NDA-1-2023-Maths-Question-Paper PDF

## Mathematics ### 1. Consider the following statements * f(x) = lnx is increasing in (0, 0∞) * g(x) = ex+e is decreasing in (0, 0∞) Which of the statements given above is/are correct? * (a) 1 only * (b) 2 only * (c) Both 1 and 2 * (d) Neither 1 nor 2 ### 2. What is the derivative of sin²x wi...

## Mathematics ### 1. Consider the following statements * f(x) = lnx is increasing in (0, 0∞) * g(x) = ex+e is decreasing in (0, 0∞) Which of the statements given above is/are correct? * (a) 1 only * (b) 2 only * (c) Both 1 and 2 * (d) Neither 1 nor 2 ### 2. What is the derivative of sin²x with respect to cos²x ? * (a) -1 * (b) 1 * (c) sin2x * (d) cos2x ### 3. For what value of m with m < 0, is the area bounded by the lines y = x, y = mx and x = 2 equal to 3? * (a) -1/2 * (b) -1 * (c) 3/2 * (d) -2 ### 4. What is the derivative of cosec(x°) ? * (a) cosec (x°) cot (x°) * (b) - cosec(x) cot (x°) * (c) π/180 cosec (x°) cot(x°) * (d) π/180 cosec(x) cot(x) ### 5. A solution of the differential equation d²y/dx² -x = 0 is * (a) y = 2x * (b) y = 2x + 4 * (c) y = x² - 1 * (d) y = (x² - 2)/2 ### 6. If f(x) = x² + 2 and g(x) = 2x - 3, then what is (fg)(1) equal to? * (a) 3 * (b) 1 * (c) -2 * (d) -3 ### 7. What is the range of the function f(x) = x + |x| if the domain is the set of real numbers? * (a) (0, ∞) * (b) [0, ∞) * (c) (-∞, ∞) * (d) [1, ∞) ## 8. If f(x) = x(4x² - 3), then what is f(sin ) equal to ? * (a) -sin30 * (b) -cos30 * (c) sin30 * (d) -sin40 ## 9. What is lim _x→5 5-x/x² - 5x equal to ? * (a) -1 * (b) 0 * (c) 1 * (d) Limit does not exist ## 10. What is lim _x→1 x²-1/x³-1 equal to? * (a) -1 * (b) -3 * (c) 3 * (d) Limit does not exist ## Consider the following for the next three (03) items that follow * Let f(x) = Pe^x + Qe^2x + Re^3x, where P, Q, R are real numbers. * Further f(0) = 6, f'(ln 3) = 282 and _0^ln3 f(x) dx = 11 ## 11. What is the value of Q? * (a) 1 * (b) 2 * (c) 3 * (d) 4 ## Consider the following for the next two (02) items that follow: * Suppose E is the differential equation representing family of curves y² = 2cx + 2c√c where c is a positive parameter. ## 12. What is the order of the differential equation? * (a) 1 * (b) 2 * (c) 3 * (d) 4 ## 13. What is f'(0) equal to? * (a) 18 * (b) 16 * (c) 15 * (d) 14 ## Consider the following for the next two (02) items that follow : * Let A(a, b, c, a) = (a a + b b a + c)/ (b a + c c a + b) * 0/0 ## 14. If A(a, b , c, a) = 0 for every a > 0, then which one of the following is correct? * (a) a, b, c are in AP * (b) a, b, c are in GP * (c) a, 2b, c are in AP * (d) a, 2b, c are in GP ## 15. If A(7, 4, 2, α) = 0, then a is a root of which one of the following equations? * (a) 7x² + 4x + 2 = 0 * (b) 7x² - 4x + 2 = 0 * (c) 7x² + 8x + 2 = 0 * (d) 7x² - 8x + 2 = 0 ## Consider the following for the next two (02) items that follow : * Given that m(θ) = cot²θ + n²tan²θ + 2n, where n is a fixed positive real number ## 16. What is the least value of m(θ)? * (a) n * (b) 2n * (c) 3n * (d) 4n ## 17. Under what condition does m attain the least value? * (a) n = tan2θ * (b) n = cot2θ * (c) n = sin2θ * (d) n = cos2θ ## Consider the following for the next two (02) items that follow : * A quadrilateral is formed by the lines x = 0, y = 0, x + y = 1 and 6x + y = 3. ## 18. What is the equation of diagonal through origin? * (a) 3x + y = 0 * (b) 2x + 3y = 0 * (c) 3x - 2y = 0 * (d) 3x + 2y = 0 ## 19. What is the equation of other diagonal? * (a) x + 2y - 1 = 0 * (b) x - 2y - 1 = 0 * (c) 2x + y + 1 = 0 * (d) 2x + y - 1 = 0 ## Consider the following for the next two (02) items that follow : * P(x, y) is any point on the ellipse x² + 4y² = 1. Let E, F be the foci of the ellipse. ## 20. What is PE + PF equal to ? * (a) 1 * (b) 2 * (c) 3 * (d) 4 ## 21. Consider the following points: * 1) (1, 0) * 2) (√3 / 2, √3 / 2 * 3) (√3 / 2, 1 / 2) * 4) (2, √3 / 4) Which of the above points lie on latus rectum of ellipse? * (a) 1 and 2 only * (b) 2 and 3 only * (c) 1 and 3 only * (d) 1, 2 and 3 ## Consider the following for the next two (02) items that follow : * The line y = x partitions the circle (x-a)² + y² = a² in two segments. ## 22. What is the area of minor segment? * (a) (π-2)a²/4 * (b) (π-1)a²/4 * (c) (π-2)a²/2 * (d) (π-1)a²/2 ## 23. What is the area of major segment? * (a) (3π-2)a²/4 * (b) (3π + 2)a²/4 * (c) (3π - 2)a²/2 * (d) (3π + 2)a²/2 ## Consider the following frequency distribution: x | f -------|------ 1 | 4 2| 6 3 | 9 4 | 7 5 | 5 ## 24. What is the value of median of the distribution? * (a) 1 * (b) 2 * (c) 3 * (d) 3.5 ## 25. For data -1, 1, 4, 3, 8, 12, 17, 19, 9, 11; if M is the median of first 5 observations and N is the median of last five observations, then what is the value of 4M - N? * (a) 7 * (b) 4 * (c) 1 * (d) 0 ## 26. Let P, Q, R represent mean, median and mode. If for some distribution 5P = 4Q = R / 2, then what is (p + Q) / (2P + Q - 7R) equal to ? * (a) 1 / 12 * (b) 1 / 7 * (c) 2 / 9 * (d) 1 / 4 ## 27. If G is the geometric mean of numbers 1, 2, 2², 2³, ..., 2^n-1, then what is the value of 1 + 2log_2G ? * (a) 1 * (b) 4 * (c) n - 1 * (d) n ## 28. If H is the harmonic mean of numbers 1, 2, 2², 2^3, ..., 2^n-1, then what is n / H equal to ? * (a) 2 - 1/ (2^n - 1) * (b) 2- 1/(2^n - 1) * (c) 2 + 1/(2^n - 1) * (d) 2 - 1/(2^n - 1) ## 29. Let P be the median, Q be the mean and R be the mode of observations x_1, x_2, x_3,... x_n. Let S = \((2x - a)^2\). S takes minimum value, when a is equal to * (a) P * (b) Q * (c) 2 * (d) R ## 30. One bag contains 3 white and 2 black balls, another bag contains 2 white and 3 black balls. Two balls are drawn from the first bag and put it into the second bag and then a ball is drawn from the second bag. What is the probability that it is white? * (a) 6/33 * (b) 7/70 * (c) 3/10 * (d) 1/70 ## 31. Three dice are thrown. What is the probability that each face shows only multiples of 3? * (a) 1/18 * (b) 1/9 * (c) 1/27 * (d) 1/3 ## 32. What is the probability that the month of December has 5 Sundays ? * (a) 1/7 * (b) 4/7 * (c) 3/7 * (d) 2/7 ## 33. A natural number n is chosen from the first 50 natural numbers. What is the probability that n + (50 / n) < 50? * (a) 23/25 * (b) 47/50 * (c) 24/25 * (d) 49 / 50 ## 34. How many real numbers satisfy the equation |x - 4| + |x - 7| = 15? * (a) Only one * (b) Only two * (c) Only three * (d) Infinitely many ## 35. A mapping f: A → B defined as f(x) = (2x + 3) / (3x + 5), x ∈ A. If f is to be onto, then what are A and B equal to ? * (a) A = R \{-1/3\} and B = R \{-3/5\} * (b) A = R and B = R \{-1/3\} * (c) A = R \{-1/3\} and B = R \{-0\} * (d) A = R \{-1/3\} and B = R \{-1 / 3\} ## 36. a and ẞ are distinct real roots of the quadratic equation x² + ax + b = 0. Which of the following statements is/are sufficient to find a? * 1. α + ẞ = 0, α² + ẞ² = 2 * 2. αβ² = 1, a = 0 Select the correct answer using the code given below: * (a) 1 only * (b) 2 only * (c) Both 1 and 2 * (d) Neither 1 nor 2 ## 37. If