Category: MCQ Questions

Free PDF Download of CBSE Physics Multiple Choice Questions for Class 12 with Answers Chapter 9 Ray Optics and Optical Instruments. Physics MCQs for Class 12 Chapter Wise with Answers PDF Download was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 12 Physics Ray Optics and Optical Instruments MCQs Pdf with Answers to know their preparation level.

Ray Optics and Optical Instruments Class 12 Physics MCQs Pdf

MCQ on ray optics class 12 Question 1. A converging lens is used to form an image on a screen. When the upper half of the lens is covered by an opaque screen.
(a) half the image will disappear.
(b) incomplete image will be formed.
(c) intensity of image will decrease but complete image is formed.
(d) intensity of image will increase but image is not distinct.

Answer/Explanation

Answer: c
Explaination:
(c) because focal length of lens does not change but amount of light passing through lens becomes half.

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2. In optical fibres, the refractive index of the core is
(a) greater than that of the cladding.
(b) equal to that of the cladding.
(c) smaller than that of the cladding.
(d) independent of that of cladding.

Answer/Explanation

Answer: a
Explaination:
(a) R.I. of core is greater than that of the cladding for total internal reflection to occur.

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Ray optics MCQ Questions pdf Question 3. An object is placed at a distance of 0.5 m in front of a plane mirror. The distance between object and image will be
(a) 0.25 m
(b) 0.5 m
(c) 1.0 m
(d) 2.0 m

Answer/Explanation

Answer: c
Explaination:
(c) Distance between object and image = 0.5 + 0.5 = 1.0 m

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4. Air bubble in water behaves as
(a) sometimes concave, sometimes convex lens
(b) concave lens
(c) convex lens
(d) always refracting surface

Answer/Explanation

Answer: b
Explaination:
(b) Air bubble in water behaves as a concave lens.

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MCQ on ray optics class 12 pdf 5. We combine two lenses, one is convex and other is concave having focal lengths /, and f2 and their combined focal length is F. Combination of the lenses will behave like concave lens, if
(a) f₁ > f₂
(b) f₁ = f₂
(c) f₁ < f₂.
(d) f₁ ≤ f₂

Answer/Explanation

Answer: a
Explaination:
(a) Focal length of the combination \(F=\frac{f_{1} f_{2}}{f_{1}+f_{2}}\), as f₂ is negative so denominator f₁ + f₂ must be positive or f₁ > f₂.

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Optics Multiple choice Questions and Answers pdf Question 6. The length of an astronomical telescope for normal vision (relaxed eye) will be

Answer/Explanation

Answer: d
Explaination:
(d) In normal vision, length of telescope L = f₀ + f_e

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Ray optics MCQ Question 7. The focal length of a biconvex lens of radii of each surface 50 cm and refractive index 1.5, is
(a) 40.4 cm
(b) 75 cm
(c) 50 cm
(d) 80 cm

Answer/Explanation

Answer: c
Explaination:

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8. A metal coin is at bottom of a beaker filled with a liquid of refractive index = 4/3 to height of 6 cm. To an observer looking from above the surface of liquid, coin will appear at a depth
(a) 1.5 cm
(b) 6.75 cm
(c) 4.5 cm
(d) 7.5 cm

Answer/Explanation

Answer: c
Explaination:

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9. Tom’ lenses of focal lengths ± 15 cm and ± 150 cm are available for making a telescope. To produce the largest magnification, the focal length of the eyepiece should be
(a) + 15 cm
(b) + 150 cm
(c) – 150 cm
(d) – 15 cm

Answer/Explanation

Answer:
Explaination:
(a) For telescope magnification,

to produce large magnification.

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10. If a convex lens of focal length 80 cm and a concave lens of focal length 50 cm are combined together, what will be their resulting power?
(a) + 6.5 D
(b) – 6.5 D
(c) + 7.5 D
(d) – 0.75 D

Answer/Explanation

Answer:
Explaination:
(d) Focal length of the combination

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Ray optics class 12 Questions and Answers pdf 11. A convex lens and a concave lens, each having the same focal length of 25 cm, are put in contact to form a combination of lenses. The power of the combination (in dioptres) is
(a) zero
(b) 25
(c) 50
(d) infinity

Answer/Explanation

Answer:
Explaination:

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12. The refractive index of the material of an equilateral prism is √3. What is the angle of minimum deviation?
(a) 45°
(b) 60°
(c) 37°
(d) 30°

Answer/Explanation

Answer:
Explaination:
(b) At minimum deviation position,

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13. In the formation of a rainbow, the light from the sun on water droplets undergoes
(a) dispersion only.
(b) only TIR.
(c) dispersion and TIR.
(d) scattering.

Answer/Explanation

Answer: b
Explaination:
(b) Rainbow is formed due to dispersion of sunlight by raindrops and total internal reflection inside the raindrop.

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14. In an experiment to find focal length of a concave mirror, a graph is drawn between the magnitude of u and v. The graph looks like

Answer/Explanation

Answer: c
Explaination: (c) u – v curve is a rectangular parabola.

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15. A convex lens of refractive index \(\frac{3}{2}\) has a power of 2.5 D in air. If it is placed in a liquid of refractive index 2 then the new power of the lens is
(a) – 1.25 D
(b) – 1.5 D
(c) 1.25 D
(d) 1.5 D

Answer/Explanation

Answer:
Explaination:

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16. A double convex lens of refractive index µ₁ is immersed in a liquid of refractive index µ₂. The lens will act as transparent plane sheet when
(a) µ₁ = µ₂
(b) µ₁ > µ₂
(c) µ₁ < µ₂
(d) µ₁ = \(\frac{1}{\mu_{2}}\)

Answer/Explanation

Answer:
Explaination:

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17. When a ray of light enters from one medium to another, then which of the following does not change?
(a) Frequency
(b) Wavelength
(c) Speed
(d) Amplitude

Answer/Explanation

Answer: a
Explaination:
(a) Only frequency of ray of light does not change when it propagates from one medium to another.

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18. A diver at a depth 12 m inside water (p = 4/3) sees the sky in a cone of semi-vertical angle

Answer/Explanation

Answer: c
Explaination:
(c) Required semi vertical angle = Critical angle i_c

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19. The astronomical telescope consists of objective and eyepiece. The focal length of the objective is
(a) equal to that of the eyepiece.
(b) shorter than that of eyepiece.
(c) greater than that of eyepiece.
(d) five times shorter than that of eyepiece.

Answer/Explanation

Answer: c
Explaination: (c) For producing large magnification focal length of objective is greater than that of the eyepiece.

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20. A rod of length 10 cm lies along the principal axis of a concave mirror of focal length 10 ’em in such a way that its end closer to the pole is 20 cm away from the mirror. The length of the image is
(a) 10 cm
(b) 15 cm
(c) 2.5 cm
(d) 5 cm

Answer

Answer: d

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21. For a total internal reflection, which of the following is correct?
(a) Light travels from rarer to denser medium.
(b) Light travels from denser to rarer medium.
(c) Light travels in air only.
(d) Light travels in water only.

