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NDA 2024 (I), 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
    Let AA and BB be matrices of order 3×33\times 3. If ∣A∣=122|A| = \dfrac{1}{2\sqrt{2}} and ∣B∣=1729|B| = \dfrac{1}{729}, then what is the value of ∣2B (adj(3A))∣\left|2B\,(\mathrm{adj}(3A))\right|?

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  2. 2
    If zz is any complex number and iz3+z2−z+i=0iz^3+z^2-z+i=0, where i=−1i=\sqrt{-1}, then what is the value of (∣z∣+1)2(|z|+1)^2?

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  3. 3
    What is the sum of all four-digit numbers formed by using all digits 0,1,4,50,1,4,5 without repetition of digits?

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  4. 4
    If x,yx,y and zz are the cube roots of unity, then what is the value of xy+yz+zxxy+yz+zx?

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  5. 5
    A man has 7 relatives (4 women and 3 men). His wife also has 7 relatives (3 women and 4 men). In how many ways can they invite 3 women and 3 men so that 3 of them are the man's relatives and 3 are his wife's relatives?

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  6. 6
    A triangle PQRPQR is such that 3 points lie on side PQPQ, 4 points on QRQR and 5 points on RPRP respectively. Triangles are constructed using these points as vertices. What is the number of triangles so formed?

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  7. 7
    If log⁡ba=p\log_b a=p, log⁡dc=2p\log_d c=2p and log⁡fe=3p\log_f e=3p, then what is (ace)1p(ace)^{\frac{1}{p}} equal to?

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  8. 8
    If −2-2 and 33 are roots of the equation a0+a1x+a2x2+a3x3+x4=0a_0+a_1 x+a_2 x^2+a_3 x^3+x^4=0 where a0,a1,a2,a3a_0,a_1,a_2,a_3 are integers, then which one of the following is correct?

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  9. 9
    Let z1z_1 and z2z_2 be two complex numbers such that ∣z1+z2z1−z2∣=1\left|\dfrac{z_1+z_2}{z_1-z_2}\right|=1, then what is Re ⁣(z1z2)+1\text{Re}\!\left(\dfrac{z_1}{z_2}\right)+1 equal to?

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  10. 10
    If 26!=n⋅8k26!=n\cdot 8^k, where kk and nn are positive integers, then what is the maximum value of kk?

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  11. 11
    Consider the following statements in respect of two nonsingular matrices AA and BB of the same order nn: A. adj(AB)=(adj A)(adj B)\text{adj}(AB)=(\text{adj }A)(\text{adj }B) B. adj(AB)=adj(BA)\text{adj}(AB)=\text{adj}(BA) C. (AB) adj(AB)−∣AB∣In(AB)\,\text{adj}(AB)-|AB|I_n is a null matrix of order nn How many of the above statements are correct?

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  12. 12
    Consider the following statements in respect of a nonsingular matrix AA of order nn: A. A adj(AT)=A(adj A)TA\,\text{adj}(A^T)=A(\text{adj }A)^T B. If A2=AA^2=A, then AA is identity matrix of order nn. C. If A3=AA^3=A, then AA is identity matrix of order nn. Which of the statements given above are correct?

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  13. 13
    How many four-digit natural numbers are there such that all of the digits are even?

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  14. 14
    If ω≠1\omega\neq1 is a cube root of unity, then what are the solutions of (z−100)3+1000=0(z-100)^3+1000=0?

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  15. 15
    What is (1+i)4+(1−i)4(1+i)^4+(1-i)^4 equal to, where i=−1i=\sqrt{-1}?

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  16. 16
    Consider the following statements in respect of a skew-symmetric matrix AA of order 3: A. All diagonal elements are zero. B. The sum of all the diagonal elements of the matrix is zero. C. AA is an orthogonal matrix. Which of the statements given above are correct?

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  17. 17
    Four digit numbers are formed by using the digits 1,2,3,51,2,3,5 without repetition of digits. How many of them are divisible by 4?

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  18. 18
    What is the remainder when 21202^{120} is divided by 7?

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  19. 19
    For what value of nn is the determinant ∣(94)(93)(10n−2)(116)(115)(12n)(m7)(m6)(m+1n+1)∣=0\begin{vmatrix}\binom{9}{4}&\binom{9}{3}&\binom{10}{n-2}\\\binom{11}{6}&\binom{11}{5}&\binom{12}{n}\\\binom{m}{7}&\binom{m}{6}&\binom{m+1}{n+1}\end{vmatrix}=0 for every m>nm>n?