the sixth term in the binomial expansion of (x^(1/3) + x^(2/3)log_10x + x^(1/3)log_10x^2)^8 is 5600, then what is the value of x ? * (a) 6 * (b) 8 * (c) 9 * (d) 10 ## 38. How many terms are there in the expansion of (3x - y)(x + 3y)^4 ? * (a) 9 * (b) 12 * (c) 15 * (d) 17 ## 39. p, q, r and s are in AP such that p + s = 8 and qr = 15. What is the difference between largest and smallest numbers? * (a) 6 * (b) 5 * (c) 4 * (d) 3 ## Consider the following statements for a fixed natural number n: * 1. C(n, r) is greatest if n = 2r * 2. C(n, r) is greatest if n = 2r-1 and n = 2r + 1 ## 40. Which of the statements given above is/are correct? * (a) 1 only * (b) 2 only * (c) Both 1 and 2 * (d) Neither 1 nor 2 ## 41. m parallel lines cut n parallel lines giving rise to 60 parallelograms. What is the value of (m + n)? * (a) 6 * (b) 7 * (c) 8 * (d) 9 ## 42. Let x be the number of permutations of the word 'PERMUTATIONS' and y be the number of permutations of the word 'COMBINATIONS'. Which one of the following is correct? * (a) x = y * (b) y = 2x * (c) x = 4y * (d) y = 4x ## 43. 5-digit numbers are formed using the digits 0, 1, 2, 4, 5 without repetition. What is the percentage of numbers which are greater than 50,000? * (a) 20% * (b) 25% * (c) 100/3% * (d) 110/3% ## 44. If 2 - i√3 where i = √-1 is a root of the equation x² + ax + b = 0, then what is the value of (a + b)? * (a) -11 * (b) -3 * (c) 0 * (d) 3 ## 45. If z = (1 + i√3) / (1- i√3) where i = √-1, then what is the argument of z ? * (a) π/3 * (b) 2π/3 * (c) 4 π/3 * (d) 5 π/6 ## 46. If a, b, c are in AP, then what is ((x+1)/(x+2) +(x+2)/(x+3) +(x+3)/(x+a)) / ((x+2)/(x+3) + (x+3)/(x+4) + (x+a)/(x+b)) equal to ? * (a) -1 * (b) 0 * (c) 1 * (d) 2 ## 47. If loga, a^x and log₁x are in GP, then what is x equal to? * (a) log(log_a b) * (b) log(log_b a) * (c) log_a (log_a b)/2 * (d) log_a (log_b _a) / 2 ## 48. If a, b, c are in GP, then which one of the following is correct? * (a) a, b, c are in AP * (b) a, b, c are in GP * (c) a, b, c are in HP * (d) ab, bc, ca are in AP ## 49. The first and the second terms of an AP are 1/2 and 2/3 respectively. If nth term is the largest negative term, what is the value of n? * (a) 5 * (b) 6 * (c) 7 * (d) n cannot be determined ## 50. For how many integral values of k, the equation x² - 4x + k = 0, where k is an integer has real roots and both of them lie in the interval (0, 5)? * (a) 3 * (b) 4 * (c) 5 * (d) 6 ## 51. In an AP, the first term is x and the sum of the first n terms is zero. What is the sum of the next m terms? * (a) mx(m+n)/(n-1) * (b) mx(m+n)/(1-n) * (c) nx(m+n)/(m-1) * (d) nx(m+n)/(1-m) ## 52. Consider the following statements: * 1. (25)! + 1 is divisible by 26 * 2. (6)! + 1 is divisible by 7 ## 53. Which of the following statements is/are correct? * (a) 1 only * (b) 2 only * (c) Both 1 and 2 * (d) Neither 1 nor 2 ## 54. If z is a complex number such that (z-1)/(z+1) is purely imaginary, then what is z equal to? * (a) 1 * (b) 1 / 2 * (c) 3 / 2 * (d) 2 ## 55. If ω is a non-real cube root of 1, then what is the value of (1-ω)/(ω+ω²) * (a) √3 * (b) √2 * (c) 1 * (d) √3 / 4 ## 56. What is the number of 6-digit numbers that can be formed only by using 0, 1, 2, 3, 4 and 5 (each once) ; and divisible by 6? * (a) 96 * (b) 120 * (c) 192 * (d) 312 ## 57. What is the binary number equivalent to decimal number 1011? * (a) 1011 * (b) 111011 * (c) 11111001 * (d) 111110011 ## 58. Let A be a matrix of order 3x3 and |A| = 4. If 2 |adj(3A)| = 