Answer

Answer: b

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22. Critical angle of glass is θ₂ and that of water is θ₂. The critical angle for water and glass surface would be (μ_g = 3/2, μ_w = 4/3).
(a) less than θ₂
(b) between θ₁ and θ₂
(c) greater than θ₂
(d) less than θ₁

Answer

Answer: c

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23. Mirage is a phenomenon due to
(a) refraction of light
(b) reflection of light
(c) total internal reflection of light
(d) diffraction of light.

Answer

Answer: c

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24. A convex lens is dipped in a liquid whose refractive index is equal to the refractive index of the lens. Then its focal length will
(a) become zero
(b) become infinite
(c) become small, but non-zero
(d) remain unchanged

Answer

Answer: b

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25. Which of the following forms a virtual and erect image for all positions of the object?
(a) Concave lens
(b) Concave mirror
(d) Convex mirror
(d) Both (a) and (c)

Answer

Answer: d

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26. Two lenses of focal lengths 20 cm and – 40 cm are held in contact. The image of an object at infinity will be formed by the combination at
(a) 10 cm
(b) 20 cm
(c) 40 cm
(d) infinity

Answer

Answer: c

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27. Two beams of red and violet color are made to pass separately through a prism (angle of the prism is 60°). In the position of minimum deviation, the angle of refraction will be
(a) 30° for both the colors
(b) greater for the violet color
(c) greater for the red color
(d) equal but not 30° for both the colors

Answer

Answer: a

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28. Which of the following colours of white light deviated most when passes through a prism?
(a) Red light
(b) Violet light
(c) Yellow light
(d) Both (a) and (b)

Answer

Answer: b

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29. An under-water swimmer cannot see very clearly even in absolutely clear water because of
(a) absorption of light in water
(b) scattering of light in water
(c) reduction of speed of light in water
(d) change in the focal length of eye lens

Answer

Answer: d

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30. An astronomical refractive telescope has an objective of focal length 20 m and an eyepiece of focal length 2 cm. Then
(a) the magnification is 1000
(b) the length of the telescope tube is 20.02 m
(c) the image formed of inverted
(d) all of these

Answer

Answer: d

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31. An object is immersed in a fluid. In order that the object becomes invisible, it should
(a) behave as a perfect reflector.
(b) absorb all light falling on it.
(c) have refractive index one.
(d) have refractive index exactly matching with that of the surrounding fluid.

Answer

Answer: d

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32. A ray of light incident at an angle θ on a refracting face of a prism emerges from the other face normally. If the angle of the prism is 5° and the prism is made of a material of refractive index 1.5, the angle of incidence is [NCERT Exemplar]
(a) 7.5°
(b) 5°
(c) 15°
(d) 2.5°

Answer

Answer: a

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33. A short pulse of white light is incident from air to a glass slab at normal incidence. After travelling through the slab, the first colour to emerge is [NCERT Exemplar]
(a) blue
(b) green
(c) violet
(d) red

Answer

Answer: d

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34. An object approaches a convergent lens from the left of the lens with a uniform speed 5 m/s and stops at the focus. The image [NCERT Exemplar]
(a) moves away from the lens with an uniform speed 5 m/s.
(b) moves away from the lens with an uniform accleration.
(c) moves away from the lens with a non-uniform acceleration.
(d) moves towards the lens with a non-uniform acceleration.

Answer

Answer: c

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35. You are given four sources of light each one providing a light of a single colour – red, blue, green and yellow. Suppose the angle of refraction for a beam of yellow light corresponding to a particular angle of incidence at the interface of two media is 90°. Which of the following statements is correct if the source of yellow light is replaced with that of other lights without changing the angle of incidence? [NCERT Exemplar]
(a) The beam of red light would undergo total internal reflection.
(b) The beam of red light would bend towards normal while it gets refracted through the second medium.
(c) The beam of blue light would undergo total internal reflection.
(d) The beam of green light would bend away from the normal as it gets refracted through the second medium.

Answer

Answer: c

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36. The radius of curvature of the curved surface of a plano-convex lens is 20 cm. If the refractive index of the material of the lens be 1.5, it will [NCERT Exemplar]
(a) act as a convex lens only for the objects that lie on its curved side.
(b) act as a concave lens for the objects that lie on its curved side.
(c) act as a convex lens irrespective of the side on which the object lies.
(d) act as a concave lens irrespective of side on which the object lies.

Answer

Answer: c

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37. The direction of ray of light incident on a concave mirror is shown by PQ while directions in which the ray would travel after reflection is shown by four rays marked 1, 2, 3 and 4. Which of the four rays correctly shows the direction of reflected ray? [NCERT Exemplar]

(a) 1
(b) 2
(c) 3
(d) 4

Answer

Answer: b

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38. The optical density of turpentine is higher than that of water while its mass density is lower. Figure shows a layer of turpentine floating over water in a container. For which one of the four rays incident on turpentine in figure the path shown is correct? [NCERT Exemplar]

(a) 1
(b) 2
(c) 3
(d) 4

Answer

Answer: b

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39. A car is moving with at a constant speed of 60 km h^-1 on a straight road. Looking at the rear view mirror, the driver finds that the car following him is at a distance of 100 m and is approaching with a speed of 5 km h^-1. In order to keep track of the car in the rear, the driver begins to glance alternatively at the rear and side mirror of his car after every 2 s till the other car overtakes. If the two cars were maintaining their speeds, which of the following statement (s) is/are correct? [NCERT Exemplar]
(a) The speed of the car in the rear is 65 km h^-1.
(b) In the side mirror the car in the rear would appear to approach with a speed of 5 km h^-1 to the driver of the leading car.
(c) In the rear view mirror the speed of the approaching car would appear to decrease as the distance between the cars decreases.
(d) In the side mirror, the speed of the approaching car would appear to increase as the distance between the cars decreases.

Answer

Answer: d

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40. Consider an extended object immersed in water contained in a plane trough. When seen from close to the edge of the trough the object looks distorted. Which of the following is not correct.
(a) the apparent depth of the points close to the edge are nearer the surface of the water compared to the points away from the edge.
(b) the angle subtended by the image of the object at the eye is smaller than the actual angle subtended by the object in air.
(c) some of the points of the object far away from the edge may not be visible because of total internal reflection.
(d) water in a trough acts as a lens and magnifies the object.

Answer

Answer: d

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41. A magnifying glass is used, as the object to be viewed can be brought closer to the eye than the normal near point. This results in
(a) a larger angle to be subtended by the object at the eye and hence viewed in greater detail.
(b) the formation of a real inverted image.
(c) increase in the field of view.
(d) infinite magnification at the near point.

Answer

Answer: a

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42. An astronomical refractive telescope has an objective of focal length 20 m and an eyepiece of focal length 2 cm. Which one of the following is not correct?
(a) The length of the telescope tube is 20.02 m.
(b) The magnification is 1000.
(c) The image formed is inverted.
(d) An objective of a larger aperture will increase the brightness and reduce chromatic aberration of the image.

Answer

Answer: d

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43. Virtual image formed by convex mirror has magnification _________ .

Answer/Explanation

Answer:
Explaination: negative

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44. Optical denseness of a medium is measured in terms of _________ .

Answer/Explanation

Answer:
Explaination: refractive index

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45. Minimum angle of incidence in the denser medium for which angle of refraction becomes 90° is called _________ .

