NDA Mathematics 10 April 2022

Q1. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next three (03) items that follow : The algebraic sum of the deviations of a set of values x~1~, x~2~, x~3~, ⋯, x~n~ measured from 100 is -20 and the algebraic sum of the deviations of the same set of values measured from 92 is 140. --- What is the algebraic sum of the deviations of the same set of values measured from 99?

Q2. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next three (03) items that follow : The algebraic sum of the deviations of a set of values x~1~, x~2~, x~3~, ⋯, x~n~ measured from 100 is -20 and the algebraic sum of the deviations of the same set of values measured from 92 is 140. --- What is the mean of the values?

Q3. mcq single +2.5 / 0.83

The frequency curve (assuming unimodal) corresponding to the data obtained in an experiment is skewed to the left. What conclusion can be drawn from the curve ?

Q4. mcq single +2.5 / 0.83

What is the coefficient of mean deviation of 21, 34, 23, 39, 26, 37, 40, 20, 33, 27 (taken from mean)?

Q5. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next three (03) items that follow : The algebraic sum of the deviations of a set of values x~1~, x~2~, x~3~, ⋯, x~n~ measured from 100 is -20 and the algebraic sum of the deviations of the same set of values measured from 92 is 140. --- If the algebraic sum of the deviations of the same set of values measured from y is 180, then what is the value of y ?

Q6. mcq single +2.5 / 0.83

**Passage:** Consider the following frequency distribution for the next three (03) items follow : Class 0-20 20-40 40-60 60-80 80-100 Frequency 17 p + q 32 p - 3q 19 The total frequency is 120. The mean is 50. --- What is the value of q ?

Q7. mcq single +2.5 / 0.83

**Passage:** Consider the following frequency distribution for the next three (03) items follow : Class 0-20 20-40 40-60 60-80 80-100 Frequency 17 p + q 32 p - 3q 19 The total frequency is 120. The mean is 50. --- If the frequency of each class is doubled, then what would be the mean?

Q8. mcq single +2.5 / 0.83

**Passage:** Consider the following frequency distribution for the next three (03) items follow : Class 0-20 20-40 40-60 60-80 80-100 Frequency 17 p + q 32 p - 3q 19 The total frequency is 120. The mean is 50. --- What is the value of p ?

Q9. mcq single +2.5 / 0.83

The variance of five positive observations is 3.6. If four of the observations are 2, 2, 4, 5 then what is the remaining observation ?

Q10. mcq single +2.5 / 0.83

If A = {{1, 2, 3}}, then how many elements are there in the power set of A?

Q11. mcq single +2.5 / 0.83

Consider the following statements in respect of the relation R in the set IN of natural numbers defined by xRy if x^(2) - 5xy + 4y^(2) = 0 : 1. R is reflexive 2. R is symmetric 3. R is transitive Which of the above statements is /are correct ?

Q12. mcq single +2.5 / 0.83

Consider all the subsets of the set A = {1, 2, 3, 4}. How many of them are supersets of the set {4}?

Q13. mcq single +2.5 / 0.83

Consider the following statements in respect of two non-empty sets A and B : 1. A ∪ B = A ∩ B if A = B 2. A Δ B = ϕ if A = B Which of the above statements is/are correct ?

Q14. mcq single +2.5 / 0.83

Consider the following statements in respect of two non-empty sets A and B : 1. x ∉ (A ∪ B) ⇒ x ∉ A or x ∉ B 2. x ∉ (A ∩ B) ⇒ x ∉ A and x ∉ B Which of the above statements is/are correct?

Q15. mcq single +2.5 / 0.83

Consider the following statements in respect of any relation R on a set A : 1. If R is reflexive, then R^(-1) is also reflexive 2. If R is symmetric, then R^(-1) is also symmetric 3. If R is transitive, then R^(-1) is also transitive Which of the above statements are correct?

Q16. mcq single +2.5 / 0.83

What is $\lim\limits_{x \to 0}x^3(\text{cosec} \ x)^2$ equal to ?

Q17. mcq single +2.5 / 0.83

Let y = [x + 1], -4 < x < -3 where [.] is the greatest integer function. What is the derivative of y with respect to x at x = -3.5?

Q18. mcq single +2.5 / 0.83

What is $\lim\limits_{x \to 1}\frac{x^3 - 1}{\sqrt x - 1}$ equal to?

