CBSE Class 11 Maths – MCQ and Online Tests – Unit 12 – Introduction to Three Dimensional Geometry
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CBSE Class 11 Maths – MCQ and Online Tests – Unit 12 – Introduction to Three Dimensional Geometry
Question 1.
The maximum distance between points (3sin ?, 0, 0) and (4cos ?, 0, 0) is
(a) 3
(b) 4
(c) 5
(d) Can not be find
Answer
Answer: (c) 5
Hint:
Given two points are (3sin ?, 0, 0) and (4cos ?, 0, 0)
Now distance = v{(4cos ? – 3sin ?)² + (0 – 0)² + (0 – 0)²}
? distance = v{(4cos ? – 3sin ?)²}
? distance = 4cos ? – 3sin ? …………….1
Now, maximum value of 4cos ? – 3sin ? = v{(4² + (-3)²}
= v(16 + 9)
= v25
= 5
From equation 1, we get
distance = 5
So, the maximum distance between points (3sin ?, 0, 0) and (4cos ?, 0, 0) is 5
Question 2.
The image of the point P(1, 3, 4) in the plane 2x – y + z = 0 is
(a) (-3, 5, 2)
(b) (3, 5, 2)
(c) (3, -5, 2)
(d) (3, 5, -2)
Answer
Answer: (a) (-3, 5, 2)
Hint:
Let image of the point P(1, 3, 4) is Q in the given plane.
The equation of the line through P and normal to the given plane is
(x-1)/2 = (y-3)/-1 = (z-4)/1
Since the line passes through Q, so let the coordinate of Q are (2r + 1, -r + 3, r + 4)
Now, the coordinate of the mid-point of PQ is
(r + 1, -r/2 + 3, r/2 + 4)
Now, this point lies in the given plane.
2(r + 1) – (-r/2 + 3) + (r/2 + 4) + 3 = 0
? 2r + 2 + r/2 – 3 + r/2 + 4 + 3 = 0
? 3r + 6 = 0
? r = -2
Hence, the coordinate of Q is (2r + 1, -r + 3, r + 4) = (-4 + 1, 2 + 3, -2 + 4)
= (-3, 5, 2)
Question 3.
If a, ß, ? are the angles made by a half ray of a line respectively with positive directions of X-axis Y-axis and Z-axis, then sin² a + sin² ß + sin² ? =
(a) 1
(b) 0
(c) -1
(d) None of these
Answer
Answer: (d) None of these
Hint:
Let l, m, n be the direction cosines of the given vector.
Then, a, ß, ?
l = cos a
m = cos ß
n = cos ?
Now, l² + m² + n² = 1
? cos² a + cos² ß + cos² ? = 1
? 1 – sin² a + 1 – sin² ß + 1 – sin² ? = 1
? 3 – sin² a – sin² ß – sin² ? = 1
? 3 – 1 = sin² a + sin² ß + sin² ?
? sin² a + sin² ß + sin² ? = 2
Question 4.
Three planes x + y = 0, y + z = 0, and x + z = 0
(a) none of these
(b) meet in a line
(c) meet in a unique point
(d) meet taken two at a time in parallel lines
Answer
Answer: (c) meet in a unique point
Hint:
Given, three planes are
x + y = 0 …….. 1
y + z = 0 …….. 2
and x + z = 0 ……… 3
add these planes, we get
2(x + y + z) = 0
? x + y + z = 0 ……… 4
From equation 1
0 + z = 0
? z = 0
From equation 2
x + 0 = 0
? x = 0
From equation 3
y + 0 = 0
? y = 0
So, (x, y, z) = (0, 0, 0)
Hence, the three planes meet in a unique point.
Question 5.
The coordinate of foot of perpendicular drawn from the point A(1, 0, 3) to the join of the point B(4, 7, 1) and C(3, 5, 3) are
(a) (5/3, 7/3, 17/3)
(b) (5, 7, 17)
(c) (5/3, -7/3, 17/3)
(d) (5/7, -7/3, -17/3)
Answer
Answer: (a) (5/3, 7/3, 17/3)
Hint:
Let D be the foot of perpendicular and let it divide BC in the ration m : 1
Then the coordinates of D are {(3m + 4)/(m + 1), (5m + 7)/(m + 1), (3m + 1)/(m + 1)}
Now, AD ? BC
? AD . BC = 0
? -(2m + 3) – 2(5m + 7) – 4 = 0
? m = -7/4
So, the coordinate of D are (5/3, 7/3, 17/3)
Question 6.