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  20. 20
    If ABCABC is a triangle, then what is the value of the determinant ∣cos⁡Csin⁡B0tan⁡A0sin⁡B0tan⁡(B+C)cos⁡C∣\begin{vmatrix}\cos C & \sin B & 0\\\tan A & 0 & \sin B\\0 & \tan(B+C) & \cos C\end{vmatrix}?

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  21. 21
    What is the number of different matrices, each having 4 entries, that can be formed using 1,2,3,41,2,3,4 (repetition is allowed)?

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  22. 22
    Let A={x∈R:−1<x<1}A=\{x\in\mathbb{R}:-1<x<1\}. Which of the following is/are bijective functions from AA to itself? (A) f(x)=x∣x∣f(x)=x|x| (B) g(x)=cos⁡(πx)g(x)=\cos(\pi x) Select the correct answer using the code given below:

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  23. 23
    Let RR be a relation on the open interval (−1,1)(-1,1) and is given by R={(x,y):∣x+y∣<2}R=\{(x,y):|x+y|<2\}. Then which of the following is correct?

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  24. 24
    For any three non-empty sets A,B,CA,B,C, what is (A∪B)−{(A−B)∪(B−A)∪(A∩B)}(A\cup B)-\{(A-B)\cup(B-A)\cup(A\cap B)\} equal to?

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  25. 25
    If a,b,ca,b,c are the sides of triangle ABCABC, then what is ∣a2bsin⁡Acsin⁡Absin⁡A1cos⁡Acsin⁡Acos⁡A1∣\begin{vmatrix}a^2 & b\sin A & c\sin A\\b\sin A & 1 & \cos A\\c\sin A & \cos A & 1\end{vmatrix} equal to?

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  26. 26
    If a,b,ca,b,c are in AP; b,c,db,c,d are in GP; c,d,ec,d,e are in HP, then which of the following is/are correct? (A) a,ca,c and ee are in GP. (B) 1a,1c,1e\dfrac{1}{a},\dfrac{1}{c},\dfrac{1}{e} are in GP. Select the correct answer using the code given below:

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  27. 27
    What is the number of solutions of log⁡4(x−1)=log⁡2(x−3)\log_4(x-1)=\log_2(x-3)?

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  28. 28
    For x≥y>1x\geq y>1, let log⁡x ⁣xy+log⁡y ⁣yx=k\log_x\!\dfrac{x}{y}+\log_y\!\dfrac{y}{x}=k, then the value of kk can never be equal to

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  29. 29
    If A=(sin⁡2θ−cos⁡2θ0cos⁡2θsin⁡2θ0001)A=\begin{pmatrix}\sin2\theta & -\cos2\theta & 0\\\cos2\theta & \sin2\theta & 0\\0 & 0 & 1\end{pmatrix}, then which of the following statements is/are correct? (A) A−1=adj AA^{-1}=\text{adj }A (B) AA is skew-symmetric matrix. (C) A−1=ATA^{-1}=A^T Select the correct answer using the code given below:

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  30. 30
    What is the coefficient of x10x^{10} in the expansion of (1−x220)−1(x2−1x2)−5\left(1-\dfrac{x^2}{20}\right)^{-1}\left(x^2-\dfrac{1}{x^2}\right)^{-5}?

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  31. 31
    If the 4th term in the expansion of (mx+1x)n\left(mx+\dfrac{1}{x}\right)^n is 52\dfrac{5}{2}, then what is the value of mnmn?

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  32. 32
    If a,ba,b and c  (a>0, c>0)c\;(a>0,\,c>0) are in GP, then consider the following in respect of the equation ax2+bx+c=0ax^2+bx+c=0: (A) The equation has imaginary roots. (B) The ratio of the roots of the equation is 1:ω1:\omega where ω\omega is a cube root of unity. (C) The product of roots of the equation is b2a2\dfrac{b^2}{a^2}. Which of the statements given above are correct?

