PYQ Vault

NDA 2021 (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
    If x2+x+1=0x^{2} + x + 1 = 0, then what is the value of x199+x200+x201x^{199} + x^{200} + x^{201}?

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  2. 2
    If xx, yy, zz are in GP, then which of the following is/are correct? 1. ln⁡(3x)\ln(3x), ln⁡(3y)\ln(3y), ln⁡(3z)\ln(3z) are in AP 2. xyz+ln⁡(x)xyz + \ln(x), xyz+ln⁡(y)xyz + \ln(y), xyz+ln⁡(z)xyz + \ln(z) are in HP Select the correct answer using the code given below.

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  3. 3
    If log⁡102\log_{10} 2, log⁡10(2x−1)\log_{10}(2^{x}-1), log⁡10(2x+3)\log_{10}(2^{x}+3) are in AP, then what is xx equal to?

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  4. 4
    Let S={2,3,4,5,6,7,9}S=\{2,3,4,5,6,7,9\}. How many different 3-digit numbers (with all digits different) from SS can be made which are less than 500?

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  5. 5
    If p=(1111⋯ up to n digits)p = (1111\cdots\text{ up to }n\text{ digits}), then what is the value of 9p2+p9p^{2}+p?

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  6. 6
    The quadratic equation 3x2−(k2+5k)x+3k2−5k=03x^{2}-(k^{2}+5k)x+3k^{2}-5k=0 has real roots of equal magnitude and opposite sign. Which one of the following is correct?

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  7. 7
    If an=n(n!)a_{n}=n(n!), then what is a1+a2+a3+⋯+a10a_{1}+a_{2}+a_{3}+\cdots+a_{10} equal to?

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  8. 8
    If pp and qq are the non-zero roots of the equation x2+px+q=0x^{2}+px+q=0, then how many possible values can qq have?

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  9. 9
    If Δ=∣abcdefghi∣\Delta=\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}, then what is ∣3d+5g4a+7g6g3e+5h4b+7h6h3f+5i4c+7i6i∣\begin{vmatrix}3d+5g&4a+7g&6g\\3e+5h&4b+7h&6h\\3f+5i&4c+7i&6i\end{vmatrix} equal to?

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  10. 10
    If 1b+c\frac{1}{b+c}, 1c+a\frac{1}{c+a}, 1a+b\frac{1}{a+b} are in HP, then which of the following is/are correct? 1. aa, bb, cc are in AP 2. (b+c)2(b+c)^{2}, (c+a)2(c+a)^{2}, (a+b)2(a+b)^{2} are in GP Select the correct answer using the code given below.

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  11. 11
    If A=(1a01)A=\begin{pmatrix}1&a\\0&1\end{pmatrix} where a∈Na\in\mathbb{N}, then what is A100−A50−2A25A^{100}-A^{50}-2A^{25} equal to?

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  12. 12
    If ∣a−ba−b−c−ab−a+b−c−a−b−a−b+c∣−kabc=0\begin{vmatrix}a&-b&a-b-c\\-a&b&-a+b-c\\-a&-b&-a-b+c\end{vmatrix}-kabc=0 (a≠0,b≠0,c≠0)(a\ne0,b\ne0,c\ne0), then what is the value of kk?

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  13. 13
    What is ∑n=18n+7in\sum_{n=1}^{8n+7}i^{n} equal to, where i=−1i=\sqrt{-1}?

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  14. 14
    If z=x+iyz=x+iy, where i=−1i=\sqrt{-1}, then what does the equation zzˉ+∣z∣2+4(z+zˉ)−48=0z\bar{z}+|z|^{2}+4(z+\bar{z})-48=0 represent?

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  15. 15
    Which one of the following is a square root of 2a+2a2+b22a+2\sqrt{a^{2}+b^{2}}, where a,b∈Ra,b\in\mathbb{R}?

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  16. 16
    If sin⁡θ\sin\theta and cos⁡θ\cos\theta are the roots of the equation ax2+bx+c=0ax^{2}+bx+c=0, then which one of the following is correct?

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  17. 17
    If C(n,4)C(n,4), C(n,5)C(n,5) and C(n,6)C(n,6) are in AP, then what is the value of nn?

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  18. 18
    How many 4-letter words (with or without meaning) containing two vowels can be constructed using only the letters (without repetition) of the word 'LUCKNOW'?

