NEET Physics Class 11 Chapter 10 Mathematical Tools Multiple Choice Question And Answers

NEET Physics Class 11 Chapter 10 Mathematical Tools Multiple Choice Question And Answers

Question 1. The surface area of a sphere as a function of its radius is A(r) = 4πr₂ the value of A(10) will be :

1. 1358 m²
2. 324 m²
3. 314 m²
4. 1256 m²

Answer: 4. 1256 m²

Question 2. If f (x) = x² –1

1. 5
2. 6
3. 7
4. 8

Answer: 4. 8

Question 3. If f(x) = \(x+\frac{1}{x}\), then the value of f(1) will be

1. 2
2. – 2
3. 1
4. – 1

Answer: 1. 2

Question 4. Find v (0), where v (t) = 3 + 2t

1. 5
2. 6
3. 3
4. None

Answer: 3. 3

Question 5. If f(θ) = sin θ, find \(f\left(\frac{\pi}{6}\right)\)

1. \(\frac{\pi}{6}\)
2. \(\frac{1}{2}\)
3. 2
4. \(\frac{\pi}{3}\)

Answer: 2. \(\frac{1}{2}\)

Question 6. If f (x) = 5, then the value of f (10) will be

1. 10
2. 5
3. 15
4. None

Answer: 2. 5

Question 7. tan15° is equivalent to :

1. \((2-\sqrt{3})\)
2. \((5+\sqrt{3})\)
3. \(\left(\frac{5-\sqrt{3}}{2}\right)\)
4. \(\left(\frac{5+\sqrt{3}}{2}\right)\)

Answer: 1. \((2-\sqrt{3})\)

Question 8. sin²θ is equivalent to:

1. \(\left(\frac{1+\cos \theta}{2}\right)\)
2. \(\left(\frac{1+\cos 2 \theta}{2}\right)\)
3. \(\left(\frac{1-\cos 2 \theta}{2}\right)\)
4. \(\left(\frac{\cos 2 \theta-1}{2}\right)\)

Answer: 3. \(\left(\frac{1-\cos 2 \theta}{2}\right)\)

Question 9. sinA. sin(A + B) is equal to

1. cos²A . cosB + sinA sin²B
2. \(\sin ^2 A \cdot \cos B+\frac{1}{2} \cos 2 A \cdot \sin B\)
3. \(\sin ^2 A \cdot \cos B+\frac{1}{2} \sin 2 A \cdot \sin B\)
4. sin²A . sinB + cosA cos²B

Answer: 3. \(\sin ^2 A \cdot \cos B+\frac{1}{2} \sin 2 A \cdot \sin B\)

Question 10. –sinθ is equivalent to :

1. \(\cos \left(\frac{\pi}{2}+\theta\right)\)
2. \(\cos \left(\frac{\pi}{2}-\theta\right)\)
3. \(\sin (\theta-\pi)\)
4. \(\sin (\pi+\theta)\)

Answer: (1,2,4)

Question 11. θ is the angle between the side CA and CB of a triangle, shown in the figure then θ is given by :

1. \(\cos \theta=\frac{2}{3}\)
2. \(\sin \theta=\frac{\sqrt{5}}{3}\)
3. \(\tan \theta=\frac{\sqrt{5}}{2}\)
4. \(\tan \theta=\frac{2}{3}\)

Answer: 2. \(\sin \theta=\frac{\sqrt{5}}{3}\)

Question 12. If tan θ = \(\frac{1}{\sqrt{5}}\) and θ lies in the first quadrant, the value of cos θ is :

1. \(\sqrt{\frac{5}{6}}\)
2. \(-\sqrt{\frac{5}{6}}\)
3. \(\frac{1}{\sqrt{6}}\)
4. \(-\frac{1}{\sqrt{6}}\)

Answer: 1. \(\sqrt{\frac{5}{6}}\)

Question 13. Calculate the slope of a shown line

1. 2/3
2. – 2/3
3. 3/2
4. –3/2

Answer: 2. – 2/3

Question 14. The speed (v) of a particle moving along a straight line is given by v = t² + 3t – 4 where v is in m/s and t in second. Find a time t at which the particle will momentarily come to rest.

1. 3
2. 4
3. 2
4. 1

Answer: 4. 1

Find the derivative of given functions w.r.t. corresponding independent variable.

Question 15. y = x² + x + 8

1. \(\frac{d y}{d x}=2 x+1\)
2. \(\frac{d y}{d x}=2+1\)
3. \(\frac{d y}{d x}=2 x-1\)
4. \(\frac{d y}{d x}=x+1\)

Answer: 1. \(\frac{d y}{d x}=2 x+1\)

Question 16. y = tan x + cot x

1. tan² x + cosec² x
2. cot² x – sin² x
3. sec² x – cosec² x
4. sec x + cosec x 2

Answer: 3. sec2 x – cosec2 x

Question 17. y = lnx + ex, then \(\frac{d^2 y}{d x^2}\) is equal to

1. \(\frac{1}{x^2}-e^x\)
2. \(\frac{1}{\mathrm{x}^2}+\mathrm{e}^{\mathrm{x}}\)
3. \(\frac{1}{x}+e^x\)
4. \(-\frac{1}{x^2}+e^x\)

Answer: 4. \(-\frac{1}{x^2}+e^x\)

Question 18. y = \(\mathrm{e}^{\mathrm{x}} \ell \mathrm{n} \mathrm{x}\)

1. \(\mathrm{e}^{\mathrm{x}} \ell \mathrm{n} x+\frac{\mathrm{e}^{\mathrm{x}}}{\mathrm{x}}\)
2. \(e^x \ell n x-\frac{e^x}{x}\)
3. \(\mathrm{e}^{\mathrm{x}} \ln x-\frac{e}{\mathrm{x}} \)
4. None of these

Answer: 1. \(\mathrm{e}^{\mathrm{x}} \ell \mathrm{n} x+\frac{\mathrm{e}^{\mathrm{x}}}{\mathrm{x}}\)

Question 19. y = sin 5 x

1. 5 cos 5 x
2. 3 cos 3 x
3. 5 cos 5x
4. 2 cos 2x

Answer: 1. 5 cos 5 x

Question 20. (x + y)² = 4

1. \(\frac{d y}{d x}=+1\)
2. \(\frac{d y}{d x}=-1\)
3. \(\frac{d}{d x}=-1\)
4. \(\frac{d y}{d}=-1\)

Answer: 2. \(\frac{d y}{d x}=-1\)

Question 21. y = 2u³, u = 8x – 1

1. \(\frac{d y}{d x}=48(8 x-1)^2\)
2. \(\frac{d y}{d x}=58(5 x-1)^2\)
3. \(\frac{d y}{d x}=48(8 x-1)^2\)
4. \(\frac{d y}{d x}=28(8 x-1)\)

Answer: 1. \(\frac{d y}{d x}=48(8 x-1)^2\)

Question 22. Given s = t² + 5t + 3, find \(\frac{\mathrm{ds}}{\mathrm{dt}}\) at t = 1

1. 7
2. 9
3. 12
4. 15

Answer: 1. 7

Question 23. If s = ut + \(s=u t+\frac{1}{2} a t^2\), where u and a are constants. Obtain the value of \(\frac{\mathrm{ds}}{\mathrm{dt}}\)

1. u – at
2. u + at
3. 2u + at
4. None of these

Answer: 2. u + at

Question 24. The minimum value of y = 5x² – 2x + 1 is

1. \(\frac{1}{5}\)
2. \(\frac{2}{5}\)
3. \(\frac{4}{5}\)
4. \(\frac{3}{5}\)

Answer: 3. \(\frac{4}{5}\)

Question 25. y = \(\frac{2 x+5}{3 x-2}\)

1. \(y^{\prime}=\frac{-19}{(3 x-2)^2}\)
2. \(y^{\prime}=\frac{19}{(3 x-2)}\)
3. \(y^{\prime}=\frac{-19}{(3 x+2)}\)
4. \(y^{\prime}=\frac{-19}{(3 x+2)^2}\)

Answer: 4. \(y^{\prime}=\frac{-19}{(3 x+2)^2}\)

Question 26. A uniform metallic solid sphere is heated uniformly. Due to thermal expansion, its radius increases at the rate of 0.05 mm/second. Find its rate of change of volume concerning the time when its radius becomes 10 mm. (take p = 3.14)

1. 31.4 mm³/second
2. 62.8 mm³/second
3. 3.14 mm³/second
4. 6.28 mm³/second

Answer: 2. 62.8 mm³/second

Question 27. If y = 3t² – 4t; then the minima of y will be at :

1. 3/2
2. 3/4
3. 2/3
4. 4/3

Answer: 3. 2/3

Question 28. If y = sin(t²) ,then \(\frac{d^2 y}{d t^2}\) will be –

1. 2t cos(t²)
2. 2 cos (t²) – 4t² sin (t²)
3. 4t² sin (t²)
4. 2 cos (t²)

Answer: 2. 2 cos (t²) – 4t² sin (t²)

Question 29. The displacement of a body at any time t after starting is given by s = 15t – 0.4t². The velocity of the body will be 7 ms^-1 after time :

1. 20 s
2. 15 s
3. 10 s
4. 5 s

Answer: 3. 10 s

Question 30. For the previous question, the acceleration of the particle at any time t is :

1. –0.8 m/s²
2. 0.8 m/s²
3. –0.6 m/s²
4. 0.5 m/s²

Answer: 1. –0.8 m/s²

Question 31. If the velocity of a particle is given by v = 2t⁴ then its acceleration (dv/dt) at any time t will be given by :

1. 8t³
2. 8t
3. –8t³
4. t²

Answer: 1. 8t³

Question 32. The maximum value of xy subject to x + y = 8, is :

1. 8
2. 16
3. 20
4. 24

Answer: 2. 16

Question 33. If y = 3t² – 4t; then the minima of y will be at :

1. 3/2
2. 3/4
3. 2/3
4. 4/3

Answer: 3. 2/3

Question 34. The slope of the graph as shown in the figure at points 1, 2 and is m₁, m₂, and m₃ respectively then

1. m₁ > m₂ > m₃
2. m₁ < m₂ < m₃
3. m₁ = m₂ = m₃
4. m₁ = m₂ > m₃

Answer: 2. m₁ < m₂ < m₃

Question 35. The magnitude of the slope of the shown graph.