Q19. mcq single +2.5 / 0.83

If a, b, c are in GP where a > 0, b > 0, c > 0, then which of the following are correct? 1. a^(2), b^(2), c^(2) are in GP 2. $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in GP 3. $\sqrt {a}, \sqrt{b}, \sqrt{c} $ are in GP Select the correct answer using the code given below :

Q20. mcq single +2.5 / 0.83

If the sum of the first 9 terms of an AP is equal to sum of the first 11 terms, then what is the sum of the first 20 terms ?

Q21. mcq single +2.5 / 0.83

If $\frac{a+b}{2}, b, \frac{b+c}{2}$ are in HP, then which one of the following is correct?

Q22. mcq single +2.5 / 0.83

**Passage:** Consider the following data for the next three (03) items that follow : The marks obtained by 51 students in a class are in AP with its first term 4 and common difference 3. --- What is the sum of the deviations measured from the median?

Q23. mcq single +2.5 / 0.83

**Passage:** Consider the following data for the next three (03) items that follow : The marks obtained by 51 students in a class are in AP with its first term 4 and common difference 3. --- What is the mean of the marks ?

Q24. mcq single +2.5 / 0.83

If the 5^(th) term of an AP is $\frac{1}{10}$ and its 10^(th) term is $\frac{1}{5},$ then what is the sum of first 50 terms ?

Q25. mcq single +2.5 / 0.83

What is the arithmetic mean of 50 terms of an AP with first term 4 and common difference 4 ?

Q26. mcq single +2.5 / 0.83

**Passage:** Consider the following data for the next three (03) items that follow : The marks obtained by 51 students in a class are in AP with its first term 4 and common difference 3. --- What is the median of the marks?

Q27. mcq single +2.5 / 0.83

Consider the following statements : 1. Dot product over vector addition is distributive 2. Cross product over vector addition is distributive 3. Cross product of vectors is associative Which of the above statements is/are correct ?

Q28. mcq single +2.5 / 0.83

Let $\vec a$ and $\vec b$ be two unit vectors such that $|\vec a - \vec b|<2.$ If 2θ is the angle between $\vec a$ and $\vec b,$ then which one of the following is correct ?

Q29. mcq single +2.5 / 0.83

If $4\hat i + \hat j - 3\hat k$ and $p\hat i + q\hat j - 2\hat k$ are collinear vectors, then what are the possible values of p and q respectively?

Q30. mcq single +2.5 / 0.83

If $\vec a, \vec b, \vec c$, are the position vectors of the vertices A, B, C respectively of a triangle ABC and G is the centroid of the triangle, then what is $\overrightarrow{AG}$ equal to ? .

Q31. mcq single +2.5 / 0.83

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-zero vectors such that $\vec{a}\times \vec{b} = \vec{c} $. Consider the following statements: 1. $\vec a$ is unique if $\vec b$ and $\vec c$ are given 2. $\vec c$ is unique if $\vec a$ and $\vec b$ are given Which of the above statements is/are correct?

Q32. mcq single +2.5 / 0.83

If 3 sin^(-1)x + cos^(-1)x = π, then what is x equal to?

Q33. mcq single +2.5 / 0.83

If $\tan^{-1} \left(\frac{1}{2}\right)+\tan^{-1} \left(\frac{x}{3}\right)=\frac{\pi}{4},$ where 0 < x < 6, then what is x equal to?

Q34. mcq single +2.5 / 0.83

**Passage:** Consider the following data for the next three (03) items that follow : There are 90 applicants for a job. Some of them are graduates. Some of them have less than three years experience. Number of graduates Number of non-graduates At least 3 years experience 18 9 Less than 3 years experience 36 27 Let G be the event that the first applicant interviewed is a graduate and T be the event that first applicant interviewed has at least 3 years experience. --- What is $P (G \cap \overline T)$ equal to?

Q35. mcq single +2.5 / 0.83

**Passage:** Consider the following data for the next three (03) items that follow : The incidence of suffering from a disease among workers in an industry has a chance of $33\frac{1}{3}\%.$ --- What is the probability that exactly 3 out of 6 workers suffer from a disease?

Q36. mcq single +2.5 / 0.83

**Passage:** Consider the following data for the next three (03) items that follow : There are 90 applicants for a job. Some of them are graduates. Some of them have less than three years experience. Number of graduates Number of non-graduates At least 3 years experience 18 9 Less than 3 years experience 36 27 Let G be the event that the first applicant interviewed is a graduate and T be the event that first applicant interviewed has at least 3 years experience. --- What is $P(\overline T | \overline G)$ equal to?