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  33. 33
    If x2+mx+nx^2+mx+n is an integer for all integral values of xx, then which of the following is/are correct? (A) mm must be an integer. (B) nn must be an integer. Select the correct answer using the code given below:

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  34. 34
    In a binomial expansion of (x+y)2n+1(x−y)2n+1(x+y)^{2n+1}(x-y)^{2n+1}, the sum of middle terms is zero. What is the value of x2y2\dfrac{x^2}{y^2}?

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  35. 35
    Let A={1,2,3,4,5}A=\{1,2,3,4,5\} and B={6,7}B=\{6,7\}. What is the number of onto functions from AA to BB?

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  36. 36
    What is 3cos⁡10°−sin⁡10°sin⁡25°cos⁡25°\dfrac{\sqrt{3}\cos10°-\sin10°}{\sin25°\cos25°} equal to?

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  37. 37
    What is sin⁡9°−cos⁡9°\sin9°-\cos9° equal to?

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  38. 38
    If in a triangle ABCABC, sin⁡3A+sin⁡3B+sin⁡3C=3sin⁡Asin⁡Bsin⁡C\sin^3A+\sin^3B+\sin^3C=3\sin A\sin B\sin C, then what is the value of the determinant ∣abcbcacab∣\begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix}?

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  39. 39
    If cos⁡−1x=sin⁡−1x\cos^{-1}x=\sin^{-1}x, then which one of the following is correct?

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  40. 40
    What is the number of solutions of (sin⁡θ−cos⁡θ)2=2(\sin\theta-\cos\theta)^2=2 where −π<θ<π-\pi<\theta<\pi?

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  41. 41
    Triangle ABCABC such that ∠C=60°\angle C=60°, then what is cos⁡A+cos⁡Bcos⁡ ⁣(A−B2)\dfrac{\cos A+\cos B}{\cos\!\left(\dfrac{A-B}{2}\right)} equal to?

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  42. 42
    What is 15+cot⁡2 ⁣(π4−2cot⁡−13)\sqrt{15+\cot^2\!\left(\dfrac{\pi}{4}-2\cot^{-1}3\right)} equal to?

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  43. 43
    What is sin⁡10°⋅sin⁡50°+sin⁡50°⋅sin⁡250°+sin⁡250°⋅sin⁡10°\sin10°\cdot\sin50°+\sin50°\cdot\sin250°+\sin250°\cdot\sin10° equal to?

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  44. 44
    What is tan⁡−1 ⁣ab−tan⁡−1 ⁣a−ba+b\tan^{-1}\!\dfrac{a}{b}-\tan^{-1}\!\dfrac{a-b}{a+b} equal to?

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  45. 45
    Under which one of the following conditions does the equation (cos⁡β−1)x2+(cos⁡β)x+sin⁡β=0(\cos\beta-1)x^2+(\cos\beta)x+\sin\beta=0 in xx have a real root for β∈[0,π]\beta\in[0,\pi]?

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  46. 46
    In a triangle ABCABC, AB=16AB=16 cm, BC=63BC=63 cm and AC=65AC=65 cm. What is the value of cos⁡2A+cos⁡2B+cos⁡2C\cos2A+\cos2B+\cos2C?

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  47. 47
    If f(θ)=11+tan⁡θf(\theta)=\dfrac{1}{1+\tan\theta} and α+β=5π4\alpha+\beta=\dfrac{5\pi}{4}, then what is the value of f(α)⋅f(β)f(\alpha)\cdot f(\beta)?

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  48. 48
    If tan⁡α\tan\alpha and tan⁡β\tan\beta are the roots of the equation x2−6x+8=0x^2-6x+8=0, then what is the value of cos⁡(2α+2β)\cos(2\alpha+2\beta)?

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  49. 49
    What is the value of tan⁡65°+2tan⁡45°−2tan⁡40°−tan⁡25°\tan65°+2\tan45°-2\tan40°-\tan25°?

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  50. 50
    Consider the following statements: A. In a triangle ABCABC, if cot⁡A⋅cot⁡B⋅cot⁡C>0\cot A\cdot\cot B\cdot\cot C>0, then the triangle is an acute angled triangle. B. In a triangle ABCABC, if tan⁡A⋅tan⁡B⋅tan⁡C>0\tan A\cdot\tan B\cdot\tan C>0, then the triangle is an obtuse angled triangle. Which of the statements given above is/are correct?

