B.Sc. (I YEAR) SEMESTER-II MATHEMATICS MODEL QUESTION PAPER ADIKAVI NANNAYA UNIVERSITY
https://www.computersprofessor.com/2017/04/bsc-i-year-semester-ii-mathematics.html
MATHEMATICS MODEL PAPER
SECOND SEMESTER – SOLID GEOME TRY
COMMON FOR B.A & B.Sc
(w.e.f. 2016-17 admitted batch)
Time: 3 Hours Maximum
Marks: 75
SECTION-A
Answer any FIVE questions. Each question carries FIVE marks.
5 x 5 = 25 Marks
1. Find
the equation of the plane through (4, 4, 0) and perpendicular to the planes x + 2y +2z = 5 and 3x + 3y + 2z – 8 =
0.
2. Find the image of a point (2, –1, 3)in the
plane 3x – 2y + z =9.
3. Find
the equation of the plane through the origin and containing the line x
– 3y +2z+3 = 0=3x–y+2z–5.
4. Find
the length of the perpendicular from the point (1, 2, 3) to the line through the point (6, 7, 7) whose d.r’s are 3, 2, –2.
5. Find
the equation to the sphere through O = (0, 0, 0) and making intercepts a, b, c on the axes.
6. Find the polar line of
w.r.t. the sphere 𝑥2 + 𝑦2 + 𝑧2 = 16.
7. Find
the equation to the cone which passes through the three coordinate axes as well as the three lines
and
8. Find
the enveloping cone of the sphere 𝑥2 + 𝑦2 + 𝑧2 + 2𝑥 − 2𝑦
= 2 with its vertex at (1, 1,
1).
SECTION-B
Answer the all FIVE
questions. Each carries TEN marks. 5 x 10 = 50 Marks
9(a). A variable plane is at a constant
distance ‘p’ from the origin and meets the coordinate
axes in A,B,C. Show that the locus of the centroid of the tetrahedron OABC is 𝑥−2 + 𝑦−2 + 𝑧−2 = 16𝑝−2
Or
(b). Find the
bisecting plane of the acute angle between the planes 3x–2y+6z = 0, –2x+y–2z–2 = 0.
10(a).
Find the S.D. between the lines
Or
(b) Prove that the lines
are coplanar. Also find their point of intersection.
11(a) Find the equations of the spheres
passing through the circle 𝑥2 + 𝑦2 = 4 , z
= 0 and is intersected by the plane x=+2y+2z = 0 in a circle of radius 3.
Or
(b) Show that the spheres 𝑥2 + 𝑦2 + 𝑧2 − 2𝑥 − 4𝑦
− 6𝑧 − 50 = 0 , 𝑥2 + 𝑦2 + 𝑧2 − 10𝑥 + 2𝑦
+ 18𝑧 + 82 = 0 touch externally at the point
12(a)
Find the limiting points of the coaxial system defined by spheres 𝑥2 + 𝑦2 + 𝑧2 + 4𝑥 − 2𝑦
+ 2𝑧 + 6 = 0 and 𝑥2 + 𝑦2 + 𝑧2 + 2𝑥 − 4𝑦
− 2𝑧 + 6 = 0
Or
(b) Find the equation of the lines of
intersection of the plane 2x+y–z=0 and the cone 4𝑥2 – 𝑦2 + 3𝑧2 = 0
13(a) Find the equation to the right circular cone whose vertex is
P(2, –3, 5), axis PQ which makes equal
angles with the axis and which passes through A(1,
–2, 3).
Or
(b) Find the equations of the tangent planes to the cone 9𝑥2 − 4𝑦2 + 16𝑧2 = 0 which
contains the line