planes

The Plane vsaqs questions and solutions

The Plane 2m Questions & Solutions || V.S.A.Q’S||

Education /

The Plane 2m Questions & Solutions || V.S.A.Q’S|| The Plane 2m Questions: The Plane: These solutions were designed by the ‘Basics in Maths’ team. These notes to do help 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 is 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 the 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 The Plane 2m Questions 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 The Plane 2m Questions 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 =                                     =                                     = Visit my Youtube Channel: Click on Below Logo

The Plane 2m Questions & Solutions || V.S.A.Q’S|| Read More »

TS Inter Maths 1B Concept

TS Inter Maths 1B Concept

Education, TS Inter Maths, TS Inter Maths1B /

TS Inter Maths 1B Concept   Ts Inter Maths 1B Concept:  designed by the ‘Basics in Maths’ team. These notes to do help the TS intermediate first-year Maths students fall in love with mathematics and overcome the fear. These notes cover all the topics covered in the TS I.P.E  first year maths 1B syllabus and include plenty of formulae and concept to help you solve all the types of Inter Math problems asked in the I.P.E and entrance examinations. 0.COORDINATE GEOMETRY( BASICS) Distance between two points A(x1, y1), B(x2, y2) is  distance between a point A(x1, y1) to the origin is The midpoint of two points A(x1, y1), B(x2, y2) is     If P divides the line segment joining the points A(x1, y1), B(x2, y2) in the ratio m:n then the coordinates of P are Area of the triangle formed by the vertices A (x1, y1), B (x2, y2) and C (x3, y3) is 1. LOCUS    Locus: The set of points that are satisfying a given condition or property is called the locus of the point. Ex:- If a point P is equidistant from the points A and B, then AP =BP Ex 2: – set of points that are at a constant distance from a fixed point. here the locus of a point is a circle. • In a right-angled triangle PAB, the right angle at P and P is the locus of the point, then AB2 = PA2 + PB2 •Area of the triangle formed by the vertices A (x1, y1), B (x2, y2), and C (x3, y3) is 2.CHANGE OF AXES   Transformation of axes: When  the origin is shifted to  (h, k), without changing the direction of axes then •To remove the first degree terms of the equation ax2  + 2hxy + by2 +2gx +2fy+ c = 0, origin should be shifted to the point    •If the equation ax2 + by2 +2gx +2fy+ c = 0, origin should be shifted to the point  Rotation of axes: When the  axes are rotated through an angle θ then •To remove the xy term of the equation ax2 + 2hxy + by2  = 0, axes should be rotated through an angle θ is given by  3.STRAIGHT LINES   Slope:-  A-line makes an angle θ with the positive direction of the X-axis, then tan θ is called the slope of the line.               It is denoted by “m”.   m= tan θ • The slope of the x-axis is zero. • Slope of any line parallel to the x-axis is zero. • The y-axis slope is undefined. • The slope of any line parallel to the y-axis is also undefined. • The slope of the line joining the points A (x1, y1) and B (x2, y2) is Slope of the line ax + by + c = 0 is  Types of the equation of a straight line: Equation of x- axis is y = 0. Equation of any line parallel to the x-axis is y = k, where k is the distance from above or below the x-axis. Equation of y- axis is x = 0. Equation of any line parallel to y-axis is x = k, where k is the distance from the left or right side of the y-axis. Slope- intercept form The equation of the line with slope m and y-intercept c is y = mx + c. Slope point form: The equation of the line passing through the point (x1, y1) with slope m is y – y1 = m (x – x1) Two points form: The equation of the line passing through the points (x1, y1) and (x2, y2) ’ is Intercept form: The equation of the line with x-intercept a, y-intercept b is • The equation of the line ∥ el    to ax +by + c = 0 is ax +by + k = 0. • The equation of the line ⊥ler   to ax +by + c = 0 is bx −ay + k = 0. Note: – If two lines are parallel then their slopes are equal m1 = m2 If two lines are perpendicular then product of their slopes is – 1 m1 × m2 = – 1 The area of the triangle formed by the line ax + by + c = 0 with the coordinate axes is The area of the triangle formed by the line   with the coordinate axes is  Perpendicular distance (Length of the perpendicular): The perpendicular distance from a point P (x1, y1) to the line ax + by + c = 0 is • The perpendicular distance from origin to the line ax + by + c = 0 is  Distance between two parallel lines: •The distance between the parallel lines ax1 + by1 + c1 = 0 and ax2 + by2 + c2 = 0 is Perpendicular form or Normal form:     The equation of the line which is at a distance of ‘p’ from the origin and α (0≤ α ≤ 3600) is the angle made by the perpendicular with the positive direction of the x-axis is x cosα + y sinα = p.       Symmetric form: The equation of the line passing through point P (x1, y1) and having inclination θ is Parametric form: if P (x, y) is any point on the line passing through A (x1, y1) and making inclination θ, then                                                                                              x = x1 + r cos θ, y = y1 + r sin θ where ‘r’  is the distance from P to A. • The ratio in which the line L ≡ ax + by + c = 0 divide the line segment joining the points A (x1, y1), B (x2, y2) is

TS Inter Maths 1B Concept Read More »

Scroll to Top