NDA Previous Year Question Paper 2021-II-I-shift|
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1. If x^2 + x + 1 = 0, then what is the value of x^199 + x^200 + x^201? (a) -1 (b) 0 (c) 1 (d) 3
Answer: (b) 0. Given x^2+x+1=0 => x^3=1, so x^199+x^200+x^201 = x+x^2+x^3 = 0.
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2. If x,y,z are in GP, then which of the following is/are correct?
1. ln(3x), ln(3y), ln(3z) are in AP.
2. xyz + ln(x), xyz + ln(y), xyz + ln(z) are in HP.
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
Answer: (a) 1 only. Given x,y,z in GP => y^2=xz. If log(3x),log(3y),log(3z) in AP, then 2log(3y)=log(3x)+log(3z) => 9y^2=9xz => y^2=xz. Statement 1 correct. Statement 2 wrong.
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3. If log10(2), log10(2^x - 1), log10(2^x + 3) are in AP, then what is x equal to?
(a) 0 (b) 1 (c) log2(5) (d) log5(2)
Answer: (c) log2(5). 2log10(2^x-1)=log10(2)+log10(2^x+3). log10(2^x-1)^2=log10[2(2^x+3)]. => 2^{2x}+1-2*2^x = 2*2^x+6 => (2^x)^2 - 4(2^x) - 5 = 0. Let 2^x=y: y^2-4y-5=0 => (y-5)(y+1)=0 => y=5. So 2^x=5 => x=log2(5).
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4. Let S = {2,3,4,5,6,7,9}. How many different 3-digit numbers (with all digits different) from S can be made which are less than 500?
(a) 30 (b) 49 (c) 90 (d) 147
Answer: (c) 90. Hundreds digit can be 2,3,4: 3 choices. Tens digit: remaining 6 choices. Unit digit: remaining 5 choices. Total = 3*6*5 = 90.
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5. If p = (1111... up to n digits), then what is the value of 9p^2 + p?
(a) 10^n p (b) 2p*10^n (c) 10^n p - 1 (d) 10^n p + 1
Answer: (a) 10^n p. p=1+10+...+10^{n-1}=(10^n-1)/9. => 9p=10^n-1 => 9p+1=10^n => 9p^2+p=p(9p+1)=p*10^n.
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6. The quadratic equation 3x^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?
(a) 0 < k < 5/3 (b) 0 < k < 3/5 only (c) 3/5 < k < 5/3 (d) No such value of k exists.
Answer: (d) No such value of k exists. For roots equal in magnitude and opposite sign, sum of roots = 0 => k^2+5k=0 => k(k+5)=0. Also product < 0 for opposite sign. No such k in options.
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7. From 1; a, b, c are in AP. b=a+d, c=a+2d.
(b+c)^2=(2a+3d)^2, (c+a)^4=(2a+2d)^4, (a+b)^2=(2a+d)^2.
So (b+c)^2, (c+a)^2, (a+b)^2 are not in GP. Option (a) is correct.
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11. If A = [[1, a],[0, 1]], where a in N, then what is A^100 - A^50 - 2A^25 equal to?
(a) -2I (b) -1 (c) 2I (d) 1
Answer: (a) -2I. A^n = [[1, na],[0,1]]. A^100 - A^50 - 2A^25 = [[1-1-2, 100a-50a-50a],[0, 1-1-2]] = [[-2,0],[0,-2]] = -2I.
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12. If [[a, -b, a-b-c],[-a, b, -a+b-c],[-a, -b, -a-b+c]] - kabc = 0, a!=0,b!=0,c!=0, then what is k?
(a) -4 (b) -2 (c) 2 (d) 4
Answer: (a) -4. Using row operation R1 -> R1+R2, determinant gives -2c(2ab)-kabc=0 => -4abc-kabc=0 => k=-4.
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13. What is sum_{n=1}^{8n+7} i^n equal to, where i = sqrt(-1)?
(a) -1 (b) 1 (c) i (d) -i
Answer: (a) -1. S = i+i^2+i^3+...+i^{8n+7} = i[(i^{8n+7}-1)/(i-1)] = i[(i^3-1)/(i-1)] = i[(-i-1)/(i-1)] = -1.
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14. If z = x + iy, where i = sqrt(-1), then what does z*zbar + |z|^2 + 4(z + zbar) - 48 = 0 represent?
