Advanced Level Problem Solving in Mathematics – ALPS JEE 2022

Vidyamandir Classes: Innovating for Your Success




ALPS Mathematics 2202| JEE 2022
Syllabus: Vectors & Three Dimensional Geometry, Probability, Matrices & Determinants
*Mark questions are more than one options correct type

Day - 1
     
1. Let a , b and c be three non zero and non coplanar vectors and p, q and r be three vectors given by
           
p  a  b  2c ; q  3a  2b  c and r  a  4b  2c . If the volume of the parallelopiped
     
determined by a , b and c is v1 and that of the parallelepiped determined by p, q and r is v2
then v2 : v1 is :
(A) 1:5 (B) 5:1 (C) 15: 1 (D) 1: 15 [  ]

2. For any two events A and B in a sample space, which of the following is not true :
 A  P  A  P  B   1
(A) P   , P  B   0 is always true [  ]
 B P B
(B) If P  A  P  A   P  A  B  , then A and B are disjoint
(C) P  A  B   1  P  A  P  B  , if A and B are independent
(D) P  A  B   1  P  A  P  B  , if A and B are disjoint
3. If a 2  b 2  c 2  ab  bc  ca  0a , b, c  R , then value of the determinant
 a  b  2 2 a2  b2 1
2
1 b  c  2 b  c2
2
equals :
2
c2  a 2 1 c  a  2
(A) 65 (B) a 2  b 2  c 2  31
(C) 
4 a2  b2  c2  (D) 0
4. Out of 3n consecutive positive integers, 3 are chosen at random without replacement. What is the
probability that the sum of these numbers is div. by 3 ?
3n 2  3n  1 3n 2  3n 3n 2  3n  2 3n 2  3n  4
(A) (B) (C) (D)
 3n  1 3n  2   3n  1 3n  2   3n  1 3n  2   3n  1 3n  2 

*5. If P (2, 3, 1) is a point and L  x  y  z  2  0 is a plane then : [  ]
(A) Origin and P lie on the same side of the plane
4
(B) distance of P from the plane is
3
 10 5 1 
(C) foot of perpendicular is  , ,  
 3 3 3
 10 5 1 
(D) image of point P by the plane  , ,  
 3 3 3


VMC | JEE-2022 | Mathematics 1 ALPS-2202 | DAY-1

, Vidyamandir Classes: Innovating for Your Success

*6. If A1 , A2 , A3 ,......, A1006 be independent events such that P  Ai   1/ 2i  i  1, 2, 3,......, 1006  and
!
probability that none of the events occurs be , then : [ ]
2 
!2
(A)  is of form 4k +2, k  I (B)   2
(C)  is a composite number (D)  is of form 4k, k  I

*7. Consider the system of equations: x sin   2 y cos   az  0, x  2 y  z  0,  x  y  z  0,   R. []
(A) The given system will have infinite solutions for a  2
(B) The number of integer values of a is 3 for the system to have nontrivial solutions
(C) For a = 1 there exists  for which the system will have infinite solutions
(D) For a  3 there exists  for which the system will have unique solution
1 0 0 
*8. If A  1 0 1  , then :
0 1 0 
 1 0 0  1 1 0
(A) 3 2
A  A  A  I (B) 
Det A 2010

 I  0 (C) A 50
  25 1 0  (D) A50
  25 1 0 
 25 0 1   25 0 1 

PARAGRAPH FOR QUESTIONS 9 - 10
 
Let two unit vectors along two lines OA and OB be â and b̂ respectively. Take their point of intersection as
the origin and let P be any point on the bisector of angle between the lines OA and OB. Draw PM parallel to AO
cutting OB at M.
AOP  POM  OPM and hence OM = PM.
 
But OM  tbˆ and MP  taˆ
 
(since OM || bˆ and MP || aˆ and their magnitudes are same)
  
Then OP  rˆ  OM  MP  t (bˆ  aˆ ) ……..(i)
For external bisector OP ' , the angle between OB and OA is the same
as the internal bisector of the angle between the unit vectors along
 
them being bˆ and a and hence the equation of OP ' be
 
 
OP '  r '  t aˆ  bˆ …….. (ii)
 
For any two vectors a and b the equations (i) and (ii) reduce to
 
  a b 
r  t    
|a | |b |
9. If the interior and exterior bisectors of the angle A of a triangle ABC meet the base BC at D and E, then
:
(A) 2 BC  BD  BE (B) BC 2  BD  BE
2 1 1
(C)   (D) None of these
BC BD BE
  
10. Let ABC be a triangle and a , b , c be the position vectors of the point A, B, C respectively. External
bisectors of B and C meet at P with the sides of the triangle as a, b, c, the position vectors of P
becomes :
   
(b)b  ( c)c aa  (b)b  (c)cˆ
(A) (B)
(b  c) (a  b  c )
  
ˆ
 aˆ  b  cˆ  aa  bb  cc
(C)   (abc) (D)
 3  (a  b  c )

VMC | JEE-2022 | Mathematics 2 ALPS-2202 | DAY-1