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# Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 4 - Pair of Straight Lines [Latest edition]

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## Chapter 4: Pair of Straight Lines

Exercise 4.1Exercise 4.2Exercise 4.3Miscellaneous Exercise 4
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Exercise 4.1 [Pages 119 - 120]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 4 Pair of Straight LinesExercise 4.1 [Pages 119 - 120]

Exercise 4.1 | Q 1.1 | Page 119

Find the combined equation of the following pair of line:

2x + y = 0 and 3x - y = 0

Exercise 4.1 | Q 1.2 | Page 119

Find the combined equation of the following pair of line:

x + 2y - 1 = 0 and x - 3y + 2 = 0

Exercise 4.1 | Q 1.3 | Page 119

Find the combined equation of the following pair of lines passing through (2, 3) and parallel to the coordinate axes.

Exercise 4.1 | Q 1.4 | Page 119

Find the combined equation of the following pair of line:

passing through (2, 3) and perpendicular to the lines 3x + 2y - 1 = 0 and x - 3y + 2 = 0

Exercise 4.1 | Q 1.5 | Page 119

Find the combined equation of the following pair of line passing through (−1, 2), one is parallel to x + 3y − 1 = 0 and other is perpendicular to 2x − 3y − 1 = 0

Exercise 4.1 | Q 2.1 | Page 119

Find the separate equation of the line represented by the following equation:

3y2 + 7xy = 0

Exercise 4.1 | Q 2.2 | Page 119

Find the separate equation of the line represented by the following equation:

5y2 + 9y2 = 0

Exercise 4.1 | Q 2.3 | Page 119

Find the separate equation of the line represented by the following equation:

x2 - 4xy = 0

Exercise 4.1 | Q 2.4 | Page 119

Find the separate equations of the lines represented by the equation  3"x"^2-10"xy"-8"y"^2=0

Exercise 4.1 | Q 2.5 | Page 119

Find the separate equation of the line represented by the following equation:

3"x"^2 - 2sqrt3"xy" - 3"y"^2 = 0

Exercise 4.1 | Q 2.6 | Page 119

Find the separate equation of the line represented by the following equation:

x2 + 2(cosec α)xy + y2 = 0

Exercise 4.1 | Q 2.7 | Page 119

Find the separate equation of the line represented by the following equation:

x2 + 2xy tan α - y2 = 0

Exercise 4.1 | Q 3.1 | Page 119

Find the combined equation of the pair of a line passing through the origin and perpendicular to the line represented by following equation:

5x2 - 8xy + 3y2 = 0

Exercise 4.1 | Q 3.2 | Page 119

Find the combined equation of the pair of a line passing through the origin and perpendicular to the line represented by the following equation:

5x2 + 2xy - 3y2 = 0

Exercise 4.1 | Q 3.3 | Page 119

Find the combined equation of the pair of a line passing through the origin and perpendicular to the line represented by the following equation:

xy + y2 = 0

Exercise 4.1 | Q 3.4 | Page 119

Find the combined equation of the pair of a line passing through the origin and perpendicular to the line represented by the following equation:

3x2 -  4xy = 0

Exercise 4.1 | Q 4.1 | Page 119

Find k, if the sum of the slopes of the lines represented by x2 + kxy − 3y2 = 0 is twice their product.

Exercise 4.1 | Q 4.2 | Page 119

Find k, the slopes of the lines represented by 3x2 + kxy - y2 = 0 differ by 4.

Exercise 4.1 | Q 4.3 | Page 119

Find k, the slope of one of the lines given by kx2 + 4xy - y2 = 0 exceeds the slope of the other by 8.

Exercise 4.1 | Q 5.1 | Page 120

Find the condition that the line 4x + 5y = 0 coincides with one of the lines given by ax2 + 2hxy + by2 = 0

Exercise 4.1 | Q 5.2 | Page 120

Find the condition that the line 3x + y = 0 may be perpendicular to one of the lines given by ax2 + 2hxy + by2 = 0

Exercise 4.1 | Q 6 | Page 120

If one of the lines given by ax2 + 2hxy + by2 = 0 is perpendicular to px + qy = 0, show that ap2 + 2hpq + bq2 = 0.

