CAREER POINT
Fresher Course for IIT JEE (Main & Advanced)–2017
Course : Fresher(XL) Batch
DAILY PRACTICE PROBLEM SHEET
Subject : Mathematics
DPPS 1
Topic : Three Dimensional Geometry
Q.1
The direction cosines of two lines are related by l + m + n = 0 and al2 + bm2 + cn2 = 0. The lines are parallel
if (B) a–1 + b–1 + c–1 = 0
(D) None of these
(A) a + b + c = 0
(C) a = b = c
Q.2
The three planes 4y + 6z = 5, 2x + 3y + 5z = 5 and 6x + 5y + 9z = 10
(A) meet in a point
(B) have a line in common
(C) form a triangular prism
(D) None of these
Q.3
Let PQ and QR be diagonals of adjacent faces a rectangular box, with its centre at O. If ∠QOR, ∠ROP and
∠POQ are θ, φ and ψ respectively then the value of ‘cos θ + cos φ + cos ψ’ is
(A) –2
Q.4
(B) − 3
(C) –1
(D) 0
A plane passes through the point P(4, 0, 0) and Q(0, 0, 4) and is parallel to the y-axis. The distance of the
plane from the origin is (A) 2
(B) 4
(C)
2
(D) 2 2
Q.5
A variable plane forms a tetrahedron of constant volume 64 K3 with the coordinate planes and the origin,
then locus of the centroid of the tetrahedron is (A) x3 + y3 + z3 = 6K2 (B) xyz = 6K3
(C) x2 + y2 + z2 = 4K2 (D) x–2 + y–2 + z–2 = 4K–2
Q.6
A parallelopiped is formed by planes drawn through the points (1, 2, 3) and (9, 8, 5) parallel to the coordinate
planes then which of the following is not the length of an edge of this rectangular parallelopiped (A) 2
(B) 4
(C) 6
(D) 8
Q.7
Q.8
x − x1
y − y1
z − z1
=
=
is 0
1
2
(A) parallel to x-axis
(C) perpendicular to YOZ plane
The line
The line
(A) ±1
(B) perpendicular to x-axis
(D) parallel to y-axis
x−2
y +1
z −1
=
=
intersects the curve xy = c2, in xy plane if c is equal to 3
2
−1
(B) ±1/3
(C) ± 5
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(D) None of these
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Q.9
The line which contains all points (x, y, z) which are of the form (x, y, z) = (2, –2, 5) + λ(1, –3, 2) intersects
the plane 2x –3y + 4z = 163 at P and intersects the YZ plane at Q. If the distance PQ is a b where a, b ∈ N
and a > 3 then (a + b) equals (A) 23
(B) 95
(C) 27
(D) None of these
Q.10
The value of m for which straight line 3x – 2y + z + 3 = 0 = 4x – 3y + 4z + 1 is parallel to the plane
2x – y + mz – 2 = 0 is (A) –2
(B) 8
(C) –18
(D) 11
Q.11
If a line makes angles α1, α2, α3, α4 with diagonals of a cube, then4
(A)
∑ cos 2α
i
i =1
Q.12
=−
4
3
4
(B)
∑ sin 2α
i =1
i
=−
4
3
∑ cos
2
αi =
i =1
4
3
4
(D)
∑ sin
2
αi = −
i =1
4
3
Let A(1, 1, 1), B(2, 3, 5), C(–1, 0, 2) be three points, then equation of a plane parallel to the plane ABC
which is at a distance 2, is
(A) 2x – 3y + z + 2 14 = 0
(C) 3x – 3y + z + 2 = 0
Q.13
4
(C)
(B) 2x – 3y + z – 2 14 = 0
(D) 2x – 3y + z – 2 = 0
P is the foot of the perpendicular dropped from origin O on the line of intersection of the planes x – 2y + 3z = 5
and 2x + 3y + z + 4 = 0, then
(A) OP, î − 2 ĵ + 3k̂ and 2î + 3 ĵ + k̂ must be co-planar
(B) Equation of acute angle bisector of the two planes is x + 5y – 2z + 9 = 0.
(C) point (1, 2, 3) lies in acute region of the two planes.
(D) Equation of the plane perpendicular to the given line and passing through (1, 1, 1) is 11x – 5y – 7z + 1 = 0
Q.14
If three vectors PA = 4î − 2 ĵ + k̂ , PB = 2î − ĵ − k̂ and PC = 2î + k̂ are all drawn from the point P with
position vector î − ĵ , then
(A) volume of parallelopiped whose coterminous edges are the vectors PA, PB and PC is equal to 6.
(B) equation of plane containing the points A,B,C is 2x + 2y – z = 3.
3
(C) the distance of plane containing points A,B,C from (0, 0, 0) is
2
(D) volume of tetrahedron OABC (where O is origin) equals 1.
ANSWERS :
1. (B)
2. (B)
3. (C)
4. (D)
5. (B)
6. (B)
7. (B)
8. (C)
9. (A)
10. (A)
11. (A,C)
12. (A,B)
13. (A,B,C,D)
14. (A,B,D)
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