Tiêu đề MP DPP Sheet 05 (1).pdf ✅

1 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 IIT JAM PHYSICS 2025 (Online Batch) SECTION: MATHEMATICAL PHYSICS Daily Practice Problem (DPP) Sheet 5: Vector Calculus (Dot & Cross Product, Scalar & Vector Triple Product) PART - A: MULTIPLE CHOICE QUESTIONS (MCQ) Q.1. Two forces whose magnitudes are in ratio of 3 : 5, gives a resultant of 35 N. If the angle of inclination is 60°, then what will be the magnitude of each force? (a) 21N, 35N (b) 15N, 25N (c) 18N, 30N (d) 9N, 15N Q.2. The projection of ˆ ˆ ˆ u i j k    3  in the direction of ˆ ˆ a j k   4 3  will be (a)   1 ˆ ˆ 4 3 25 j k  (b)   1 ˆ ˆ 4 3 5 j k  (c)   1 ˆ ˆ 4 3 25  j k (d)   1 ˆ ˆ 4 3 5  j k Q.3. A boat sails 5km south-east then 3km due west. Approximately how far from its starting position is it now? (a) 8km (b) 4km (c) 5.8km (d) 3.6km Q.4. If a  and b  are unit vectors inclined to x-axis at angles 300 and 1200 respectively. The value of a b    is (a) 1 (b) 2 (c) 3 (d) 2 Q.5. Given: ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ a i j k b i j k c = 2i + j – k       3 2 , 3 5 and 4    are the sides of (a) an isosceles triangle (b) a right-angled triangle (c) an equilateral triangle (d) none of these Q.6. The value of the constant ‘m’ for which ˆ ˆ ˆ A mi j k    5  subtends an angle 600 with ˆ ˆ ˆ B i j k    4 5 5  is equal to (a) 46 / 3 (b) 29 / 4 (c) 41/ 3 (d) 17 / 3 Q.7. The range of x for which the angle between the vectors 2 ˆ ˆ ˆ A x i xj k    2 4  and ˆ ˆ ˆ B i j xk    7 2  is obtuse, is equal to (a) x < 0 (b) x > 1/2 (c) 0 < x < 1/2 (d) None of these Q.8. If a b c    0     and a b c    3, 5, 7    , then the angle between  a and  b is (a)   1 cos 15 22  (b)   1 cos 225 2  (c)  2 (d)  3
2 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 Q.9. For any vector A  , which of the following statements is CORRECT? (a)       A A i i A j j A k k ˆ ˆ ˆ ˆ ˆ ˆ           (b)       ˆ ˆ ˆ ˆ ˆ ˆ A A i i A j j A k k       . . .     (c)       ˆ ˆ ˆ ˆ ˆ ˆ A Ai i A j j A k k    . . .     (d)       ˆ ˆ ˆ ˆ ˆ ˆ A Ai i A j j A k k       . . .     Q.10. Let ˆ ˆ ˆ a i j k    2 2  and b i j  ˆ ˆ  . If c  is a vector such that a c c c a . , 2 2         and the angle between a b    and c  is 300 , then a b c      is (a) 2/3 (b) 3/2 (c) 2 (d) 3 Q.11. The area of the triangle PQR having vertices P(1,3,2),Q(2, –1, 1) , R 1, 2,3  is equal to (a) 1 107 2 (b) 1 77 2 (c) 1 117 2 (d) 1 117 2 Q.12. Given: ˆ ˆ ˆ ˆ ˆ ˆ ˆ P i j Q i j k R i k     2 – 3 , - , 3 -    , the vector V,  which is orthogonal to P Q and   and having unit scalar product with R  , can be estimated to be (a)   ˆ ˆ ˆ 3 2 5 / 4 i j k   (b)   ˆ ˆ ˆ 3 2 5 / 4 i j k   (c)   ˆ ˆ ˆ 3 2 5 / 4 i j k   (d)   ˆ ˆ ˆ 3 2 5 / 4 i j k   Q.13. If 2 a b   9   , a  2  and b  3  , then a b.   is equal to (a) 27 (b) 3 3 (c) 8 (d) 2 2 Q.14. If sum of two unit vectors a  and b  is also an unit vector, the angle between the vectors a  and b  is (a) 0 30 (b) 0 60 (c) 0 90 (d) 0 120 Q.15. The vector ˆ ˆ ˆ a i j k      2  lies in the plane of the vectors b i j ˆ  ˆ ˆ and ˆ ˆ c j k    . Then the possible values of   and will be (a) 2, 2 (b) 1, 2 (c) 2, 1 (d) 1, 1 Q.16. If a b. 3    and a b   4   , then the angle between a  and b  is (a) 1 4 cos 3        (b) 1 3 cos 4        (c) 1 4 cos 5        (d) 1 3 cos 5        Q.17. The unit vector which is perpendicular to vector ˆ ˆ ˆ 2 3 i j k   and lie in the plane of vectors ˆ ˆ ˆ 7i j k   and ˆ ˆ ˆ i j k   5 3 is equal to (a)   ˆ ˆ ˆ 3 3 / 19 i j k   (b)   ˆ ˆ ˆ 3 3 / 19 i j k   (c)   ˆ ˆ ˆ 3 3 i j k   (d)   ˆ ˆ ˆ 3 3 i j k   Q.18. If       ˆ ˆ ˆ ˆ ˆ ˆ              b i a i j a j k a k , then b  can be simplified to (a) 0 (b)  a (c) 2  a (d) None of these