1. First increases then decreases
2. First decrease then increases
3. Increase
4. Decrease

Answer: 2. First decrease then increases

Question 36. y = – x² + 3

1. \(\frac{d y}{d x}=-2 x, \frac{d^2 y}{d x^2}=-2\)
2. \(\frac{d y}{d x}=2 x, \frac{d^2 y}{d x^2}=-2\)
3. \(\frac{d y}{d x}=-2 x, \frac{d^2 y}{d x^2}=2\)
4. None of these

Answer: 1. \(\frac{d y}{d x}=-2 x, \frac{d^2 y}{d x^2}=-2\)

Question 37. y = \(\frac{x^3}{3}+\frac{x^3}{2}+\frac{x}{4}\)

1. \(\frac{d y}{d x}=x^2-x+\frac{1}{4}, \frac{d^2 y}{d x^2}=2 x+3\)
2. \(\frac{d y}{d x}=x^2+x-\frac{1}{4}, \frac{d^2 y}{d x^2}=2 x+1\)
3. \(\frac{d y}{d x}=x^2+x+\frac{1}{4}, \frac{d^2 y}{d x^2}=2 x+1\)
4. \(\frac{d y}{d x}=x^2+x+\frac{1}{4}, \frac{d^2 y}{d x^2}=2 x-1\)

Answer: 3. \(\frac{d y}{d x}=x^2+x+\frac{1}{4}, \frac{d^2 y}{d x^2}=2 x+1\)

Question 38. y = 4 – 2x – x^-3

1. \(\frac{d y}{d x}=2+3 x^{-4}, \frac{d^2 y}{d x^2}=-12 x^{-5}\)
2. \(\frac{d y}{d x}=-2+3 x^{-4}, \frac{d^2 y}{d x^2}=-12 x^{-5}\)
3. \(\frac{d y}{d x}=-2+3 x^{-4}, \frac{d^2 y}{d x^2}=12 x^{-5}\)
4. \(\frac{d y}{d x}=-2-3 x^{-4}, \frac{d^2 y}{d x^2}=-12 x^{-5}\)

Answer: 2. \(\frac{d y}{d x}=-2+3 x^{-4}, \frac{d^2 y}{d x^2}=-12 x^{-5}\)

Question 39. y = – 10x + 3 cos x

1. 10 – 3 sin x
2. – 10 + 3 sin x
3. – 10 + 5 sin x
4. – 10 – 3 sin x

Answer: 4. – 10 – 3 sin x

Question 40. y = \(\frac{3}{x}+5 \sin x\)

1. \(-\frac{3}{x^2}+5 \cos x\)
2. \(\frac{3}{x^2}+5 \cos \mathrm{x}\)
3. \(-\frac{3}{x^2}-\cos x\)
4. \(-\frac{3}{x^2}-5 \cos x\)

Answer: 1. \(-\frac{3}{x^2}+5 \cos x\)

Question 41. y = cosec \(x-4 \sqrt{x}+7\)

1. \(-\csc x \cot x-\frac{2}{\sqrt{x}}\)
2. \(\csc x \cot x+\frac{2}{\sqrt{x}}\)
3. \(-\csc x \cot x+\frac{2}{\sqrt{x}}\)
4. \(\csc x \cot x+\frac{2}{\sqrt{x}}\)

Find \(\frac{\mathrm{ds}}{\mathrm{dt}}\)

Answer: 1. \(-\csc x \cot x-\frac{2}{\sqrt{x}}\)

Question 42. s = tan t – t

1. sec²t + t
2. sec²t
3. sec t – 1
4. sec²t – 1

Answer: 4. sec²t – 1

Question 43. s = t² – sec t + t

1. 2t + sec t tan t + 1
2. 2t – sec t tan t + 1
3. 2t – sec t tan t –1
4. 2t + sec²tan t – 1

Answer: 2. 2t – sec t tan t + 1

Question 44. p = \(5+\frac{1}{\cot q}\) find \(\frac{d p}{d q}\)

1. sec² q
2. sec³ q
3. sec q
4. tan² q

Answer: 1. sec² q

Question 45. p = (1 + cosec q) cos q, find \(\frac{d p}{d q}\)

1. sin q – cosec² q
2. – sin q – cosec² q
3. – sin q – cos² q
4. sec q – cosec² q

Answer: 2. – sin q – cosec² q

Question 46. y = sin³ x , find the \(\frac{d y}{d x}\)

1. 3 sin² x (cosx)
2. 3 sin³ x (cosx)
3. 3 sin x (cos x)²
4. sin x (cos x)

Answer: 1. 3 sin² x (cosx)

Question 47. y = 5 cos^-4 x, find \(\frac{d y}{d x}\)

1. 20 in x cos^-5 x
2. 10 in x cos^-5 x
3. 20 in x cos^-3 x
4. 20 in x sin^-5 x

Answer: 1. 20 sin x cos^-5 x

Find the derivatives of the functions

Question 48. s = \(\frac{4}{3 \pi} \sin 3 t+\frac{4}{5 \pi} \cos 5 t\)

1. \(\frac{4}{\pi}\)(cos 3t – sin 5t)
2. \(\frac{4}{\pi}\) 4(cos 3t + sin 5t)
3. \(\frac{4}{\pi}\)(cos t – sin t)
4. \(\frac{4}{\pi}\)(cot 3t – sec 5t)

Answer: 1. \(\frac{4}{\pi}\)(cos 3t – sin 5t)

Question 49. s = \(\sin \left(\frac{3 \pi t}{2}\right)+\cos \left(\frac{3 \pi t}{2}\right)\)

1. \(\frac{3 \pi}{2}\left[\cos \left(\frac{3 \pi t}{2}\right)-\sin \left(\frac{3 \pi t}{2}\right)\right]\)
2. \(\frac{3 \pi}{2}\left[\cos \left(\frac{3 \pi t}{2}\right)+\sin \left(\frac{3 \pi t}{2}\right)\right]\)
3. \(\frac{3 \pi}{2}\left[\cot \left(\frac{3 \pi t}{2}\right)+\sin \left(\frac{3 \pi t}{2}\right)\right]\)
4. None of these

Answer: 1. \(\frac{3 \pi}{2}\left[\cos \left(\frac{3 \pi t}{2}\right)-\sin \left(\frac{3 \pi t}{2}\right)\right]\)

Find integrals of given functions

Question 50. \(\int\left(x^2-2 x+1\right) d x\)

1. \(\frac{x^3}{3}+x^2-x-c\)
2. \(\frac{x^3}{3}+x+x+c\)
3. \(\frac{x}{3}+x^2+x-c\)
4. \(\frac{x^3}{3}-x^2+x+c\)

Answer: 4. \(\frac{x^3}{3}-x^2+x+c\)

Question 51. \(\int\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right) d x\)

1. \(\frac{2 \sqrt{x}}{3}+2 \sqrt{x}-c\)
2. \(\frac{2 \sqrt{x^2}}{3}-2 \sqrt{x}+c\)
3. \(\frac{2 \sqrt{x^3}}{3}+2 \sqrt{x}+c\)
4. \(\frac{2 \sqrt{x}}{2}+2 \sqrt{x}-c\)

Answer: 3. \(\frac{2 \sqrt{x^3}}{3}+2 \sqrt{x}+c\)

Question 52. \(\int \frac{1}{3 x} d x\)

1. \(\frac{1}{3} \ln x+x\)
2. \(\frac{1}{3} \ln x\)
3. \(\frac{1}{2} \ln x+x\)
4. \(\frac{1}{3} \ln x+x\)

Answer: 4. \(\frac{1}{3} \ln x+x\)

Question 53. \(\int x \sin \left(2 x^2\right) d x\), (use,u = 2x²)

1. \(-\frac{1}{4} \cos \left(2 x^2\right)+C\)
2. \(\frac{1}{4} \cos \left(2 x^2\right)+C\)
3. \(-\frac{1}{2} \cos (2 x)+C\)
4. \(-\frac{1}{3} \cos \left(3 x^2\right)+C\)

Answer: 1. \(-\frac{1}{4} \cos \left(2 x^2\right)+C\)

Question 54. \(\int \frac{3}{(2-x)^2} d x\)

1. \(\frac{3}{2-x}+C\)
2. \(\frac{2}{2-x}+C\)
3. \(\frac{3}{2-x}+C\)
4. \(\frac{3}{2+x}+C\)

Answer: 1. \(\frac{3}{2-x}+C\)

Question 55. \(\int_{-4}^{-1} \frac{\pi}{2} d \theta\)

1. \(\frac{3 \pi}{3}\)
2. \(\frac{3 \pi}{2}\)
3. \(\frac{2 \pi}{3}\)
4. \(\frac{\pi}{2}\)

Answer: 1. \(\frac{3 \pi}{3}\)

Question 56. \(\int_0^1 e^x d x\)

1. e – 1 m
2. e + 1
3. e – 2
4. None of these

Answer: 1. e – 1 m

Question 57. y = 2x, the area under the curve from x = 0 to x = b will be

1. b²/2 units
2. b² units
3. 2b² units
4. b/2 units

Answer: 2. b2 units

Question58. y = \(\int_0^\pi \sin x d x\)

1. 2 units
2. 3 units
3. 4 units
4. 5 units

Answer: 1. 2 units

Question 59. The integral \(\int_1^5 x^2 d x\) is equal to

1. \(\frac{125}{3}\)
2. \(\frac{124}{3}\)
3. \(\frac{1}{3}\)
4. 45

Answer: 2. \(\frac{124}{3}\)

Question 60. \(\int x^{-\frac{3}{2}} d x\) is equal to :

1. \(\frac{-2}{\sqrt{x}}+C\)
2. \(\frac{2}{\sqrt{x}}+C\)
3. \(2 \sqrt{x}+C\)
4. \(-2 \sqrt{x}+C\)

Answer: 1. \(\frac{-2}{\sqrt{x}}+C\)

Question 61. \(\int x^{-\frac{5}{3}} d x\) is equal to :

1. \(\frac{3}{2} x^{\frac{2}{3}}+C\)
2. \(-\frac{3}{2} x^{\frac{2}{3}}+C\)
3. \(\frac{3}{2} x^{-\frac{2}{3}}+C\)
4. \(-\frac{3}{2} x^{-\frac{2}{3}}+C\)

Answer: 4. \(-\frac{3}{2} x^{-\frac{2}{3}}+C\)

Question 62. \(\int x^{2019} d x\) dx ∫is equal to :

