Navodaya Vidyalaya Samiti, Bhopal Region
Sample paper of Mathematics for 2015-16
Chapters: Vectors, Three Dimensional Geometry
Time allotted 3:00 hrs
Max. marks 100
General instructions:
1.All questions are compulsory.
2.In section A questions 1-6 are very short answer type questions carrying 1 mark each.
3. In section B questions 7-19 are long answer type I questions carrying 4 marks each.
4. In section C questions 20-26 are long answer type II questions carrying 6 marks each.
5.There is no overall choice in the question paper internal choice has been given to particular
questions itself.
Section (A)
Q1. Find λ when projection of a = λ î+ĵ+4k On b=2î+6ĵ+3k is 4 units.
Q2.If |a|=√3 , |b|=2 and a.b=3 find angle between them.
Q3.What is the value of î.( ĵ x k)+ ĵ.( î x k)+k.( î x ĵ).
Q4.The equation of a line is (2x-5)/4= (y+4)/3 = (6-z)/6.Find the direction cosines of a line
parallel to this line.
Q5. Find the equation of a line parallel to x-axis and passing through origin.
Q6. If a unit vector a makes an angle π/3 with i, π/4 with j and an acute angle ∅ with k,then find
the value of ∅ .
Section (B)
Q7. Prove that the sum of the vertices directed from the vertices to the mid points of opposite
sides of a triangle is zero.
Q8.Show that the points A(1,-2,-8) ; B(5,0,-2) &C(11,3,7) are collinear and find the ratio in
which B divides AC.
OR
Find x such that the four points A(3,2,1,);B(4,x,5);C(4,2,-2)&D(6,5,-1) are coplanar.
Q9. If a,b and c are three vectors such that |a|=3 , |b|=4 and |c|=5 And each one of these is
perpendicular to sum of other two find |a+b-c|.
Q10. Dot products of a vector with vectors 3 î-5k, 2 î+7ĵ and î+ĵ+k are respectively -1, 6,5.
Find the vector.
OR
If a= î+ĵ+k and b= î-k, Find a c such that a x c=b and a.c = 3.
Q11. Show that the angle between two diagonals of a cube is cos-1(1/3)
Q12. Find the angle between the lines
r =3i-2j+6k+ (2i+j+2k) and r=2i-5k+µ(6i+3j+2k)
Q13. Find the distance of a point (2,4,-1) from the line
(x+5)/1=(y+3)/4=(z-6)/-9.
Q14. Find the coordinates of a point, where the (x-2)/3= (y+1)/4= (z-2)/2 intersect the plane
x-y+z-5=0. Also find the angle between the line and the plane
Q15. Find the equation of a plane passing through the points (2,1,-1) and (-1,3,4) and
perpendicular to the plane x-2y+4z=10.
Q16. If the points (1,1,p) and (-3,0,1) be equidistant from the plane r(3i+4j-12k)+13=0, then find
the value of p.
Q17. Find the coordinates of the point where the line through (5,1,6) and (3,4,1) cross the YZ
plane.
Q18. Show that the lines (x+3)/-3= (y-1)/1= (z-5)/5 and ( x+1)/-1= (y-2)/2= (z-5)/5 are coplanar.
OR
Find the value of p, so that the line (1-x)/3=(7y-14)/2p= (z-3)/2 and (7-7x)/3p= (y-5) /1=
(6-z)/5 are at right angles.
Q19. Find the shortest distance between the lines l1 and l2 whose vector equations are
r= i+j+λ(2i-j+k) and r=2i+j-k+µ(3i-5j+2k).
OR
Find the Cartesian as well as vector equation of the plane through the intersection of the
planes
r.(2i+6j)+12=0 and r.(3i-j+4k)=0 which are a unit distance from the origin.
Section (C)
Q20. If a,b,c determine the vertices of triangle so that ½[bxc+cxa+axb] gives the vector area of
the triangle. Hence deduce the condition that the three points a,b and c are collinear. Also,
find the unit vector normal to the plane of the triangle.
Q21. If a,b and c are three vectors such that a+b+c=0 then prove that axb = bxc = cxa.
OR
Show that (axb)² = a.a
a.b
a.b
b.b
Q 22. If a,b and c are three mutually perpendicular vectors of equal magnitude , Prove that a+b+c
is equally inclined with vector a,b,c.
Q23. A line makes angles α, β, γ and δ with the diagonals of a cube. Prove that
cos2α+cos2β+cos2 γ+cos2δ=4/3.
Q24. Find the shortest distance between the lines (x-8)/3= (y+9)/-16= (z-10)/7 and
(x-15)/3= (y-29)/8= (z-5)/-5.Also find the equation of shortest distance.
Q25. Find the equation of the plane passing through the line of intersections of the planes
r.(i+3j)-6=0 and r.(3i-j-4k)=0, whose perpendicular distance from the origin is unity.
Q26. Write the vector equation of the following lines and hence determine the distance between
them
(x-1)/2= (y-2)/3= (z+4)/6 and (x-3)/4= (y-3)/6= (z+5)/12.
OR
Find the vector equation of the plane passing through the points A(2,2,-1), B(3,4,2) and
C(7,0,6). Also find the cartesian equation of the plane.