the following is not\textbf{\text{not}} correct? (where A' is the complement of A)

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  6. 6
    Let x be the number of integers lying between 2999 and 8001 which have at least two digits equal. Then x is equal to

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  7. 7
    The sum of the series 3−1+13−19+⋯3 - 1 + \frac{1}{3} - \frac{1}{9} + \cdots is equal to

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  8. Consider the information given below and answer the two (02) items that follow: A survey was conducted among 300 students. It was found that 125 students like to play cricket, 145 students like to play football and 90 students like to play tennis. 32 students like to play exactly two games out of the three games.
    8
    How many students like to play all the three games?

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  9. 9
    How many students like to play exactly only one game?

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  10. 10
    If α\alpha and β\beta (≠0)(\neq 0) are the roots of the quadratic equation x2+αx−β=0x^2 + \alpha x - \beta = 0, then the quadratic expression −x2+αx+β-x^2 + \alpha x + \beta where x∈Rx \in \mathbb{R} has

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  11. 11
    What is the coefficient of the middle term in the binomial expansion of (2+3x)4(2 + 3x)^4?

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  12. 12
    For a square matrix A, which of the following properties hold? 1. (A−1)−1=A(A^{-1})^{-1} = A 2. det⁡(A−1)=1det⁡A\det(A^{-1}) = \frac{1}{\det A} 3. (λA)−1=λA−1(\lambda A)^{-1} = \lambda A^{-1} where λ\lambda is a scalar Select the correct answer using the code given below:

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  13. 13
    Which one of the following factors does the expansion of the determinant ∣xy3x25y39x310y527∣\begin{vmatrix} x & y & 3 \\ x^2 & 5y^3 & 9 \\ x^3 & 10y^5 & 27 \end{vmatrix} contain?

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  14. 14
    What is the adjoint of the matrix (cos⁡(−θ)−sin⁡(−θ)−sin⁡(−θ)cos⁡(−θ))\begin{pmatrix} \cos(-\theta) & -\sin(-\theta) \\ -\sin(-\theta) & \cos(-\theta) \end{pmatrix}?

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  15. 15
    What is the value of (−1+i32)3n+(−1−i32)3n\left(\frac{-1+i\sqrt{3}}{2}\right)^{3n} + \left(\frac{-1-i\sqrt{3}}{2}\right)^{3n}, where i=−1i = \sqrt{-1}?

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  16. 16
    There are 17 cricket players, out of which 5 players can bowl. In how many ways can a team of 11 players be selected so as to include 3 bowlers?

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  17. 17
    What is the value of log⁡927+log⁡832\log_9 27 + \log_8 32?

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  18. 18
    If A and B are two invertible square matrices of same order, then what is (AB)−1(AB)^{-1} equal to?

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  19. 19
    If a+b+c=0a + b + c = 0, then one of the solutions of ∣a−xcbcb−xabac−x∣=0\begin{vmatrix} a-x & c & b \\ c & b-x & a \\ b & a & c-x \end{vmatrix} = 0 is

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  20. 20
    What should be the value of x so that the matrix (24−8x)\begin{pmatrix} 2 & 4 \\ -8 & x \end{pmatrix} does not\textbf{\text{not}} have an inverse?

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  21. 21
    The system of equations 2x+y−3z=52x + y - 3z = 5, 3x−2y+2z=53x - 2y + 2z = 5 and 5x−3y−z=165x - 3y - z = 16

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  22. 22
    Which one of the following is correct in respect of the cube roots of unity?

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  23. 23
    If u, v and w (all positive) are the pthp^{\text{th}}, qthq^{\text{th}} and rthr^{\text{th}} terms of a GP, then the determinant of the matrix (ln⁡up1ln⁡vq1ln⁡wr1)\begin{pmatrix} \ln u & p & 1 \\ \ln v & q & 1 \\ \ln w & r & 1 \end{pmatrix} is

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  24. 24
    Let the coefficient of the middle term of the binomial expansion of (1+x)2n(1+x)^{2n} be α\alpha and those of two middle terms of the binomial expansion of (1+x)2n−1(1+x)^{2n-1} be β\beta and γ\gamma. Which one of the following relations is correct?