Exercise 4.1 | Q 7 | Page 120

Find the combined equation of the pair of lines through the origin and making an equilateral triangle with the line y = 3.

Exercise 4.1 | Q 8 | Page 120

If the slope of one of the lines given by ax2 + 2hxy + by2 = 0 is four times the other, show that 16h2 = 25ab.

Exercise 4.1 | Q 9 | Page 120

If one of the lines given by ax2 + 2hxy + by2 = 0 bisect an angle between the coordinate axes, then show that (a + b)2 = 4h2 .

Exercise 4.2 [Page 124]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 4 Pair of Straight LinesExercise 4.2 [Page 124]

Exercise 4.2 | Q 1 | Page 124

. Show that the lines represented by 3x2 - 4xy - 3y2 = 0 are perpendicular to each other.

Exercise 4.2 | Q 2 | Page 124

Show that the lines represented by x2 + 6xy + 9y2 = 0 are coincident.

Exercise 4.2 | Q 3 | Page 124

Find the value of k if lines represented by kx2 + 4xy - 4y2 = 0 are perpendicular to each other.

Exercise 4.2 | Q 4.1 | Page 124

Find the measure of the acute angle between the line represented by 3"x"^2 - 4sqrt3"xy" + 3"y"^2 = 0

Exercise 4.2 | Q 4.2 | Page 124

Find the measure of the acute angle between the line represented by:

4x2 + 5xy + y2 = 0

Exercise 4.2 | Q 4.3 | Page 124

Find the measure of the acute angle between the line represented by:

2x2 + 7xy + 3y2 = 0

Exercise 4.2 | Q 4.4 | Page 124

Find the measure of the acute angle between the line represented by:

(a2 + 3b2)x2 + 8abxy + (b2 - 3a2)y2 = 0

Exercise 4.2 | Q 5 | Page 124

Find the combined equation of lines passing through the origin each of which making an angle of 30° with the line 3x + 2y - 11 = 0

Exercise 4.2 | Q 6 | Page 124

If the angle between the lines represented by ax2 + 2hxy + by2 = 0 is equal to the angle between the lines 2x2 - 5xy + 3y2 = 0, then show that 100 (h2 - ab) = (a + b)2

Exercise 4.2 | Q 7 | Page 124

Find the combined equation of lines passing through the origin and each of which making an angle of 60° with the Y-axis.

Exercise 4.3 [Pages 127 - 128]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 4 Pair of Straight LinesExercise 4.3 [Pages 127 - 128]

Exercise 4.3 | Q 1.1 | Page 127

Find the joint equation of the pair of the line through the point (2, -1) and parallel to the lines represented by 2x2 + 3xy - 9y2 = 0.

Exercise 4.3 | Q 1.2 | Page 127

Find the joint equation of the pair of the line through the point (2, -3) and parallel to the lines represented by x2 + xy - y2 = 0.

Exercise 4.3 | Q 2 | Page 127

Show that the equation x2 + 2xy + 2y2 + 2x + 2y + 1 = 0 does not represent a pair of lines.

Exercise 4.3 | Q 3 | Page 127

Show that the equation 2x2 - xy - 3y2 - 6x + 19y - 20 = 0 represents a pair of lines.

Exercise 4.3 | Q 4 | Page 127

Show that the equation 2x2 + xy - y2 + x + 4y - 3 = 0 represents a pair of lines. Also, find the acute angle between them.

Exercise 4.3 | Q 5.1 | Page 127

Find the separate equation of the line represented by the following equation:

(x - 2)2 - 3(x - 2)(y + 1) + 2(y + 1)2 = 0

Exercise 4.3 | Q 5.2 | Page 127

Find the separate equation of the line represented by the following equation:

10(x + 1)2 + (x + 1)(y - 2) - 3(y - 2)2 = 0

Exercise 4.3 | Q 6.1 | Page 127

Find the value of k, if the following equations represent a pair of line:

3x2 + 10xy + 3y2 + 16y + k = 0

Exercise 4.3 | Q 6.2 | Page 127

Find the value of k, if the following equations represent a pair of line:

kxy + 10x + 6y + 4 = 0

Exercise 4.3 | Q 6.3 | Page 127

Find the value of k, if the following equations represent a pair of line:

x2 + 3xy + 2y2 + x - y + k = 0

Exercise 4.3 | Q 7 | Page 128

Find p and q, if the equation px2 - 8xy + 3y2 + 14x + 2y + q = 0 represents a pair of perpendicular lines.