3 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 Q.19. For three non-zero vectors a b c a b c                will be valid if (a) a b and   are collinear (b) b c and   are collinear (c) a c and   are collinear (d) Can never be valid Q.20. The vector which is coplanar with vectors ˆ ˆ ˆ ˆ ˆ ˆ i j k i j k     2 and 2 & perpendicular to the vector ˆ ˆ ˆ i j k   , will be (a) ˆ ˆ j k  (b) ˆ ˆ i j  (c) ˆ ˆ  i j (d) ˆ ˆ i k  PART - B: Numerical Answer Type (NAT) Questions Q.21. Let a b   2   and p a b q a b     ,       . If   1/2 2 p q k a b 2 .              , then k will be _____________ (Your answer should be AN INTEGER) Q.22. If the projection of the vector A i mj k ˆ ˆ ˆ     on vector ˆ ˆ ˆ B i j k    4 4 7  is equal to 19/9, then the value of ‘m’ is equal to ___________________________________ (Your answer should be AN INTEGER) Q.23. If ˆ ˆ ˆ    2 2 3  a i j k , ˆ ˆ ˆ     2  b i j k and   3ˆ ˆ  c i j , then    a tb is perpendicular to  c if ‘t’ is equal to ____________________________________________ (Your answer should be AN INTEGER) Q.24. If the resultant of two forces P and Q   is equal in magnitude to one of the components  P  and perpendicu- lar to it in direction. Then Q P  ______ (Your answer should be upto TWO DECIMAL PLACES) Q.25. Given: ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ A i j k B i j k C i j mk           2 , 3 5 6 , 4    . The value of ‘m’ such that the vectors will be co- planar is equal to ___________________ (Your answer should be upto ONE DECIMAL PLACES) Q.26. If a b c , and    are such that   a b c  9      , then   a b b c c a            is equal to ___________________ (Your answer should be AN INTEGER) Q.27. The volume of the parallelopiped whose edges are represented by ˆ ˆ ˆ a i j k    2 3 4  , ˆ ˆ ˆ b i j k    2  and ˆ ˆ ˆ c i j k    3 2  is ___________________________ (Your answer should be AN INTEGER) PART - C: Multiple Select Questions (MSQ) Q.28. If . .      a b a c and        a b a c , then which of the following statements is/are CORRECT? (a)  a is parallel to ( )    b c (b)    b c (c)  0  a (d)  a is perpendicular to ( )    b c
4 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 Q.29. If a  and b  are unit vectors, which of the following is/are CORRECT? (a) 1 θ sin =2 2 a b    (b) 1 θ sin =2 2 a b    (c) 1 θ cos =2 2 a b    (d) 1 θ cos =2 2 a b    Q.30. Three unit vectors a b c , ,    (b c and   are not parallel) are such that   1 2 a b c c        . Which of the following statements is/are CORRECT? (a) Angle which a  makes with b  is 0 60 . (b) Angle which a  makes with b  is 0 120 (c) Angle which a  makes with c  is 0 90 (d) Angle which a  makes with c  is 0 120 Answer Key PART - A: MULTIPLE CHOICE QUESTIONS (MCQ) 1. (b) 2. (a) 3. (d) 4. (b) 5. (b) 6. (a) 7. (c) 8. (d) 9. (c) 10. (b) 11. (a) 12. (a) 13. (b) 14. (d) 15. (d) 16. (d) 17. (b) 18. (c) 19. (c) 20. (a) PART - B: NUMERICAL ANSWER TYPE (NAT) QUESTIONS 21. (16) 22. (-2) 23. (8) 24. (1.41 to 1.47) 25.(6.5) 26. (9) 27. (7) PART - C: MULTIPLE SELECT QUESTIONS (MSQ) 28. (b, c) 29. (a, c) 30. (b, c)