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  25. 25
    Let A={x∈R:−1≤x≤1}A = \{x \in \mathbb{R} : -1 \leq x \leq 1\}, B={y∈R:−1≤y≤1}B = \{y \in \mathbb{R} : -1 \leq y \leq 1\} and S be the subset of A×BA \times B, defined by S={(x,y)∈A×B:x2+y2=1}S = \{(x,y) \in A \times B : x^2 + y^2 = 1\}. Which one of the following is correct?

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  26. 26
    Let TrT_r be the rthr^{\text{th}} term of an AP for r=1,2,3,…r = 1, 2, 3, \ldots. If for some distinct positive integers m and n we have Tm=1nT_m = \frac{1}{n} and Tn=1mT_n = \frac{1}{m}, then what is TmnT_{mn} equal to?

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  27. 27
    Suppose f(x) is such a quadratic expression that it is positive for all real x. If g(x)=f(x)+f′(x)+f′′(x)g(x) = f(x) + f'(x) + f''(x), then for any real x

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  28. 28
    Consider the following in respect of matrices A, B and C of same order: 1. (A+B+C)′=A′+B′+C′(A+B+C)' = A' + B' + C' 2. (AB)′=A′B′(AB)' = A'B' 3. (ABC)′=C′B′A′(ABC)' = C'B'A' where A' is the transpose of the matrix A. Which of the above are correct?

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  29. 29
    The sum of the binary numbers (11011)2(11011)_2, (10110110)2(10110110)_2 and (10011x0y)2(10011x0y)_2 is the binary number (101101101)2(101101101)_2. What are the values of x and y?

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  30. 30
    Let matrix B be the adjoint of a square matrix A, ll be the identity matrix of same order as A. If kk (≠0)(\neq 0) is the determinant of the matrix A, then what is AB equal to?

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  31. 31
    If (0.2)x=2(0.2)^x = 2 and log⁡102=0.3010\log_{10} 2 = 0.3010, then what is the value of x to the nearest tenth?

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  32. 32
    The total number of 5-digit numbers that can be composed of distinct digits from 0 to 9 is

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  33. 33
    What is the determinant of the matrix (xyy+zzxz+xyzx+y)\begin{pmatrix} x & y & y+z \\ z & x & z+x \\ y & z & x+y \end{pmatrix}?

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  34. 34
    If A, B and C are the angles of a triangle and ∣1111+sin⁡A1+sin⁡B1+sin⁡Csin⁡A+sin⁡2Asin⁡B+sin⁡2Bsin⁡C+sin⁡2C∣=0\begin{vmatrix} 1 & 1 & 1 \\ 1+\sin A & 1+\sin B & 1+\sin C \\ \sin A+\sin^2 A & \sin B+\sin^2 B & \sin C+\sin^2 C \end{vmatrix} = 0, then which one of the following is correct?

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  35. 35
    Consider the following in respect of matrices A and B of same order: 1. A2−B2=(A+B)(A−B)A^2-B^2=(A+B)(A-B) 2. (A−I)(I+A)=O⇔A2=I(A-I)(I+A)=O \Leftrightarrow A^2=I where I is the identity matrix and O is the null matrix. Which of the above is/are correct?

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  36. 36
    What is 2tan⁡θ1+tan⁡2θ\frac{2\tan\theta}{1+\tan^2\theta} equal to?

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  37. 37
    If sec⁡(θ−α)\sec(\theta-\alpha), sec⁡θ\sec\theta and sec⁡(θ+α)\sec(\theta+\alpha) are in AP, where cos⁡α≠1\cos\alpha \neq 1, then what is the value of sin⁡2θ+cos⁡α\sin^2\theta + \cos\alpha?