1. \(\frac{x^{2020}}{2020}+C\)
2. \(\frac{x^{2018}}{2018}+C\)
3. \(2019 \mathrm{X}^{2018}+\mathrm{C}\)
4. \(-2012 X^{2011}+C\)

Answer: 1. \(\frac{x^{2020}}{2020}+C\)

Question 63. ∫2sin(x)dx is equal to :

1. –2cos x + C
2. 2 cosx + C
3. –2 cos x
4. 2 cox

Answer: 1. –2cos x + C

Question 64. \(\int(\sin x+\cos x) d x\) is equal to :

1. –cox + sinx
2. – cox + sinx + C
3. cosx – sinx + C
4. – cosx – sinx + C

Answer: 2. – cost + sinx + C

Question 65. \(\int\left(x+x^2+x^3+x^4\right) d x\) is equal to :

1. 1+2x+3x²+4x³ + C
2. 1+2x+3x²+4x³
3. \(\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\frac{x^5}{5}+C\)
4. \(\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\frac{x^5}{5}\)

Answer: 3. \(\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\frac{x^5}{5}+C\)

Question 66. If y = sin(ax+b), ∫y dx will be :

1. \(\frac{\cos (a x+b)}{a}+C\)
2. \(-\frac{\cos (a x+b)}{a}+c\)
3. a cos(ax+b)+C
4. – a cos(ax+b)+C

Answer: 2. \(-\frac{\cos (a x+b)}{a}+c\)

Question 67. If y = x²sin(x³), then ∫y dx will be :

1. \(-\cos \left(x^3\right)+C\)
2. \(\left(-\frac{\cos x^3}{3}\right)+C\)
3. \(\cos \left(x^3\right)+C\)
4. \(\frac{\cos x^3}{3}+C\)

Answer: 2. \(\left(-\frac{\cos x^3}{3}\right)+C\)

Question 68. If y = x², then area of curve y v/s x from x = 0 to 2 will be :

1. 1/3
2. 8/3
3. 4/3
4. 2/3

Answer: 2. 8/3

Question 69. If y = t sin (t²) then ∫ydt will be : 

1. \(\frac{\cos \left(\mathrm{t}^2\right)}{2}+\mathrm{c}\)
2. \(\frac{\cos \left(\mathrm{t}^2\right)}{2}+\mathrm{c}\)
3. \(\frac{-\cos \left(\mathrm{t}^2\right)}{2}+\mathrm{c}\)
4. \(\cos \left(t^2\right)\)

Answer: 3. \(\frac{-\cos \left(\mathrm{t}^2\right)}{2}+\mathrm{c}\)

Question 70. If x = (6y + 4) (3y² + 4y + 3) then ∫x day will be :

1. \(\frac{1}{3 y^2+4 y+3}\)
2. \(\frac{\left(3 y^2+4 y+3\right)^2}{2}+C\)
3. \(\left(3 y^2+4 y+3\right)\)
4. \(\frac{(6 y+4)}{\left(3 y^2+4 y+3\right)}\)

Answer: 2. \(\frac{\left(3 y^2+4 y+3\right)^2}{2}+C\)

Question 71. Value of \(\int_0^{\pi / 2} \cos 3 t\) it is

1. \(\frac{2}{3}\)
2. \(-\frac{1}{3}\)
3. \(-\frac{2}{3}\)
4. \(\frac{1}{3}\)

Answer: 2. \(-\frac{1}{3}\)

Question 72. \(\int_0^1\left(t^2+9 t+c\right) d t=\frac{9}{2}\). Then the value of ‘c’.

1. \(\frac{2}{3}\)
2. \(-\frac{1}{3}\)
3. \(-\frac{2}{3}\)
4. \(\frac{1}{3}\)

Answer: 2. \(-\frac{1}{3}\)

Question 73. Find the value of the following integration. \(\int_0^{2 \pi} \sin ^2 \theta d \theta\) Here c,a are constants.

1. π
2. 2π
3. 3 π
4. 4 π

Answer: 1. π

Question 74. If y = \(\frac{1}{a x+b}\), then ∫y dx will be :

1. \(\frac{1}{(a x+b)^2}+C\)
2. \(a x+b+C\)
3. \(a \ln (a x+b)+C\)
4. \(\frac{\ln (a x+b)}{a}+C\)

Answer: 4. \(\frac{\ln (a x+b)}{a}+C\)

Question 75. \(\int_\pi^{2 \pi} \theta d \theta\)

1. \(\frac{3 \pi^2}{2}\)
2. \(\frac{3 \pi^3}{2}\)
3. \(\frac{\pi^3}{2}\)
4. π

Answer: 1. \(\frac{3 \pi^2}{2}\)

Question 76. \(\int_0^{\sqrt[3]{7}} x^2 d x\)

1. \(\frac{7}{3}\)
2. \(\frac{7}{4}\)
3. \(\frac{5}{4}\)
4. 0

Answer: 1. \(\frac{7}{3}\)

Question 77. \(\int_0^1 \frac{d x}{3 x+2}\)

1. \(\ln \left(\frac{5}{2}\right)^{1 / 3}\)
2. \(\ln \left(\frac{5}{2}\right)^{1 / 2}\)
3. \(\ln \left(\frac{5}{2}\right)^{1 / 4}\)
4. None of these

Answer: 1. \(\ln \left(\frac{5}{2}\right)^{1 / 3}\)

Question 78. \(\int(x+1) d x\)

1. \(\frac{x^2}{2}+2 x-C\)
2. \(\frac{x^2}{2}+x+C\)
3. \(\frac{x^2}{2}-x+C\)
4. \(\frac{x^2}{2}-x-C\)

Answer: 2. \(\frac{x^2}{2}+2 x-C\)

Question 79. ∫ (5–6x) dx

1. 5x – x² + C
2. x – 3x²– C
3. 5x + 3x² + C
4. 5x – 3x² + C

Answer: 4. 5x – 3x² + C

Question 80. \(\int\left(3 t^2+\frac{t}{2}\right) d t\)

1. \(t^2+\frac{t^2}{4}-C\)
2. \(t^2+\frac{t^2}{4}+C\)
3. \(t^3-\frac{t^2}{4}-C\)
4. \(\frac{t^3}{6}+t^4+C\)

Answer: 2. \(t^2+\frac{t^2}{4}+C\)

Question 81. \(\int\left(\frac{t^2}{2}+4 t^3\right) d t\)

1. \(\frac{t^3}{6}+t^2+C\)
2. \(\frac{t^3}{6}+t+C\)
3. \(\frac{t^3}{6}-t+C\)
4. \(\frac{t^3}{6}+t^4+C\)

Answer: 4. \(\frac{t^3}{6}+t^4+C\)

Question 82. \(\int x^{-1 / 3} d x\)

1. \(\frac{3}{2} x^{2 / 3}+C\)
2. \(\frac{3}{2} x^{2 / 5}+C\)
3. \(\frac{3}{2} x^{1 / 3}+C\)
4. \(\frac{3}{2} x^{2 / 7}+C\)

Answer: 1. \(\frac{3}{2} x^{2 / 3}+C\)

Question 83. \(\int\left(\frac{\sqrt{x}}{2}+\frac{2}{\sqrt{x}}\right) d x\)

1. \(\frac{x^{3 / 2}}{3}+4 x^{1 / 2}+C\)
2. \(\frac{x^{3 / 2}}{3}+x^{1 / 2}+C\)
3. \(\frac{x^{3 / 2}}{3}+4 x^{2 / 5}+C\)
4. \(\frac{x^{3 / 2}}{3}+4 x^2+C\)

Answer: 1. \(\frac{x^{3 / 2}}{3}+4 x^{1 / 2}+C\)

Question 84. \(\int\left(8 y-\frac{2}{y^{1 / 4}}\right) d y\)

1. \(4 y^2-\frac{8}{3} y^{3 / 4}+C\)
2. \(4 y^2+\frac{8}{3} y^{3 / 4}+C\)
3. \(y^2-\frac{8}{3} y^{3 / 4}+C\)
4. \(4 y^2-\frac{8}{3} y^{1 / 3}+C\)

Answer: 1. \(4 y^2-\frac{8}{3} y^{3 / 4}+C\)

Question 85. \(\int 2 x\left(1-x^{-3}\right) d x\)

1. \(x+\frac{2}{x}-C\)
2. \(x^2+\frac{2}{x}+C\)
3. \(2 x^2+\frac{2}{x}+C\)
4. \(5 x^2+\frac{2}{x}+C\)

Answer: 2. \(x^2+\frac{2}{x}+C\)

Question 86. ∫(– 2cost) dt

1. – 2 sin t + C
2. – 3 sin t + C
3. – 5 sin t + C
4. – 7 sin t + C

Answer: 1. – 2 sin t + C

Question 87. ∫(– 5 sint) dt

1. 5 cos t + C
2. 2 cos t – C
3. 5 cosec t + C
4. 5 tan t + C

Answer: 1. 5 cos t + C

Question 88. \(\int 7 \sin \frac{\theta}{3} d \theta\)

1. \(-21 \cos \frac{\theta}{3}+\mathrm{C}\)
2. \(-14 \cos \frac{\theta}{3}+C\)
3. \(-42 \cos \frac{\theta}{3}\)
4. \(-7 \cos \frac{\theta}{3}+C\)

Answer: 1. \(-21 \cos \frac{\theta}{3}+\mathrm{C}\)

Question 89. ∫3cos 5θ+C

1. \(\frac{3}{5} \sin 5 \theta+C\)
2. \(\frac{3}{5} \sin 3 \theta+C\)
3. \(\frac{3}{5} \cos 5 \theta+C\)
4. \(\frac{3}{5} \sec 5 \theta+C\)

Answer: 1. \(\frac{3}{5} \sin 5 \theta+C\)

Question 90. \(\int\left(-3 \csc ^2 x\right) d x\) dx

1. 3 cot x + C
2. cot x + C
3. 3 tan x + C
4. 5 cot x + C

Answer: 1. 3 cot x + C

Question 91. \(\int\left(-\frac{\sec ^2 x}{3}\right) d x\)

1. \(\frac{-\tan x}{3}+x\)
2. \(\frac{-\tan x}{3}+C\)
3. \(\frac{-\tan x}{3}+C\)
4. None

Answer: 2. \(\frac{-\tan x}{3}+C\)