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  19. 19
    Suppose 20 distinct points are placed randomly on a circle. Which of the following statements is/are correct? 1. The number of straight lines that can be drawn by joining any two of these points is 380. 2. The number of triangles that can be drawn by joining any three of these points is 1140. Select the correct answer using the code given below.

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  20. 20
    How many terms are there in the expansion of (a2b2+b2a2+2)21\left(\frac{a^{2}}{b^{2}}+\frac{b^{2}}{a^{2}}+2\right)^{21} where a≠0a\ne0, b≠0b\ne0?

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  21. 21
    For what values of kk is the system of equations 2k2x+3y−1=02k^{2}x+3y-1=0, 7x−2y+3=07x-2y+3=0, 6kx+y+1=06kx+y+1=0 consistent?

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  22. 22
    The inverse of a matrix AA is given by (−2132−12)\begin{pmatrix}-2&1\\\frac{3}{2}&-\frac{1}{2}\end{pmatrix}. What is AA equal to?

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  23. 23
    What is the period of the function f(x)=ln⁡(2+sin⁡2x)f(x)=\ln(2+\sin^{2}x)?

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  24. 24
    If sin⁡(A+B)=1\sin(A+B)=1 and 2sin⁡(A−B)=12\sin(A-B)=1, where 0<A,B<π20<A,B<\frac{\pi}{2}, then what is tan⁡A:tan⁡B\tan A:\tan B equal to?

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  25. 25
    Consider a regular polygon with 10 sides. What is the number of triangles that can be formed by joining the vertices which have no common side with any of the sides of the polygon?

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  26. 26
    Consider all the real roots of the equation x4−10x2+9=0x^{4}-10x^{2}+9=0. What is the sum of the absolute values of the roots?

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  27. 27
    Consider the expansion of (1+x)n(1+x)^{n}. Let pp, qq, rr and ss be the coefficients of first, second, nnth and (n+1)(n+1)th terms respectively. What is (ps+qr)(ps+qr) equal to?

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  28. 28
    Let sin⁡−1x+sin⁡−1y+sin⁡−1z=3π2\sin^{-1}x+\sin^{-1}y+\sin^{-1}z=\frac{3\pi}{2} for 0≤x,y,z≤10\le x,y,z\le1. What is the value of x1000+y1001+z1002x^{1000}+y^{1001}+z^{1002}?

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  29. 29
    Let sin⁡x+sin⁡y=cos⁡x+cos⁡y\sin x+\sin y=\cos x+\cos y for all x,y∈Rx,y\in\mathbb{R}. What is tan⁡ ⁣(x2+y2)\tan\!\left(\frac{x}{2}+\frac{y}{2}\right) equal to?

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  30. 30
    Let A=(02−20)A=\begin{pmatrix}0&2\\-2&0\end{pmatrix} and (mI+nA)2=A(mI+nA)^{2}=A where mm, nn are positive real numbers and II is the identity matrix. What is (m+n)(m+n) equal to?

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  31. 31
    What is the value of cot⁡ ⁣[sin⁡−1 ⁣35+cot⁡−1 ⁣32]\cot\!\left[\sin^{-1}\!\frac{3}{5}+\cot^{-1}\!\frac{3}{2}\right]?

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  32. 32
    Let 4sin⁡2x=34\sin^{2}x=3, where 0≤x≤π0\le x\le\pi. What is tan⁡3x\tan 3x equal to?

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  33. 33
    Let pp, qq and 33 be respectively the first, third and fifth terms of an AP. Let dd be the common difference. If the product (pq)(pq) is minimum, then what is the value of dd?

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  34. 34
    Consider the following statements in respect of the roots of the equation x3−8=0x^{3}-8=0: 1. The roots are non-collinear. 2. The roots lie on a circle of unit radius. Which of the above statements is/are correct?

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  35. 35
    Let the equation sec⁡x⋅cosec⁡x=p\sec x\cdot\cosec x=p have a solution, where pp is a positive real number. What should be the smallest value of pp?

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  36. 36
    For what value of θ\theta, where 0<θ<π20<\theta<\frac{\pi}{2}, does sin⁡θ+sin⁡θcos⁡θ\sin\theta+\sin\theta\cos\theta attain maximum value?

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  37. 37
    Consider the following statements in respect of sets: 1. The union over intersection of sets is distributive. 2. The complement of union of two sets is equal to intersection of their complements. 3. If the difference of two sets is equal to empty set, then the two sets must be equal. Which of the above statements are correct?