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  38. 38
    If A+B+C=180°A + B + C = 180°, then what is sin⁡2A−sin⁡2B−sin⁡2C\sin 2A - \sin 2B - \sin 2C equal to?

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  39. 39
    A balloon is directly above one end of a bridge. The angle of depression of the other end of the bridge from the balloon is 48°. If the height of the balloon above the bridge is 122 m, then what is the length of the bridge?

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  40. 40
    A is an angle in the fourth quadrant. It satisfies the trigonometric equation 3(3−tan⁡2A−cot⁡A)2=13(3-\tan^2 A - \cot A)^2 = 1. Which one of the following is a value of A?

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  41. 41
    The top of a hill observed from the top and bottom of a building of height h is at angles of elevation π6\frac{\pi}{6} and π3\frac{\pi}{3} respectively. What is the height of the hill?

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  42. 42
    What is/are the solution(s) of the trigonometric equation csc⁡x+cot⁡x=3\csc x + \cot x = \sqrt{3}, where 0<x<2π0 < x < 2\pi?

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  43. 43
    If θ=π8\theta = \frac{\pi}{8}, then what is the value of (2cos⁡θ+1)10(2cos⁡2θ−1)10(2cos⁡θ−1)10(2cos⁡4θ−1)10(2\cos\theta+1)^{10}(2\cos 2\theta-1)^{10}(2\cos\theta-1)^{10}(2\cos 4\theta-1)^{10}?

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  44. 44
    If cos⁡α\cos\alpha and cos⁡β\cos\beta (0<α<β<π)(0 < \alpha < \beta < \pi) are the roots of the quadratic equation 4x2−3=04x^2 - 3 = 0, then what is the value of sec⁡α×sec⁡β\sec\alpha \times \sec\beta?

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  45. 45
    Consider the following values of x: 1. 8 2. −4-4 3. 16\frac{1}{6} 4. −14-\frac{1}{4} Which of the above values of x is/are the solution(s) of the equation tan⁡−1(2x)+tan⁡−1(3x)=π4\tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}?

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  46. 46
    If the second term of a GP is 2 and the sum of its infinite terms is 8, then the GP is

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  47. 47
    If a, b, c are in AP or GP or HP, then a−bb−c\frac{a-b}{b-c} is equal to

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  48. 48
    What is the sum of all three-digit numbers that can be formed using all the digits 3, 4 and 5, when repetition of digits is not\textbf{\text{not}} allowed?

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  49. 49
    The ratio of roots of the equations ax2+bx+c=0ax^2 + bx + c = 0 and px2+qx+r=0px^2 + qx + r = 0 are equal. If D1D_1 and D2D_2 are respective discriminants, then what is D1D2\frac{D_1}{D_2} equal to?

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  50. 50
    If A=sin⁡2θ+cos⁡4θA = \sin^2\theta + \cos^4\theta, then for all real θ\theta, which one of the following is correct?

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  51. 51
    The equation of a circle whose end points of a diameter are (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is

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  52. 52
    The second degree equation x2+4y2−2x−4y+2=0x^2 + 4y^2 - 2x - 4y + 2 = 0 represents

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  53. 53
    The angle between the two lines lx+my+n=0lx + my + n = 0 and l′x+m′y+n′=0l'x + m'y + n' = 0 is given by tan⁡−1θ\tan^{-1}\theta. What is θ\theta equal to?

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  54. 54
    Consider the following statements: 1. The distance between the lines y=mx+c1y=mx+c_1 and y=mx+c2y=mx+c_2 is ∣c1−c2∣1+m2\frac{|c_1-c_2|}{\sqrt{1+m^2}}. 2. The distance between the lines ax+by+c1=0ax+by+c_1=0 and ax+by+c2=0ax+by+c_2=0 is ∣c1−c2∣a2+b2\frac{|c_1-c_2|}{\sqrt{a^2+b^2}}. 3. The distance between the lines x=c1x=c_1 and x=c2x=c_2 is ∣c1−c2∣|c_1-c_2|. Which of the above statements are correct?