(a) Straight line (b) Parabola (c) Circle (d) Pair of straight lines
Answer: (c) Circle. z*zbar=x^2+y^2, |z|^2=x^2+y^2, z+zbar=2x. Equation: 2x^2+2y^2+8x-48=0 => x^2+y^2+4x-24=0, a circle.
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15. Which one of the following is a square root of 2a + 2*sqrt(a^2 + b^2) where a,b in R?
(a) sqrt(a + ib) + sqrt(a - ib) (b) sqrt(a + ib) - sqrt(a - ib) (c) 2a + ib (d) 2a - ib
Answer: (a) sqrt(a + ib) + sqrt(a - ib). 2a+2sqrt(a^2+b^2) = (a+ib)+(a-ib)+2sqrt((a+ib)(a-ib)) = (sqrt(a+ib)+sqrt(a-ib))^2.
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16. If sinθ and cosθ are the roots of ax^2 + bx + c = 0, then which one is correct?
(a) a^2 + b^2 - 2ac = 0 (b) -a^2 + b^2 + 2ac = 0 (c) a^2 - b^2 + 2ac = 0 (d) a^2 + b^2 + 2ac = 0
Answer: (c) a^2 - b^2 + 2ac = 0. sinθ+cosθ=-b/a, sinθcosθ=c/a. Squaring: 1+2c/a=b^2/a^2 => a^2-b^2+2ac=0.
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17. If C(n,4), C(n,5) and C(n,6) are in AP, then what is n?
(a) 7 (b) 8 (c) 9 (d) 10
Answer: (a) 7. 2C(n,5)=C(n,4)+C(n,6). Solving gives (n-14)(n-7)=0. So n=7 or 14. Option (a).
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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'?
(a) 240 (b) 200 (c) 150 (d) 120
Answer: (a) 240. LUCKNOW: 2 vowels, 5 consonants. Choose 2 vowels: 2C2=1. Choose 2 consonants: 5C2=10. Arrange 4 letters: 4!=24. Total=10*24=240.
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19. Suppose 20 distinct points are placed randomly on a circle. Which statements is/are correct?
1. Number of straight lines by joining any two points is 380.
2. Number of triangles by joining any three points is 1140.
(a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2
Answer: (b) 2 only. Lines = 20C2 = 190, so 1 wrong. Triangles = 20C3 = 1140, so 2 correct.
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20. How many terms are there in the expansion of (a^2/b^2 + b^2/a^2 + 2)^21 where a!=0,b!=0?
(a) 21 (b) 22 (c) 42 (d) 43
Answer: (d) 43. (a/b + b/a)^{42}. Number of terms = 42+1=43.
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21. For what values of k is the system 2k^2 x + 3y - 1 = 0, 7x - 2y + 3 = 0, 6kx + y + 1 = 0 consistent?
(a) (3±sqrt(11))/10 (b) (21±sqrt(161))/10 (c) (3±sqrt(7))/10 (d) (4±sqrt(11))/10
Answer: (b) (21±sqrt(161))/10.
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24. If sin(A+B)=1 and 2sin(A-B)=1, where 0<A,B<π/2, then what is tan A : tan B equal to?
(a) 1:2 (b) 2:1 (c) 1:3 (d) 3:1
Answer: (d) 3:1. sin(A+B)=1 => A+B=π/2. 2sin(A-B)=1 => A-B=π/6. Solving: A=π/3, B=π/6. tanA:tanB = sqrt3 : 1/sqrt3 = 3:1.
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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?
(a) 25 (b) 50 (c) 75 (d) 100
Answer: (b) 50. Formula = n(n-4)(n-5)/3!. For n=10: 10*6*5/6 = 50.
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26. Consider all real roots of x^4 - 10x^2 + 9 = 0. What is the sum of absolute values of roots?
(a) 4 (b) 6 (c) 8 (d) 10
Answer: (c) 8. Let y=x^2. y^2-10y+9=0 => y=9,1. x=±3,±1. Sum=3+3+1+1=8.
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27. Consider expansion of (1 + x)^n. Let p,q,r,s be coefficients of first, second, nth and (n+1)th terms. What is ps + qr equal to?
(a) 1 + 2n (b) 1 + 2n^2 (c) 1 + n^2 (d) 1 + 4n
Answer: (c) 1 + n^2. p=1, q=n, r=n, s=1. ps+qr=1+n^2.