Q37. mcq single +2.5 / 0.83

Two digits out of 1, 2, 3, 4, 5 are chosen at random and multiplied together. What is the probability that the last digit in the product appears as 0?

Q38. mcq single +2.5 / 0.83

**Passage:** Consider the following data for the next three (03) items that follow : There are 90 applicants for a job. Some of them are graduates. Some of them have less than three years experience. Number of graduates Number of non-graduates At least 3 years experience 18 9 Less than 3 years experience 36 27 Let G be the event that the first applicant interviewed is a graduate and T be the event that first applicant interviewed has at least 3 years experience. --- What is $P(G | \overline T) $ equal to?

Q39. mcq single +2.5 / 0.83

**Passage:** Consider the following data for the next three (03) items that follow : The incidence of suffering from a disease among workers in an industry has a chance of $33\frac{1}{3}\%.$ --- What is the probability that at least one out of 6 workers suffer from a disease ?

Q40. mcq single +2.5 / 0.83

**Passage:** Consider the following data for the next three (03) items that follow : The incidence of suffering from a disease among workers in an industry has a chance of $33\frac{1}{3}\%.$ --- What is the probability that no one out of 6 workers suffers from a disease?

Q41. mcq single +2.5 / 0.83

A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height h. At a point on the plane the angles of elevation of the bottom and top of the flagstaff are θ and 2θ respectively. What is the height of the tower ?

Q42. mcq single +2.5 / 0.83

The shadow of a tower becomes x metre longer, when the angle of elevation of sun changes from 60° to θ. If the height of the tower is $\sqrt{3}x$ metre, then which one of the following is correct ?

Q43. mcq single +2.5 / 0.83

Consider the inequations 5x - 4y + 12 < 0, x + y < 2, x < 0 and y > 0. Which one of the following points lies in the common region?

Q44. mcq single +2.5 / 0.83

If the points with coordinates (-5, 0), (5p^(2), 10p) and (5q^(2), 10q) are collinear, then what is the value of pq where p ≠ q ?

Q45. mcq single +2.5 / 0.83

What is the equation of the locus of the mid-point of the line segment obtained by cutting the line x + y = p, (where p is a real number) by the coordinate axes ?

Q46. mcq single +2.5 / 0.83

What is the equation of the straight line which passes through the point (1, -2) and cuts off equal intercepts from the axes ?

Q47. mcq single +2.5 / 0.83

If the point (x, y) is equidistant from the points (2a, 0) and (0, 3a) where a > 0, then which one of the following is correct ?

Q48. mcq single +2.5 / 0.83

A straight line passes through the point of intersection of x + 2y + 2 = 0 and 2x - 3y - 3 = 0. It cuts equal intercepts in the fourth quadrant. What is the sum of the absolute values of the intercepts?

Q49. mcq single +2.5 / 0.83

Under which one of the following conditions are the lines ax + by + c = 0 and bx + ay + c = 0 parallel (a ≠ 0, b ≠ 0)?

Q50. mcq single +2.5 / 0.83

What is value of cot^(2) 15° + tan^(2) 15° ?

Q51. mcq single +2.5 / 0.83

If (1 + tan θ) (1 + tan 9θ) = 2, then what is the value of tan(10θ)?

Q52. mcq single +2.5 / 0.83

What is $\sin^2\left(\frac{\pi}{4}+\theta\right)-\sin^2\left(\frac{\pi}{4}-\theta\right)$ equal to?

Q53. mcq single +2.5 / 0.83

Let sin 2θ = cos 3θ, where θ is acute angle. What is the value of 1 + 4 sin θ? (given that $\sin 18^\circ =\frac{\sqrt 5 - 1}{4}$)

Q54. mcq single +2.5 / 0.83

If tan α + tan β = 1 - tan α.tan β, where tan α tan β ≠ 1, then which of the following is one of the values of (α + β) ?

Q55. mcq single +2.5 / 0.83

In a triangle ABC, sin A - cos B - cos C = 0. What is angle B equal to?

Q56. mcq single +2.5 / 0.83

What is the maximum value of the functions $f(x) = \frac{1}{\tan x+\cot x},$ where $0 < x < \frac{\pi}{2}?$

Q57. mcq single +2.5 / 0.83

What is the value of sin 0° + sin 10° + sin 20° + sin 30° + ⋯ + sin 360°?