The locus of a point which moves so that the difference of the squares of its distances from two given points is constant, is a
(a) Straight line
(b) Plane
(c) Sphere
(d) None of these
Answer
Answer: (b) Plane
Hint:
Let the position vectors of the given points A and B be a and b respectively and that of the variable point be r.
Now, given that
PA² – PB² = k (constant)
? |AP|² – |BP|² = k
? |r – a|² – |r – b|² = k
? (|r|² + |a|² – 2r.a) – (|r|² + |b|² – 2r.b) = k
? 2r.(b – a) = k + |b|² – |a|²
? r.(b – a) = (k + |b|² – |a|²)/2
? r.(b – a) = C where C = (k + |b|² – |a|²)/2 = constant
So, it represents the equation of a plane.
Question 7.
The equation of the set of point P, the sum of whose distance from A(4, 0, 0) and B(-4, 0, 0) is equal to 10 is
(a) 9x² + 25y² + 25z² + 225 = 0
(b) 9x² + 25y² + 25z² – 225 = 0
(c) 9x² + 25y² – 25z² – 225 = 0
(d) 9x² – 25y² – 25z² – 225 = 0
Answer
Answer: (b) 9x² + 25y² + 25z² – 225 = 0
Hint:
Let the point P is (x, y, z)
Now given that
PA + PB = 10
? v{(x-4)² + y² + z²} + v{(x+4)² + y² + z²} = 10
? v{(x-4)² + y² + z²} = 10 – v{(x+4)² + y² + z²}
Now square both side
[v{(x-4)² + y² + z²}]² = (10)² + [{(x+4)² + y² + z²}]² – 2 ×10×v{(x+4)² + y² + z²}
? {(x-4)² + y² + z²} = 100 + {(x+4)² + y² + z²} – 20×v{(x+4)² + y² + z²}
? x² + 16 – 8x + y² + z² = 100 + x² + 16 + 8x + y² + z² – 20×v{(x+4)² + y² + z²}
? – 8x = 100 + 8x – 20×v{(x+4)² + y² + z²}
? -8x -8x – 100 = – 20×v{(x+4)² + y² + z²}
? -16x -100 = – 20×v{(x+4)² + y² + z²}
? 4x + 25 = 5×v{(x+4)² + y² + z²}
Again square both side,
(4x + 25)² = 25 ×[v{(x+4)² + y² + z²}]²
? 16x² + 625 + 200x = 25×{(x+4)² + y² + z²}
? 16x² + 625 + 200x = 25×(x² + 16 + 8x + y² + z²)
? 16x² + 625 + 200x = 25x² + 400 + 200x + 25y² + 25z²
? 25x² + 400 + 200x + 25y² + 25z² – 16x² – 625 – 200x = 0
? 9x² + 25y² + 25z² – 225 = 0
Question 8.
The cartesian equation of the line is 3x + 1 = 6y – 2 = 1 – z then its direction ratio are
(a) 1/3, 1/6, 1
(b) -1/3, 1/6, 1
(c) 1/3, -1/6, 1
(d) 1/3, 1/6, -1
Answer
Answer: (a) 1/3, 1/6, 1
Hint:
Given 3x + 1 = 6y – 2 = 1 – z
= (3x + 1)/1 = (6y – 2)/1 = (1 – z)/1
= (x + 1/3)/(1/3) = (y – 2/6)/(1/6) = (1 – z)/1
= (x + 1/3)/(1/3) = (y – 1/3)/(1/6) = (1 – z)/1
Now, the direction ratios are: 1/3, 1/6, 1
Question 9.
The image of the point P(1, 3, 4) in the plane 2x – y + z = 0 is
(a) (-3, 5, 2)
(b) (3, 5, 2)
(c) (3, -5, 2)
(d) (3, 5, -2)
Answer
Answer: (a) (-3, 5, 2)
Hint:
Let image of the point P(1, 3, 4) is Q in the given plane.
The equation of the line through P and normal to the given plane is
(x-1)/2 = (y-3)/-1 = (z-4)/1
Since the line passes through Q, so let the coordinate of Q are (2r + 1, -r + 3, r + 4)
Now, the coordinate of the mid-point of PQ is
(r + 1, -r/2 + 3, r/2 + 4)
Now, this point lies in the given plane.
2(r + 1) – (-r/2 + 3) + (r/2 + 4) + 3 = 0
? 2r + 2 + r/2 – 3 + r/2 + 4 + 3 = 0
? 3r + 6 = 0
? r = -2
Hence, the coordinate of Q is (2r + 1, -r + 3, r + 4) = (-4 + 1, 2 + 3, -2 + 4)
= (-3, 5, 2)
Question 10.