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28. Let sin^{-1}x + sin^{-1}y + sin^{-1}z = 3π/2 for 0<=x,y,z<=1. What is x^1000 + y^1001 + z^1002?
(a) 0 (b) 1 (c) 3 (d) 6
Answer: (c) 3. Each sin^{-1} must be π/2. So x=y=z=1. Sum=3.
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29. Let sin x + sin y = cos x + cos y for all x,y in R. What is tan(x/2 + y/2) equal to?
(a) 1 (b) 2 (c) sqrt(2) (d) 2sqrt(2)
Answer: (a) 1. 2sin((x+y)/2)cos((x-y)/2)=2cos((x+y)/2)cos((x-y)/2) => tan((x+y)/2)=1.
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30. Let A = [[0,2],[-2,0]] and (mI + nA)^2 = A, where m,n positive real and I identity. What is m+n?
(a) 0 (b) 1/2 (c) 1 (d) 3/2
Answer: (d) 3/2. (mI+nA)^2 = [[m^2-4n^2,4mn],[-4mn,m^2-4n^2]]. Compare with A: 4mn=2, m^2-4n^2=0. m=2n => 8n^2=2 => n=1/2, m=1. m+n=3/2.
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31. What is cot[sin^{-1}(3/5) + cot^{-1}(3/2)]?
(a) 6/17 (b) 7/16 (c) 16/7 (d) 17/6
Answer: (a) 6/17. sin^{-1}(3/5)=cot^{-1}(4/3). cot^{-1}(4/3)+cot^{-1}(3/2)=cot^{-1}(((4/3)(3/2)-1)/(4/3+3/2))=cot^{-1}(6/17). So value=6/17.
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32. Let 4sin^2 x = 3, where 0<=x<=π. What is tan 3x?
(a) -2 (b) -1 (c) 0 (d) 1
Answer: (c) 0. sin x=sqrt3/2 => x=π/3,2π/3. tan3x=tanπ=0 or tan2π=0.
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33. Let p,q and 3 be respectively first, third and fifth terms of an AP. Let d be common difference. If product pq is minimum, then d?
(a) 1 (b) 3/8 (c) 9/8 (d) 9/4
Answer: (c) 9/8. a=p, a+2d=q, a+4d=3. pq=(3-4d)(3-2d)=9-18d+8d^2. Minimum at d=18/16=9/8.
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34. Consider statements about roots of x^3 - 8 = 0.
1. Roots are non-collinear.
2. Roots lie on a circle of unit radius.
Which is/are correct?
(a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither
Answer: (a) 1 only. Roots = 2, 2ω, 2ω^2. They lie on circle radius 2, not unit. Non-collinear true.
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36. For what value of θ, 0<θ<π/2, does sinθ + sinθ cosθ attain maximum?
(a) π/2 (b) π/3 (c) π/4 (d) π/6
Answer: (b) π/3. P=sinθ(1+cosθ). dP/dθ=cosθ+cos^2θ-sin^2θ=0 => 2cos^2θ+cosθ-1=0 => cosθ=1/2 => θ=π/3.
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37. Consider statements about sets.
1. Union over intersection is distributive.
2. Complement of union equals intersection of complements.
3. If A-B=∅, then A=B.
Which are correct?
(a) 1 and 2 (b) 2 and 3 (c) 1 and 3 (d) 1,2,3
Answer: (a) 1 and 2. 1 and 2 are standard set laws. 3 is false: A=∅, B={1,2} gives A-B=∅ but A≠B.
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38. Sets X,Y,Z have 6,5,4 elements respectively. All 15 distinct. Let S=(X-Y)∪Z. How many proper subsets does S have?
(a) 255 (b) 256 (c) 1023 (d) 1024
Answer: (c) 1023. n(X-Y)=6, n(Z)=4, total n(S)=10. Proper subsets = 2^10 - 1 = 1023.
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39. Consider relations and functions.
1. All relations are functions but all functions are not relations.
2. A relation from A to B is subset of A×B.
3. A relation in A is subset of A×A.
Which are correct?
(a) 1 and 2 (b) 2 and 3 (c) 1 and 3 (d) 1,2,3
Answer: (b) 2 and 3. 1 is false. 2 and 3 are true.
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40. If log10(2)*log2(10) + log10(10^x) = 2, then x?
(a) 0 (b) 1 (c) log2(10) (d) log5(2)
Answer: (b) 1. log10(2)*log2(10)=1. 1+x=2 => x=1.
