IMPORTANT MATHEMATICS MCQ FOR JEE MAIN 2017 Part - 1

IMPORTANT MATHEMATICS MCQ

1.  Consider the system of linear equations
             x1 + 2x2 + x3 = 3
            2x1 + 3x2 + x3 = 3
           3x1 +5x2 +2x3 = 1
     The system has --- a) infinite number of solutions   b) exactly 3 solutions c) a unique solution
                                 d) no solution

2. There are two Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn, two
    balls are taken out at random and then transferred to the other . The number of ways in which this can
    be can be done is
    a) 3    b) 36   c) 66   d) 108

3. A line AB in three-dimensional space makes angles 45* and 120* with the positive X-axis and the
    positive Y-axis respectively . If AB makes an acute angle θ with the positive Z-axis , then θ equals
    a) 30*   b) 45 *   c) 60*    d)75*

4. An urn contains nine balls of which three are red, four are blue and two are green.Three balls are 
    drawn at random without replacement from the urn . The probability that the three balls have different
    colours , is 
     a) 1 / 3    b) 2 / 7   c) 1 / 21    d) 2 / 23

5. For two data sets each of size 5 , the variances are given to be 4 and 5 and the corresponding means
    are given to be 2 and 4 respectively . The variance of the combined data set is
     a)5 / 2    b) 11 / 2   c) 6     d) 13 / 2

6. Let  cos ( α + β ) = 4 /5 and sin(α +β ) = 5 / 13 , where 0 ≤ α , β ≤ π /4 . then , tan 2α is equal to
     a) 25 / 16   b) 56 / 33   c) 19 / 12  d) 20 / 7

7. Consider the following relations 
     R = { ( x , y )  x , y are real numbers and  x = wy for some rational number w } ;

     S = { ( m / n , p / q )|  m ,n ,p and q are integers such that n , q  ≠ 0 and qm = pn }. Then,
     a) R is an equivalence relation but S is not an equivalent relation
     b) Neither R and S is an equivalence relation 
     c) S is an equivalence relation but R is not an equivalence relation
     d) R and S both are equivalence relations

8. The number of complex numbers z such that | z -- 1 | = |z + 1 | = |z - i |  equals    
       a)  0      b)  1    c)  2   d)  ∞    

9. The number of 3 x 3 non -singular matrices , with four entries as 1 and all other entries as 0 , is
     a) less than 4   b) 5   c) 6    d) atleast 7

10. Let f : R ---> r be defined by
                          k - 2x , if x  ≤  -- 1         
             f(x) ={                             
                     2x + 3 , if x > --1
If f has a local minimum at x = --1 then a possible value of k is
       a) 1     b) 0     c) - 1 / 2  d) - 1

11. If the mean deviation of number  1 , 1 + d , 1 + 2d , .... , 1 + 100 d from their mean is 255 , then d is
       is equal to
        a) 10.0   b) 20.0     c) 10.1      d) 20.2

12. Let A and B denote the statements
        A : cos α  + cos β + cos γ  = 0
        B :  sin α  +  sin β   +  sin γ  = 0
        If
        cos ( β--γ ) + cos  (γ--α)  + cos( α--β) = -- 3 / 2
         then
         a) A is true and B is false     b) A is false abd B is true    c) Both A and B are true
         d) Both A and B are false

13. If u , v , w are non-coplanar vectors and p , q are real numbers then the equality
      [3u pv pw ] -- [pv w qv ] -- [ 2w qv qu ] = 0 holds for
      a) exactly two values of ( p , q )
      b) more than two but not all values of ( p , q )
      c) all values of ( p , q)
      d) exactly one value of ( p , q )

14. Let the line x- 2 / 3 = y - 1 / - 5  = z + 2 / 2 lies in the plane x +3y - αz + β = 0. Then ( α ,β ) equals
        a) ( 6 , - 17 )    b) ( - 6 , 7 )   c) (5 , - 15 )    d) ( -5 , 15 )