Exercise 4.3 | Q 8 | Page 128

Find p and q, if the equation 2x2 + 8xy + py2 + qx + 2y - 15 = 0 represents a pair of parallel lines.

Exercise 4.3 | Q 9 | Page 128

Equations of pairs of opposite sides of a parallelogram are x2 - 7x + 6 = 0 and y2 - 14y + 40 = 0. Find the joint equation of its diagonals.

Exercise 4.3 | Q 10 | Page 128

ΔOAB is formed by the lines x2 - 4xy + y2 = 0 and the line AB. The equation of line AB is 2x + 3y - 1 = 0. Find the equation of the median of the triangle drawn from O.

Exercise 4.3 | Q 11 | Page 128

Find the coordinates of the points of intersection of the lines represented by x2 - y2 - 2x + 1 = 0

Miscellaneous Exercise 4 [Pages 129 - 130]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 4 Pair of Straight LinesMiscellaneous Exercise 4 [Pages 129 - 130]

Miscellaneous Exercise 4 | Q 1.01 | Page 129

Choose correct alternatives:

If the equation 4x2 + hxy + y2 = 0 represents two coincident lines, then h = _______

• ± 2

• ± 3

• ± 4

• ± 5

Miscellaneous Exercise 4 | Q 1.02 | Page 129

If the lines represented by kx2 − 3xy + 6y2 = 0 are perpendicular to each other, then

• k = 6

• k = − 6

• k = 3

• k = − 3

Miscellaneous Exercise 4 | Q 1.03 | Page 129

Choose correct alternatives:

Auxiliary equation of 2x2 + 3xy - 9y2 = 0 is

• 2m2 + 3m - 9 = 0

• 9m2 - 3m - 2 = 0

• 2m2 - 3m + 9 = 0

• - 9m2 - 3m + 2 = 0

Miscellaneous Exercise 4 | Q 1.04 | Page 129

Choose correct alternatives:

The difference between the slopes of the lines represented by 3x2 - 4xy + y2 = 0 is 2

• 2

• 1

• 3

• 4

Miscellaneous Exercise 4 | Q 1.05 | Page 129

Choose correct alternatives:

If two lines ax2 + 2hxy + by2 = 0 make angles α and β with X-axis, then tan (α + β) = _____.

• "h"/("a + b")

• "h"/("a - b")

• "2h"/("a + b")

• "2h"/("a - b")

Miscellaneous Exercise 4 | Q 1.06 | Page 129

Choose correct alternatives:

If the slope of one of the two lines given by "x"^2/"a" + "2xy"/"h" + "y"^2/"b" = 0 is twice that of the other, then ab : h2 = ______.

• 1 : 2

• 2 : 1

• 8 : 9

• 9 : 8

Miscellaneous Exercise 4 | Q 1.07 | Page 130

Choose correct alternatives:

The joint equation of the lines through the origin and perpendicular to the pair of lines 3x2 + 4xy - 5y2 = 0 is _______.

• 5x2 + 4xy - 3y2 = 0

• 3x2 + 4xy - 5y2 = 0

• 3x2 - 4xy + 5y2 = 0

• 5x2 + 4xy + 3y2 = 0

Miscellaneous Exercise 4 | Q 1.08 | Page 130

Choose correct alternatives:

If acute angle between lines ax2 + 2hxy + by2 = 0 is, pi/4, then 4h2 = ______.

• a2 + 4ab + b2

• a2 + 6ab + b2

• (a + 2b)(a + 3b)

• (a - 2b)(2a + b)

Miscellaneous Exercise 4 | Q 1.09 | Page 130

Choose correct alternatives:

If the equation 3x2 - 8xy + qy2 + 2x + 14y + p = 1 represents a pair of perpendicular lines, then the values of p and q are respectively.