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  38. 38
    Consider three sets XX, YY and ZZ having 6, 5 and 4 elements respectively. All these 15 elements are distinct. Let S=(X−Y)∪ZS=(X-Y)\cup Z. How many proper subsets does SS have?

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  39. 39
    Consider the following statements in respect of relations and functions: 1. All relations are functions but all functions are not relations. 2. A relation from AA to BB is a subset of Cartesian product A×BA\times B. 3. A relation in AA is a subset of Cartesian product A×AA\times A. Which of the above statements are correct?

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  40. 40
    If log⁡102⋅log⁡210+log⁡10(10x)=2\log_{10}2\cdot\log_{2}10+\log_{10}(10^{x})=2, then what is the value of xx?

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  41. 41
    Let ABCABC be a triangle. If cos⁡2A+cos⁡2B+cos⁡2C=−1\cos 2A+\cos 2B+\cos 2C=-1, then which one of the following is correct?

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  42. 42
    What is the value of the following determinant? ∣cos⁡Ctan⁡A0sin⁡B0−tan⁡A0sin⁡Bcos⁡C∣\begin{vmatrix}\cos C&\tan A&0\\\sin B&0&-\tan A\\0&\sin B&\cos C\end{vmatrix}

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  43. 43
    Suppose set AA consists of first 250 natural numbers that are multiples of 3 and set BB consists of first 200 even natural numbers. How many elements does A∪BA\cup B have?

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  44. 44
    Let SkS_{k} denote the sum of first kk terms of an AP. What is S30S20−S10\frac{S_{30}}{S_{20}-S_{10}} equal to?

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  45. 45
    If the roots of the equation 4x2−(5k+1)x+5k=04x^{2}-(5k+1)x+5k=0 differ by unity, then which one of the following is a possible value of kk?

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  46. 46
    Consider the digits 3, 5, 7, 9. What is the number of 5-digit numbers formed by these digits in which each of these four digits appears?

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  47. 47
    How many distinct 2×22\times2 matrices exist with all four entries taken from {1,2}\{1,2\}?

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  48. 48
    If i=−1i=\sqrt{-1}, then how many values does i−2ni^{-2n} have for different n∈Zn\in\mathbb{Z}?

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  49. 49
    If x=ab−cx=\frac{a}{b-c}, y=bc−ay=\frac{b}{c-a}, z=ca−bz=\frac{c}{a-b}, then what is the value of ∣1−xx11−y1z1∣\begin{vmatrix}1&-x&x\\1&1&-y\\1&z&1\end{vmatrix}?

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  50. 50
    Consider the following in respect of the matrix A=(111111111)A=\begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix}: 1. Inverse of AA does not exist 2. A3=AA^{3}=A 3. 3A=A23A=A^{2} Which of the above are correct?

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  51. Direction: Consider the following for the next two (02) items that follow. A circle is passing through the points (5, -8), (-2, 9) and (2, 1).
    51
    What are the coordinates of the centre of the circle?

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  52. 52
    If rr is the radius of the circle, then which one of the following is correct?

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  53. Direction: Consider the following for the next two (02) items that follow. The two vertices of an equilateral triangle are (0, 0) and (2, 2).
    53
    Consider the following statements: 1. The third vertex has at least one irrational coordinate. 2. The area is irrational. Which of the above statements is/are correct?

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  54. 54
    The difference of coordinates of the third vertex is

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  55. Direction: Consider the following for the next two (02) items that follow. The coordinates of three consecutive vertices of a parallelogram ABCD are A(1, 3), B(-1, 2) and C(3, 5).
    55
    What is the equation of the diagonal BDBD?

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  56. 56
    What is the area of the parallelogram?

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  57. Direction: Consider the following for the next two (02) items that follow. The equations of the sides AB, BC and CA of a triangle ABC are x - 2 = 0, y + 1 = 0 and x + 2y - 4 = 0 respectively.
    57
    What is the equation of the altitude through BB on ACAC?

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  58. 58
    What are the coordinates of circumcentre of the triangle?

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  59. Direction: Consider the following for the next two (02) items that follow. The two ends of the latus rectum of a parabola are (-2, 4) and (-2, -4).
    59
    What is the maximum number of parabolas that can be drawn through these two points as end points of latus rectum?