Question 92. \(\int \frac{\csc \theta \cot \theta}{2} d \theta\)

1. \(-\frac{1}{2} \csc \theta+C\)
2. \(-\frac{1}{2} \tan \theta+C\)
3. \(-\frac{1}{2} \cot \theta+C\)
4. \(-\frac{1}{2} \sec \theta+C\)

Answer: 1. \(-\frac{1}{2} \csc \theta+C\)

Question 93. \(\int \frac{2}{5} \sec \theta \tan \theta \mathrm{d} \theta\)

1. \(\frac{2}{5} \sec \theta+C\)
2. \(\frac{2}{5} \cos \theta+\mathrm{C}\)
3. \(\frac{2}{5} \tan \theta+C\)
4. \(\frac{2}{5}{cosec} \theta+C\)

Answer: 1. \(\frac{2}{5} \sec \theta+C\)

Question 94. \(\int\left(4 \sec x \tan -2 \sec ^2 x\right) d x\)

1. 4 sec x – 2 tan x + C
2. 2 sec x – 2 tan x + C
3. 4 sec x – 3 tan x + C
4. 4 sec x–5 tan x + C

Answer: 1. 4 sec x – 2 tan x + C

Question 95. \(\int \frac{1}{2}\left(\csc ^2 x-{cxc} x \cot x\right) d x\)

1. \(-\frac{1}{2} \cot x+\frac{1}{2} \csc x+C\)
2. \(\frac{1}{2} \tan x+\frac{1}{2} \csc x+C\)
3. \(-\frac{1}{2} \sec x+\frac{1}{2} \csc x+C\)
4. \(-\frac{1}{2} \sin x+\frac{1}{2} \csc x+C\)

Answer: 1. \(-\frac{1}{2} \cot x+\frac{1}{2} \csc x+C\)

Question 96. \(\int\left(\sin 2 x-\csc ^2 x\right) d x\)

1. \(-\frac{1}{2} \cos 2 x-\cot \mathrm{x}+C\)
2. \(-\frac{1}{2} \cos 2 x+\cot \mathrm{x}+\mathrm{C}\)
3. \(-\frac{1}{2} \cos 3 x-\cot x+C\)
4. \(-\frac{1}{2} \cos 2 x+\tan x+C\)

Answer: 2. \(-\frac{1}{2} \cos 2 x+\cot \mathrm{x}+\mathrm{C}\)

Question 97. \(\int(2 \cos 2 x-3 \sin 3 x) d x\)

1. sin 2x + cos 3x + C
2. sin 2x + cos 5x + C
3. sin 2x + cot 3x + C
4. sin 3x + cos 3x + C

Answer: 1. sin 2x + cos 3x + C

Question 98. \(\int \frac{1+\cos 4 t}{2} d t\)

1. \(\frac{t}{2}+\frac{\sin 4 t}{8}+C\)
2. \(\frac{t}{2}-\frac{\sin 4 t}{8}-C\)
3. \(\frac{t}{3}+\frac{\sin 4 t}{8}+C\)
4. All of these

Answer: 1. \(\frac{t}{2}+\frac{\sin 4 t}{8}+C\)

Question 99. \(\int \frac{1-\cos 6 \mathrm{t}}{2} \mathrm{dt}\)

1. \(\frac{t}{2}+\frac{\sin 6 t}{12}+C\)
2. \(\frac{t}{2}-\frac{\sin 6 t}{12}+C\)
3. \(2 \times \frac{t}{2}-\frac{\sin 6 t}{12}+C\)
4. \(\frac{t}{2}-\frac{\sin 6 t}{12}+C\)

Answer: 2. \(\frac{t}{2}-\frac{\sin 6 t}{12}+C\)

Question 100. \(\int\left(1+\tan ^2 \theta\right) d \theta\)

1. tan θ + C
2. cot θ + C
3. sec θ + C
4. cosec θ + C

Answer: 1. tan θ + C

Question 101. \(\int_{1 / 2}^{3 / 2}(-2 x+4) d x\)

1. 2 square units
2. 4 square units
3. 6 square units
4. 8 square units

Answer: 1. 2 square units

Evaluate definite integrals of the following functions

Question 102. \(\int_0^{\pi / 2} \theta^2 d \theta\)

1. \(\frac{\pi^3}{24}\)
2. \(\frac{\pi^2}{24}\)
3. \(\frac{\pi^2}{36}\)
4. \(\frac{\pi^2}{48}\)

Answer: 1. \(\frac{\pi^3}{24}\)

Question 103. \(\int_0^{3 b} x^2 d x\)

1. 9b³
2. 3b³
3. 27b³
4. 81 b³

Answer: 1. 9b³

Question 104. The forces, each numerically equal to 5N, are acting as shown in the Figure. Find the angle between forces.

1. 90º
2. 180º
3. 120º
4. 160º

Answer: 3. 120º

Question 105. The vector joining the points A (1, 1, –1) and B (2, –3, 4) and pointing from A to B is –

1. \(-\hat{i}+4 \hat{j}-5 \hat{k}\)
2. \(\hat{i}+4 \hat{j}+5 \hat{k}\)
3. \(\hat{i}-4 \hat{j}+5 \hat{k}\)
4. \(-\hat{i}-4 \hat{j}-5 \hat{k}\)

Answer: 4. \(-\hat{i}-4 \hat{j}-5 \hat{k}\)

Question 106. A vector of magnitude 30 and direction eastwards is added with another vector of magnitude 40 and direction Northwards. Find the magnitude and direction of the resultant with the east.

1. 45, 50º with East
2. 53, 75 with East
3. 53, 50º with East
4. 50, 53º with East

Answer: 4. 50, 53º with East

Question 107. The vector sum of the forces of 10 N and 6 N can be

1. 2 N
2. 8 N
3. 18 N
4. 20 N.

Answer: 2. 8 N

Question 108. The vector sum of two forces P and Q is minimum when the angle θ between their positive directions, is

1. \(\frac{\pi}{4}\)
2. \(\frac{\pi}{3}\)
3. \(\frac{\pi}{2}\)
4. \(\pi\)

Answer: 4. \(\pi\)

Question 109. The vector sum of two vectors \(\overrightarrow{\mathrm{A}} \text { and } \overrightarrow{\mathrm{B}}\) is maximum, then the angle θ between two vectors is –

1. 0º
2. 30°
3. 45°
4. 60°

Answer: 1. 0º

Question 110. Find the magnitude of \(3 \hat{i}+2 \hat{j}+\hat{k}\)?

1. \(\sqrt{10}\)
2. \(\sqrt{11}\)
3. \(\sqrt{13}\)
4. \(\sqrt{14}\)

Answer: 4. \(\sqrt{14}\)

Question 111. If \(\overrightarrow{\mathrm{A}}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}\) then find \(\hat{A}\)

1. \(\frac{3 \hat{i}+4 \hat{j}}{5}\)
2. \(\frac{2 \hat{i}+3 \hat{j}}{5}\)
3. \(\frac{2 \hat{i}+4 \hat{j}}{5}\)
4. \(\frac{3 \hat{i}-2 \hat{j}}{5}\)

Answer: 1. \(\frac{3 \hat{i}+4 \hat{j}}{5}\)

Question 112. One of the rectangular components of a velocity of 60 km h–1 is 30 km h–1. Find another rectangular component.

1. \(30 \sqrt{3} \mathrm{~km} \mathrm{~h}^{-1}\)
2. \(20 \sqrt{3} \mathrm{~km} \mathrm{~h}^{-1}\)
3. \(30 \sqrt{2} \mathrm{~km} \mathrm{~h}^{-1}\)
4. \(30 \sqrt{2} \mathrm{~km} \mathrm{~h}^{-1}\)

Answer: 1. \(30 \sqrt{3} \mathrm{~km} \mathrm{~h}^{-1}\)

Question 113. The x and y components of a force are 2 N and – 3N. The force is

1. \(2 \hat{i}-3 \hat{j}\)
2. \(2 \hat{i}+3 \hat{j}\)
3. \(-2 \hat{i}-3 \hat{j}\)
4. \(3 \hat{i}+2 \hat{j}\)

Answer: 1. \(2 \hat{i}-3 \hat{j}\)

Question 114. A force of 30 N is inclined at an angle θ to the horizontal. If its vertical component is 18 N, find the horizontal component & the value of θ

1. 24 N ; 370 approx
2. 20 N ; 470 approx
3. 25 N ; 350 approx
4. 37 N ; 240 approx

Answer: 1. 24 N; 370 approx

Question 115. The angle θ between directions of forces \(\vec{A} \text { and } \vec{B}\) is 90º where A = 8 dyne and B = 6 dyne. If the resultant \(\vec{R}\) makes an angle α with \(\vec{A}\) then find the value of ‘α’?

1. 47º
2. 37º
3. 75º
4. 120º

Answer: 2. 37º

Question 116. If \(\vec{A}=3 \hat{i}+4 \hat{j} \text { and } \vec{B}=\hat{i}+\hat{j}+2 \hat{k}\) then find out unit vector along \(\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}\)

1. \(\frac{4 \hat{i}+5 \hat{j}-2 \hat{k}}{\sqrt{45}}\)
2. \(\frac{2 \hat{i}-5 \hat{j}-2 \hat{k}}{\sqrt{45}}\)
3. \(\frac{4 \hat{i}-2 \hat{j}+2 \hat{k}}{\sqrt{45}}\)
4. \(\frac{4 \hat{i}+5 \hat{j}+2 \hat{k}}{\sqrt{45}}\)

Answer: 4. \(\frac{4 \hat{i}+5 \hat{j}+2 \hat{k}}{\sqrt{45}}\)

Question 117. The x and y components of the vector \(\overrightarrow{\mathrm{A}}\) are 4m and 6m respectively. The x,y components of vector are \(\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}\) 10m and 9m respectively. Find the length of \(\overrightarrow{\mathrm{B}}\) and the angle that \(\overrightarrow{\mathrm{B}}\) makes with the x-axis.

1. \(3 \sqrt{3}, \tan ^{-1} \frac{1}{2}\)
2. \(3 \sqrt{5}, \tan ^{-1} \frac{1}{2}\)
3. \(3 \sqrt{5}, \tan \frac{1}{3}\)
4. \(2 \sqrt{3}, \tan ^{-1} \frac{1}{2}\)

Answer: 2. \(3 \sqrt{5}, \tan ^{-1} \frac{1}{2}\)

Question 118. A vector is not changed if

1. It is displaced parallel to itself
2. It is rotated through an arbitrary angle
3. It is cross-multiplied by a unit vector
4. It is multiplied by an arbitrary scalar.