• - 3 and - 7

• - 7 and - 3

• 3 and 7

• - 7 and 3

Miscellaneous Exercise 4 | Q 1.1 | Page 130

Choose correct alternatives:

The area of triangle formed by the lines x2 + 4xy + y2 = 0 and x - y - 4 = 0 is

• 4/sqrt3 sq units

• 8/sqrt3 sq units

• 16/sqrt3 sq units

• 15/sqrt3 sq units

Miscellaneous Exercise 4 | Q 1.11 | Page 130

Choose correct alternatives:

The combined equation of the coordinate axes is

• x + y = 0

• xy = k

• xy = 0

• x - y = k

Miscellaneous Exercise 4 | Q 1.12 | Page 130

Choose correct alternatives:

If h2 = ab, then slopes of lines ax2 + 2hxy + by2 = 0 are in the ratio

• 1:2

• 2:1

• 2:3

• 1:1

Miscellaneous Exercise 4 | Q 1.13 | Page 130

Choose correct alternatives:

If slope of one of the lines ax2 + 2hxy + by2 = 0 is 5 times the slope of the other, then 5h2 = ______

• ab

• 2ab

• 7ab

• 9ab

Miscellaneous Exercise 4 | Q 1.14 | Page 130

Choose correct alternatives:

If distance between lines (x - 2y)2 + k(x - 2y) = 0 is 3 units, then k = ______.

• ± 3

• ± 5sqrt5

• 0

• ±3sqrt5

Miscellaneous Exercise 4 [Pages 130 - 132]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 4 Pair of Straight LinesMiscellaneous Exercise 4 [Pages 130 - 132]

Miscellaneous Exercise 4 | Q 1.01 | Page 130

Find the joint equation of the line:

x - y = 0 and x + y = 0

Miscellaneous Exercise 4 | Q 1.02 | Page 130

Find the joint equation of the line:

x + y - 3 = 0 and 2x + y - 1 = 0

Miscellaneous Exercise 4 | Q 1.03 | Page 130

Find the joint equation of the line passing through the origin having slopes 2 and 3.

Miscellaneous Exercise 4 | Q 1.04 | Page 130

Find the joint equation of the line passing through the origin and having inclinations 60° and 120°.

Miscellaneous Exercise 4 | Q 1.05 | Page 130

Find the joint equation of the line passing through (1, 2) and parallel to the coordinate axes

Miscellaneous Exercise 4 | Q 1.06 | Page 130

Find the joint equation of the line passing through (3, 2) and parallel to the lines x = 2 and y  = 3.

Miscellaneous Exercise 4 | Q 1.07 | Page 131

Find the joint equation of the line passing through (-1, 2) and perpendicular to x + 2y + 3 = 0 and 3x - 4y - 5 = 0

Miscellaneous Exercise 4 | Q 1.08 | Page 131

Find the joint equation of the line passing through the origin and having slopes 1 + sqrt3 and 1 - sqrt3

Miscellaneous Exercise 4 | Q 1.09 | Page 131

Find the joint equation of the line which are at a distance of 9 units from the Y-axis.

Miscellaneous Exercise 4 | Q 1.1 | Page 131

Find the joint equation of the line passing through the point (3, 2), one of which is parallel to the line x - 2y = 2, and other is perpendicular to the line y = 3.