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  60. 60
    Consider the following statements in respect of such parabolas: 1. One of the parabolas passes through the origin (0, 0). 2. The focus of one of the parabolas lies at (-2, 0). Which of the above statements is/are correct?

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  61. 61
    The locus of a point P(x,y,z)P(x,y,z) which moves in such a way that z=7z=7 is a

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  62. 62
    Consider the following statements: 1. A line in space can have infinitely many direction ratios. 2. It is possible for certain line that the sum of the squares of direction cosines can be equal to sum of its direction cosines. Which of the above statements is/are correct?

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  63. 63
    The xyxy-plane divides the line segment joining the points (−1,3,4)(-1,3,4) and (2,−5,6)(2,-5,6)

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  64. 64
    The number of spheres of radius rr touching the coordinate axes is

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  65. 65
    ABCDEFGH is a cuboid with base ABCD. Let A(0, 0, 0), B(12, 0, 0), C(12, 6, 0) and G(12, 6, 4) be the vertices. If α\alpha is the angle between AB and AG; β\beta is the angle between AC and AG, then what is the value of cos⁡2α+cos⁡2β\cos 2\alpha+\cos 2\beta?

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  66. 66
    Let a⃗\vec{a}, b⃗\vec{b} and c⃗\vec{c} be unit vectors such that a⃗×b⃗\vec{a}\times\vec{b} is perpendicular to c⃗\vec{c}. If θ\theta is the angle between a⃗\vec{a} and b⃗\vec{b}, then which of the following is/are correct? 1. a⃗×b⃗=sin⁡θ c⃗\vec{a}\times\vec{b}=\sin\theta\,\vec{c} 2. a⃗⋅(b⃗×c⃗)=0\vec{a}\cdot(\vec{b}\times\vec{c})=0 Select the correct answer using the code given below.

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  67. 67
    If a⃗+3b⃗=3i^−j^\vec{a}+3\vec{b}=3\hat{i}-\hat{j} and 2a⃗+b⃗=i^−2j^2\vec{a}+\vec{b}=\hat{i}-2\hat{j}, then what is the angle between a⃗\vec{a} and b⃗\vec{b}?

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  68. 68
    If (a⃗+b⃗)(\vec{a}+\vec{b}) is perpendicular to a⃗\vec{a} and magnitude of b⃗\vec{b} is twice that of a⃗\vec{a}, then what is the value of (4a⃗+b⃗)⋅b⃗(4\vec{a}+\vec{b})\cdot\vec{b} equal to?

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  69. 69
    Let a⃗\vec{a}, b⃗\vec{b} and c⃗\vec{c} be three vectors such that a⃗\vec{a}, b⃗\vec{b} and c⃗\vec{c} are coplanar. Which of the following is/are correct? 1. (a⃗×b⃗)×c⃗(\vec{a}\times\vec{b})\times\vec{c} is coplanar with a⃗\vec{a} and b⃗\vec{b} 2. (a⃗×b⃗)×c⃗(\vec{a}\times\vec{b})\times\vec{c} is perpendicular to a⃗×b⃗\vec{a}\times\vec{b} Select the correct answer using the code given below.

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  70. 70
    If the position vectors of AA and BB are (2−1)i^−j^(\sqrt{2}-1)\hat{i}-\hat{j} and i^+(2+1)j^\hat{i}+(\sqrt{2}+1)\hat{j} respectively, then what is the magnitude of AB→\overrightarrow{AB}?

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  71. 71
    If y=(1+x)(1+x2)(1+x4)(1+x8)(1+x16)y=(1+x)(1+x^{2})(1+x^{4})(1+x^{8})(1+x^{16}), then what is dydx\frac{dy}{dx} at x=0x=0 equal to?

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  72. 72
    If y=cos⁡x⋅cos⁡4x⋅cos⁡8xy=\cos x\cdot\cos 4x\cdot\cos 8x, then what is 1ydydx\frac{1}{y}\frac{dy}{dx} at x=π4x=\frac{\pi}{4} equal to?

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  73. 73
    Let f(x)f(x) be a polynomial function such that f∘f(x)=x4f\circ f(x)=x^{4}. What is f′(1)f'(1) equal to?

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  74. 74
    What is lim⁡n→∞an+bnan−bn\lim_{n\to\infty}\frac{a^{n}+b^{n}}{a^{n}-b^{n}} where a>b>1a>b>1 equal to?