Answer/Explanation

Answer:
Explaination: critical angle

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46. Relation between critical angle and refractive index is _________ .

Answer/Explanation

Answer:
Explaination: \(\mu=\frac{1}{\sin i_{c}}\)

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47. Optical fibre works on the principle of

Answer/Explanation

Answer:
Explaination: Total internal reflection

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48. The splitting of white light into its constituent colours when it passes through a glass prism is called _________ .

Answer/Explanation

Answer:
Explaination: dispersion

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49. Blue colour of sky is due to phenomenon of _________ of sunlight.

Answer/Explanation

Answer:
Explaination: scattering

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50. A ray of monochromatic light passes from medium (1) to medium (2). If the angle of incidence in medium (1) is θ and the corresponding angle of refraction in medium (2) is θ/2, which of the two media is optically denser? Give reason. [Foreign 2013]

Answer/Explanation

Answer:
Explaination:

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51. For the same value of angle of incidence, the angles of refraction in three media A, B and C are 15°, 25° and 35° respectively. In which medium would the velocity of light be minimum? [AI2012]

Answer/Explanation

Answer:
Explaination:

Thus, the medium for which angle of refraction is of 15°, the speed of light is minimum.

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52. A concave lens of refractive index 1.5 is immersed in a medium of refractive index 1.65. What is the nature of the lens? [Delhi 2015]

Answer/Explanation

Answer:
Explaination: The nature of the lens is converging.

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53. A lens behaves as a converging lens in air and a diverging lens in water (μ = 4/3). What will be the condition on the value of refractive index (μ) of the material of the lens? [Delhi 2011C]

Answer/Explanation

Answer:
Explaination:
The refractive index p of the lens is less than the refractive index of water, i.e. \(\frac{4}{3}\) > μ_L > 1.

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54. An air bubble is formed inside water. Does it act as a converging lens or a diverging lens? [DoE]

Answer/Explanation

Answer:
Explaination: An air bubble behaves as a diverging lens inside the water.

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55. The image of an object formed by a lens on the screen is not in sharp focus. Suggest a method to get the clear focussing of the image on the screen without disturbing the position of the object, the lens or the screen.

Answer/Explanation

Answer:
Explaination:
The image of an object formed by a lens can be brought to a sharp focus on a unfixed screen by changing the focal length of the lens by any of the following methods:
(i) By placing another lens of suitable focal length in contact with the previous lens.
(ii) By immersing the given lens in a liquid of appropriate refractive index.

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56. Can absolute value of refractive index of a medium be less than unity?

Answer/Explanation

Answer:
Explaination:
As the speed of light is maximum in vacuum, therefore absolute value of refractive index cannot be less than unity as it is given by the relation n = \(\frac{c}{v}\).

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57. For which material the value of refractive index is (i) minimum and (ii) maximum?

Answer/Explanation

Answer:
Explaination:
(i) Refractive index is minimum for vacuum (μ = 1).
(ii) Refractive index is maximum for diamond.

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58. Can virtual image be photographed?

Answer/Explanation

Answer:
Explaination:
Yes, because the lens in the camera produces a real image of the virtual image being formed.

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59. How does the angle of minimum deviation of a glass prism vary, if the incident violet light is replaced with red light? [Chennai 2019, AI 2017]

Answer/Explanation

Answer:
Explaination: It decreases.

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60. An object is first seen in red light and then in violet light through a simple microscope. In which case is the magnifying power larger?

Answer/Explanation

Answer:
Explaination:
As we know m = 1 + \(\frac{D}{f}\)
∵ f_V < f_R
So, the magnifying power is larger when the object is seen in violet light.

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61. Name the phenomenon due to which one cannot see through the fog. [DoE]

Answer/Explanation

Answer:
Explaination: Due to scattering of light, one cannot see through the fog.

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62. How does the angle of minimum deviation of a glass prism of refractive index 1.5 change, if it is immersed in a liquid of refractive index 1.3?

Answer/Explanation

Answer:
Explaination: It will decrease.

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63. A convex lens is placed in contact with a plane mirror. A point object at a distance of 20 cm on the axis of this combination has its image coinciding with itself. What is the focal length of the lens? [Delhi 2014]

Answer/Explanation

Answer:
Explaination: The focal length of convex lens is 20 cm.

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64. If an object is moved from infinity to convex mirror, then in which direction will the image shift?

Answer/Explanation

Answer:
Explaination: The image will shift from focus to the convex mirror.

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65. A green light is incident from water to the air-water interface at the critical angle (θ). Which part of the spectrum will come out in the air medium?

Answer/Explanation

Answer:
Explaination: The spectrum of visible light whose frequency is less than that of green light will come out of the water into the air.

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66. How do the increasing (i) wavelength and (ii) intensity of light affect the speed of light in glass?

Answer/Explanation

Answer:
Explaination:
(i) ∵ v ∝ λ; ∴ speed of light increases on increasing the wavelength in glass.
(ii) There is no effect on speed of light on changing the intensity.

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67. Name the principle on which an optical fibre works.

Answer/Explanation

Answer:
Explaination: Total internal reflection.

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68. Why is there no dispersion in the light refracted through a rectangular glass slab?

Answer/Explanation

Answer:
Explaination:
A glass slab can be considered as two glass prisms placed together and the position of one prism is inverted w.r.t. another. Therefore, the various colours of white light dispersed by the first prism are again combined to form white light.

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69. The line AB in the ray diagram represents a lens. State whether the lens is convex or concave.

Answer/Explanation

Answer:
Explaination:
It is a diverging (concave lens) and the refracted rays are bending away from the principal axis.

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70. A ray of light incident on one of the faces of a glass prism of angle A has angle of incidence 2A. The refracted ray in the prism strikes the opposite face which is silvered, the reflected ray from it retracing its path. Trace the ray diagram and find the relation between the refractive index of the material of the prism and the angle of the prism.

Answer/Explanation

Answer:
Explaination:
Given: i = 2 A, r = 90° – (90° – A) = A

∴ n = 2 cos A

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71. Does the magnifying power of a microscope depend on the colour of the light used? Justify your answer. [Foreign 2017]

Answer/Explanation

Answer:
Explaination:
Magnifying power of a microscope,

Since the focal length of a convex lens depends on the refractive index, and refractive indices for different colours are different, so according to the lens maker’s formula

The magnifying power of a microscope depends on the colour of the light used.

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Free PDF Download of CBSE Maths Multiple Choice Questions for Class 12 with Answers Chapter 13 Probability. Maths MCQs for Class 12 Chapter Wise with Answers PDF Download was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 12 Maths Probability MCQs Pdf with Answers to know their preparation level.

Probability Class 12 Maths MCQs Pdf

1. Let A and B be two given events such that P(A) = 0.6, P(B) = 0.2 and P(A/B) = 0.5. Then P(A’/B’) is

Answer/Explanation

Answer: c
Explaination:

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2. Let A and B be two given independent events such that P(A) =p and P(B) = q and P(exactly one of A, B) = \(\frac{2}{3}\), then value of 3p + 3q – 6pq is
(a) 2
(b) -2
(c) 4
(d) -4

Answer/Explanation

Answer: a
Explaination:

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3. If P(A ∩ B) = 70% and P(B) = 85%, then P(A/B) is equal to

Answer/Explanation

Answer: a
Explaination:

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4. Two dice are thrown once. If it is known that the sum of the numbers on the dice was less than 6 the probability of getting a sum 3 is

Answer/Explanation

Answer: c
Explaination: (c), as favourable cases for sum less than 6 are 10 and favourable for a total of 3 is 2.