Q58. mcq single +2.5 / 0.83

If tan (45° + θ) = 1 + sin 2θ, where $-\frac{\pi}{4}< θ <\frac{\pi}{4},$ then what is the value of cos 2θ?

Q59. mcq single +2.5 / 0.83

If $α + β = \frac{\pi}{4}$ and 2tan α = 1, then what is tan 2β equal to?

Q60. mcq single +2.5 / 0.83

If $\tan θ = - \frac{5}{12},$ then what can be the value of sin θ?

Q61. mcq single +2.5 / 0.83

What is the value of $\cos^4\frac{7\pi}{8}+\cos^4\frac{5\pi}{8}?$

Q62. mcq single +2.5 / 0.83

If the derivative of the function$f(x) =\frac{m}{x} +2nx + 1$ vanishes at x = 2, then what is the value of m + 8n ?

Q63. mcq single +2.5 / 0.83

If xy = 4225 where x, y are natural numbers, then what is the minimum value of x + y ?

Q64. mcq single +2.5 / 0.83

How many extreme values does sin4x + 2x, where $0 < x < \frac{\pi}{2} $ have ?

Q65. mcq single +2.5 / 0.83

What is the area included in the first quadrant between the curves y = x and y = x^(3 )?

Q66. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (2) items that follow : Let the plane $\frac{2x}{k} + \frac{2y}{3} + \frac{z}{3} = 2 $ pass through the point (2, 3, -6). --- If p, q and r are the intercepts made by the plane on the coordinate axes respectively, then what is (p + q + r) equal to ?

Q67. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next three (03) items that follow The Plane 6x + ky + 3z - 12 = 0 where k ≠ 0 meets the coordinate axes at A, B and C respectively. The equation of the sphere passing through the origin and A, B, C is x^(2 )+ y^(2) + z^(2) - 2x - 3y - 4z = 0. --- What is the value of k?

Q68. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next three (03) items that follow The Plane 6x + ky + 3z - 12 = 0 where k ≠ 0 meets the coordinate axes at A, B and C respectively. The equation of the sphere passing through the origin and A, B, C is x^(2 )+ y^(2) + z^(2) - 2x - 3y - 4z = 0. --- If p is the perpendicular distance from the centre of the sphere to the plane, then which one of the following is correct ?

Q69. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (2) items that follow : Let the plane $\frac{2x}{k} + \frac{2y}{3} + \frac{z}{3} = 2 $ pass through the point (2, 3, -6). --- What are the direction ratios of the normal to the plane ?

Q70. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next three (03) items that follow The Plane 6x + ky + 3z - 12 = 0 where k ≠ 0 meets the coordinate axes at A, B and C respectively. The equation of the sphere passing through the origin and A, B, C is x^(2 )+ y^(2) + z^(2) - 2x - 3y - 4z = 0. --- What is the equation of the line through the origin and the centre of the sphere ?

Q71. mcq single +2.5 / 0.83

What is the equation of the parabola with focus (-3, 0) and directrix x - 3 = 0 ?

Q72. mcq single +2.5 / 0.83

The area of the region bounded by the parabola y^(2) = 4kx, where k > 0 and its latus rectum is 24 square units. What is the value of k ?

Q73. mcq single +2.5 / 0.83

In the expansion of $\left(x + \frac{1}{x}\right)^{2n} $, what is the (n + 1)th term from the end (when arranged in descending powers of x) ?

Q74. mcq single +2.5 / 0.83

How many terms are there in the expansion of $\left(1 + \frac{2}{x}\right)^9\left(1-\frac{2}{x}\right)^9 ? $

Q75. mcq single +2.5 / 0.83

What is the value of C(51, 21) - C(51, 22) + C(51, 23) - C(51, 24) + C(51, 25) - C(51, 26) + C(51, 27) - C(51, 28) + C(51, 29) - C(51, 30) ?

Q76. mcq single +2.5 / 0.83

Consider the following statements in respect of the expansion of (x + y)^(10) 1. Among all the coefficients of the terms, the coefficient of the 6th term has the highest value 2. The coefficient of the 3rd term is equal to coefficient of the 9th term Which of the above statements is /are correct ?

Q77. mcq single +2.5 / 0.83

A radioactive substance decays at a rate proportional to the amount of substance present. If half of the substance decays in 100 years, then what is the decay constant (proportionality constant) ?