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  55. 55
    What is the equation of straight line passing through the point of intersection of the lines x2+y3=1\frac{x}{2}+\frac{y}{3}=1 and x3+y2=1\frac{x}{3}+\frac{y}{2}=1, and parallel to the line 4x+5y−6=04x+5y-6=0?

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  56. 56
    What is the distance of the point (2,3,4)(2, 3, 4) from the plane 3x−6y+2z+11=03x - 6y + 2z + 11 = 0?

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  57. 57
    Coordinates of the points O, P, Q and R are respectively (0,0,0)(0,0,0), (4,6,2m)(4,6,2m), (2,0,2n)(2,0,2n) and (2,4,6)(2,4,6). Let L, M, N and K be points on the sides OR, OP, PQ and QR respectively such that LMNK is a parallelogram whose two adjacent sides LK and LM are each of length 2\sqrt{2}. What are the values of m and n respectively?

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  58. 58
    The line x−12=y−23=z−34\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} is given by

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  59. 59
    Consider the following statements: 1. The angle between the planes 2x−y+z=12x-y+z=1 and x+y+2z=3x+y+2z=3 is π3\frac{\pi}{3}. 2. The distance between the planes 6x−3y+6z+2=06x-3y+6z+2=0 and 2x−y+2z+4=02x-y+2z+4=0 is 109\frac{10}{9}. Which of the above statements is/are correct?

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  60. 60
    Consider the following statements: Statement I: If the line segment joining the points P(m, n) and Q(r, s) subtends an angle α\alpha at the origin, then cos⁡α=ms−nr(m2+n2)(r2+s2)\cos\alpha=\frac{ms-nr}{\sqrt{(m^2+n^2)(r^2+s^2)}}. Statement II: In any triangle ABC, it is true that a2=b2+c2−2bccos⁡Aa^2=b^2+c^2-2bc\cos A. Which one of the following is correct in respect of the above two statements?

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  61. 61
    What is the area of the triangle with vertices (x1,1x1),(x2,1x2),(x3,1x3)\left(x_1,\frac{1}{x_1}\right), \left(x_2,\frac{1}{x_2}\right), \left(x_3,\frac{1}{x_3}\right)?

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  62. 62
    If y-axis touches the circle x2+y2+gx+fy+c4=0x^2+y^2+gx+fy+\frac{c}{4}=0, then the normal at this point intersects the circle at the point

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  63. 63
    Let ∣a⃗∣≠0,∣b⃗∣≠0|\vec{a}|\neq 0, |\vec{b}|\neq 0. (a⃗+b⃗)⋅(a⃗+b⃗)=∣a⃗∣2+∣b⃗∣2(\vec{a}+\vec{b})\cdot(\vec{a}+\vec{b})=|\vec{a}|^2+|\vec{b}|^2 holds if and only if

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  64. 64
    If r⃗=xi^+yj^+zk^\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}, then what is r⃗⋅(i^+j^+k^)\vec{r}\cdot(\hat{i}+\hat{j}+\hat{k}) equal to?

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  65. 65
    A unit vector perpendicular to each of the vectors 2i^−j^+k^2\hat{i}-\hat{j}+\hat{k} and 3i^−4j^−k^3\hat{i}-4\hat{j}-\hat{k} is

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  66. 66
    If ∣a⃗∣=3,∣b⃗∣=4|\vec{a}|=3, |\vec{b}|=4 and ∣a⃗−b⃗∣=5|\vec{a}-\vec{b}|=5, then what is the value of ∣a⃗+b⃗∣|\vec{a}+\vec{b}|?

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  67. 67
    Let a⃗,b⃗\vec{a}, \vec{b} and c⃗\vec{c} be three mutually perpendicular vectors each of unit magnitude. If A⃗=a⃗+b⃗+c⃗\vec{A}=\vec{a}+\vec{b}+\vec{c}, B⃗=a⃗−b⃗+c⃗\vec{B}=\vec{a}-\vec{b}+\vec{c} and C⃗=a⃗−b⃗−c⃗\vec{C}=\vec{a}-\vec{b}-\vec{c}, then which one of the following is correct?