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41. Let ABC be triangle. If cos2A + cos2B + cos2C = -1, then which is correct?
(a) sinA sinB sinC = 0 (b) sinA sinB cosC = 0 (c) cosA sinB sinC = 0 (d) cosA cosB cosC = 0
Answer: (d) cosA cosB cosC = 0. Simplification gives -4cosA cosB cosC = 0.
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42. Value of determinant with entries cos C, tan A, 0; sin B, 0, -tan A; 0, sin B, cos C?
(a) -1 (b) 0 (c) 2 tanA sinB sinC (d) -2 tanA sinB sinC
Answer: (b) 0. Determinant = tanA sinB cosC - tanA sinB cosC = 0.
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43. Set A consists of first 250 natural numbers multiples of 3; set B consists of first 200 even natural numbers. How many elements in A∪B?
(a) 324 (b) 364 (c) 384 (d) 400
Answer: (c) 384. n(A)=250, n(B)=200. A∩B: multiples of 6 up to 750: 125? Actually up to 400: 66. n(A∪B)=250+200-66=384.
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44. Let S_k denote sum of first k terms of AP. What is S_30/(S_20 - S_10)?
(a) 1 (b) 2 (c) 3 (d) 4
Answer: (c) 3. For natural numbers, S_k=k(k+1)/2. S_30=465, S_20=210, S_10=55. Denominator=155. Ratio=3.
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45. If roots of 4x^2 - (5k + 1)x + 5k = 0 differ by unity, then possible k?
(a) -3 (b) -1 (c) -1/5 (d) -3/5
Answer: (c) -1/5. α-β=1. (α+β)^2-4αβ=1. Solving gives 5k^2-14k-3=0 => k=3,-1/5. Option (c).
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46. Consider digits 3,5,7,9. Number of 5-digit numbers formed by these digits in which each of these four digits appears?
(a) 240 (b) 180 (c) 120 (d) 60
Answer: (a) 240. One digit repeated. Total 4*(5!/2!)=4*60=240.
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47. How many distinct matrices exist with all four entries taken from {1,2}?
(a) 16 (b) 24 (c) 32 (d) 48
Answer: (a) 16. Each entry 2 choices. 2^4=16.
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48. If i=sqrt(-1), how many values does i^{-2n} have for different n in Z?
(a) One (b) Two (c) Four (d) Infinite
Answer: (b) Two. i^{-2n}=(-1)^n, values 1 or -1.
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49. If x=a/(b-c), y=b/(c-a), z=c/(a-b), then value of some expression?
Answer: (a) abc as per OCR.
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50. Consider matrix A=[[1,1,1],[1,1,1],[1,1,1]].
1. Inverse of A does not exist.
2. A^3=A.
3. 3A=A^2.
Which correct?
(a) 1 and 2 only (b) 2 and 3 only (c) 1 and 3 only (d) 1,2,3
Answer: (c) 1 and 3 only. |A|=0, so 1 true. A^2=3A, so 3 true. A^3=A^2A=3A*A=3A^2=9A≠A, so 2 false.
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Directions (Q.Nos. 51 and 52): A circle passes through (5,-8), (2,9), (2,1).
51. Coordinates of centre?
(a) (-2,-50) (b) (-50,-20) (c) (-24,-58) (d) (-58,-24)
Answer: (d) (-58,-24).
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52. If r is radius, then?
(a) r<10 (b) 10<r<30 (c) 30<r<60 (d) r>60
Answer: (d) r>60. r=sqrt(58^2+24^2+285)=sqrt(4225)=65.
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Directions (Q.Nos. 53 and 54): Two vertices of equilateral triangle are (0,0) and (2,2).
53. Statements:
1. Third vertex has at least one irrational coordinate.
2. Area is irrational.
Which correct?
(a) 1 only (b) 2 only (c) Both (d) Neither
Answer: (c) Both. Third vertex coordinates involve sqrt3, area=2sqrt3, both irrational.
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55. Vertices of parallelogram ABCD: A(1,3), B(-1,2), C(3,5). Equation of diagonal BD?
(a) 2x-3y+2=0 (b) 3x-2y+5=0 (c) 2x-3y+8=0 (d) 3x-2y-5=0
Answer: (c) 2x-3y+8=0. Midpoint of AC = midpoint of BD. D=(5,6). Line BD: 2x-3y+8=0.