Q78. mcq single +2.5 / 0.83

What is the principal argument of $\frac{1}{1 + i}$ where $i = \sqrt{-1}?$

Q79. mcq single +2.5 / 0.83

What is the modulus of $\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right)^{200}?$

Q80. mcq single +2.5 / 0.83

What is the domain of the function $f(x) =\sqrt{1 - (x-1)^2} \ ?$

Q81. mcq single +2.5 / 0.83

In which one of the following intervals is the function $f(x) = \frac{x^3}{3}-\frac{7x^2}{2} + 6x + 5$ decreasing?

Q82. mcq single +2.5 / 0.83

If $4f(x) - f \left(\frac{1}{x}\right)=\left(2x+\frac{1}{x}\right)\left(2x-\frac{1}{x}\right),$ then what is f(2) equal to?

Q83. mcq single +2.5 / 0.83

Consider the following in respect of the function f(x) = 10^(x) : 1. Its domain is (-∞, ∞) 2. It is a continuous function 3. It is differentiable at x = 0 Which of the above statements are correct?

Q84. mcq single +2.5 / 0.83

Consider the following statements in respect of the function y = [x], x ∈ (-1, 1) where [.] is the greatest integer function: 1. Its derivative is 0 at x = 0.5 2. It is continuous at x = 0 Which of the above statements is/are correct?

Q85. mcq single +2.5 / 0.83

If f(x) = 4x + 3, then what is f o f o f(-1) equal to?

Q86. mcq single +2.5 / 0.83

What does the equation $x \frac{dy}{dx}-2y= 0$ represent ?

Q87. mcq single +2.5 / 0.83

If $\frac{dy}{dx} = (\ln 5)y $ with y(0) = ln 5, then what is y(1) equal to?

Q88. mcq single +2.5 / 0.83

What is the degree of the differential equation? $1+\left(\frac{dy}{dx}\right)^2 =\left(\frac{d^2y}{dx^2}\right)^{\frac{4}{3}}?$

Q89. mcq single +2.5 / 0.83

What is the equation of the circle which touches both the axes in the first quadrant and the line y - 2 = 0?

Q90. mcq single +2.5 / 0.83

If the roots of the equation x^(2) - ax - bx - cx + bc + ca = 0 are equal, then which one of the following is correct?

Q91. mcq single +2.5 / 0.83

Let α and β be the roots of the equation x^(2) - ax - bx + ab - c = 0. What is the quadratic equation whose roots are a and b?

Q92. mcq single +2.5 / 0.83

Let α and β be the roots of the equation x^(2) + px + q = 0. If α^(3) and β^(3) are the roots of the equation x^(2 )+ mx + n = 0, then what is the value of m + n ?

Q93. mcq single +2.5 / 0.83

Let α and β (α > β) be the roots of the equation x^(2) - 8x + q = 0. If α^(2) - β^(2) = 16, then what is the value of q?

Q94. mcq single +2.5 / 0.83

What is the distance between the foci of the ellipse x^(2) + 2y^(2) = 1 ?

Q95. mcq single +2.5 / 0.83

If x^(y)y^(x) = 1, then what is $\frac{dy}{dx}$ at (1, 1) equal to?

Q96. mcq single +2.5 / 0.83

If y = (x^(x))^(x), then what is the value of $\frac{dy}{dx}$ at x = 1?

Q97. mcq single +2.5 / 0.83

What is the maximum value of n such that 5^(n) divides (30! + 35!), where n is a natural number?

Q98. mcq single +2.5 / 0.83

Consider the following statements: 1. $\frac{n!}{3!}$ is divisible by 6, where n > 3 2. $\frac{n!}{3!}+3 $ is divisible by 7, where n > 3 Which of the above statements is/are correct?

Q99. mcq single +2.5 / 0.83

How many permutations are there of the letters of the word 'TIGER' in which the vowels should not occupy the even positions ?

Q100. mcq single +2.5 / 0.83

In how many ways can a team of 5 players be selected out of 9 players so as to exclude two particular players ?

Q101. mcq single +2.5 / 0.83

What is the value of 2(2 × 1) + 3(3 × 2 × 1) + 4(4 × 3 × 2 × 1) + 5(5 × 4 × 3 × 2 × 1) + .................. + 9(9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) + 2 ?

Q102. mcq single +2.5 / 0.83

How many odd numbers between 300 and 400 are there in which none of the digits is repeated ?