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  68. 68
    What is (a⃗−b⃗)×(a⃗+b⃗)(\vec{a}-\vec{b})\times(\vec{a}+\vec{b}) equal to?

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  69. 69
    A spacecraft located at i^+2j^+3k^\hat{i}+2\hat{j}+3\hat{k} is subjected to a force λk^\lambda\hat{k} by firing a rocket. The spacecraft is subjected to a moment of magnitude

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  70. 70
    In a triangle ABC, if taken in order, consider the following statements: 1. AB⃗+BC⃗+CA⃗=0⃗\vec{AB}+\vec{BC}+\vec{CA}=\vec{0} 2. AB⃗+BC⃗−CA⃗=0⃗\vec{AB}+\vec{BC}-\vec{CA}=\vec{0} 3. AB⃗−BC⃗+CA⃗=0⃗\vec{AB}-\vec{BC}+\vec{CA}=\vec{0} 4. BA⃗−BC⃗+CA⃗=0⃗\vec{BA}-\vec{BC}+\vec{CA}=\vec{0} How many of the above statements are correct?

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  71. 71
    Let the slope of the curve y=cos⁡−1(sin⁡x)y=\cos^{-1}(\sin x) be tan⁡θ\tan\theta. Then the value of θ\theta in the interval (0,π)(0,\pi) is

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  72. 72
    If f(x)=x−1x−4f(x)=\frac{\sqrt{x-1}}{x-4} defines a function on R\mathbb{R}, then what is its domain?

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  73. 73
    Consider the function f(x)={sin⁡2x5xif x≠0215if x=0f(x) = \begin{cases} \frac{\sin 2x}{5x} & \text{if } x \neq 0 \\ \frac{2}{15} & \text{if } x = 0 \end{cases}. Which one of the following is correct in respect of the function?

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  74. 74
    For the function f(x)=∣x−3∣f(x)=|x-3|, which one of the following is not\textbf{\text{not}} correct?

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  75. 75
    If the function f(x)=2x−sin⁡−1x2x+tan⁡−1xf(x)=\frac{2x-\sin^{-1}x}{2x+\tan^{-1}x} is continuous at each point in its domain, then what is the value of f(0)?

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  76. 76
    If f(x)=25−x2f(x)=\sqrt{25-x^2}, then what is lim⁡x→1f(x)−f(1)x−1\lim_{x\to1}\frac{f(x)-f(1)}{x-1} equal to?

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  77. 77
    If y=tan⁡−1(5−2tan⁡x2+5tan⁡x)y=\tan^{-1}\left(\frac{5-2\tan\sqrt{x}}{2+5\tan\sqrt{x}}\right), then what is dydx\frac{dy}{dx} equal to?

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  78. 78
    Which one of the following is correct in respect of the function f(x)=xsin⁡x+cos⁡x+12cos⁡2xf(x)=x\sin x+\cos x+\frac{1}{2}\cos^2 x?

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  79. 79
    What is lim⁡θ→01−cos⁡θθ\lim_{\theta\to0}\frac{\sqrt{1-\cos\theta}}{\theta} equal to?

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  80. 80
    A function f:A→Rf: A\to\mathbb{R} is defined by the equation f(x)=x2−4x+5f(x)=x^2-4x+5 where A=(1,4)A=(1,4). What is the range of the function?

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  81. 81
    What is ∫ab[x] dx+∫ab[−x] dx\int_a^b [x]\,dx + \int_a^b [-x]\,dx equal to, where [.][.] is the greatest integer function?

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  82. 82
    What is ∫28∣x−5∣ dx\int_2^8 |x-5|\,dx equal to?

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  83. 83
    What is ∫sin⁡3xcos⁡x dx\int\sin^3 x\cos x\,dx equal to?

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  84. 84
    What is ∫eln⁡(tan⁡x) dx\int e^{\ln(\tan x)}\,dx equal to?