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  51. 51
    If (a,b)(a,b) is the centre and cc is the radius of the circle x2+y2+2x+6y+1=0x^2+y^2+2x+6y+1=0, then what is the value of a2+b2+c2a^2+b^2+c^2?

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  52. 52
    If (1,−1,2)(1,-1,2) and (2,1,−1)(2,1,-1) are the endpoints of a diameter of a sphere x2+y2+z2+2ux+2vy+2wz−1=0x^2+y^2+z^2+2ux+2vy+2wz-1=0, then what is u+v+wu+v+w equal to?

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  53. 53
    The number of points represented by the equation x=5x=5 on the xyxy-plane is

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  54. 54
    If ⟨l,m,n⟩\langle l,m,n\rangle are the direction cosines of a normal to the plane 2x−3y+6z+4=02x-3y+6z+4=0, then what is the value of 49(7l2+m2−n2)49(7l^2+m^2-n^2)?

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  55. 55
    A line through (1,−1,2)(1,-1,2) with direction ratios ⟨3,2,2⟩\langle3,2,2\rangle meets the plane x+2y+3z=18x+2y+3z=18. What is the point of intersection?

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  56. 56
    If pp is the perpendicular distance from the origin to the plane passing through (1,0,0),(0,1,0)(1,0,0),(0,1,0) and (0,0,1)(0,0,1), then what is 3p23p^2 equal to?

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  57. 57
    If the direction cosines ⟨l,m,n⟩\langle l,m,n\rangle of a line are connected by relation l+2m+n=0l+2m+n=0, 2l−2m+3n=02l-2m+3n=0, then what is the value of l2+m2−n2l^2+m^2-n^2?

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  58. 58
    If a variable line passes through the point of intersection of the lines x+2y−1=0x+2y-1=0 and 2x−y−1=02x-y-1=0 and meets the coordinate axes at AA and BB, then what is the locus of the mid-point of ABAB?

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  59. 59
    What is the equation of the straight line passing through the point (−sin⁡θ,cos⁡θ)(-\sin\theta,\cos\theta) and perpendicular to the line xcos⁡θ+ysin⁡θ=9x\cos\theta+y\sin\theta=9?

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  60. 60
    Two points PP and QQ lie on line y=2x+3y=2x+3. These two points PP and QQ are at a distance of 2\sqrt{2} units from another point R(1,5)R(1,5). What are the coordinates of PP and QQ?

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  61. 61
    If two sides of a square lie on the lines 2x+y−3=02x+y-3=0 and 4x+2y+5=04x+2y+5=0, then what is the area of the square in square units?

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  62. 62
    Triangle ABCABC has A(3,5)A(3,5). The mid-points of sides ABAB and ACAC are at (−1,2)(-1,2) and (6,4)(6,4) respectively. What are the coordinates of the centroid of the triangle ABCABC?

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  63. 63
    Triangle ABCABC is an acute angled isosceles triangle. Two equal sides ABAB and ACAC lie on the lines 7x−y−3=07x-y-3=0 and x+y−5=0x+y-5=0. If θ\theta is one of the equal angles, then what is cot⁡θ\cot\theta equal to?

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  64. 64
    In the parabola y2=8xy^2=8x, the focal distance of a point PP lying on it is 8 units. Which of the following statements is/are correct? (A) The coordinates of PP can be (6,43)(6,4\sqrt{3}). (B) The perpendicular distance of PP from the directrix of the parabola is 8 units. Select the correct answer using the code given below:

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  65. 65
    What is the eccentricity of the ellipse if the angle between the straight lines joining the foci to an extremity of the minor axis is 90°90°?

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  66. 66
    Let a⃗=i^−j^+k^\vec{a}=\hat{i}-\hat{j}+\hat{k} and b⃗=i^+2j^−k^\vec{b}=\hat{i}+2\hat{j}-\hat{k}. If a⃗×(b⃗×a⃗)=αi^−βj^+γk^\vec{a}\times(\vec{b}\times\vec{a})=\alpha\hat{i}-\beta\hat{j}+\gamma\hat{k}, then what is the value of α+β+γ\alpha+\beta+\gamma?

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  67. 67
    If a vector of magnitude 2 units makes an angle π3\dfrac{\pi}{3} with 2i^2\hat{i}, π4\dfrac{\pi}{4} with 3j^3\hat{j} and an acute angle θ\theta with 4k^4\hat{k}, then what are the components of the vector?