Answer: 1. It is displaced parallel to itself

Question 119. If the angle between two forces increases, the magnitude of their resultant

1. Decreases
2. Increases
3. Remains unchanged
4. First decreases and then increases

Answer: 1. Decreases

Question 120. Which of the following sets of displacements might be capable of bringing a car to its returning point?

1. 5, 10, 30 and 50 km
2. 5, 9, 9 and 16 km
3. 40, 40, 90 and 200 km
4. 10, 20, 40 and 90 km

Answer: 21. 5, 9, 9 and 16 km

Question 121. When two vectors \(\overrightarrow{\mathrm{a}} \text { and } \overrightarrow{\mathrm{b}}\) are added, the magnitude of the resultant vector is always

1. Greater than (a + b)
2. Less than or equal to (a + b)
3. Less than (a + b)
4. Equal to (a + b)

Answer: 2. Less than or equal to (a + b)

Question 122. If \(|\vec{A}+\vec{B}|=|\vec{A}|=|\vec{B}|\), then the angle between \(\overrightarrow{\mathrm{A}} \text { and } \overrightarrow{\mathrm{B}}\) is

1. 0º
2. 60º
3. 90º
4. 120º

Answer: 4. 120º

Question 123. Vector \(\overrightarrow{\mathrm{A}}\) is of length 2 cm and is 60º above the x-axis in the first quadrant. Vector \(\overrightarrow{\mathrm{B}}\) is of length 2 cm and 60º below the x-axis in the fourth quadrant. The sum \(\) is a vector of magnitude –

1. 2 along + y-axis
2. 2 along + x-axis
3. 1 along – x-axis
4. 2 along – x-axis

Answer: 2. 2 along + x-axis

Question 124. Which of the following is a true statement?

1. A vector cannot be divided by another vector
2. Angular displacement can either be a scalar or a vector.
3. Since the addition of vectors is commutative therefore vector subtraction is also commutative.
4. The resultant of two equal forces of magnitude F acting at a point is F if the angle between the two forces is 120º.

Answer: 1. A vector cannot be divided by another vector

Question 125. In the Figure which of the ways indicated for combining the x and y components of vector a is proper to determine that vector?

Answer: 1

Question 126. Two vectors having an equal magnitude of 5 units have an angle of 60º between them. Find the magnitude of their resultant vector and its angle from one of the vectors.

1. 5, 20º
2. \(5 \sqrt{3}\), 30º
3. 3, 40º
4. 3, 50º

Answer: 2. \(5 \sqrt{3}\), 30º

Question 127. Two forces each numerically equal to 10 dynes are acting as shown in the figure, then find the resultant of these two vectors.

1. 5 dyne
2. 10 dyne
3. 15 dyne
4. 25 dyne

Answer: 2. 10 dyne

Question 128. The magnitude of pairs of displacement vectors is given. Which pairs of displacement vectors cannot be added to give a resultant vector of magnitude 13 cm?

1. 4 cm, 16 cm
2. 20 cm, 7 cm
3. 1 cm, 15 cm
4. 6 cm, 8 cm

Answer: 3. 1 cm, 15 cm

Question 129. If \(\overrightarrow{\mathrm{A}}=3 \hat{\mathrm{i}}+2 \hat{\mathrm{i}} \text { and } \overrightarrow{\mathrm{B}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{i}}-\hat{\mathrm{k}}\), then find a unit vector along \((\vec{A}-\vec{B})\)

1. \(\frac{\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}}{\sqrt{3}}\)
2. \(\frac{\hat{i}-\hat{j}-\hat{k}}{\sqrt{3}}\)
3. \(\frac{\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}}{\sqrt{3}}\)
4. \(\frac{\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}}{\sqrt{3}}\)

Answer: 3. \(\frac{\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}}{\sqrt{3}}\)

Question 130. If \(\hat{n}\) is a unit vector in the direction of the vector \(\overrightarrow{\mathrm{A}}\), then –

1. \(\hat{n}=\frac{\vec{A}}{|A|}\)
2. \(\hat{n}=\vec{A}|\vec{A}|\)
3. \(\hat{n}=\frac{|\vec{A}|}{\vec{A}}\)
4. \(\hat{n}=\hat{n} \times \vec{A}\)

Answer: 1. \(\hat{n}=\frac{\vec{A}}{|A|}\)

Question 131. The resultant of \(\vec{A} \text { and } \vec{B}\) makes an angle α with \(\vec{A} \text { and } \vec{B}\), then :

1. α < β
2. α < β
3. α < β if A > B
4. α < β if A = B

Answer: 3. α < β if A > B

Question 132. If \(\vec{P}+\vec{Q}=\vec{P}-\vec{Q}\) and θ is the angle between \(\overrightarrow{\mathrm{P}} \text { and } \overrightarrow{\mathrm{Q}}\), then

1. θ = 0º
2. θ = 90º
3. P =0
4. Q = 0

Answer: 4. Q = 0

Question 133. The magnitudes of the sum and difference of two vectors are the same, then the angle between them is

1. 90º
2. 40º
3. 45º
4. 60º

Answer: 1. 90º

Question 134. The projection of a vector \(3 \hat{i}+4 \hat{k}\) on the y-axis is:

1. 5
2. 4
3. 3
4. 3

Answer: 4. 3

Question 135. Two forces of 12N and 8N act up the n body. The resultant force on the body has a maximum value of-

1. 4N
2. 0N
3. 20 N
4. 8 N

Answer: 3. 20 N

Question 136. In figure, \(\vec{E}\) equals

1. \(\vec{A}\)
2. \(\vec{B}\)
3. \(\vec{A}+\vec{B}\)
4. \(-(\vec{A}+\vec{B})\)

Answer: 4. \(-(\vec{A}+\vec{B})\)

Question 137. In figure, \(\overrightarrow{\mathrm{D}}-\overrightarrow{\mathrm{C}}\) equals

1. \(\vec{A}\)
2. \(-\overrightarrow{\mathrm{A}}\)
3. \(\vec{B}\)
4. \(-\vec{B}\)

Answer: 1. \(\vec{A}\)

Question 138. In the figure, \(\vec{E}+\vec{D}-\vec{C}\) equals

1. \(\vec{A}\)
2. \(-\overrightarrow{\mathrm{A}}\)
3. \(\vec{B}\)
4. \(-\vec{B}\)

Answer: 4. \(-\vec{B}\)

Question 139. Forces proportional to AB, BC, and 2CA act along the sides of triangle ABC in order. They’re resultant represented in magnitude and direction as

1. CA
2. AC
3. BC
4. CB

Answer: 1. CA

Question 140. A given force is resolved into components P and Q equally inclined to it. Then :

1. P = 2Q
2. 2P = Q
3. P = Q
4. None of these

Answer: 3. P = Q

Question 141. A particle starting from the origin (0,0) moves in a straight line in the (x, y) plane. Its coordinates at a later time are (\(\sqrt{3}\), 3). The path of the particle makes with the x-axis an angle of :

1. 30º
2. 45º
3. 60º
4. 0º

Answer: 3. 60º

Question 142. If \(\overrightarrow{\mathrm{A}}=\hat{\mathrm{i}}+\hat{\mathrm{J}}+\hat{\mathrm{k}} \text { and } \overrightarrow{\mathrm{B}}=2 \hat{\mathrm{i}}+\hat{\mathrm{J}}\) find (a) \(\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}\) (b) \(\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}\)

1. 3 and \(-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}\)
2. 5 and \(-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}\)
3. 1 and \(-\hat{i}+2 \hat{j}+\hat{k}\)
4. 3 and \(-\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\)

Answer: 1. 3 and \(-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}\)

Question 143. If \(|\vec{A}|=4,|\vec{B}|\) = 3 and θ = 60º in figure, Find (a) \(\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}\) (b) \(\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}\)

1. 3 and \(6 \sqrt{3}\)
2. 6 and \(3 \sqrt{3}\)
3. 6 and \(3 \sqrt{6}\)
4. 6 and \(6 \sqrt{3}\)

Answer: 4. 6 and \(6 \sqrt{3}\)

Question 144. Three non zero vector \(\overrightarrow{\mathrm{A}}, \overrightarrow{\mathrm{B}} and \overrightarrow{\mathrm{C}}\) satisfy the relation \(\vec{A} \cdot \vec{B}=0 \text { and } \vec{A}. \vec{C}=0\). Then \(\overrightarrow{\mathrm{A}}\)= 0 can be parallel to :

1. \(\vec{B}\)
2. \(\overrightarrow{\mathrm{C}}\)
3. \(\vec{B} \cdot \vec{C}\)
4. \(\overrightarrow{\mathrm{B}} \times \overrightarrow{\mathrm{C}}\)

Answer: 4. \(\overrightarrow{\mathrm{B}} \times \overrightarrow{\mathrm{C}}\)

Question 145. If \(\vec{A}=4 \hat{i}+n \hat{J}-2 \hat{k} \text { and } \vec{B}=2 \hat{i}+3 \hat{j}+\hat{k}\), then find the value of n so that \(\overrightarrow{\mathrm{A}} \perp \overrightarrow{\mathrm{B}}\)

1. n = 2
2. n = – 1
3. n + 1
4. n = – 2

Answer: 4. n = – 2

Question 146. If \(\vec{F}=(4 \hat{i}-10 \hat{j}) \text { and } \vec{r}=(5 \hat{i}-3 \hat{j})\), then calculate torque \((\vec{\tau}=\vec{r} \times \vec{F})\)

1. \(-38 \hat{k}\)
2. \(-35 \hat{k}\)
3. \(-55 \hat{k}\)
4. \(-28 \hat{k}\)

Answer: 1. \(-38 \hat{k}\)

Question 147. Find a unit vector perpendicular to both the vectors) \((2 \hat{i}+3 \hat{j}+\hat{k}) \text { and }(\hat{i}-\hat{j}+2 \hat{k})\)

1. \(\hat{n}= \pm \frac{1}{\sqrt{83}}(7 \hat{i}+3 \hat{j}+5 \hat{k})\)
2. \(\hat{n}= \pm \frac{1}{\sqrt{83}}(-7 \hat{i}-3 \hat{j}+5 \hat{k})\)
3. \(\hat{n}= \pm \frac{1}{\sqrt{83}}(7 \hat{i}-3 \hat{j}-5 \hat{k})\)
4. \(\hat{n}= \pm \frac{1}{\sqrt{58}}(7 \hat{i}-3 \hat{j}-5 \hat{k})\)

Answer: 3. \(\hat{n}= \pm \frac{1}{\sqrt{83}}(7 \hat{i}-3 \hat{j}-5 \hat{k})\)

Question 148. Which of the following vector identities is false?