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5. Three balls are drawn from a bag containing 2 red and 5 black balls, if the random variable X represents the number of red balls drawn, then X can take values
(a) 0, 1, 2
(b) 0, 1, 2, 3
(c) 0
(d) 1, 2

Answer/Explanation

Answer: a
Explaination: (a), as there are 2 red balls, so maximum ’ red balls can be 2.

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6. The probability distribution of the discrete variable X is given as:

The value of k is
(a) 8
(b) 16
(c) 32
(d) 48

Answer/Explanation

Answer:
Explaination:

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7. Let A and B be two given mutually exclusive events. Then P(A/B) is _________ .

Answer/Explanation

Answer:
Explaination: 0, as P(A ∩ B) = 0 and P(A/B) = \(\frac{P(A \cap B)}{P(B)}\)

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8. If A and B are two independent events such that P(A) = \(\frac{1}{7}\) and P(B) = \(\frac{1}{6}\) then P(A’ ∩ B’) is _________ .

Answer/Explanation

Answer:
Explaination:

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9. Three persons A, B and C, fire a target in turn. Their probabilities of hitting the target are 0.2,0.3 and 0.5 respectively, the probability that target hit is _________ .

Answer/Explanation

Answer:
Explaination:
0.72, as /^target hit)
= P(at least one hits the target)
= 1 – P( none hits) =1 \(-P(\bar{A} \bar{B} \bar{C})\)

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10. If A and B are independent events, then P(A and B) = P(A) + P(B). State true or false.

Answer/Explanation

Answer:
Explaination: False, as P(A and B) = P(A) P(B).

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11. A speaks truth in 70% cases and B speaks truth in 85% cases. The probability that they speak the same fact is ______ .

Answer/Explanation

Answer:
Explaination:

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12. Given P(A) = 0.4, P(B) = 0.7 and P(B/A) = 0.6. Find P(A ∪ B).

Answer/Explanation

Answer:
Explaination: .
P(B/A) = \(\frac{P(A \cap B)}{P(A)}\)
⇒ 0.6 × 0.4 = P (A ∩ B)
⇒ P(A ∩ B) = 0.24
Now, P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
= 0.4 + 0.7 – 0.24
= 0.86

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13. Given P(A) = \(\frac{1}{2}\) p(B) = \(\frac{1}{3}\) and P(A ∩ B) = \(\frac{1}{6}\). Are the events A and B independent?

Answer/Explanation

Answer:
Explaination:

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14. Prove that if E and F are independent events, then the events E and F are also independent. [Delhi 2017]

Answer/Explanation

Answer:
Explaination:
As E and F are independent events
∴ P(E ∩ F) = P(E)P(F) …(i)
Consider,
P(E)P(F) = P(E)[1 – P(F)]
= P(E) – P(E)P(F)
= P(E) – P(E ∩ F)
P(E)P(F’) = P(E ∩ F’)
⇒ E and F’ are independent events.

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15. A die, whose faces are marked 1, 2, 3 in red and 4,5,6 in green, is tossed. Let A be the event “number obtained is even” and B be the event “number obtained is red”. Find if A and B are independent events. [AI 2017]

Answer/Explanation

Answer:
Explaination:
A : number is even, i.e. 2, 4, 6; P(A) = \(\frac{3}{6}\)
= \(\frac{1}{2}\)
B : number is red, i.e. 1, 2, 3; P(B) = \(\frac{3}{6}\)
= \(\frac{1}{2}\)
A ∩ B : number is even and red, i.e. 2;
P(A ∩ B) = \(\frac{1}{6}\)
P(A) × P(B) = \(\frac{1}{2} \times \frac{1}{2}=\frac{1}{4}\)
As P(A ∩ B) ≠ P(A) . P(B)
Hence, not independent.

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16. A black and a red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4. [CBSE 2018]

Answer/Explanation

Answer:
Explaination:
A : sum 8, i.e. (2, 6), (3, 5), (4, 4), (5, 3), (6, 2)
B : red die number less than 4.
B : (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3).
A ∩ B : (6, 2), (5, 3)
∴ Conditional probability of obtaining sum 8, given that red die resulted in number less than 4.

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17. 12 cards numbered 1 to 12 are placed in a box, mixed up thoroughly and then a card is drawn at random from the box. If it is known that the number on the drawn card is more than 3, find the probability that it is an even number.

Answer/Explanation

Answer:
Explaination:
Total cards are 12
A : number drawn is more than 3, i.e. 4, 5, 6, …, 12.
B : getting an even number, i.e. 2, 4, 6, 8, 10, 12.
A ∩ B : 4, 6, 8, 10, 12

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18. One card is drawn from a well shuffled pack of 52 cards. If E is the event “the card drawn is a king or a queen” and F is the event “the card drawn is an ace or a queen,” then find the probability of the conditional event E/F. [HOTS]

Answer/Explanation

Answer:
Explaination:

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19. Two cards are drawn from a well shuffled pack of 52 cards one after the other without replacement. Find the probability that one of them is a queen and the other is a king of opposite colour.

Answer/Explanation

Answer:
Explaination:

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20. A coin is tossed thrice and all eight outcomes are assumed equally likely. Let the event E be “the first throw results in head” and event F be “the last throw results in tail”. Find whether events E and F are independent.

Answer/Explanation

Answer:
Explaination:
Sample space
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
E: first throw results in head = {HHH,HHT,HTH,HTT}
F: Last throw result in tail = {HHT, HTT, THT, TTT}

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21. Two dice are thrown. Find the probability that the numbers appeared have a sum 8 if it is known that the second die always exhibits 4. [HOTS]

Answer/Explanation

Answer:
Explaination:
A: Sum of the numbers is 8, i.e. {(2,6), (3,5), (4,4), (5,3), (6,2)}
B: the second die always exhibits 4, i.e. {(1,4), (2,4), (3,4), (4,4), (5,4), (6,4)} A ∩ B= {(4, 4)}
Required probability,

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22. If P(A)=0.8,P(B)=0.5 andP(B/A)=0.4, find (i) P(A ∩ B) (ii) P(A/B) (iii) P(A ∪ B). [NCERT]

Answer/Explanation

Answer:
Explaination:

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23. The value of k, for which the following distribution is a probability distribution

Answer/Explanation

Answer:
Explaination:

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24. Find mean (µ) and variance (a2) for the following probability distribution:

Answer/Explanation

Answer:
Explaination:

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25. A random variable X has a probability distribution P(X) of the following form where k is some number:Determine (i) k (ii) P(X ≤ 2) (iii) P(X ≤ 2) (iv) P(X ≥ 2). [NCERT; Delhi 2019]

Answer/Explanation

Answer:
Explaination:

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We hope the given Maths MCQs for Class 12 with Answers Chapter 13 Probability will help you. If you have any query regarding CBSE Class 12 Maths Probability MCQs Pdf, drop a comment below and we will get back to you at the earliest.