15. From 6 different novels and 3 different dictionaries , 4 novels and 1 dictionary are to be selected and
      arranged on the shelf so that the dictionary is always in the middle . Then the number of such arrange-
      ment is
         a) atleast 500 but less than 750    b) atleast 750 but less than 1000    c) atleast 1000
         d) less than 500

16. The projection of a vector on the three coordinate axes are 6 , - 3 , 2 respectively. The direction
      cosines of the vector are
      a) 6 , - 3 , 2      b) 6/5 , -3 /2 , 2 /5   c) 6/7 , 3 /7 , 2 /7    d) -6/7 , - 3/7 , 2 /7

17. Three distinct points A , B, and C given in the 2 dimensional coordinate plane such that the ratio of
       the distance of any one of them from the point ( 1 , 0) to the distance from the point ( - 1 , 0) is
      equal to 1 /3 . Then the circumference of the    Δ ABC is at the point
      a) (5/4 , 0 )  b) ( 5/2 , 0 )    c) ( 5/3 , 0 )   d) ( 0 ,  0)

18.  The shortest distance between the line y - x = 1 and the curve x = y 2 is
         a) 3 √2 / 8   b) 2 √3 / 8   c) 3√2 / 5  d) √3 / 4

19. For any two statements p and q  ~ ( p v q )  v ( ~ p ^ q ) is logically equivalent to
      a) p     b)  ~ p    c) q    d) ~ q

20. If C is the mid point of AB and P is any point outside AB then
      a) PA + PB + PC = 0       b) PA + PB + 2PC = 0    c) PA + PB = PC d) PA +PB=2PC

21. If in a distribution the mean and median are 21 and 22 respectively then its mode is approximately
      a) 24.0      b) 25.5     c) 20.5    d) 22.0

22. Let R ={ ( 3 , 3) ,(6 ,6) , (9 ,9) ,(12 ,12),(6,12) ,(3,9) ,( 3,12), (3 ,6) be a relation on the
      A ={3,6,9,12}. The relation is
      a) reflective and symmetric only    b) an equivalence relation  c) reflective only
      d) reflective and transitive only

23. ABC is a triangle. Forces P ,Q ,R acting along I A , I B and IC respectively are in equilibrium ,
      where I is the incentre of  Δ ABC . Then P:Q:R is
       a) cos A : cos B : cos C     b) cos A/2 : cos B/2 : cos C/2 c) sin A/2: sin B/2 : sin C/2
       d) sin A : sin B : sin C

24. In a ΔPQR , ∠R = π / 2 . If tan ( P/2) and tan (Q/2) are the roots of αx² + bx + c = 0 , a ≠ 0 ,
       then
       a) b = a + c   b) b = c  c) c = a + b   d) a = b +c

25. If the letters of the word SACHIN are arranged in all possible ways and these words are written out
       as in dictionary , then the word SACHIN appears at serial number
       a) 602    b) 603    c) 600    d) 601

26. If in a  Δ ABC , the altitudes from the vertices A , B ,C on opposite sides are in HP, then sin A , sin B
       sin C are in
       a) HP   b) Arithmetic-Geometric progression   c) AP    d) GP

27. The normal to the curve x = α (cosΘ  + ΘsinΘ ) , y =α (sinΘ - ΘcosΘ) at any point Θ is such that
       a) it is at a constant distance from the origin
       b) it passes through (a π/2 , -a )  c) it makes angle π / 2 + Θ with the x - axis
       d) it passes through the origin

28.  A circle touches the X -axis and also touches the circle with centre at (0 , 3 ) and radius 2 . The locus
        of the circle is 
        a) a parabola   b) a hyperbola  c) a circle   d) an ellipse

29. The line parallel to the X - axis and passing through the intersection of the lines 
        ax + 2 by + 3 b = 0 and bx - 2 ay - 3a =0 , where (a ,b) ≠ (0 , 0) is
        a) above the X axis at a distance of  (2 /3) from it
        b) above the X axis at a distance of ( 3 / 2) from it
        c) below the X- axis at a distance of  ( 2 / 3) from it
        d) below the X - axis at a distance of ( 3 / 2 ) from it