Q103. mcq single +2.5 / 0.83

If C(3n, 2n) = C(3n, 2n - 7), then what is the value of C(n, n - 5)?

Q104. mcq single +2.5 / 0.83

What is $\int\limits^{\frac{\pi}{4}}_0 \frac{dx}{(\sin x + \cos x)^2}$ equal to?

Q105. mcq single +2.5 / 0.83

What is $\int\limits^{-1}_{-2}\frac{x}{|x|}dx$ equal to?

Q106. mcq single +2.5 / 0.83

Let A and B be non-singular matrices of the same order such that AB = A and BA = B. Which of the following statements is/are correct ? 1. A^(2 )= A 2. AB^(2 )= A^(2)B Select the correct answer using the code given below:

Q107. mcq single +2.5 / 0.83

If $A = \begin{bmatrix} 2 \sin \theta & \cos \theta & 0 \\ -2\cos \theta & \sin \theta & 0 \\ -1 & 1 & 1 \end{bmatrix},$ then what is A(adj A) equal to? (where) I is the identity matrix.

Q108. mcq single +2.5 / 0.83

What is (1110011)~2~ ÷ (10111)~2~ equal to ?

Q109. mcq single +2.5 / 0.83

Let A be a non-singular matrix and B = adj A. Which of the following statements is/are correct? 1. AB = BA 2. AB is a scalar matrix 3. AB can be a null matrix Select the correct answer using the code given below:

Q110. mcq single +2.5 / 0.83

Consider the following in respect of the matrices: A = [m n], B = [-n -m] & $ C = \begin{bmatrix} m \\ -m\end{bmatrix} $ 1. CA = CB 2. AC = BC 3. C(A + B) = CA + CB Which of the above statements is/are correct?

Q111. mcq single +2.5 / 0.83

If x^(3 )+ y^(3) = (100010111)~2~ and x + y = (11111)~2~, then what is (x - y)^(2 )+ xy equal to ?

Q112. mcq single +2.5 / 0.83

Consider the following statements in respect of square matrices A and B of same order : 1. If AB is a null matrix, then at least one of A and B is a null matrix. 2. If AB is an identity matrix, then BA = AB. Which of the above statements is/are correct?

Q113. mcq single +2.5 / 0.83

Let a, b, c be the lengths of sides BC, CA, AB respectively of a triangle ABC. If p is the perimeter and q is the area of the triangle, then what is $p(p-2a) \tan \left(\frac{A}{2}\right)$ equal to ?

Q114. mcq single +2.5 / 0.83

If (a - b) (b - c) (c - a) = 2 and abc = 6, then what is the value of $\begin{vmatrix}a & b & c \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix}?$

Q115. mcq single +2.5 / 0.83

Under which of the following conditions does the determinant $\begin{vmatrix}a & b & c \\ b & c & a \\ c & a & b\end{vmatrix}$ vanish? 1. a + b + c = 0 2. a^(3) + b^(3) + c^(3) = 3abc 3. a^(2) + b^(2) + c^(2) - ab - bc - ca = 0 Select the correct answer using the code given below:

Q116. mcq single +2.5 / 0.83

If $Δ_1 = \begin{vmatrix} 1 & p & q \\ 1 & q & r \\ 1 & r & p\end{vmatrix} \ \text{and} \ Δ_2 = \begin{vmatrix} 1 & 1 & 1 \\ q & r & p \\ r & p & q \end{vmatrix}$ where p ≠ q ≠ r, then Δ~1~ + Δ~2~ is

Q117. mcq single +2.5 / 0.83

For what value of k is the matrix $\begin{bmatrix} 2\cos 2\theta & 2\cos 2\theta & 6 \\ 1 -2 \sin^2\theta & 2 \cos^2\theta -1 & 3 \\ k & 2k & 1 \end{bmatrix}$ singular?

Q118. mcq single +2.5 / 0.83

If A is the identity matrix of order 3 and B is its transpose, then what is the value of the determinant of the matrix C = A + B?

Q119. mcq single +2.5 / 0.83

If $I_1 =\int\frac{e^x dx}{e^x + e^{-x}}$ and $I_2 =\int\frac{dx}{e^{2x} + 1},$then what is I~1~ + I~2~ equal to?

Q120. mcq single +2.5 / 0.83

What is $\int (\sin x)^{-1/2} (\cos x)^{-3/2}dx$ equal to?

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