Free PDF Download of CBSE Maths Multiple Choice Questions for Class 12 with Answers Chapter 12 Linear Programming. Maths MCQs for Class 12 Chapter Wise with Answers PDF Download was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 12 Maths Linear Programming MCQs Pdf with Answers to know their preparation level.

Linear Programming Class 12 Maths MCQs Pdf

1. Of all the points of the feasible region, for maximum or minimum of objective function, the point lies
(a) inside the feasible region
(b) at the boundary line of the feasible region
(c) vertex point of the boundary of the feasible region
(d) none of these

Answer

Answer: c

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2. A dealer wishes to purchase a number of fans and sewing machines. He has only ₹ 5,760 to invest and has space for at most 20 items. A fan costs him ₹ 360 and a sewing machine ₹ 240. He expects to sell a fan at a profit of ₹ 22 and a sewing machine at a profit of ₹ 18. Assinning that he can sell all the items that he buys, how should he invests his money to maximise the profit? The LPP for above question is
(a) x → fans, y → sewing machines
To maximise Z = 22x + 18y
subject to constraints
x ≥ 0, y ≥ 0, x + y ≤ 20
360x + 240y ≥ 5760
(b) x → fans, y → sewing machines
To maximise Z= 18x + 22y subject to constraints
x ≥ 0, y ≥ 0, x + y ≤ 20 360x + 240y ≥ 5760
(c) x → fans, y → sewing machines
To maximise Z = 22x + 18y
x ≥ 0, y ≥ 0, x + y ≥ 0
360x + 240y ≥ 5760
(d) x → fans, y → sewing machines
To maximise Z = 22x + 18j
x ≥ 0, y ≥ 0, x + y ≤ 20 360x + 240y ≤ 5760

Answer

Answer: d

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3. Feasible region is the set of points which satisfy
(a) the objective functions
(b) some of the given constraints
(c) all of the given constraints
(d) none of these

Answer

Answer: c

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4. An aeroplane can carry a maximum of 200 passangers. A profit of ₹ 400 is made on each first class ticket and a profit of ₹ 300 is made on each economy class ticket. An airline reserves at least 20 seats for first class. However, at least 4 times as many passengers prefer to travel by economy class than by first class. Find how many tickets of each type must be sold to maximise the profit? The LPP for the given situation is
(a) x → first class, y → economy class
To maximise Z = 400x + 300y
subject to constraints
x ≥ 0, y ≥ 0, x+y ≤ 200
x ≥ 20, y ≥ 80
(b) x → first class, y → economy class
To maximise Z 400x + 300y
subject to constraints
x ≥ 0, y ≥ 0, x+y ≥ 200
x ≥ 20, y ≥ 80
(c) x → first class, y → economy class
To maximise Z = 400x + 300y
subject to constraints
x ≥ 0, y ≥ 0, x ≥ 20
x + y ≤ 200
x ≥ 4y
(d) x → first class, y → economy class
To maximise Z = 400x + 300y
subject to constraints
x ≥ 20, y ≥ 0
x + y ≤ 200
y ≥ 4x

Answer

Answer: d

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5. A manufacturer makes two types of furniture, chairs and tables. Both the products are processed on three machines A₁ A₂ and A₃. Machine A₁ requires 3 hours for a chair and 3 hours for a table, machine A₂ requires 5 hours for a chair and 2 hours for a table and machine A₃ requires 2 hours for a chair and 6 hours for a table. Maximum time available on machine A₁, A₂ and A₃ is 36 hours, 50 hours and 60 hours respectively. Profits are ₹ 20 per chair and ₹ 30 per table. Formulate the above as a linear programming problem to maximise the profit. [HOTS]

Answer/Explanation

Answer:
Explaination:

Let x chairs and y tables are manufactured.
Then LPP is Maximise P = 20x + 30y
subject to the constraints, x ≥ 0, y ≥ 0, 3x + 3y ≤ 36, 5x + 2y ≤ 50, 2x + 6y ≤ 60.

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6. One kind of cake requires 200 g of flour and 25 g of fat and another kind of cake requires 100 g of flour and 50 g of fat, 5 kg of flour and 1 kg of fat is available, formulate the problem to find the maximum number of cakes which can be made, assuming that there is no shortage of the other ingredients used in making the cakes.

Answer/Explanation

Answer:
Explaination:
Let x cakes of kind I and y cakes of kind II are made.
Then LPP is
Maximise Z = x + y
subject to the constraints x ≥ 0, y ≥ 0
200x + 100y ≤ 5000
25x + 50y ≤ 1000

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7. A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at mo£ 24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is ₹ 100 and that on a bracelet is ₹ 300. Formulate an LPP for finding how many of each should be produced daily to maximise the profit? It is being given that at lea£ one of each muft be produced. [Delhi 2017]

Answer/Explanation

Answer:
Explaination:
Let x necklaces and y bracelets be produced.
Then LPP is
To maximise profit
Z =100x + 300y
subject to constraints
x ≥ 1, y ≥ 1
x + y ≤ 24; \(\frac{1}{2}\)x + y ≤ 16

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8. Two tailors, A and B, earn ₹ 300 and ₹ 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP. [AI 2017]

Answer/Explanation

Answer:
Explaination:
Let A works for x days and B for y days
∴ LPP is to minimise Z = 300x + 400y
subject to the constraints 6x + 10y ≥ 60
4x + 4y ≥ 32
x ≥ 0, y ≥ 0

---

We hope the given Maths MCQs for Class 12 with Answers Chapter 12 Linear Programming will help you. If you have any query regarding CBSE Class 12 Maths Linear Programming MCQs Pdf, drop a comment below and we will get back to you at the earliest.

Free PDF Download of CBSE Maths Multiple Choice Questions for Class 12 with Answers Chapter 11 Three Dimensional Geometry. Maths MCQs for Class 12 Chapter Wise with Answers PDF Download was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 12 Maths Three Dimensional Geometry MCQs Pdf with Answers to know their preparation level.

Three Dimensional Geometry Class 12 Maths MCQs Pdf

1. The distance of point (2, 5, 7) from the x-axis is
(a) 2
(b) √74
(c) √29
(d) √53

Answer/Explanation

Answer: b
Explaination:
(b), as distance of point (2, 5, 7) from the x-axis is

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2. P is a point on the line segment joining the points (3, 5, -1) and (6, 3, -2). If y-coordinate of point P is 2, then its x-coordinate will be
(a) 2
(b) \(\frac{17}{3}\)
(c) \(\frac{15}{2}\)
(d) -5

Answer/Explanation

Answer: c
Explaination:
(c), as let P divides the join of (3, 5, -1) and (6, 3, -2) in the ratio k : 1

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3. Direction ratios of a line are 2, 3, -6. Then direction cosines of a line making obtuse angle with the y-axis are

Answer/Explanation

Answer: c
Explaination:
(c), as direction cosines of a line whose direction ratio are 2,3, -6 are \(\frac{2}{7}, \frac{3}{7}, \frac{-6}{7}\).
As angle with the y-axis is obtuse,
∴ cos β < 0,
Therefore direction ratios are \(\frac{-2}{7}, \frac{-3}{7}, \frac{6}{7}\).