The plane 2x – (1 + a)y + 3az = 0 passes through the intersection of the planes
2x – y = 0 and y + 3z = 0
2x – y = 0 and y – 3z = 0
2x + 3z = 0 and y = 0
2x – 3z = 0 and y = 0
Answer
Answer: (d) A
Hint:
Given, equation of plane is:
2x – (1 + a)y + 3az = 0
=> (2x – y) + a(-y + 3z) = 0
which is passing through the intersection of the planes
2x – y = 0 and -y + 3z = 0
2x – y = 0 and y – 3z = 0
Question 11.
If the end points of a diagonal of a square are (1, -2, 3) and (2, -3, 5) then the length of the side of square is
(a) v3 unit
(b) 2v3 unit
(c) 3v3 unit
(d) 4v3 unit
Answer
Answer: (a) v3 unit
Hint:
Let a is the length of the side of a square.
Given, the diagonal of a square are (1,–2,3) and (2, -3, 5)
Now, length of the diagonal of square = v{(1 – 2)² + (-2 + 3)² + (3 – 5)²}
= v{1 + 1 + 4}
= v6
Again length of the diagonal of square is v2 times the length of side of the square.
? av2 = v6
? av2 = v3×v2
? a = v3
So, the length of side of square is v3 unit
Question 12.
The coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the YZ plane is
(a) (0, 17/2, 13/2)
(b) (0, -17/2, -13/2)
(c) (0, 17/2, -13/2)
(d) None of these
Answer
Answer: (c) (0, 17/2, -13/2)
Hint:
The line passing through the points (5, 1, 6) and (3, 4, 1) is given as
(x-5)/(3-5) = (y-1)/(4-1) = (z-6)/(1-6)
? (x-5)/(-2) = (y-1)/3 = (z-6)/(-5) = k(say)
? (x-5)/(-2) = k
? x – 5 = -2k
? x = 5 – 2k
(y-1)/3 = k
? y – 1 = 3k
? y = 3k + 1
and (z-6)/(-5) = k
? z – 6 = -5k
? z = 6 – 5k
Now, any point on the line is of the form (5 – 2k, 3k + 1, 6 – 5k)
The equation of YZ-plane is x = 0
Since the line passes through YZ-plane
So, 5 – 2k = 0
? k = 5/2
Now, 3k + 1 = 3 × 5/2 + 1 = 15/2 + 1 = 17/2
and 6 – 5k = 6 – 5×5/2 = 6 – 25/2 = -13/2
Hence, the required point is (0, 17/2, -13/2)
Question 13.
The angle between the vectors with direction ratios are 4, -3, 5 and 3, 4, 5 is
(a) p/2
(b) p/3
(c) p/4
(d) p/6
Answer
Answer: (b) p/3
Hint:
Let a is a vector parallel to the vector having direction ratio is 4, -3, 5
? a = 4i – 3j + 5k
Let b is a vector parallel to the vector having direction ratio is 3 ,4, 5
? b = 3i + 4j + 5k
Let ? be the angle between the given vectors.
Now, cos ? = (a . b)/(|a|×|b|)
? cos ? = (12 – 12 + 25)/{v(16 + 9 + 25)×v(9 + 16 + 25)}
? cos ? = 25/{v(50)×v(50)}
? cos ? = 25/50
? cos ? = 1/2
? cos ? = p/3
? ? = p/3
So, the angle between the vectors with direction ratios are 4, -3, 5 and 3, 4, 5 is p/3
Question 14.
The equation of plane passing through the point i + j + k and parallel to the plane r . (2i – j + 2k) = 5 is
(a) r . (2i – j + 2k) = 2
(b) r . (2i – j + 2k) = 3
(c) r . (2i – j + 2k) = 4
(d) r . (2i – j + 2k) = 5
Answer
Answer: (b) r . (2i – j + 2k) = 3
Hint:
The equation of plane parallel to the plane r . (2i – j + 2k) = 5 is
r . (2i – j + 2k) = d
Since it passes through the point i + j + k, therefore
(i + j + k) . (2i – j + 2k) = d
? d = 2 – 1 + 2
? d = 3
So, the required equation of the plane is
r . (2i – j + 2k) = 3
Question 15.