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56. Area of parallelogram?
(a) 1 (b) 3/2 (c) 2 (d) 5/2
Answer: (c) 2 sq. units.
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Directions (Q.Nos. 57 and 58): Sides AB, BC, CA of triangle are x-2=0, y+1=0, x+2y-4=0.
57. Equation of altitude through B on AC?
(a) x-3y+1=0 (b) x-3y+4=0 (c) 2x-y+4=0 (d) 2x-y-5=0
Answer: (d) 2x-y-5=0. B=(2,-1). Slope AC=-1/2, altitude slope=2. Line: 2x-y-5=0.
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58. Coordinates of circumcentre?
(a) (4,0) (b) (2,1) (c) (0,4) (d) (2,-1)
Answer: (a) (4,0). Triangle right angled at B. Circumcentre midpoint of AC=(4,0).
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Directions (Q.Nos. 59 and 60): Ends of latus rectum of parabola are (-2,4) and (-4,2).
59. Maximum number of parabolas?
(a) Only one (b) Two (c) Four (d) Infinite
Answer: (b) Two.
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60. Statements:
1. One parabola passes through origin.
2. Focus of one parabola lies at (-2,0).
Which correct?
(a) 1 only (b) 2 only (c) Both (d) Neither
Answer: (a) 1 only.
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61. Locus of point P(x,y,z) with z=7 is?
(a) line parallel X-axis (b) line parallel Y-axis (c) line parallel Z-axis (d) plane parallel xy-plane
Answer: (d) plane parallel xy-plane.
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62. Statements:
1. A line in space can have infinitely many direction ratios.
2. It is possible for certain line that sum of squares of direction cosines equals sum of direction cosines.
Which correct?
(a) 1 only (b) 2 only (c) Both (d) Neither
Answer: (c) Both.
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63. xy-plane divides line joining (-1,3,4) and (2,-5,6)
(a) internally 2:3 (b) internally 3:2 (c) externally 2:3 (d) externally 2:1
Answer: (c) externally 2:3. Ratio -z1:z2 = -4:6 = -2:3.
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64. Number of spheres of radius r touching coordinate axes?
(a) 4 (b) 6 (c) 8 (d) infinite
Answer: (c) 8.
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65. Cuboid ABCDEFGH: A(0,0,0), B(12,0,0), C(12,6,0), G(12,6,4). If α angle AB & AG, β angle AC & AG, then cos2α + cos2β?
(a) 40/49 (b) 64/49 (c) 120/49 (d) 160/49
Answer: (b) 64/49.
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66. Let a,b,c unit vectors such that a×b perpendicular to c. θ angle between a and b. Statements:
1. a×b = sinθ c
2. a·(b×c)=0
Which correct?
(a) 1 only (b) 2 only (c) Both (d) Neither
Answer: (c) Both.
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67. If a+3b=3i-j and 2a+b=i-2j, angle between a and b?
(a) 0 (b) π/6 (c) π/3 (d) π/2
Answer: (d) π/2. Solving: b=i, a=-j. Dot product 0.
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68. If (a+b) perpendicular to a, and |b|=2|a|, then (4a+b)·b?
(a) 0 (b) 1 (c) 8|a|^2 (d) 8|b|^2
Answer: (a) 0. a·b=-|a|^2. (4a+b)·b=4a·b+b·b=-4|a|^2+4|a|^2=0.
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69. a,b,c coplanar. Statements:
1. (a×b)×c coplanar with a and b.
2. (a×b)×c perpendicular to a×b.
Which correct?
(a) 1 only (b) 2 only (c) Both (d) Neither
Answer: (c) Both.
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70. Position vectors of A and B are (sqrt2-1)i-j and i+(sqrt2+1)j. Magnitude of AB?
(a) 2sqrt2 (b) 3sqrt2 (c) 2sqrt3 (d) 3sqrt3
Answer: (c) 2sqrt3. AB=(2-sqrt2)i+(sqrt2+2)j. |AB|=sqrt(12)=2sqrt3.
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71. If y=(1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^16), then dy/dx at x=0?
(a) 0 (b) 1 (c) 2 (d) 4
Answer: (b) 1. Only derivative of first factor contributes at x=0. Value=1.
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72. If y=cos x cos4x cos8x, then dy/dx at x=π/4?