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4. A line makes angle α, β, γ with x-axis, y-axis and z-axis respectively then cos 2α + cos 2β + cos 2γ is equal to
(a) 2
(b) 1
(c) -2
(d) -1

Answer/Explanation

Answer: d
Explaination:

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5. The equations of y-axis in space are
(a) x = 0, y = 0
(b) x = 0, z = 0
(c) y = 0, z = 0
(d) y = 0

Answer/Explanation

Answer: b
Explaination: (b), as on the y-axis, x-coordinate and z-coordinate are zeroes.

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6. If the direction cosines of a line are \(\frac{k}{3}, \frac{k}{3}, \frac{k}{3}\), then value of k is
(a) k > 0
(b) 0 < k < 1.
(c) k = \(\frac{1}{3}\)
(d) k = ± 73

Answer/Explanation

Answer: d
Explaination:

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7. Distance of plane \(\vec{r} \cdot(2 \hat{i}+3 \hat{i}-6 \hat{k})+2=0\), from origin is
(a) 2
(b) 14
(c) \(\frac{2}{7}\)
(d) –\(\frac{2}{7}\)

Answer/Explanation

Answer: c
Explaination:

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8. Distance between planes

Answer/Explanation

Answer: c
Explaination:

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9. The line joining the points (0, 5, 4) and (1, 3, 6) meets XY-plane at the point ________ .

Answer/Explanation

Answer:
Explaination:
(-2, 9, 0), as line is \(\frac{x-1}{1}=\frac{y-3}{-2}=\frac{z-6}{2}=\lambda\)
General point on line is (λ + 1, -2λ + 3, 2λ + 6)
If it meets AT-plane, then 2λ + 6 = 0
⇒ λ = – 3
∴ Point is (-2, 9, 0)

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10. A line makes angles α, β, γ with z-axis, x-axis and y-axis respectively. Then direction cosines of line are cos β, cos γ, cos α. State true or false.

Answer/Explanation

Answer:
Explaination: True, as direction cosines of a line are cosines of the angles which a line makes with x, y and z-axes respectively.

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11. A line makes angles \(\frac{\pi}{4}, \frac{3 \pi}{4}\) with x-axis and y-axis respectively. Then the angle which it makes with z-axis can be ________ .

Answer/Explanation

Answer:
Explaination:

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12. The vector equation of the line

State true or false.

Answer/Explanation

Answer:
Explaination:

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13. The Cartesian equation of a line AB is

Find the direction cosines of a line parallel to AB.

Answer/Explanation

Answer:
Explaination:

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14. Find the direction cosines of the line passing through the following points: (-2, 4, -5), (1, 2, 3) [NCERT]

Answer/Explanation

Answer:
Explaination:

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15. Find the Cartesian equation of the line which passes through the point (-2,4, -5) and is parallel to the line \(\frac { x+3 }{ 3 } =\frac { 4-y }{ 5 } =\frac { z+8 }{ 6 } \) [Delhi 2013]

Answer/Explanation

Answer:
Explaination:

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16. Write the vector equation of the following line: \(\frac { x-5 }{ 3 } =\frac { y+4 }{ 7 } =\frac { 6-z }{ 2 } \)

Answer/Explanation

Answer:
Explaination:
The line passes through the point (5, -4, 6) and dr’s of the line are 3, 7, – 2.
∴ vector equation is

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17. Write the Cartesian equation of the following line given in vector form:

Answer/Explanation

Answer:
Explaination:
Point through which line passes is (2, 1, -4) and dr’s: 1, – 1, – 1.
∴ Cartesian equation of line

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18. What are the direction cosines of a line, which makes equal angles with the coordinate axes? [NCERT; Foreign 2011]

Answer/Explanation

Answer:
Explaination:

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19. If the direction cosines of a given line are \(\frac{1}{k}, \frac{1}{k}, \frac{1}{k}\) then, find the value of k.

Answer/Explanation

Answer:
Explaination:

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20. If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction ofz-axis. [Delhi 2017]

Answer/Explanation

Answer:
Explaination:
Let angle with z-axis be γ.
cos²90° + cos²60° + cos² γ = 1
⇒ 0 + \(\frac{1}{4}\) + cos² γ = 1
⇒ cos² γ = \(\frac{3}{4}\)
cos γ = \(\pm \frac{\sqrt{3}}{2}\)
γ = 30°, 150°

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21. Find the vector equation of the line passing through the point A(1, 2,-1) and parallel to the line 5x – 25 = 14 – 7y = 35z. [Delhi 2017]

Answer/Explanation

Answer:
Explaination:
Given line is 5x – 25 = 14 – 7y = 35z
⇒ 5(x – 5) = – 7(y – 2) = 35z

DR’s of line are 7, – 5 and 1
dr’s of line parallel to the given line are 7,-5, 1.
vector equation of line through the point (1, 2, – 1) and having dr’s 7,-5 and 1 is

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22. Write the distance of the point (3, – 5, 12) from the x-axis. [Foreign 2017]

Answer/Explanation

Answer:
Explaination:
Distance of the point (3, – 5, 12) from the x-axis

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23. Find the angle between the following pair of lines:

and check whether the lines are parallel or perpendicular. [Delhi 2011]

Answer/Explanation

Answer:
Explaination:
DR’s of lines are 2, 7, – 3 and – 1, 2, 4
As 2 × (- 1) + 7 × 2 – 3 × 4 = 0, so lines are perpendicular. Angle = 90°

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24. The x-coordinate of a point on the line joining the points P(2, 2, 1) and Q(5, 1, -2) is 4. Find its z-coordinate. [AI2017]

Answer/Explanation

Answer:
Explaination:
Let point R(4, y, z) lies on the line joining P(2, 2, 1) and Q(5, 1, -2). Let R divides PQ in ratio k: 1

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25. If P(1, 5, 4) and Q(4, 1, – 2), find the direction ratios of \(\overrightarrow{P Q}\).

Answer/Explanation

Answer:
Explaination:
Direction ratios of \(\overrightarrow{P Q}\) =4 – 1, 1 – 5 and -2 -4, i.e. 3, -4 and – 6.

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26. The equations of a line are 5x – 3 = 15y + 7 = 3 – 10z. Write the direction cosines of the line. [All India]

Answer/Explanation

Answer:
Explaination:
The equation of a line are 5x – 3 = 15y + 7 = -10z + 3

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27. Equation of the perpendicular drawn from the point with position vector \(2 \hat{i}- \hat{j}+ \hat{k}\) to the plane \(\vec{r} \cdot(\hat{i}-3 \hat{k})=5\) is ________ .

Answer/Explanation

Answer:
Explaination:

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28. General equation of a plane passing through the intersection of two given

Answer/Explanation

Answer:
Explaination:

---

29. Cartesian equation of the plane

State true or false.