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  85. 85
    What is ∫−11{ddx(tan⁡−11x)}dx\int_{-1}^{1}\left\{\frac{d}{dx}\left(\tan^{-1}\frac{1}{x}\right)\right\}dx equal to?

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  86. 86
    In which one of the following intervals is the function f(x)=x2−5x+6f(x)=x^2-5x+6 decreasing?

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  87. 87
    The differential equation of the family of curves y=pcos⁡(ax)+qsin⁡(ax)y=p\cos(ax)+q\sin(ax), where p, q are arbitrary constants, is

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  88. 88
    The equation of the curve passing through the point (−1,−2)(-1,-2) which satisfies dydx=−x2−1x3\frac{dy}{dx}=-x^2-\frac{1}{x^3}, is

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  89. 89
    What is the order of the differential equation whose solution is y=acos⁡x+bsin⁡x+ce−x+dy=a\cos x+b\sin x+ce^{-x}+d, where a, b, c and d are arbitrary constants?

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  90. 90
    What is the solution of the differential equation ln⁡(dydx)=ax+by\ln\left(\frac{dy}{dx}\right)=ax+by?

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  91. 91
    If u=eaxsin⁡bxu=e^{ax}\sin bx and v=eaxcos⁡bxv=e^{ax}\cos bx, then what is ududx+vdvdxu\frac{du}{dx}+v\frac{dv}{dx} equal to?

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  92. 92
    If y=sin⁡(ln⁡x)y=\sin(\ln x), then which one of the following is correct?

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  93. 93
    A flower-bed in the form of a sector has been fenced by a wire of 40 m length. If the flower-bed has the greatest possible area, then what is the radius of the sector?

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  94. 94
    What is the minimum value of [x(x−1)+1]13[x(x-1)+1]^{\frac{1}{3}}, where 0≤x≤10\leq x\leq 1?

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  95. 95
    If y=∣sin⁡x∣∣x∣y=|\sin x|^{|x|}, then what is the value of dydx\frac{dy}{dx} at x=−π6x=-\frac{\pi}{6}?

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  96. 96
    What is d1−sin⁡2xdx\frac{d\sqrt{1-\sin 2x}}{dx} equal to, where π4<x<π2\frac{\pi}{4}<x<\frac{\pi}{2}?

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  97. 97
    What is ∫dxa2sin⁡2x+b2cos⁡2x\int\frac{dx}{a^2\sin^2x+b^2\cos^2x} equal to?

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  98. 98
    Let f(x+y)=f(x)f(y)f(x+y)=f(x)f(y) and f(x)=1+xg(x)ϕ(x)f(x)=1+xg(x)\phi(x), where lim⁡x→0g(x)=a\lim_{x\to0}g(x)=a and lim⁡x→0ϕ(x)=b\lim_{x\to0}\phi(x)=b. Then what is f′(x)f'(x) equal to?

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  99. 99
    What is the solution of the differential equation dxdy=x+y+1x+y−1\frac{dx}{dy}=\frac{x+y+1}{x+y-1}?

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  100. 100
    What is lim⁡x→π62sin⁡2x+sin⁡x−12sin⁡2x−3sin⁡x+1\lim_{x\to\frac{\pi}{6}}\frac{2\sin^2x+\sin x-1}{2\sin^2x-3\sin x+1} equal to?

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  101. 101
    If two dice are thrown and at least one of the dice shows 5, then the probability that the sum is 10 or more is

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  102. 102
    The correlation coefficient computed from a set of 30 observations is 0.8. Then the percentage of variation not explained by linear regression is

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  103. 103
    The average age of a combined group of men and women is 25 years. If the average age of the group of men is 26 years and that of the group of women is 21 years, then the percentage of men and women in the group is respectively