(a) -1 (b) 0 (c) 1 (d) 3
Answer: (a) -1. log y derivative: -tan x -4tan4x -8tan8x. At π/4: -1-0-0=-1.
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73. Let f be polynomial such that f(f(x))=x^4. What is f'(1)?
(a) 0 (b) 1 (c) 2 (d) 4
Answer: (c) 2. f(x)=x^2, f'(x)=2x, f'(1)=2.
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74. lim_{n→∞} (a^n+b^n)/(a^n-b^n), a>b>1?
(a) -1 (b) 0 (c) 1 (d) does not exist
Answer: (c) 1. Divide by a^n: (1+(b/a)^n)/(1-(b/a)^n) → 1.
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75. f(x)=1+x/(2k) for 0<x<2, kx for 2≤x<4. If limit at x=2 exists, k?
(a) -2 (b) -1 (c) 0 (d) 1
Answer: (d) 1. 1+2/(2k)=2k => 1+1/k=2k => 2k^2-k-1=0 => k=1.
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77. If f(x)=[x]/[x], x≠0, right-hand limit at x=1?
(a) -1 (b) 0 (c) 1 (d) does not exist
Answer: (c) 1. For x→1+, [x]=1, f=1.
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78. f(x)=sin(1/x^2), x≠0.
1. Continuous at x=0 if f(0)=0.
2. Continuous at x=2/sqrt(π).
Which correct?
(a) 1 only (b) 2 only (c) Both (d) Neither
Answer: (b) 2 only.
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79. Range of f(x)=1-sin x?
(a) (0,2) (b) [0,2] (c) (-1,1) (d) [-1,1]
Answer: (b) [0,2].
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80. Slope of tangent of y=cos^{-1}(cos x) at x=-π/4?
(a) -1 (b) 0 (c) 1 (d) 2
Answer: (a) -1. For x in (-π,0), y=-x, slope -1.
NDA Previous Year Question Paper
81. Integral of f(x)=1+x^2+x^4 with respect to x^2?
(a) x+x^3/3+x^5/5+C (b) x^3/3+x^5/5+C (c) x^2+x^4/4+x^6/6+C (d) x^2+x^4/2+x^6/3+C
Answer: (d) x^2+x^4/2+x^6/3+C.
NDA Previous Year Question Paper
82. f(x)=x^2+1 in (1,2).
1. Maximum value 5.
2. Minimum value 2.
Which correct?
(a) 1 only (b) 2 only (c) Both (d) Neither
Answer: (c) Both.
NDA Previous Year Question Paper
83. If f(1)=f(4), then ∫_1^4 f'(x) dx?
(a) -1 (b) 0 (c) 1 (d) 2
Answer: (b) 0. = f(4)-f(1)=0.
NDA Previous Year Question Paper
84. ∫_0^π e^{ln(cos x)} dx?
(a) -1 (b) 0 (c) 1 (d) 2
Answer: (b) 0. = ∫_0^π cos x dx = 0.
NDA Previous Year Question Paper
85. If ∫ sqrt(1-sin2x) dx = A sin x + B cos x + C, 0<x<π/4, then A+B-2?
(a) -1 (b) 0 (c) 1 (d) 2
Answer: (b) 0. sqrt(1-sin2x)=cos x - sin x. So A=1,B=1. A+B-2=0.
NDA Previous Year Question Paper
86. Order of differential equation of all ellipses whose axes are along coordinate axes?
(a) 1 (b) 2 (c) 3 (d) 4
Answer: (b) 2. Two arbitrary constants a,b.
NDA Previous Year Question Paper
87. Degree of differential equation of all circles touching both coordinate axes in first quadrant?
(a) 1 (b) 2 (c) 3 (d) 4
Answer: (b) 2.
NDA Previous Year Question Paper
88. Differential equation of y=A-B/x?
(a) x y_2+y_1=0 (b) x y_2+2y_1=0 (c) x y_2-2y_1=0 (d) 2x y_2+y_1=0
Answer: (b) x y_2+2y_1=0. y'=B/x^2, y''=-2B/x^3. x y''+2y'=0.
NDA Previous Year Question Paper
89. ∫_0^π ln(tan(x/2)) dx?
(a) 0 (b) 1/2 (c) 1 (d) 2
Answer: (a) 0. Using property, 2I=0 => I=0.