Answer/Explanation

Answer:
Explaination:

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30. Find the distance of the point (2,3,4) from the plane

Answer/Explanation

Answer:
Explaination:

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31. Write the intercept cut off by the plane 2x + y – z = 5 on the x-axis. [Delhi 2011]

Answer/Explanation

Answer:
Explaination:
For intercept on the x-axis, put y = 0 and z = 0
⇒ 2x = 5
⇒ x = \(\frac{5}{2}\)
∴ x-intercept = \(\frac{5}{2}\)

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32. Find the distance of the plane 3x – 4y + 12z = 3 from the origin. [AI 2012]

Answer/Explanation

Answer:
Explaination:

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33. Find the angle between the planes

Answer/Explanation

Answer:
Explaination:

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34. Find the distance between the planes 2x – y + 2z – 5 and 5x – 2.5y + 5z = 20. [AI 2017]

Answer/Explanation

Answer:
Explaination:
Planes are 2x – y + 2z = 5
⇒ 2x – y + 2z – 5 = 0
and 5x – 2.5y + 5z = 20
⇒ 2x – y + 2z – 8 = 0

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35. A line passes through the point with position vector \(2 \hat{i}-3 \hat{j}+4 \hat{k}\) and is perpendicular to the plane \(\vec{r} \cdot(3 \hat{i}+4 \hat{j}-5 \hat{k})=7\). Find the

Answer/Explanation

Answer:
Explaination:

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We hope the given Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry will help you. If you have any query regarding CBSE Class 12 Maths Three Dimensional Geometry MCQs Pdf, drop a comment below and we will get back to you at the earliest.

Free PDF Download of CBSE Maths Multiple Choice Questions for Class 12 with Answers Chapter 10 Vector Algebra. Maths MCQs for Class 12 Chapter Wise with Answers PDF Download was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 12 Maths Vector Algebra MCQs Pdf with Answers to know their preparation level.

Vector Algebra Class 12 Maths MCQs Pdf

1. A vector equally inclined to axes is

Answer/Explanation

Answer: a
Explaination:
(a), as direction ratios are 1, 1, 1 and
direction cosines \(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\)
⇒ cos α = cos β = cos γ
⇒ α = β = γ

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2. The position vector of a point which divides the join of points with position vectors \(\vec{a}+\vec{b}\) and \(2 \vec{a}-\vec{b}\) in the ratio 1:2 internally is

Answer/Explanation

Answer: d
Explaination:

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3. A vector in the direction of vector \(\hat{i}-2 \hat{j}+2 \hat{k}\) that has magnitude 15 is

Answer/Explanation

Answer: d
Explaination:

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4. If \(|\vec{a}|\) = 4 and -3 ≤ λ ≤ 2 then the range of \(|\lambda \vec{a}|\) is
(a) [0, 8]
(b) [-12, 8]
(c) [0, 12]
(d) [8, 12]

Answer/Explanation

Answer: c
Explaination:

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5.

(a) 30°
(b) 45°
(c) 60°
(d) 90°

Answer/Explanation

Answer: a
Explaination:

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6.

Answer/Explanation

Answer: c
Explaination:

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7.

Answer/Explanation

Answer: b
Explaination:

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8.

Answer/Explanation

Answer: a
Explaination:

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9. Mathematically a vector is defined as a “directed line segment.” State true or false.

Answer/Explanation

Answer:
Explaination: True

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10. If vectors are equal then their magnitudes are equal but the converse may not be true. State true or false.

Answer/Explanation

Answer:
Explaination:

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11. Given vector \(\overrightarrow{P Q}=2 \hat{i}+\hat{j}-3 \hat{k}\) and position vector of point P is \(3 \hat{j}-2 \hat{k}\), then position vector of point Q is _________ .

Answer/Explanation

Answer:
Explaination:

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12. Find a unit vector in the direction of \(\overrightarrow{a}=3 \hat{i}-2 \hat{j}+6 \hat{k}\)

Answer/Explanation

Answer:
Explaination:

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13. For what value of p, is (\(\hat{i}+ \hat{j}- \hat{k}\))p a irnit vector?

Answer/Explanation

Answer:
Explaination:

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14. If \(\overrightarrow{A B}=2 \hat{i}+\hat{j}-2 \hat{k}\) and \(\overrightarrow{B C}=6 \hat{i}+3\hat{j}-6 \hat{k}\), can we say that the points A, B, C are collinear?

Answer/Explanation

Answer:
Explaination:

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15. In the given figure, find
(i) parallel or collinear vectors

(ii) co-initial vectors

Answer/Explanation

Answer:
Explaination:

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16.

Answer/Explanation

Answer:
Explaination:

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17.

Answer/Explanation

Answer:
Explaination:

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18. Find a vector in the direction of \(\vec{a}=\vec{i}-2 \hat{j}\) whose magnitude is 7. [NCERT]

Answer/Explanation

Answer:
Explaination:

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19. For what value of ‘a’ the vectors \(2 \hat{i}-3 \hat{j}+4 \hat{k}\) and \(\hat{a} \hat{i}+\hat{6} \hat{j}-8 \hat{k}\) are collinear?[Delhi2011]

Answer/Explanation

Answer:
Explaination:
For vectors to be collinear \(\frac{2}{a}=\frac{-3}{6}=\frac{4}{-8}\)
⇒ a = -4

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20. ABCDEF is a regular hexagon,

Answer/Explanation

Answer:
Explaination:

---

21. The position vectors A, B, C, D are

Answer/Explanation

Answer:
Explaination:

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22. Write the direction cosines of the vector \(-2 \hat{i}+\hat{j}-5 \hat{k}\) [Delhi 2011]

Answer/Explanation

Answer:
Explaination:

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23. Show that the vector \( \hat{i}+\hat{j}+ \hat{k}\) is equally inclined to axes.

Answer/Explanation

Answer:
Explaination:

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24. Write a unit vector in the direction of the

Answer/Explanation

Answer:
Explaination:

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25. The value of X for which vectors

are orthogonal is __________ .

Answer/Explanation

Answer:
Explaination:

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26. For any non zero vector \(\vec{a}\),

State true or false.

Answer/Explanation

Answer:
Explaination:

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27.
are perpendicular. State true or false.

Answer/Explanation

Answer:
Explaination:

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28. Find the angle between the vectors

Answer/Explanation

Answer:
Explaination:

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29.

Answer/Explanation

Answer:
Explaination:

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30.

Answer/Explanation

Answer:
Explaination:

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31. If \(\vec{p}\) is a unit vector and

Answer/Explanation

Answer:
Explaination:

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32.

Answer/Explanation

Answer:
Explaination:

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33.

Answer/Explanation

Answer:
Explaination:

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34.
\(\vec{b}\) are along adjacent sides of a rectangle.

Answer/Explanation

Answer:
Explaination:

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35.

Answer/Explanation

Answer:
Explaination:

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36. Find the magnitude of each of the two vectors \(\vec{a}\) and \(\vec{b}\), having the same magnitude such that the angle between them is 60° and their scalar product is \(\frac{9}{2}\). [CBSE 2018]

Answer/Explanation

Answer:
Explaination:

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37.

Answer/Explanation

Answer:
Explaination:

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38.

Answer/Explanation

Answer:
Explaination:

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39.

Answer/Explanation

Answer:
Explaination:

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40.

Answer/Explanation

Answer:
Explaination:

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41.

Answer/Explanation

Answer:
Explaination:

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42.

Answer/Explanation

Answer:
Explaination:

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43.