1. \(\vec{P}+\vec{Q}=\vec{Q}+\vec{P}\)
2. \(\vec{P}+\vec{Q}=\vec{Q} \times \vec{P}\)
3. \(\vec{P} \cdot \vec{Q}=\vec{Q} \cdot \vec{P}\)
4. \(\vec{P} \times \vec{Q} \neq \vec{Q} \times \vec{P}\)

Answer: 2. \(\vec{P}+\vec{Q}=\vec{Q} \times \vec{P}\)

Question 149. The area of a parallelogram, whose diagonals are \(\) will be

1. 14 unit
2. \(5 \sqrt{3}\)
3. \(10 \sqrt{3}\)
4. \(20 \sqrt{3}\)

Answer: 3. \(10 \sqrt{3}\)

Question 150. If \(\overrightarrow{\mathrm{A}}=\hat{\mathrm{i}}+\hat{\mathrm{j}} \) and \(\overrightarrow{\mathrm{B}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}\) The value of \((\vec{A}+\vec{B}) \cdot(\vec{A}-\vec{B})\) is :

1. \(\sqrt{2}\)
2. 0
3. \(\frac{1}{2}\)
4. 2

Answer: 2. 0

Question 151. Vectors \(\vec{A}=\hat{i}+\hat{j}-2 \hat{k} \text { and } \vec{B}=3 \hat{i}+3 \hat{j}-6 \hat{k}\) are:

1. Parallel
2. Antiparallel
3. Perpendicular
4. At acute angle with each other

Answer: 3. Perpendicular

Question 152. If two vectors are given as : \(\overrightarrow{\mathrm{A}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}\) and \( \overrightarrow{\mathrm{B}}=-\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}\) then the vector is not perpendicular to \(\) is:

1. \(-2 \hat{i}+4 \hat{j}+2 \hat{k}\)
2. \(\hat{i}+\hat{j}+\hat{k}\)
3. \(25 \hat{i}-625 \hat{j}-25 \hat{k}\)
4. \(3 \hat{i}-2 \hat{j}-3 \hat{k}\)

Answer: 1. \(-2 \hat{i}+4 \hat{j}+2 \hat{k}\)

Question 153. If a vector \(2 \hat{i}+3 \hat{j}+8 \hat{k}\) is perpendicular to the vector \(4 \hat{i}-4 \hat{j}+\alpha \hat{k}\), then the value of α is :

1. –1
2. \(\frac{1}{2}\)
3. \(-\frac{1}{2}\)
4. 1

Answer: 3. \(-\frac{1}{2}\)

Question 154. If the angle between the vectors \(\overrightarrow{\mathrm{A}} \text { and } \overrightarrow{\mathrm{B}}\)is θ, the value of the product \((\overrightarrow{\mathrm{B}} \times \overrightarrow{\mathrm{A}}) \cdot \overrightarrow{\mathrm{A}}\) is equal to :

1. BA² cos θ
2. BA² sin θ
3. BA² sin θ cos θ
4. Zero

Answer: 4. Zero

Question 155. \(\overrightarrow{\mathrm{A}} \text { and } \overrightarrow{\mathrm{B}}\) are two vectors and θ is the angle between them, if \(|\vec{A} \times \vec{B}|=\sqrt{3}(\vec{A} \cdot \vec{B})\) the value of θ is :

1. 60º
2. 45º
3. 30º
4. 90º

Answer: 1. 60º

Question 156. Two forces P and Q acting at a point are such that if P is reversed, the direction of the resultant is turned through 90º. Then

1. P = Q
2. P =2Q
3. P = \(\frac{Q}{2}\)
4. No relation between P and Q

Answer: 1. P = Q

Question 157. The vector sum of two forces is perpendicular to their vector differences. In that case, the forces :

1. Are not equal to each other in magnitude
2. Cannot be predicted
3. Are equal to each other
4. Are equal to each other in magnitude

Answer: 4. Are equal to each other in magnitude

Question 158. If \(|\vec{A} \times \vec{B}|=\sqrt{3} \cdot \vec{A}, \vec{B}\) then the value of \(|\vec{A}+\vec{B}|\) is :

1. \(\left(\mathrm{A}^2+\mathrm{B}^2+\mathrm{AB}\right)^{1 / 2}\)
2. \(\left(A^2+B^2+\frac{A B}{\sqrt{3}}\right)^{1 / 2}\)
3. \(A+B\)
4. \(\left(\mathrm{A}^2+\mathrm{B}^2+\sqrt{3} A B\right)^{1 / 2}\)

Answer: 1. \(\left(\mathrm{A}^2+\mathrm{B}^2+\mathrm{AB}\right)^{1 / 2}\)

Question 159. Velocity as a function of time is V(t) = sin²t – cos(2t). Then the value of \(\left(\frac{\pi}{3}\right)\) will be :

1. \(\frac{2}{3}\)
2. \(\frac{1}{4}\)
3. \(\frac{1}{4}\)
4. \(\frac{5}{4}\)

Answer: 4. \(\frac{5}{4}\)

Question 160. If f = \(2 \pi \frac{x^3 y^5}{\sqrt{z}}\) then log f is equal to :

1. \(\log 2 \pi+3 \log x+5 \log y+\frac{1}{2} \log z\)
2. \(\log 2 \pi+3 \log x+5 \log y-\frac{1}{2} \log z\)
3. \(\log 2 \pi-3 \log x+5 \log y+\frac{1}{2} \log z\)
4. \(\log 2 \pi+3 \log x+5 \log y+\log z\)

Answer: 2. \(\log 2 \pi+3 \log x+5 \log y-\frac{1}{2} \log z\)

Question 161. Which of the following is true

1. sin37° + cos37° = sin53° + cos53°
2. sin37° – cos37° = cos53° – sin53°
3. tan37° + 1 = tan 53° – 1
4. tan37° × tan53° = 1

Answer: 1. sin37° + cos37° = sin53° + cos53°

Question 162. If y₁ = A sinθ₁ and y₂ = A sin θ₂ then

1. \(y_1+y_2=2 A \sin \left(\frac{\theta_1+\theta_2}{2}\right) \cos \left(\frac{\theta_1-\theta_2}{2}\right)\)
2. \(y_1+y_2=2 A \sin \theta_1 \sin \theta_1\)
3. \(y_1-y_2=2 A \sin \left(\frac{\theta_1-\theta_2}{2}\right) \cos \left(\frac{\theta_1+\theta_2}{2}\right)\)
4. \(y_1, y_2=-2 A^2 \cos \left(\frac{\pi}{2}+\theta_1\right) \cdot \cos \left(\frac{\pi}{2}-\theta_2\right)\)

Answer: (1,3)

Question 163. If R² = A² + B² + 2AB cosθ , if |A| = |B| then value of magnitude of R is equivalent to :

1. 2Acosθ
2. \(A \cos \frac{\theta}{2}\)
3. \(2 A \cos \frac{\theta}{2}\)
4. \(2 B \cos \frac{\theta}{2}\)

Answer: (3,4)

Question 164. A particle starting from the origin (0, 0) moves in a straight line in the (x, y) plane. Its coordinates at a later time are (\(\sqrt{3}\), 3). The path of the particle makes with the x-axis an angle of :

1. 30º
2. 45º
3. 60º
4. 0º

Answer: 3. 60º

Question 165. Find the value of an if the distance between the point (–9cm, a cm) and (3cm, 3 cm) is 13 cm.

1. 6 cm
2. 8 cm
3. 10 cm
4. 12 cm

Answer: 2. 8 cm

Question 166. y = lnx² + sin x

1. \(\frac{d y}{d x}=\frac{2}{x}+\cos x, \frac{d^2 y}{d x^2}=\frac{-2}{x^2}-\sin x\)
2. \(\frac{d y}{d x}=\frac{2}{x}-\cos x, \frac{d^2 y}{d x^2}=\frac{-2}{x^2}+\sin x\)
3. \(\frac{d y}{d x}=-\frac{2}{x}+\cos x, \frac{d^2 y}{d x^2}=\frac{-2}{x^2}-\sin x\)
4. \(\frac{d y}{d x}=-\frac{2}{x}-\cos x, \frac{d^2 y}{d x^2}=\frac{2}{x^2}-\sin x\)

Answer: 1. \(\frac{d y}{d x}=\frac{2}{x}+\cos x, \frac{d^2 y}{d x^2}=\frac{-2}{x^2}-\sin x\)

Question 167. y = \(\sqrt[7]{x}+\tan x\)

1. \(\frac{d y}{d x}=-\frac{x^{-\frac{6}{7}}}{7}+\sec ^2 x, \frac{d^2 y}{d x^2}=\frac{-6}{49} x^{\frac{-13}{7}}-2 \tan x \sec ^2 x\)
2. \(\frac{d y}{d x}=\frac{x^{-\frac{6}{7}}}{7}+\sec ^2 x, \frac{d^2 y}{d x^2}=\frac{-6}{49} x^{\frac{-13}{7}}+2 \tan x \sec ^2 x\)
3. \(\frac{d y}{d x}=\frac{x^{-\frac{6}{7}}}{7}-\sec ^2 x, \frac{d^2 y}{d x^2}=\frac{-6}{49} x^{\frac{-13}{7}}-2 \tan x \sec ^2 x\)
4. \(\frac{d y}{d x}=\frac{x^{-\frac{6}{7}}}{7}+\sec ^2 x, \frac{d^2 y}{d x^2}=\frac{6}{49} x^{\frac{-13}{7}}+2 \tan x \sec ^2 x\)

Answer: 2. \(\frac{d y}{d x}=\frac{x^{-\frac{6}{7}}}{7}+\sec ^2 x, \frac{d^2 y}{d x^2}=\frac{-6}{49} x^{\frac{-13}{7}}+2 \tan x \sec ^2 x\)

Find the derivative of given functions w.r.t. the corresponding independent variable.

