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 (x₁, y₁, z₁) and dr’s (a, b, c) is

   a(x – x₁) + b (y – y₁) + c (z – z₁) = 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 (x₁, y₁, z₁) (x₂, y₂, z₂) (x₃, y₃, z₃) is

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

   =0

⟹  = 0

   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 a₁ x + b₁ y + c₁ z + d₁ = 0 and a₂x + b₂ y + c₂ z + d₂ = 0, then cos θ =

            Cos θ =

                        =

                          =

            Cos θ = cos 60⁰

                    θ = 60⁰ =

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

         dividing into both sides by 

         the normal form is

        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

       ⟹

      ⟹

       It is in the form of

      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)

       The midpoint of AB =

                                    =

                                    =

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