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  75. 75
    Let f(x)={1+x2k,0<x<2kx,2≤x<4f(x)=\begin{cases}1+\frac{x}{2k},&0<x<2\\kx,&2\le x<4\end{cases}. If lim⁡x→2f(x)\lim_{x\to2}f(x) exists, then what is the value of kk?

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  76. 76
    Consider the following statements in respect of f(x)=∣x∣−1f(x)=|x|-1: 1. f(x)f(x) is continuous at x=1x=1. 2. f(x)f(x) is differentiable at x=0x=0. Which of the above statements is/are correct?

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  77. 77
    If f(x)=[x]∣x∣f(x)=\frac{[x]}{|x|}, x≠0x\ne0, where [⋅][\cdot] denotes the greatest integer function, then what is the right-hand limit of f(x)f(x) at x=1x=1?

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  78. 78
    Consider the following statements in respect of the function f(x)=sin⁡ ⁣(1x2)f(x)=\sin\!\left(\frac{1}{x^{2}}\right), x≠0x\ne0: 1. It is continuous at x=0x=0, if f(0)=0f(0)=0. 2. It is continuous at x=2πx=\frac{2}{\sqrt{\pi}}. Which of the above statements is/are correct?

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  79. 79
    What is the range of the function f(x)=1−sin⁡xf(x)=1-\sin x defined on entire real line?

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  80. 80
    What is the slope of the tangent of y=cos⁡−1(cos⁡x)y=\cos^{-1}(\cos x) at x=−π4x=-\frac{\pi}{4}?

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  81. 81
    What is the integral of f(x)=1+x2+x4f(x)=1+x^{2}+x^{4} with respect to x2x^{2}?

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  82. 82
    Consider the following statements in respect of the function f(x)=x2+1f(x)=x^{2}+1 in the interval (1,2)(1,2): 1. The maximum value of the function is 5. 2. The minimum value of the function is 2. Which of the above statements is/are correct?

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  83. 83
    If f(x)f(x) satisfies f(1)=f(4)f(1)=f(4), then what is ∫14f′(x) dx\int_{1}^{4}f'(x)\,dx equal to?

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  84. 84
    What is ∫0π/2eln⁡(cos⁡x) dx\int_{0}^{\pi/2}e^{\ln(\cos x)}\,dx equal to?

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  85. 85
    If ∫1−sin⁡2x dx=Asin⁡x+Bcos⁡x+C\int\sqrt{1-\sin 2x}\,dx=A\sin x+B\cos x+C, where 0<x<π40<x<\frac{\pi}{4}, then which one of the following is correct?

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  86. 86
    What is the order of the differential equation of all ellipses whose axes are along the coordinate axes?

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  87. 87
    What is the degree of the differential equation of all circles touching both the coordinate axes in the first quadrant?

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  88. 88
    What is the differential equation of y=A−Bxy=A-\frac{B}{x}?

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  89. 89
    What is ∫0πln⁡ ⁣(tan⁡x2)dx\int_{0}^{\pi}\ln\!\left(\tan\frac{x}{2}\right)dx equal to?

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  90. 90
    Where does the tangent to the curve y=exy=e^{x} at the point (0,1)(0,1) meet the xx-axis?

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  91. 91
    Consider the following statements in respect of the function f(x)=x+1xf(x)=x+\frac{1}{x}: 1. The local maximum value of f(x)f(x) is less than its local minimum value. 2. The local maximum value of f(x)f(x) occurs at x=1x=1. Which of the above statements is/are correct?

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  92. 92
    What is the maximum area of a rectangle that can be inscribed in a circle of radius 2 units?

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  93. 93
    What is ∫dxx(x2+1)\int\frac{dx}{x(x^{2}+1)} equal to?

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  94. 94
    What is the derivative of eexe^{e^{x}} with respect to exe^{x}?

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  95. 95
    What is the condition that f(x)=x3+x2+kxf(x)=x^{3}+x^{2}+kx has no local extremum?

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  96. 96
    If f(x)=2xf(x)=2^{x}, then what is ∫210f′(x)f(x) dx\int_{2}^{10}\frac{f'(x)}{f(x)}\,dx equal to?

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  97. 97
    If ∫−20f(x) dx=k\int_{-2}^{0}f(x)\,dx=k, then ∫−20∣f(x)∣ dx\int_{-2}^{0}|f(x)|\,dx is

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  98. 98
    If the function f(x)=x2−kxf(x)=x^{2}-kx is monotonically increasing in the interval (1,∞)(1,\infty), then which one of the following is correct?