NDA Previous Year Question Paper
90. Tangent to y=e^x at (0,1) meets X-axis where?
(a) (1,0) (b) (-1,0) (c) (2,0) (d) (-2,0)
Answer: (b) (-1,0). Tangent: y-1=x. Put y=0, x=-1.
NDA Previous Year Question Paper
91. f(x)=x+1/x.
1. Local maximum value < local minimum value.
2. Local maximum occurs at x=1.
Which correct?
(a) 1 only (b) 2 only (c) Both (d) Neither
Answer: (a) 1 only. Local max at x=-1, value -2; local min at x=1, value 2.
NDA Previous Year Question Paper
92. Maximum area of rectangle inscribed in circle radius 2?
(a) 4 (b) 6 (c) 8 (d) 16
Answer: (c) 8. Square of diagonal 4: side 2sqrt2, area 8.
NDA Previous Year Question Paper
93. ∫ dx/[x(x^2+1)]?
(a) (1/2)ln|x^2/(x^2+1)|+C (b) ln|x^2/(x^2+1)|+C (c) (3/2)ln|x^2/(x^2+1)|+C (d) (1/2)ln|(x^2+1)/x^2|+C
Answer: (a) (1/2)ln|x^2/(x^2+1)|+C.
NDA Previous Year Question Paper
94. Derivative of e^{e^x} with respect to e^x?
(a) e^{e^x} (b) e^x (c) e^{e^x} e^x (d) e e^x
Answer: (a) e^{e^x}.
NDA Previous Year Question Paper
95. f(x)=x^3+x^2+kx has no local extremum. Condition?
(a) 4k<1 (b) 3k>1 (c) 3k<1 (d) 3k≤1
Answer: (b) 3k>1. f'(x)=3x^2+2x+k. Discriminant <0: 4-12k<0 => 3k>1.
NDA Previous Year Question Paper
96. If f(x)=2^x, then ∫_2^10 f'(x)/f(x) dx?
(a) 4ln2 (b) ln4 (c) ln5 (d) 8ln2
Answer: (d) 8ln2. f'(x)/f(x)=ln2. Integral=8ln2.
NDA Previous Year Question Paper
97. If ∫_0^2 f(x) dx = k, then ∫_{-2}^0 |f(x)| dx?
(a) less than k (b) greater than k (c) less or equal (d) greater or equal
Answer: (d) greater or equal.
NDA Previous Year Question Paper
98. If f(x)=x^2-kx monotonically increasing in (1,∞), then?
(a) k<2 (b) 2<k<3 (c) 3<k<4 (d) k>4
Answer: (a) k<2. f'(x)=2x-k≥0 for x>1 => k≤2.
NDA Previous Year Question Paper
99. Area bounded by y=[x], X-axis, x=-1.5 and x=-1.8?
(a) 0.3 (b) 0.4 (c) 0.6 (d) 0.8
Answer: (c) 0.6. [x]=-2 in interval. Area=0.3*2=0.6.
NDA Previous Year Question Paper
100. Tangent to x^2=y at (1,1) makes angle θ with positive X-axis. Which correct?
(a) 0<θ<π/6 (b) π/6<θ<π/4 (c) π/4<θ<π/3 (d) π/3<θ<π/2
Answer: (d) π/3<θ<π/2. dy/dx=2x=2, tanθ=2. tanπ/3=sqrt3≈1.732, so π/3<θ<π/2.
NDA Previous Year Question Paper
101. For events E,F:
1. P(E∩F) ≥ P(E)+P(F)-1
2. P(E∪F) = P(E)+P(F)+P(E∩F)
3. P(E∪F) ≤ P(E)+P(F)
Which correct?
(a) 1 only (b) 3 only (c) 1 and 3 only (d) 1,2,3
Answer: (c) 1 and 3 only.
NDA Previous Year Question Paper
102. If P(A|B) < P(A), then?
(a) P(B|A) < P(B) (b) P(B|A) > P(B) (c) equal (d) > P(A)
Answer: (a) P(B|A) < P(B).
NDA Previous Year Question Paper
103. Most reliable measure of variability when mean is available?
(a) Range (b) Mean deviation (c) Standard deviation (d) Quartile deviation
Answer: (c) Standard deviation.
NDA Previous Year Question Paper
104. A,B,C solve independently with probabilities 1/2, 3/4, p. Problem solved probability 29/32. p?
(a) 2/5 (b) 2/3 (c) 1/3 (d) 1/4
Answer: (d) 1/4. 1 - (1/2)(1/4)(1-p) = 29/32 => (1-p)/8 = 3/32 => p=1/4.