Miscellaneous Exercise 4 | Q 1.11 | Page 131

Find the joint equation of the line passing through the origin and perpendicular to the lines x + 2y = 19 and 3x + y = 18

Miscellaneous Exercise 4 | Q 2.1 | Page 131

Show that the following equations represents a pair of line:

x2 + 2xy - y2 = 0

Miscellaneous Exercise 4 | Q 2.2 | Page 131

Show that the following equations represents a pair of line:

4x2 + 4xy + y2 = 0

Miscellaneous Exercise 4 | Q 2.3 | Page 131

Show that the following equations represent a pair of line:

x2 - y2 = 0

Miscellaneous Exercise 4 | Q 2.4 | Page 131

Show that the following equations represent a pair of line:

x2 + 7xy - 2y2 = 0

Miscellaneous Exercise 4 | Q 2.5 | Page 131

Show that the following equations represent a pair of line:

"x"^2 - 2sqrt3"xy" - "y"^2 = 0

Miscellaneous Exercise 4 | Q 3.1 | Page 131

Find the separate equation of the line represented by the following equation:

6x2 - 5xy - 6y2 = 0

Miscellaneous Exercise 4 | Q 3.2 | Page 131

Find the separate equation of the line represented by the following equation:

x2 - 4y2 = 0

Miscellaneous Exercise 4 | Q 3.3 | Page 131

Find the separate equation of the line represented by the following equation:

3x2 - y2 = 0

Miscellaneous Exercise 4 | Q 3.4 | Page 131

Find the separate equation of the line represented by the following equation:

2x2 + 2xy - y2 = 0

Miscellaneous Exercise 4 | Q 4.1 | Page 131

Find the joint equation of the pair of a line through the origin and perpendicular to the lines given by

x2 + 4xy - 5y2 = 0

Miscellaneous Exercise 4 | Q 4.2 | Page 131

Find the joint equation of the pair of a line through the origin and perpendicular to the lines given by

2x2 - 3xy - 9y2 = 0

Miscellaneous Exercise 4 | Q 4.3 | Page 131

Find the joint equation of the pair of a line through the origin and perpendicular to the lines given by

x2 + xy - y2 = 0

Miscellaneous Exercise 4 | Q 5.1 | Page 131

Find k, if the sum of the slopes of the lines given by 3x2 + kxy - y2 = 0 is zero.

Miscellaneous Exercise 4 | Q 5.2 | Page 131

Find k, if the sum of the slopes of the lines given by x2 + kxy - 3y2 = 0 is equal to their product.

Miscellaneous Exercise 4 | Q 5.3 | Page 131

Find k, if the slope of one of the lines given by 3x2 - 4xy + ky2 = 0 is 1.

Miscellaneous Exercise 4 | Q 5.4 | Page 131

Find k, if one of the lines given by 3x2 - kxy + 5y2 = 0 is perpendicular to the line 5x + 3y = 0.

Miscellaneous Exercise 4 | Q 5.5 | Page 131

Find k if the slope of one of the lines given by 3x2 + 4xy + ky2 = 0 is three times the other.

Miscellaneous Exercise 4 | Q 5.6 | Page 131

Find k, if the slopes of lines given by kx2 + 5xy + y2 = 0 differ by 1.

Miscellaneous Exercise 4 | Q 5.7 | Page 131

Find k, if one of the lines given by 6x2 + kxy + y2 = 0 is 2x + y = 0.

Miscellaneous Exercise 4 | Q 6 | Page 131

Find the joint equation of the pair of lines which bisect angles between the lines given by x2 + 3xy + 2y2 = 0

Miscellaneous Exercise 4 | Q 7 | Page 131

Find the joint equation of the pair of lines through the origin and making an equilateral triangle with the line x = 3.

Miscellaneous Exercise 4 | Q 8 | Page 131

Show that the lines x2 - 4xy + y2 = 0 and x + y = 10 contain the sides of an equilateral triangle. Find the area of the triangle.

Miscellaneous Exercise 4 | Q 9 | Page 131

If the slope of one of the lines given by ax2 + 2hxy + by2 = 0 is three times the other, prove that 3h2 = 4ab.

Miscellaneous Exercise 4 | Q 10 | Page 132

Find the combined equation of bisectors of angles between the lines represented by 5x2 + 6xy - y2 = 0.

Miscellaneous Exercise 4 | Q 11 | Page 132

Find an if the sum of the slope of lines represented by ax2 + 8xy + 5y2 = 0 is twice their product.