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  68. 68
    Consider the following in respect of moment of a force: (A) The moment of force about a point is independent of point of application of force. (B) The moment of a force about a line is a vector quantity. Which of the statements given above is/are correct?

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  69. 69
    For any vector r⃗\vec{r}, what is (r⃗⋅i^)(r⃗×i^)+(r⃗⋅j^)(r⃗×j^)+(r⃗⋅k^)(r⃗×k^)(\vec{r}\cdot\hat{i})(\vec{r}\times\hat{i})+(\vec{r}\cdot\hat{j})(\vec{r}\times\hat{j})+(\vec{r}\cdot\hat{k})(\vec{r}\times\hat{k}) equal to?

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  70. 70
    Let a⃗\vec{a} and b⃗\vec{b} be two vectors of magnitude 4 inclined at an angle π3\dfrac{\pi}{3}, then what is the angle between a⃗\vec{a} and a⃗−b⃗\vec{a}-\vec{b}?

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  71. 71
    Let y1(x)y_1(x) and y2(x)y_2(x) be two solutions of the differential equation dydx=x\dfrac{dy}{dx}=x. If y1(0)=0y_1(0)=0 and y2(0)=4y_2(0)=4, then what is the number of points of intersection of the curves y1(x)y_1(x) and y2(x)y_2(x)?

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  72. 72
    The differential equation representing the curve y=ex(acos⁡x+bsin⁡x)y=e^x(a\cos x+b\sin x), where aa and bb are arbitrary constants, is

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  73. 73
    If f(x)=ax−bf(x)=ax-b and g(x)=cx+dg(x)=cx+d are such that f(g(x))=g(f(x))f(g(x))=g(f(x)), then which one of the following holds?

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  74. 74
    What is ∫−11(3sin⁡x−sin⁡3x)cos⁡2x dx\displaystyle\int_{-1}^{1}(3\sin x-\sin3x)\cos^2x\,dx equal to?

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  75. 75
    What are the order and degree respectively of the differential equation (2−(dydx)2)0.6=d2ydx2\left(2-\left(\dfrac{dy}{dx}\right)^2\right)^{0.6}=\dfrac{d^2y}{dx^2}?

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  76. 76
    If dydx=2exy3,  y(0)=12\dfrac{dy}{dx}=2e^x y^3,\;y(0)=\dfrac{1}{2}, then what is 4y2(2−ex)4y^2(2-e^x) equal to?

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  77. 77
    Let p=∫abf(x) dxp=\displaystyle\int_a^b f(x)\,dx and q=∫ab∣f(x)∣ dxq=\displaystyle\int_a^b|f(x)|\,dx. If f(x)=e−xf(x)=e^{-x}, then which one of the following is correct?

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  78. 78
    What is ∫0π/2a+sin⁡x2(a+sin⁡x+cos⁡x) dx\displaystyle\int_0^{\pi/2}\dfrac{a+\sin x}{2(a+\sin x+\cos x)}\,dx equal to?

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  79. 79
    The non-negative values of bb for which the function 16x33−4bx2+x\dfrac{16x^3}{3}-4bx^2+x has neither maximum nor minimum in the range x>0x>0 is

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  80. 80
    Which one of the following is correct in respect of f(x)=1∣x∣−xf(x)=\dfrac{1}{|x|-x} and g(x)=1x−∣x∣g(x)=\dfrac{1}{x-|x|}?

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  81. Given that ∫3cos⁡x+4sin⁡x2cos⁡x+5sin⁡x dx=αx29+β29ln⁡∣2cos⁡x+5sin⁡x∣+c\displaystyle\int\dfrac{3\cos x+4\sin x}{2\cos x+5\sin x}\,dx=\dfrac{\alpha x}{29}+\dfrac{\beta}{29}\ln|2\cos x+5\sin x|+c.
    81
    What is the value of α\alpha?

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  82. 82
    What is the value of β\beta?

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  83. 83
    Let f(x)=xln⁡x  (x>1)f(x)=\dfrac{x}{\ln x}\;(x>1). Consider the following statements: (A) f(x)f(x) is increasing in the interval (e,∞)(e,\infty). (B) f(x)f(x) is decreasing in the interval (1,e)(1,e). (C) 9ln⁡7>7ln⁡99\ln7>7\ln9. Which of the statements given above are correct?