Question 168. y = \(\left(x+\frac{1}{x}\right)\left(x-\frac{1}{x}+1\right)\)

1. \(\frac{d y}{d x}=1+2 x+\frac{2}{x^3}-\frac{1}{x^2}\)
2. \(\frac{d y}{d x}=1+2 x-\frac{2}{x^3}-\frac{1}{x^2}\)
3. \(\frac{d y}{d x}=1+2 x+\frac{2}{x^3}-\frac{1}{x^2}\)
4. \(\frac{d y}{d x}=1+2 x+\frac{2}{x^3}+\frac{1}{x^2}\)

Answer: 3. \(\frac{d y}{d x}=1+2 x+\frac{2}{x^3}-\frac{1}{x^2}\)

Question 169. r = (1 + sec θ) sin θ, r′ is

1. \(\frac{d r}{d \theta}=\cos \theta+\sec ^2 \theta\)
2. \(\frac{d r}{d \theta}=\cos \theta-\sec ^2 \theta\)
3. \(\frac{d r}{d \theta}=\cos \theta+\tan ^2 \theta\)
4. \(\frac{d r}{d \theta}=\cos \theta+\sec ^2 \theta\)

Answer: 1. \(\frac{d r}{d \theta}=\cos \theta+\sec ^2 \theta\)

Question 170. q = \(\sqrt{2 r-r^2}\), find \(\frac{\mathrm{dq}}{\mathrm{dr}}\)

1. \(\frac{1-r}{\sqrt{2 r-r^2}}\)
2. \(\frac{1+r}{\sqrt{2 r+r^2}}\)
3. \(\frac{1-r}{\sqrt{3 r+r}}\)
4. \(\frac{1-r}{\sqrt{2 r-r^2}}\)

Answer: 1. \(\frac{1-r}{\sqrt{2 r-r^2}}\)

Find \(\frac{d y}{d x}\)

Question 171. y = \(\frac{\cot x}{1+\cot x}\)

1. \(\frac{-\csc ^2 x}{(1+\cot x)^2}\)
2. \(\frac{-\csc ^2 x}{(1-\cot x)^2}\)
3. \(\frac{-\csc ^2 x}{(1+\cot x)^2}\)
4. \(\frac{-\csc ^2 x}{(1+\tan x)^2}\)

Answer: 1. \(\frac{-\csc ^2 x}{(1+\cot x)^2}\)

Question 172. y = \(\frac{\ell \mathrm{nx}+\mathrm{e}^{\mathrm{x}}}{\tan \mathrm{x}}\), then \(\frac{d y}{d x}\) is

1. \(\frac{\tan x\left(e^x+\frac{1}{x}\right)+\sec ^2 x\left(e^x+\ell \ln x\right)}{\tan ^2 x}\)
2. \(\frac{\tan x\left(e^x+\frac{1}{x}\right)-\sec ^2 x\left(e^x+\ell \ln x\right)}{\tan ^2 x}\)
3. \(\frac{\tan x\left(e^x-\frac{1}{x}\right)-\sec ^2 x\left(e^x+\ell n x\right)}{\tan ^2 x}\)
4. \(\frac{\tan x\left(e^x+\frac{1}{x}\right)-\sec ^2 x\left(e^x-\ell n x\right)}{\tan ^2 x}\)

Answer: 1. \(\frac{\tan x\left(e^x+\frac{1}{x}\right)+\sec ^2 x\left(e^x+\ell \ln x\right)}{\tan ^2 x}\)

Find \(\frac{\mathrm{dy}}{\mathrm{dx}}\) as a function of x

Question 173. x³ + y³ = 18 xy

1. \(\frac{d y}{d x}=\frac{18 y+3 x^2}{3 y^2+18 x}\)
2. \(\frac{d y}{d x}=\frac{15 y+3 x^2}{3 y^2-18 x}\)
3. \(\frac{d y}{d x}=\frac{18 y-3 x^2}{3 y^2-18 x}\)
4. \(\frac{d y}{d x}=\frac{18 y-3 x^2}{3 y^2+12 x}\)

Answer: 3. \(\frac{d y}{d x}=\frac{18 y-3 x^2}{3 y^2-18 x}\)

Question 174. Find two positive numbers x & y such that x + y = 60 and xy is maximum –

1. 15, 45
2. 30, 30
3. 20, 40
4. 10, 50

Answer: 2. 30, 30

Find integrals of given functions.

Question 175. \(\int x^{-3}(x+1) d x\)

1. \(-\frac{1}{x}-\frac{1}{2 x^2}+C\)
2. \(\frac{1}{x}+\frac{1}{2 x^2}+C\)
3. \(3-\frac{1}{2 x^2}+C\)
4. \(-\frac{1}{x}+\frac{1}{2 x^2}+C\)

Answer: 1. \(-\frac{1}{x}-\frac{1}{2 x^2}+C\)

Question 176. \(\int\left(1-\cot ^2 x\right) d x\)

1. 2x + cot x + C
2. x + cot x + C
3. 2x – cot x + C
4. 2x + tan x + C

Answer: 1. 2x + cot x + C

Question 177. ∫cos (tanθ + secθ)dθ

1. – cos θ + θ + C
2. – cos θ – θ + C
3. – cosec θ + θ + C
4. – 2cos θ + θ + C

Answer: 1. – cos θ + θ + C

Question 178. \(\int \sqrt{3-2 s} d s\)

1. \(-\frac{1}{3}(3-2 s)^{2 / 3}+C\)
2. \(-\frac{1}{3}(3-2 s)^{3 / 2}+C\)
3. \(-\frac{1}{3}(3+2 s)^{3 / 2}+C\)
4. None of these

Answer: 2. \(-\frac{1}{3}(3-2 s)^{3 / 2}+C\)

Question 179. \(\int \frac{d x}{\sqrt{5 x+8}}\)

1. \(\left[\frac{2}{5} \sqrt{5 x+8}\right]+C\)
2. \(\left[\frac{2}{5} \sqrt{3 x-8}\right]+C\)
3. \(\left[\frac{2}{5} \sqrt{5 x-4}\right]-C\)
4. \(\left[\frac{2}{5} \sqrt{5 x-4}\right]\)

Answer: 1. \(\left[\frac{2}{5} \sqrt{5 x+8}\right]+C\)

Question 180. \(\int_0^{\sqrt{\pi}} x \sin x^2 d x\)

1. 1
2. 2
3. 3
4. 1

Answer: 1. 1

Use a definite integral to find the area of the region between the given curve and the x-axis on the interval [0,b],

Question 181. y = 3x²

1. b³
2. b²
3. b
4. b⁵

Answer: 1. b³

Question 182. Two vectors \(\overrightarrow{\mathrm{a}} \text { and } \overrightarrow{\mathrm{b}}\) inclined at an angle θ w.r.t. each other have a resultant \(\overrightarrow{\mathrm{c}}\) which makes an angle β with \(\overrightarrow{\mathrm{a}}\). If the directions of \(\overrightarrow{\mathrm{a}} \text { and } \overrightarrow{\mathrm{b}}\) are interchanged, then the resultant will have the same

1. Magnitude
2. Direction
3. Magnitude as well as direction
4. Neither magnitude nor direction.

Answer: 1. Magnitude

Question 183. Two vectors \(\overrightarrow{\mathrm{A}} \text { and } \overrightarrow{\mathrm{B}}\) lie in a plane. Another vector \(\overrightarrow{\mathrm{C}}\) lies outside this plane. The resultant \(\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}+\overrightarrow{\mathrm{C}}\) of these three vectors

1. Can be zero
2. Cannot be zero
3. Lies in the plane of \(\vec{A} and \vec{B}\)
4. Lies in the plane of \(\overrightarrow{\mathrm{A}} and \overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}\)

Answer: 2. Cannot be zero

Question 184. The rectangular components of a vector are (2, 2). The corresponding rectangular components of another vector are (1, \(\sqrt{3}\)). Find the angle between the two vectors

1. 10º
2. 15º
3. 20º
4. 25º

Answer: 2. 15º

Question 185. Given : \(\vec{a}+\vec{b}+\vec{c}\)= 0. Out of the three vectors \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \text { and } \overrightarrow{\mathrm{c}}\) two are equal in magnitude. The magnitude of the vector \(\sqrt{2}\) times that of either of the two having equal magnitude. The angles between vectors are:

1. 90º, 135º,. 135º
2. 30º, 60º, 90º
3. 45º, 45º, 90º
4. 45º, 60º, 90º

Answer: 1. 90º, 135º,. 135º

Question 186. Let \(\vec{a} \text { and } \vec{b}\) be two non-null vectors such that \(|\vec{a}+\vec{b}|=|\vec{a}-2 \vec{b}|\). Then the value of \(\frac{|\vec{a}|}{|\vec{b}|}\) may be :

1. \(\frac{1}{4}\)
2. \(\frac{1}{8}\)
3. 1
4. 2

Answer: (3,4)

Question 187. A truck traveling due north at 20 ms^-1 turns west and travels with the same speed. What is the change in velocity?

1. \(20 \sqrt{2} \mathrm{~ms}^{-1}\) south-west
2. \(40 \mathrm{~ms}^{-1}\) south-west
3. \(20 \sqrt{2} \mathrm{~ms}^{-1}\) north-west
4. \(40 \mathrm{~ms}^{-1}\) north-west

Answer: 1. \(20 \sqrt{2} \mathrm{~ms}^{-1}\) south-west

Question 188. Determine that vector which when added to the resultant of \(\overrightarrow{\mathrm{P}}=2 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}-10 \hat{\mathrm{k}}\) and \(\vec{Q}=\hat{i}+2 \hat{j}+3 \hat{k}\) a unit vector along X-axis.

1. \(-2 \hat{i}-9 \hat{j}+7 \hat{k}\)
2. \(+2 \hat{i}+9 \hat{j}-7 \hat{k}\)
3. \(-2 \hat{i}+7 \hat{j}+9 \hat{k}\)
4. \(+2 \hat{i}-5 \hat{j}+3 \hat{k}\)

Answer: 1. \(-2 \hat{i}-9 \hat{j}+7 \hat{k}\)

Question 189. Two vectors acting in opposite directions have a resultant of 10 units. If they act at right angles to each other, then the result is 50 units. Calculate the magnitude of two vectors.