Answer/Explanation

Answer:
Explaination:

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44.

Answer/Explanation

Answer:
Explaination:

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45.

Answer/Explanation

Answer:
Explaination:

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46.

Answer/Explanation

Answer:
Explaination:

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47.

Answer/Explanation

Answer:
Explaination:

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48.

Answer/Explanation

Answer:
Explaination:

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49.

Answer/Explanation

Answer:
Explaination:

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50. Write the value of \((\hat{i} \times \hat{j}) . \hat{k}+\hat{i} \hat{j}\) [AI 2012]

Answer/Explanation

Answer:
Explaination:

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51. Write a unit vector perpendicular to both the
[AI 2015]

Answer/Explanation

Answer:
Explaination:

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52.

Answer/Explanation

Answer:
Explaination: coplanr

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53.

(a) 4
(b) 0
(c) -3
(d) 2

Answer/Explanation

Answer: b
Explaination:

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54.

Answer/Explanation

Answer:
Explaination:

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55.

Answer/Explanation

Answer:
Explaination:

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We hope the given Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra will help you. If you have any query regarding CBSE Class 12 Maths Vector Algebra MCQs Pdf, drop a comment below and we will get back to you at the earliest.

Free PDF Download of CBSE Maths Multiple Choice Questions for Class 12 with Answers Chapter 9 Differential Equations. Maths MCQs for Class 12 Chapter Wise with Answers PDF Download was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 12 Maths Differential Equations MCQs Pdf with Answers to know their preparation level.

Differential Equations Class 12 Maths MCQs Pdf

1. Order of differential equation correspon- ding to family of curves y = Ae^2x + Be^2x is ______ .

Answer/Explanation

Answer:
Explaination: 2, as there are two arbitrary constants and we have to differentiate twice.

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2. The order of the differential equation corresponding to the family of curves y = c(x – c)², c is constant is ______ .

Answer/Explanation

Answer:
Explaination: One, as there is one arbitrary constant.

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3. The degree of differential equation

is not defined. State true or false.

Answer/Explanation

Answer:
Explaination: Three, as equation cannot be represented as polynomial of derivatives.

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4. If p and q are the degree and order of the differential equation

then the value of 2p – 3q is
(a) 7
(b) -7
(c) 3
(d) -3

Answer/Explanation

Answer: b
Explaination:
(b), as degree p = 1 and order q = 3
∴ 2p – 3q = 2 – 9 = -7

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5. The degree of the differential equation

(a) 1
(b) 2
(c) 3
(d) 4

Answer/Explanation

Answer: c
Explaination:
(c), as differential equation is

Exponent of highest order derivative is 3.

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6. The degree of the differential equation

(a) 1
(b) 2
(c) 3
(d) not defined

Answer/Explanation

Answer: d
Explaination:
(d), as equation cannot be represented as a polynomial of derivatives.

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7. The order of the differential equation of all the circles of given radius 4 is
(a) 1
(b)2
(c) 3
(d) 4

Answer/Explanation

Answer: b
Explaination:
(b), as centre is arbitrary (h, k), two arbitrary constants so we have to differentiate twice to eliminate h, k
∴ order is 2.

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8. Degree of the differential equation

is not defined. State true or false.

Answer/Explanation

Answer:
Explaination:
False, as equation can be written as

Further it can be written as a polynomial of derivatives. 9.

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9. Write the degree of the differential equation

Answer/Explanation

Answer:
Explaination: Degree 1

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10. Write the degree of the differential equation

Answer/Explanation

Answer:
Explaination: Degree 3.

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11. Find the value of m and n, where m and n are order and degree of differential equation

Answer/Explanation

Answer:
Explaination:

Order of differential equation (m) = 3
Degree of differential equation (n) = 2

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12. Write the order and degree of the differential equation \(\frac{d y}{d x}+\sin \left(\frac{d y}{d x}\right)\) = 0. [HOTS]

Answer/Explanation

Answer:
Explaination:
Highest order derivative is \(\frac{dy}{dx}\). Hence, order of differential equation is 1. Equation cannot be written as a polynomial’ in derivatives. Hence, degree is not defined.

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13. The differential equation of the family of lines passing through ongrn is
(a) y = mx
(b) \(\frac{dy}{dx}\) = m
(c) x dy – y dx = 0
(d) \(\frac{dy}{dx}\) = 0

Answer/Explanation

Answer: c
Explaination:
(c), as general equation of line through origin is
y = mx
⇒ \(\frac{dy}{dx}\) = m
Substituting in (i), we get dy
y = \(\frac{dy}{dx}\).x
⇒ x dy – y dx = 0

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14. Find the differential equation representing the family of curves y = ae^{bx + 5}, where a and b are arbitrary constants. [CBSE 2018]

Answer/Explanation

Answer:
Explaination:
Consider y = ae^{bx + 5}
.On differentiating both sides, w.r.t, x
\(\frac{dy}{dx}\) = abe^{bx + 5} = by …..(i)
Again differentiating w.r.t. x, we get
\(\frac{d^{2} y}{d x^{2}}=b \cdot \frac{d y}{d x}\) …..(ii)
From (i) and (ii), eliminating b, we get
\(y \cdot \frac{d^{2} y}{d x^{2}}=\left(\frac{d y}{d x}\right)^{2}\) as required equation.

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15. Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of the x-axis. [NCERT; DoE]

Answer/Explanation

Answer:
Explaination:
General equation of parabola is y² = 4ax …..(i)
Differentiating, we get 2yy’ = 4a
⇒ yy’ = 2a.
Substituting in (i), we gety2 = 2xyy’.

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16. Form the differential equation of the family of parabolas having vertex at the origin and axis along positive y-axis. [Delhi 2011]

Answer/Explanation

Answer:
Explaination:
x² = 4 ay
⇒ 2x = 4 ay’
⇒ \(\frac{x^{2}}{2 x}=\frac{4 a y}{4 a y^{\prime}}\)
⇒ xy’ – 2y = 0 is the required equation.

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17. General solution of the differential equation log\(\frac{dy}{dx}\) = 2x +y is _______ .

Answer/Explanation

Answer:
Explaination:

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18. Solve the differential equation \(\frac{dy}{dx}\) = e^{x – y} + x³e^-y.

Answer/Explanation

Answer:
Explaination:

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19. Find the particular solution of the differential equation \(\frac{dy}{dx}\) =y tanx, given that y= 1 when x = 0.

Answer/Explanation

Answer:
Explaination:
∫ \(\frac{dy}{y}\) = ∫tan x dx
⇒ log |y| = log|sec x| + log C
⇒ y = C sec x ….(i)
Given y = 1, x = 0
⇒ 1 = C sec 0
⇒ C = 1
∴ solution is y = sec x [from (i)]

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20. Find the general solution of the differential equation \(\frac{dy}{dx}\) = \(\frac{x+1}{2-y}\), (y ≠ 2). [NCERT]

Answer/Explanation

Answer:
Explaination:

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We hope the given Maths MCQs for Class 12 with Answers Chapter 9 Differential Equations will help you. If you have any query regarding CBSE Class 12 Maths Differential Equations MCQs Pdf, drop a comment below and we will get back to you at the earliest.