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  84. 84
    Let f(x)=xln⁡x  (x>1)f(x)=\dfrac{x}{\ln x}\;(x>1). Consider the following statements: (A) f′′(e)=1ef''(e)=\dfrac{1}{e}. (B) f(x)f(x) attains a local minimum value at x=ex=e. (C) A local minimum value of f(x)f(x) is ee. Which of the statements given above are correct?

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  85. Let f(x)f(x) and g(x)g(x) be two functions such that g(x)=x−1xg(x)=x-\dfrac{1}{x} and f ⁣∘ ⁣g(x)=x3−1x3f\!\circ\! g(x)=x^3-\dfrac{1}{x^3}.
    85
    What is g[f(x)−3x]g[f(x)-3x] equal to?

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  86. 86
    What is f′′(x)f''(x) equal to?

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  87. 87
    Let f(x)=∣x∣+1f(x)=|x|+1 and g(x)=[x]−1g(x)=[x]-1, where [⋅][\cdot] is the greatest integer function. Let h(x)=f(x)⋅g(x)h(x)=f(x)\cdot g(x). Consider the following statements: (A) f(x)f(x) is differentiable for all x<0x<0. (B) g(x)g(x) is continuous at x=0.0001x=0.0001. (C) The derivative of g(x)g(x) at x=2.5x=2.5 is 1. Which of the statements given above are correct?

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  88. 88
    Let f(x)=∣x∣+1f(x)=|x|+1, g(x)=[x]−1g(x)=[x]-1, h(x)=f(x)g(x)h(x)=\dfrac{f(x)}{g(x)}, where [⋅][\cdot] is the greatest integer function. What is lim⁡x→0−h(x)+lim⁡x→0+h(x)\displaystyle\lim_{x\to0^-}h(x)+\lim_{x\to0^+}h(x) equal to?

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  89. Let ϕ(a)=∫aa+100π∣sin⁡x∣ dx\phi(a)=\displaystyle\int_a^{a+100\pi}|\sin x|\,dx.
    89
    What is ϕ(a)\phi(a) equal to?

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  90. 90
    What is ϕ′(a)\phi'(a) equal to?

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  91. 91
    A differentiable function f(x)f(x) has a local maximum at x=0x=0. Let y=2f(x)+ax−by=2f(x)+ax-b. Which of the following is/are correct? (A) f′(0)=0f'(0)=0 (B) f′′(0)<0f''(0)<0 Select the correct answer using the code given below:

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  92. 92
    A differentiable function f(x)f(x) has a local maximum at x=0x=0. Let y=2f(x)+ax−by=2f(x)+ax-b. The function yy has a relative maxima at x=0x=0 for

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  93. Let f(x)=∣x−1∣f(x)=|x-1|, g(x)=[x]g(x)=[x] and h(x)=f(x)⋅g(x)h(x)=f(x)\cdot g(x) where [⋅][\cdot] is the greatest integer function.
    93
    What is ∫−10h(x) dx\displaystyle\int_{-1}^{0}h(x)\,dx equal to?

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  94. 94
    What is ∫02h(x) dx\displaystyle\int_{0}^{2}h(x)\,dx equal to?

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  95. Let ∫dxx+1−x−1=α(x+1)3/2+β(x−1)3/2+c\displaystyle\int\dfrac{dx}{\sqrt{x+1}-\sqrt{x-1}}=\alpha(x+1)^{3/2}+\beta(x-1)^{3/2}+c.
    95
    What is the value of α\alpha?

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  96. 96
    What is the value of β\beta?

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  97. The circle x2+y2−2x=0x^2+y^2-2x=0 is partitioned by line y=xy=x in two segments. Let A1,A2A_1,A_2 be the areas of major and minor segments respectively.
    97
    What is the value of A1A_1?

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  98. 98
    What is the value of 2(A1+A2)A1−3A2\dfrac{2(A_1+A_2)}{A_1-3A_2}?

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  99. Let 3f(x)+f ⁣(1x)=1x+13f(x)+f\!\left(\dfrac{1}{x}\right)=\dfrac{1}{x}+1.
    99
    What is f(x)f(x) equal to?

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  100. 100
    What is 8∫12f(x) dx8\displaystyle\int_{1}^{2}f(x)\,dx equal to?