1. P = 40 ; Q = 30
2. P = 30 ; Q = 40
3. P = 80 ; Q = 50
4. P = 30 ; Q = 40

Answer: 1. P = 40 ; Q = 30

Question 160. Find the resultant of the three vectors \(\overrightarrow{\mathrm{OA}}, \overrightarrow{\mathrm{OB}} \text { and } \overrightarrow{\mathrm{OC}}\) each of magnitude r as shown in the figure?

1. \(r(1+\sqrt{2})\)
2. \(r(1-\sqrt{2})\)
3. \((1+\sqrt{2})\)
4. \(r(1+\sqrt{2})^2\)

Answer: 1. \(r(1+\sqrt{2})\)

Question 161. A car is moving on a straight road due north with a uniform speed of 50 km h^-1 when it turns left through 90º. If the speed remains unchanged after turning, the change in the velocity of the car in the turning process is

1. Zero
2. \(50 \sqrt{2} \mathrm{~km} \mathrm{~h}^{-1}\) S-W direction
3. \(50 \sqrt{2} \mathrm{~km} \mathrm{~h}^{-1}\) N-W direction
4. 50 km h^-1 due west.

Answer: 2. \(50 \sqrt{2} \mathrm{~km} \mathrm{~h}^{-1}\) S-W direction

Question 162. Six forces, 9.81 N each, acting at a point are coplanar. If the angles between neighboring forces are equal, then the resultant is

1. 0 N
2. 9.81 N
3. 2 × 9.81 N
4. 3 × 9.81 N.

Answer: 1. 0 N

Question 163. At what angle must the two forces (x + y) and (x – y) act so that the resultant may be \(\sqrt{\left(x^2+y^2\right)}\)?

1. \(\cos ^{-1}\left[\frac{-\left(x^2+y^2\right)}{2\left(x^2-y^2\right)}\right]\)
2. \(\cos ^{-1}\left[\frac{-2\left(x^2-y^2\right)}{x^2+y^2}\right]\)
3. \(\cos ^{-1}\left[\frac{-\left(x^2+y^2\right)}{x^2-y^2}\right]\)
4. \(\cos ^{-1}\left[\frac{\left(x^2-y^2\right)}{x^2+y^2}\right]\)

Answer: 1. \(\cos ^{-1}\left[\frac{-\left(x^2+y^2\right)}{2\left(x^2-y^2\right)}\right]\)

Question 164. The magnitude of the scalar product of two vectors is 8 and that of the vector product is \(8 \sqrt{3}\). The angle between them is :

1. 30º
2. 60º
3. 120º
4. 150º

Answer: (2,3)

Question 165. A vector \(\vec{A}\) points vertically downward and \(\vec{B}\) points towards the east, then the vector product \(\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}\) is

1. Along west
2. Along east
3. Zero
4. Along south

Answer: 4. Along south

Question 166. Which of the arrangement of axes in a figure? can be labeled “right-handed coordinate system”? As usual, each axis label indicates the positive side of the axis.

1. (1), (2)
2. (3), (4)
3. (6)
4. (5)

Answer: (1,2,3)

Question 167. The unit vector perpendicular to each of the vectors \(3 \hat{i}+\hat{j}+2 \hat{k}\) and \(2 \hat{i}-2 \hat{j}+\hat{k}\) is given by

1. \(\frac{1}{\sqrt{3}}(\hat{i}-\hat{j}-\hat{k})\)
2. \(\frac{1}{\sqrt{3}}(\hat{\mathbf{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})\)
3. \(\frac{5 \hat{i}+\hat{j}+4 \hat{k}}{\sqrt{46}}\)
4. \(\pm \frac{5 \hat{i}+\hat{j}-4 \hat{k}}{\sqrt{42}}\)

Answer: 4. \(\pm \frac{5 \hat{i}+\hat{j}-4 \hat{k}}{\sqrt{42}}\)

Question 168. Three vectors \(\overrightarrow{\mathrm{A}}, \overrightarrow{\mathrm{B}} \text { and } \overrightarrow{\mathrm{C}}\) are such that \(\overrightarrow{\mathrm{A}}=\overrightarrow{\mathrm{B}}+\overrightarrow{\mathrm{C}}\) and their magnitudes are in ratio 5: 4 : 3 respectively. Find the angle between vector \(\overrightarrow{\mathrm{A}} \text { and } \overrightarrow{\mathrm{C}}\)

1. 35º
2. 53º
3. 60º
4. 75º

Answer: 2. 53º

Question 169. A car travels 6 km towards the north at an angle of 45° to the east and then travels a distance of 4 km towards the north at an angle of 135° to the east. How far is the point from the starting point? What angle does the straight line joining its initial and final position make with the east?

1. \(\sqrt{50} \mathrm{~km} and \tan ^{-1}(5)\)
2. \(10 \mathrm{~km} and \tan ^{-1}(\sqrt{5})\)
3. \(\sqrt{52} \mathrm{~km} and \tan ^{-1}(5)\)
4. \(\sqrt{52} \mathrm{~km} and \tan ^{-1}(\sqrt{5})\)

Answer: 3. \(\sqrt{52} \mathrm{~km} and \tan ^{-1}(5)\)

Question 170. The vectors \(\vec{A} \text { and } \vec{B}\) are such that: 

⇒\(|\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}|=|\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}|\)

The angle between the two vectors is :

1. 90°
2. 60°
3. 75°
4. 45°

Answer: 1. 90°

Question 171. Six vectors, \(\vec{a}\) through \(\vec{f}\) have the magnitudes and directions indicated in figure e. Which of the following statements is true?

1. \(\vec{b}+\vec{c}=\vec{f}\)
2. \(\vec{d}+\vec{c}=\vec{f}\)
3. \(\vec{d}+\vec{e}=\vec{f}\)
4. \(\vec{b}+\vec{e}=\vec{f}\)

Answer: 3. \(\vec{d}+\vec{e}=\vec{f}\)

Question 172. If dimensions of critical velocity υc of a liquid flowing through a tube are expressed as \(\left[\eta^{\mathrm{x}} \rho^{\mathrm{y}} \mathrm{r}^{\mathrm{x}}\right]\), where η, ρ, and r are the coefficient of viscosity of the liquid, density of a liquid, and radius of the tube respectively, then the values of x, y, and z are given by :

1. –1, –1, 1
2. –1, –1, –1
3. 1,1,1
4. 1, –1, –1

Answer: 4. 1, –1, –1

Question 173. If vectors \(\vec{A}=\cos \omega t \hat{i}+\sin \omega t \hat{j} \text { and } \vec{B}=\cos \frac{\omega t}{2} \hat{i}+\sin \frac{\omega t}{2} \hat{j}\) are functions of time, then the value of t at which they are orthogonal to each other is :

1. \(t=\frac{\pi}{2 \omega}\)
2. \(t=\frac{\pi}{\omega}\)
3. \(t=0\)
4. \(t=\frac{\pi}{4 \omega}\)

Answer: 2. \(t=\frac{\pi}{\omega}\)

Question 174. If the magnitude of the sum of two vectors is equal to the magnitude of the difference of the two vectors, the angle between these vectors is :

1. 180°
2. 0°
3. 90°
4. 45°

Answer: 3. 90°

Question 175. A particle moves so that its position vector is given by \(\overrightarrow{\mathrm{r}}=\cos \omega t \hat{x}+\sin \omega t \hat{y}\). Where ω is a constant. Which of the following is true?

1. Velocity and acceleration both are perpendicular to \(\overrightarrow{\mathrm{r}}\)
2. Velocity and acceleration both are parallel to \(\overrightarrow{\mathrm{r}}\)
3. Velocity is perpendicular to \(\overrightarrow{\mathrm{r}}\) and acceleration is directed towards the origin
4. Velocity is perpendicular to \(\overrightarrow{\mathrm{r}}\) and acceleration is directed away from the origin

Answer: 3. Velocity is perpendicular to \(\overrightarrow{\mathrm{r}}\) and acceleration is directed towards the origin

Question 176. The x and y coordinates of the particles at any time are x = 5t -2t² and y = 10t respectively, where x and y are in meters and t in seconds. The acceleration of the particle at t = 2s is

1. 5m/s²
2. -4 m/s²
3. – 8 m/s²
4. 0

Answer: 2. -4 m/s²

Question 177. The moment of the force, \(\overrightarrow{\mathrm{F}}=4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}\) at (2, 0, –3), about the point (2, –2, –2), is given by

1. \(-8 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}-7 \hat{\mathrm{k}}\)
2. \(-4 \hat{\mathrm{i}}-\hat{\mathrm{j}}-8 \hat{\mathrm{k}}\)
3. \(-7 \hat{i}-4 \hat{j}-8 \hat{k}\)
4. \(-7 \hat{i}-8 \hat{j}-4 \hat{k}\)

Answer: 3. \(-7 \hat{i}-4 \hat{j}-8 \hat{k}\)

Question 178. In the cube of side ‘as shown in the figure, the vector from the central point of the face ABOD to the central point of the face BEFO will be :

1. \(\frac{1}{2} a(\hat{j}-\hat{k})\)
2. \(\frac{1}{2} a(\hat{j}-\hat{i})\)
3. \(\frac{1}{2} \mathrm{a}(\hat{\mathrm{k}}-\hat{\mathrm{i}})\)
4. \(\frac{1}{2} a(\hat{i}-\hat{k})\)

Answer: 2. \(\frac{1}{2} a(\hat{j}-\hat{i})\)

Question 179. Two forces P and Q, of magnitude 2F and 3F, respectively are at an angle θ with each other. If the force Q is doubled, then their resultant also gets doubled. Then, the angle θ is :

1. 30°
2. 60°
3. 90°
4. 120°

Answer: 4. 120°

Question 180. Two vectors \(\vec{A} \text { and } \vec{B}\) have equal magnitudes. The magnitude of \((\vec{A}+\vec{B})\) is ‘n’ times the magnitude of \((\vec{A}-\vec{B})\). The angle between \(\vec{A} \text { and } \vec{B}\) is

1. \(\sin ^{-1}\left[\frac{n-1}{n+1}\right]\)
2. \(\cos ^{-1}\left[\frac{n^2-1}{n^2+1}\right]\)
3. \(\sin ^{-1}\left[\frac{n^2-1}{n^2+1}\right]\)
4. \(\cos ^{-1}\left[\frac{n-1}{n+1}\right]\)

Answer: 2. \(\cos ^{-1}\left[\frac{n^2-1}{n^2+1}\right]\)