238, then what is the value of (α+β)? * (a) 12 * (b) 13 * (c) 17 * (d) 24 ## 59. If a and β are the distinct roots of equation x² - x + 1 = 0, then what is the value of ( α^100 + β^100 ) / ( α^100 * β^100 ) * (a) √3 * (b) √2 * (c) 1 * (d) 1/√3 ## 60. Let A and B be symmetric matrices of same order, then which one of the following is correct regarding (AB - BA)? * 1. Its diagonal entries are equal but nonzero. * 2. The sum of its non-diagonal entries is zero. Select the correct answer using the code given below: * (a) 1 only * (b) 2 only * (c) Both 1 and 2 * (d) Neither 1 nor 2 ## 61. Consider the following statements in respect of square matrices A, B, C each of same order n: * 1. AB = AC = C if A is non-singular. * 2. If BX = CX for every column matrix X having n rows then B = C. ## 62. Which of the statements given above is/are correct? * (a) 1 only * (b) 2 only * (c) Both 1 and 2 * (d) Neither 1 nor 2 ## 63. The system of linear equations x + 2y + z = 4, 2x + 4y + 2z = 8 and 3x + 6y + 3z = 10 has * (a) a unique solution * (b) infinite many solutions * (c) no solution * (d) exactly three solutions ## 64. Let AX = B be a system of 3 linear equations with 3-unknowns. Let X_1 and X_2 be its two distinct solutions. If the combination aX_1 + bX_2 is solution of AX = B; where a, b are real numbers, then which one of the following is correct? * (a) a = b * (b) a + b = 1 * (c) a + b = 0 * (d) a - b = 1 ## 65. What is the sum of the roots of the equation 0/ (x - a)(x - b) + 0/(x + b)(x + c) + 1/ (x - c) = 0? * (a) a + b + c * (b) a - b + c * (c) a + b - c * (d) a - b - c ## Consider the following for the next two (02) items that follow: * Let A(1, -1, 2) and B(2, 1, -1) be the end points of the diameter of the sphere x² + y² + z² + 2ux + 2vy + 2wz - 1 = 0 ## 66. What is u + v + w equal to? * (a) -2 * (b) -1 * (c) 1 * (d) 2 ## 67. If P(x, y, z) is any point on the sphere, then what is PA² + PB² equal to? * (a) 15 * (b) 14 * (c) 13 * (d) 6.5 ## Consider the following for the next two (02) items that follow: * Consider two lines whose direction ratios are (2, -1, 2) and (k, 3, 5). They are inclined at an angle π/4 ## 68. What is the value of k? * (a) 4 * (b) 2 * (c) 1 * (d) -1 ## 69. What are the direction ratios of a line which is perpendicular to both the lines? * (a) (1, 2, 10) * (b) (-1, -2, 10) * (c) (11, 12, -10) * (d) (11, 2, -10) ## Consider the following for the next two (02) items that follow : * Let a = 3î + 3ĵ + 3k and c = ĵ -k. Let b be such that a.b = 27 and a x b = 9c ## 70. What is b equal to? * (a) 3î + 4ĵ + 2k * (b) 5î + 2ĵ + 2k * (c) 5î - 2ĵ + 6k * (d) 3î + 3ĵ + 4k ## 71. What is the angle between (a + b) and c? * (a) π/2 * (b) π/3 * (c) π/4 * (d) π/6 ## Consider the following for the next two (02) items that follow: * Let a vector a = 4î - 8ĵ + k make angles α, β, γ with the positive directions of x, y, z axes respectively. ## 72. What is cos α equal to? * (a) 1/3 * (b) 4/9 * (c) 5/9 * (d) 2/3 ## 73. What is cos²β + cos²γ equal to? * (a) 32/81 * (b) 16/81 * (c) 16 * (d) 32 ## Consider the following for the next two (02) items that follow: * The position vectors of two points A and B are î - ĵ and î + k respectively. ## 74. Consider the following points: * 1. (-1, -3, 1) * 2. (-1, 3, 2) * 3. (-2, 5, 3) ## 75. Which of the above points lie on the line joining A and B? * (a) 1 and 2 only * (b) 2 and 3 only * (c) 1 and 3 only * (d) 1, 2 and 3 ## 76. What is the magnitude of AB? * (a) 2 * (b) 3 * (c) √6 * (d) √3 ## 