A vector r is equally inclined with the coordinate axes. If the tip of r is in the positive octant and |r| = 6, then r is
(a) 2v3(i – j + k)
(b) 2v3(-i + j + k)
(c) 2v3(i + j – k)
(d) 2v3(i + j + k)
Answer
Answer: (d) 2v3(i + j + k)
Hint:
Let l, m, n are DCs of r.
Given, l = m = n
? l² + m² + n² = 1
? 3l² = 1
? l² = 1/3
? l = m = n = 1/v3
So, r = |r|(li + mj + nk)
? r = 6(i/v3 + j/v3 + k/v3)
? r = 2v3(i + j + k)
Question 16.
If P(x, y, z) is a point on the line segment joining Q(2, 2, 4) and R(3, 5, 6) such that the projections of OP on the axes are 13/5, 19/5, 26/5 respectively, then P divides QR in the ration
(a) 1 : 2
(b) 3 : 2
(c) 2 : 3
(d) 1 : 3
Answer
Answer: (b) 3 : 2
Hint:
Since OP has projections 13/5, 19/5 and 26/5 on the coordinate axes, therefore
OP = 13i/5 + 19j/5 + 26/5k
Let P divides the join of Q(2, 2, 4) and R(3, 5, 6) in the ratio m : 1
Then the position vector of P is
{(3m + 2)/(m + 1), (5m + 2)/(m + 1), (6m + 4)/(m + 1)}
So, 13i/5 + 19j/5 + 26/5k = (3m + 2)/(m + 1)+ (5m + 2)/(m + 1)+ (6m + 4)/(m + 1)
? (3m + 2)/(m + 1) = 13/5
? 2m = 3
? m = 3/2
? m : 1 = 3 : 2
Hence, P divides QR in the ration 3 : 2
Question 17.
In a three dimensional space, the equation 3x – 4y = 0 represents
(a) a plane containing Y axis
(b) none of these
(c) a plane containing Z axis
(d) a plane containing X axis
Answer
Answer: (c) a plane containing Z axis
Hint:
Given, equation is 3x – 4y = 0
Here z = 0
So, the given equation 3x – 4y = 0 represents a plane containing Z axis.
Question 18.
The points on the y- axis which are at a distance of 3 units from the point (2, 3, -1) is
(a) either (0, -1, 0) or (0, -7, 0)
(b) either (0, 1, 0) or (0, 7, 0)
(c) either (0, 1, 0) or (0, -7, 0)
(d) either (0, -1, 0) or (0, 7, 0)
Answer
Answer: (d) either (0, -1, 0) or (0, 7, 0)
Hint:
Let the point on y-axis is O(0, y, 0)
Given point is A(2, 3, -1)
Given OA = 3
? OA² = 9
? (2 – 0)² + (3 – y)² + (-1 – 0)² = 9
? 4 + (3 – y)² + 1 = 9
? 5 + (3 – y)² = 9
? (3 – y)² = 9 – 5
? (3 – y)² = 4
? 3 – y = v4
? 3 – y = ±4
? 3 – y = 4 and 3 – y = -4
? y = -1, 7
So, the point is either (0, -1, 0) or (0, 7, 0)
Question 19.
The maximum distance between points (3sin ?, 0, 0) and (4cos ?, 0, 0) is
(a) 3
(b) 4
(c) 5
(d) Can not be find
Answer
Answer: (c) 5
Hint:
Given two points are (3sin ?, 0, 0) and (4cos ?, 0, 0)
Now distance = v{(4cos ? – 3sin ?)² + (0 – 0)² + (0 – 0)²}
? distance = v{(4cos ? – 3sin ?)²}
? distance = 4cos ? – 3sin ? ……………. 1
Now, maximum value of 4cos ? – 3sin ? = v{(4² + (-3)²}
= v(16 + 9)
= v25
= 5
From equation 1, we get
distance = 5
So, the maximum distance between points (3sin ?, 0, 0) and (4cos ?, 0, 0) is 5
Question 20.
A vector r is equally inclined with the coordinate axes. If the tip of r is in the positive octant and |r| = 6, then r is
(a) 2v3(i – j + k)
(b) 2v3(-i + j + k)
(c) 2v3(i + j – k)
(d) 2v3(i + j + k)
Answer
Answer: (d) 2v3(i + j + k)
Hint:
Let l, m, n are DCs of r.
Given, l = m = n
? l² + m² + n² = 1
? 3l² = 1
? l² = 1/3
? l = m = n = 1/v3
So, r = |r|(li + mj + nk)
? r = 6(i/v3 + j/v3 + k/v3)
? r = 2v3(i + j + k)
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