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  99. 99
    What is the area bounded by y=[x]y=[x], where [⋅][\cdot] is the greatest integer function, the xx-axis and the lines x=−1.5x=-1.5 and x=−1.8x=-1.8?

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  100. 100
    The tangent to the curve x2=yx^{2}=y at (1,1)(1,1) makes an angle θ\theta with the positive direction of xx-axis. Which one of the following is correct?

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  101. 101
    Consider the following relations for two events EE and FF: 1. P(E∩F)≥P(E)+P(F)−1P(E\cap F)\ge P(E)+P(F)-1 2. P(E∪F)=P(E)+P(F)+P(E∩F)P(E\cup F)=P(E)+P(F)+P(E\cap F) 3. P(E∪F)≤P(E)+P(F)P(E\cup F)\le P(E)+P(F) Which of the above relations is/are correct?

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  102. 102
    If P(A∣B)<P(A)P(A|B)<P(A), then which one of the following is correct?

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  103. 103
    When the measure of central tendency is available in the form of mean, which one of the following is the most reliable and accurate measure of variability?

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  104. 104
    A problem is given to three students AA, BB and CC, whose probabilities of solving the problem independently are 12\frac{1}{2}, 34\frac{3}{4} and pp respectively. If the probability that the problem can be solved is 2932\frac{29}{32}, then what is the value of pp?

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  105. 105
    In a cricket match, a batsman hits a six 8 times out of 60 balls he plays. What is the probability that on a ball played he does not hit a six?

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  106. Direction: Consider the following for the next two (02) items that follow. Two regression lines are given as 3x - 4y + 8 = 0 and 4x - 3y - 1 = 0.
    106
    Consider the following statements: 1. The regression line of yy on xx is y=34x+2y=\frac{3}{4}x+2. 2. The regression line of xx on yy is x=34y+14x=\frac{3}{4}y+\frac{1}{4}. Which of the above statements is/are correct?

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  107. 107
    Consider the following statements: 1. The coefficient of correlations rr is 34\frac{3}{4}. 2. The means of xx and yy are 3 and 4 respectively. Which of the above statements is/are correct?

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  108. Direction: Consider the following for the next two (02) items that follow. The marks obtained by 60 students in a certain subject out of 75 are given below: Marks: 15-20 (4), 20-25 (5), 25-30 (11), 30-35 (6), 35-40 (5), 40-45 (8), 45-50 (9), 50-55 (6), 55-60 (4), 60-65 (2).
    108
    What is the median?

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  109. 109
    What is the mode?

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  110. 110
    What is the mean of natural numbers contained in the interval [15,64][15,64]?

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  111. 111
    For the set of numbers x,x,x+2,x+3,x+10x,x,x+2,x+3,x+10, where xx is a natural number, which of the following is/are correct? 1. Mean >> Mode 2. Median >> Mean Select the correct answer using the code given below.

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  112. 112
    The mean of 10 observations is 5·5. If each observation is multiplied by 4 and subtracted from 44, then what is the new mean?

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  113. 113
    If gg is the geometric mean of 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, then which one of the following is correct?

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  114. 114
    If the harmonic mean of 60 and xx is 48, then what is the value of xx?

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  115. 115
    What is the mean deviation of first 10 even natural numbers?

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  116. 116
    If ∑i=110xi=110\sum_{i=1}^{10}x_{i}=110 and ∑i=110xi2=1540\sum_{i=1}^{10}x_{i}^{2}=1540, then what is the variance?

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  117. 117
    3-digit numbers are formed using the digits 1, 3, 7 without repetition of digits. A number is randomly selected. What is the probability that the number is divisible by 3?

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  118. 118
    What is the probability that the roots of the equation x2+x+n=0x^{2}+x+n=0 are real, where n∈Nn\in\mathbb{N} and n<4n<4?

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  119. 119
    If AA and BB are two events such that P(not A)=710P(\text{not }A)=\frac{7}{10}, P(not B)=310P(\text{not }B)=\frac{3}{10} and P(A∣B)=314P(A|B)=\frac{3}{14}, then what is P(B∣A)P(B|A) equal to?

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  120. 120
    Seven white balls and three black balls are randomly placed in a row. What is the probability that no two black balls are placed adjacently?

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