Miscellaneous Exercise 4 | Q 12 | Page 132

If the line 4x - 5y = 0 coincides with one of the lines given by ax2 + 2hxy + by2 = 0, then show that 25a + 40h + 16b = 0

Miscellaneous Exercise 4 | Q 13.1 | Page 132

Show that the following equation represents a pair of line. Find the acute angle between them:

9x2 - 6xy + y2 + 18x - 6y + 8 = 0

Miscellaneous Exercise 4 | Q 13.2 | Page 132

Show that the following equation represents a pair of line. Find the acute angle between them:

2x2 + xy - y2 + x + 4y - 3 = 0

Miscellaneous Exercise 4 | Q 13.3 | Page 132

Show that the following equation represents a pair of line. Find the acute angle between them:

(x - 3)2 + (x - 3)(y - 4) - 2(y - 4)2 = 0

Miscellaneous Exercise 4 | Q 14 | Page 132

Find the combined equation of lines passing through the origin and each of which making an angle of 60° with the Y-axis.

Miscellaneous Exercise 4 | Q 15 | Page 132

If the lines represented by ax2 + 2hxy + by2 = 0 make angles of equal measure with the coordinate axes, then show that a ± b.

OR

Show that, one of the lines represented by ax2 + 2hxy + by2 = 0 will make an angle of the same measure with the X-axis as the other makes with the Y-axis, if a = ± b.

Miscellaneous Exercise 4 | Q 16 | Page 132

Show that the combined equation of the pair of lines passing through the origin and each making an angle α with the line x + y = 0 is x2 + 2(sec 2α)xy + y2 = 0

Miscellaneous Exercise 4 | Q 17 | Page 132

Show that the line 3x + 4y + 5 = 0 and the lines (3x + 4y)2 - 3(4x - 3y)2 = 0 form the sides of an equilateral triangle.

Miscellaneous Exercise 4 | Q 18 | Page 132

Show that the lines x2 - 4xy + y2 = 0 and the line x + y = sqrt6 form an equilateral triangle. Find its area and perimeter.

Miscellaneous Exercise 4 | Q 19 | Page 132

If the slope of one of the lines given by ax2 + 2hxy + by2 = 0 is square of the slope of the other line, show that a2b + ab2 + 8h3 = 6abh.

Miscellaneous Exercise 4 | Q 20 | Page 132

Prove that the product of length of perpendiculars drawn from P(x1, y1) to the lines represented by ax2 + 2hxy + by2 = 0 is |("ax"_1^2 + "2hx"_1"y"_1 + "by"_1^2)/(sqrt("a - b")^2 + "4h"^2)|

Miscellaneous Exercise 4 | Q 21 | Page 132

Show that the difference between the slopes of the lines given by (tan2θ + cos2θ)x2 - 2xy tan θ + (sin2θ)y2 = 0 is two.

Miscellaneous Exercise 4 | Q 22 | Page 132

Find the condition that the equation ay2 + bxy + ex + dy = 0 may represent a pair of lines.

Miscellaneous Exercise 4 | Q 23 | Page 132

If the lines given by ax2 + 2hxy + by2 = 0 form an equilateral triangle with the line lx + my = 1, show that (3a + b)(a + 3b) = 4h2.

Miscellaneous Exercise 4 | Q 24 | Page 132

If the line x + 2 = 0 coincides with one of the lines represented by the equation x2 + 2xy + 4y + k = 0, then prove that k = - 4.

Miscellaneous Exercise 4 | Q 25 | Page 132

Prove that the combined of the pair of lines passing through the origin and perpendicular to the lines ax2 + 2hxy + by2 = 0 is bx2 - 2hxy + ay2 = 0.

Miscellaneous Exercise 4 | Q 26 | Page 132

If equation ax2 - y2 + 2y + c = 1 represents a pair of perpendicular lines, then find a and c.

## Chapter 4: Pair of Straight Lines

Exercise 4.1Exercise 4.2Exercise 4.3Miscellaneous Exercise 4 ## Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 4 - Pair of Straight Lines

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 4 (Pair of Straight Lines) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 4 Pair of Straight Lines are Combined Equation of a Pair Lines, Homogeneous Equation of Degree Two, Angle Between Lines, General Second Degree Equation, Equation of a Line in Space.

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