NDA Previous Year Question Paper
105. Batsman hits six 8 times out of 60 balls. Probability he does not hit six?
(a) 2/3 (b) 1/15 (c) 2/15 (d) 13/15
Answer: (d) 13/15. 52/60 = 13/15.
NDA Previous Year Question Paper
Directions (Q.Nos. 106 and 107): Regression lines 3x-4y+8=0 and 4x-3y-1=0.
106. Statements:
1. Regression line of y on x is y=(3/4)x+2.
2. Regression line of x on y is x=(3/4)y+1/4.
Which correct?
(a) 1 only (b) 2 only (c) Both (d) Neither
Answer: (a) 1 only. Statement 2 is wrong.
NDA Previous Year Question Paper
108-109. Marks distribution table.
Median?
(a) 36.8 (b) 38.3 (c) 39.5 (d) 40.3
Answer: (c) 39.5. N=60, median class 35-40. Median=35+((30-26)/5)*5=39.
NDA Previous Year Question Paper
109. Mode?
(a) 27.27 (b) 27.73 (c) 27.93 (d) 28.27
Answer: (b) 27.73. Modal class 25-30. Mode=25+((11-5)/(22-5-6))*5=27.73.
NDA Previous Year Question Paper
110. Mean of natural numbers in [15,64]?
(a) 36.8 (b) 38.3 (c) 39.5 (d) 40.3
Answer: (c) 39.5.
NDA Previous Year Question Paper
111. For numbers x,x,x+2,x+3,x+10, x natural.
1. Mean > Mode
2. Median > Mean
Which correct?
(a) 1 only (b) 2 only (c) Both (d) Neither
Answer: (a) 1 only. Mean=x+3, Mode=x, Median=x+2.
NDA Previous Year Question Paper
112. Mean of 10 observations is 5.5. Each multiplied by 4 and subtracted from 44. New mean?
(a) 20 (b) 22 (c) 34 (d) 44
Answer: (b) 22. New mean=44-4*5.5=22.
NDA Previous Year Question Paper
113. Geometric mean of 2,4,8,...,1024. Which correct?
(a) 8<g<16 (b) 16<g<32 (c) 32<g<64 (d) g>64
Answer: (c) 32<g<64. GM=2^{5.5}.
NDA Previous Year Question Paper
114. Harmonic mean of 60 and x is 48. x?
(a) 32 (b) 36 (c) 40 (d) 44
Answer: (c) 40. 48=2*60*x/(60+x) => x=40.
NDA Previous Year Question Paper
115. Mean deviation of first 10 even natural numbers?
(a) 5 (b) 5.5 (c) 10 (d) 10.5
Answer: (a) 5.
NDA Previous Year Question Paper
116. If Σx_i=110, Σx_i^2=1540, variance?
(a) 22 (b) 33 (c) 44 (d) 55
Answer: (b) 33. Variance=1540/10-(110/10)^2=154-121=33.
NDA Previous Year Question Paper
117. 3-digit numbers formed using 1,3,7 without repetition. Probability divisible by 3?
(a) 0 (b) 1/3 (c) 1/4 (d) 1/8
Answer: (a) 0. Sum digits=11, never divisible by 3.
NDA Previous Year Question Paper
118. Probability roots of x^2+x+n=0 are real, n in N, n<4?
(a) 0 (b) 1/4 (c) 1/3 (d) 1/2
Answer: (a) 0. n=1,2,3. Discriminants negative.
NDA Previous Year Question Paper
119. If P(not A)=7/10, P(not B)=3/10, P(A|B)=3/14, then P(B|A)?
(a) 11/14 (b) 9/14 (c) 1/4 (d) 1/2
Answer: (d) 1/2. P(A)=3/10, P(B)=7/10. P(A∩B)=3/20. P(B|A)=1/2.
NDA Previous Year Question Paper
120. Seven white and three black balls randomly placed in row. Probability no two black adjacent?
(a) 7/15 (b) 8/15 (c) 13/15 (d) 1/15
Answer: (a) 7/15. Total arrangements=10!/(7!3!)=120. Favorable=8C3=56. Probability=56/120=7/15.
NDA Previous Year Question Paper
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NDA Previous Year Question Paper 2021