77. The mean and variance of five observations are 14 and 13.2 respectively. Three of the five observations are 11, 16 and 20. What are the other two observations? * (a) 8 and 15 * (b) 9 and 14 * (c) 10 and 13 * (d) 11 and 12 ## 78. Let A and B be two independent events such that P(A) = 0.7, P(B) = k, P(AUB) = 0.8. What is the value of k? * (a) 5/7 * (b) 4/7 * (c) 2/7 * (d) 1/7 ## 79. A biased coin with the probability of getting head equal to 1/4 is tossed five times. What is the probability of getting tail in all the first four tosses followed by head? * (a) 81/512 * (b) 81/1024 ## 80. Let X and Y be two random variables such that X + Y = 100. If X follows Binomial distribution with parameters n = 100 and p = 3/5 , what is the variance of Y? * (a) 1 * (b) 1/2 * (c) 16 * (d) 1/16 ## 81. If two lines of regression are x + 4y + 1 = 0 and 4x + 9y + 7 = 0, then what is the value of x when y = -3? * (a) -13 * (b) -5 * (c) 5 * (d) 7 ## 82. The central angles p, q, r and s (in degrees) of four sectors in a Pie Chart satisfy the relation 9p = 3q = 2r = 6s. What is the value of 4p - q? * (a) 12 * (b) 24 * (c) 30 * (d) 36 ## 83. The observations 4, 1, 4, 3, 6, 2, 1, 3, 4, 5, 1, 6 are outputs of 12 dices thrown simultaneously. If m and M are means of lowest 8 observations and highest 4 observations respectively, then what is (2m + M) equal to? * (a) 10 * (b) 12 * (c) 17 * (d) 21 ## 84. A bivariate data set contains only two points (-1, 1) and (3, 2). What will be the line of regression of y on x? * (a) x - 4y + 5 = 0 * (b) 3x + 2y - 1 = 0 * (c) x + 4y + 1 = 0 * (d) 5x - 4y + 1 = 0 ## 85. A die is thrown 10 times and obtained the following outputs: 1, 2, 1, 1, 2, 1, 4, 6, 5, 4 What will be the mode of data so obtained? * (a) 6 * (b) 4 * (c) 2 * (d) 1 ## Consider the following for the next three (03) items that follow: * Let I_1 = _0^π/2 \((1 + cos^2 (x))/(1 + cos(x))\) dx and * I_2 = _0^(π/2) \((1+ sin(x))/(1 + sin^2(x))\) dx ## 86. What is the value of (I_1 + I_2) / (I_1 - I_2)? * (a) 1 * (b) π * (c) π² * (d) (π + 1) / (π - 1) * ## 87. What is the value of 81? * (a) π * (b) π² * (c) π³ * (d) π^4 ## 88. What is the value of I_2? * (a) π/√2 * (b) π / (2√2) * (c) 3π / (2√2) * (d) π / (4√2) ## Consider the following for the next two (02) items that follow: * Let I = _a^b |x| / x dx, a < 0 < b. ## 89. What is I equal to when a < 0 < b? * (a) a + b * (b) a - b * (c) b - a * (d) (a + b)/2 ## 90. What is I equal to when a < b < 0 ? * (a) a + b * (b) a - b * (c) b - a * (d) (a + b) / 2 ## Consider the following for the next three (03) items that follow: * Let f(x) = \ln x, x = 1 ## 91. What is the derivative of f(x) at x = 0.5? * (a) -2 * (b) -1 * (c) 1 * (d) 2 ## 92. What is the derivative of f(x) at x = 2? * (a) -2 * (b) -1 * (c) 1/2 * (d) 2 ## 93. What is the derivative of fof(x), where 1<x<2? * (a) 1/ln(x) * (b) 1/(xln(x)) * (c) -1/ln(x) * (d) -1/(xln(x)) ## Consider the following for the next two (02) items that follow: * Consider the function f(x) = \x-2\ + \3-x\ + \4-x\, where x ∈ R. ## 94. At what value of x does the function attain minimum value? * (a) 2 * (b)3 * (c) 4 * (d) 0 ## 95. What is the minimum value of the function? * (a) 2 * (b) 3 * (c) 4 * (d) 0 ## Consider the following for the next two (02) items that follow: * Consider the sum S = 0! + 1! + 2! + 3! + 4! +...+ 100! ## 96. If the sum S is divided by 8, what is the remainder? * (a) 0 * (b) 1 * (c) 2 * (d) Cannot be determined ## 97. If the sum S is divided by 60