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  101. 101
    A bag contains 5 black and 4 white balls. A man selects two balls at random. What is the probability that both of these are of the same colour?

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  102. 102
    If a random variable XX follows binomial distribution with mean 5 and variance 4523\frac{45}{23}, P(X=3)=λ⋅4λP(X=3)=\lambda\cdot4^{\lambda}, then what is the value of λ\lambda?

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  103. 103
    From data (−4,1),(−1,2),(2,7)(-4,1),(-1,2),(2,7) and (3,1)(3,1), the regression line of yy on xx is obtained as y=a+bxy=a+bx, then what is the value of 2a+15b2a+15b?

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  104. 104
    Let x+2y+1=0x+2y+1=0 and 2x+3y+4=02x+3y+4=0 be two lines of regression computed from some bivariate data. If θ\theta is the acute angle between them, then what is the value of 488tan⁡3θ488\tan 3\theta?

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  105. 105
    If two random variables XX and YY are connected by relation 2X−3Y5X+4Y=4\dfrac{2X-3Y}{5X+4Y}=4 and XX follows Binomial distribution with parameters n=10n=10 and p=12p=\dfrac{1}{2}, then what is the variance of YY?

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  106. 106
    If a,b,ca,b,c are in HP, then what is 1b−a+1b−c\dfrac{1}{b-a}+\dfrac{1}{b-c} equal to?

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  107. 107
    An edible oil is sold at 150, 200, 250, 300 rupees per litre in four consecutive years. Assuming that an equal amount of money is spent on oil by a family every year, what is the average price of oil (approximately) per litre?

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  108. 108
    If the letters of the word "TIRUPATI" are written down at random, then what is the probability that both Ts are always consecutive?

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  109. 109
    Let m=77nm=77^n. The index nn is given a positive integral value at random. What is the probability that the value of mm will have 1 in the units place?

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  110. 110
    Three different numbers are selected at random from the first 15 natural numbers. What is the probability that the product of two of the numbers is equal to the third?

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  111. Let AA and BB be two events such that P(A∪B)≥0.75P(A\cup B)\geq0.75 and 0.125≤P(A∩B)≤0.3750.125\leq P(A\cap B)\leq0.375.
    111
    What is the minimum value of P(A)+P(B)P(A)+P(B)?

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  112. 112
    What is the maximum value of P(A)+P(B)P(A)+P(B)?

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  113. Events A,B,CA,B,C: P(A)=0.6,P(B)=0.4,P(C)=0.5,P(A∪B)=0.8,P(A∩C)=0.3,P(A∩B∩C)=0.2P(A)=0.6,P(B)=0.4,P(C)=0.5,P(A\cup B)=0.8,P(A\cap C)=0.3,P(A\cap B\cap C)=0.2 and P(A∪B∪C)≥0.85P(A\cup B\cup C)\geq0.85.
    113
    What is the minimum value of P(B∩C)P(B\cap C)?

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  114. 114
    What is the maximum value of P(B∩C)P(B\cap C)?

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  115. An unbiased coin is tossed nn times. The probability of getting at least one tail is pp and the probability of at least two tails is qq and p−q=532p-q=\dfrac{5}{32}.
    115
    What is the value of nn?

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  116. An unbiased coin is tossed nn times. The probability of getting at least one tail is pp, at least two tails is qq, and p−q=532p-q=\dfrac{5}{32}. With n=5n=5:
    116
    What is the value of p+qp+q?

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  117. Distribution: xi=1,2,3,…,nx_i=1,2,3,\ldots,n with fi=1,2−1,2−2,…,2−(n−1)f_i=1,2^{-1},2^{-2},\ldots,2^{-(n-1)}.
    117
    What is ∑ixifi\displaystyle\sum_i x_i f_i equal to?

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  118. Distribution: xi=1,2,…,nx_i=1,2,\ldots,n with fi=1,2−1,…,2−(n−1)f_i=1,2^{-1},\ldots,2^{-(n-1)}.
    118
    What is the mean of the distribution?

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  119. Marks obtained by 10 students: 24,47,18,32,19,15,21,35,50,41. Largest five observations: 32,35,41,47,50.
    119
    What is the mean deviation of the largest five observations?

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  120. 120
    What is the variance of the largest five observations?

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