These solutions designed by the ‘Basics in Maths’ team. These notes to do help the intermediate First-year Maths students.

Inter Maths – 1B two mark questions and solutions are very useful in IPE examinations.

These notes cover all the topics covered in the intermediate First-year Maths syllabus and include plenty of solutions to help you solve all the major types of Math problems asked in the IPE examinations.

# The Plane

**Question 1**

Find the equation of the plane if the foot of the perpendicular from the origin to the plane is (2, 3, – 5).

Sol:

The plane passes through A and perpendicular to OA, then the line segment OA is normal to the plane.

Dr’s of OA = (2, 3, – 5)

The equation of the plane passing through point (x_{1}, y_{1}, z_{1}) and dr’s (a, b, c) is

a(x – x_{1}) + b (y – y_{1}) + c (z – z_{1}) = 0

⟹ 2(x – 2) + 3 (y – 3) – 5 (z + 5) = 0

2x – 4 + 3y – 9 – 5z – 25 = 0

2x + 3y – 5z – 38 = 0

**Question 2**

Find the equation of the plane passing through the points (0, – 1, – 1), (4, 5, 1) and (3, 9, 4)

Sol:

The equation of the plane passing through the points (x_{1}, y_{1}, z_{1}) (x_{2}, y_{2}, z_{2}) (x_{3}, y_{3}, z_{3}) is

The plane passing through the points (0, – 1, – 1), (4, 5, 1) and (3, 9, 4) is

x (30 – 20) – (y + 1) (20 – 6) + (z + 1) (40 – 18) = 0

x (10) – (y + 1) (14) + (z + 1) (22) = 0

10x – 14y – 14 + 22z + 22 = 0

10x – 14y + 22z + 8 = 0

2(5x – 7y + 11z + 4) = 0

∴ the equation of the plane is 5x – 7y + 11z + 4 = 0

**Question 3**

Find the equation to the plane parallel to the ZX-plane and passing through (0, 4, 4).

Sol:

Equation of ZX-plane is y = 0

The equation of the plane parallel to the ZX-plane is y = k

But it is passing through (0, 4, 4)

⟹ y = 4

**Question 4**

Find the equation to the plane passing through the point (α, β, γ) and parallel to the plane axe + by + cz + d = 0.

Sol:

The equation of the plane parallel to the plane ax + by + cz + d = 0 is ax + by + cz + k = 0

But it is passing through the point (α, β, γ)

a α + b β + c γ + k = 0

⟹ k = – a α – b β – c γ

The equation of the plane is ax + by + cz – a α – b β – c γ = 0

⟹ a(x – α) + b (y – β)+ c (z – γ) = 0

**Question 5**

Find the angle between the planes 2x – y + z = 6 and x + y + 2z = 7.

Sol: If θ is the angle between the planes a_{1} x + b_{1} y + c_{1} z + d_{1} = 0 and a_{2}x + b_{2} y + c_{2} z + d_{2} = 0, then cos θ =

Cos θ = cos 60^{0}

θ = 60^{0} =

**Question 6**

Reduce the equation x + 2y – 2z – 9 = 0 to the normal form and hence find the dc’s of the normal to the plane.

Sol: Given plane is x + 2y – 2z – 9 = 0

x + 2y – 2z = 9

dc’s of the normal to the plane are

**Question 7**

Suppose a plane makes intercepts 2, 3, 4 on X, Y, Z axes respectively. Find the equation of the plane in the intercept form.

Sol: Given a = 2, b = 3, c = 4

The equation of the line in the intercept form is

**Question 8**

Express x – 3y + 2z = 9 in the intercept form

Sol: Given plane is x – 3y + 2z = 9

a = 9, b = – 3, c = 9/2

**Question 9**

Find the direction cosine of the normal to the plane x + 2y + 2z – 4 = 0.

Sol: Given plane is x + 2y + 2z – 4 = 0

We know that Dr’s of the normal to the plane ax + by + cz + d = 0 are (a, b, c)

⟹ dc’s of the normal to the plane =

⟹ dr’s of the normal to the plane x + 2y + 2z – 4 = 0 are (1, 2, 2)

⟹ dc’s of the normal to the plane are

**Question 10**

Find the midpoint of the line joining the points (1, 2, 3) and (–2, 4, 2)

Sol: Given points are A (1, 2, 3), B (–2, 4, 2)