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  104. 104
    If sin⁡β\sin\beta is the harmonic mean of sin⁡α\sin\alpha and cos⁡α\cos\alpha, and sin⁡θ\sin\theta is the arithmetic mean of sin⁡α\sin\alpha and cos⁡α\cos\alpha, then which of the following is/are correct? 1. 2sin⁡(α+π4)sin⁡β=sin⁡2α\sqrt{2}\sin\left(\alpha+\frac{\pi}{4}\right)\sin\beta=\sin 2\alpha 2. 2sin⁡θ=cos⁡(α−π4)\sqrt{2}\sin\theta=\cos\left(\alpha-\frac{\pi}{4}\right) Select the correct answer using the code given below:

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  105. 105
    Let A, B and C be three mutually exclusive and exhaustive events associated with a random experiment. If P(B)=1.5 P(A)P(B)=1.5\,P(A) and P(C)=0.5 P(B)P(C)=0.5\,P(B), then P(A) is equal to

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  106. 106
    In a bolt factory, machines X, Y, Z manufacture bolts that are respectively 25%, 35% and 40% of the factory's total output. The machines X, Y, Z respectively produce 2%, 4% and 5% defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it was manufactured by machine X?

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  107. 107
    8 coins are tossed simultaneously. The probability of getting at least 6 heads is

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  108. 108
    Three groups of children contain 3 girls and 1 boy; 2 girls and 2 boys; 1 girl and 3 boys. One child is selected at random from each group. The probability that the three selected consist of 1 girl and 2 boys is

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  109. 109
    Consider the following statements: 1. If 10 is added to each entry on a list, then the average increases by 10. 2. If 10 is added to each entry on a list, then the standard deviation increases by 10. 3. If each entry on a list is doubled, then the average doubles. Which of the above statements are correct?

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  110. 110
    The variance of 25 observations is 4. If 2 is added to each observation, then the new variance of the resulting observations is

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  111. 111
    If xi>0,yi>0x_i>0, y_i>0 (i=1,2,3,…,n)(i=1,2,3,\ldots,n) are the values of two variables X and Y with geometric means P and Q respectively, then the geometric mean of XY\frac{X}{Y} is

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  112. 112
    If the probability of simultaneous occurrence of two events A and B is p and the probability that exactly one of A, B occurs is q, then which of the following is/are correct? 1. P(Aˉ)+P(Bˉ)=2−2p−qP(\bar{A})+P(\bar{B})=2-2p-q 2. P(Aˉ∩Bˉ)=1−p−qP(\bar{A}\cap\bar{B})=1-p-q Select the correct answer using the code given below:

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  113. 113
    If the regression coefficient of Y on X is −6-6, and the correlation coefficient between X and Y is −12-\frac{1}{2}, then the regression coefficient of X on Y would be

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  114. 114
    The set of bivariate observations (x1,y1),(x2,y2),…,(xn,yn)(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n) are such that all the values are distinct and all the observations fall on a straight line with non-zero slope. Then the possible values of the correlation coefficient between x and y are

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  115. 115
    Two integers x and y are chosen with replacement from the set {0,1,2,…,10}\{0,1,2,\ldots,10\}. The probability that ∣x−y∣>5|x-y|>5 is

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  116. 116
    An analysis of monthly wages paid to the workers in two firms A and B belonging to the same industry gives the following result: Number of workers: Firm A = 500, Firm B = 600; Average monthly wage: Firm A = ₹1860, Firm B = ₹1750; Variance of distribution of wages: Firm A = 81, Firm B = 100. The average of monthly wage and variance of distribution of wages of all the workers in the firms A and B taken together are

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  117. 117
    Three dice having digits 1, 2, 3, 4, 5 and 6 on their faces are marked I, II and III and rolled. Let x, y and z represent the number on die-I, die-II and die-III respectively. What is the number of possible outcomes such that x>y>zx>y>z?

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  118. 118
    Which one of the following can be obtained from an ogive?

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  119. 119
    In any discrete series (when all values are not same), if x represents mean deviation about mean and y represents standard deviation, then which one of the following is correct?

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  120. 120
    In which one of the following cases would you expect to get a negative correlation?

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