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Exercise 4.3 - Chapter 4 Determinants class 12 ncert solutions Maths - SaraNextGen [2024]


Question 1:

Find area of the triangle with vertices at the point given in each of the following:

(i) (1, 0), (6, 0), (4, 3) (ii) (2, 7), (1, 1), (10, 8)

(iii) (−2, −3), (3, 2), (−1, −8)

Answer:

(i) The area of the triangle with vertices (1, 0), (6, 0), (4, 3) is given by the relation,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6480/Chapter%204_html_27997392.gif

(ii) The area of the triangle with vertices (2, 7), (1, 1), (10, 8) is given by the relation,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6480/Chapter%204_html_1ae5503b.gif

(iii) The area of the triangle with vertices (−2, −3), (3, 2), (−1, −8)

is given by the relation,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6480/Chapter%204_html_m477135a4.gif

Hence, the area of the triangle ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6480/Chapter%204_html_mbd4265a.gif .

Question 2:

Show that points

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6481/Chapter%204_html_m6ee10de5.gif are collinear

Answer:

Area of ΔABC is given by the relation,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6481/Chapter%204_html_4574753c.gif

Thus, the area of the triangle formed by points A, B, and C is zero.

Hence, the points A, B, and C are collinear.

Question 3:

Find values of k if area of triangle is 4 square units and vertices are

(i) (k, 0), (4, 0), (0, 2) (ii) (−2, 0), (0, 4), (0, k)

Answer:

We know that the area of a triangle whose vertices are (x1y1), (x2y2), and

(x3y3) is the absolute value of the determinant (Δ), where

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6486/Chapter%204_html_m7d88ee23.gif

It is given that the area of triangle is 4 square units.

∴Δ = ± 4.

(i) The area of the triangle with vertices (k, 0), (4, 0), (0, 2) is given by the relation,

Δ =https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6486/Chapter%204_html_5c597026.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6486/Chapter%204_html_5271de10.gif

k + 4 = ± 4

When −k + 4 = − 4, k = 8.

When −k + 4 = 4, k = 0.

Hence, k = 0, 8.

(ii) The area of the triangle with vertices (−2, 0), (0, 4), (0, k) is given by the relation,

Δ =https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6486/Chapter%204_html_48f576c9.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6486/Chapter%204_html_19a19c67.gif

k − 4 = ± 4

When k − 4 = − 4, k = 0.

When k − 4 = 4, k = 8.

Hence, k = 0, 8.

Question 4:

(i) Find equation of line joining (1, 2) and (3, 6) using determinants

(ii) Find equation of line joining (3, 1) and (9, 3) using determinants

Answer:

(i) Let P (xy) be any point on the line joining points A (1, 2) and B (3, 6). Then, the points A, B, and P are collinear. Therefore, the area of triangle ABP will be zero.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6490/Chapter%204_html_m4c94f002.gif

Hence, the equation of the line joining the given points is y = 2x.

(ii) Let P (xy) be any point on the line joining points A (3, 1) and

B (9, 3). Then, the points A, B, and P are collinear. Therefore, the area of triangle ABP will be zero.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6490/Chapter%204_html_2932c449.gif

Hence, the equation of the line joining the given points is x − 3y = 0.

Question 5:

If area of triangle is 35 square units with vertices (2, −6), (5, 4), and (k, 4). Then k is

A. 12 B. −2 C. −12, −2 D. 12, −2

Answer:

Answer: D

The area of the triangle with vertices (2, −6), (5, 4), and (k, 4) is given by the relation,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6501/Chapter%204_html_m1ffb3930.gif

It is given that the area of the triangle is ±35.

Therefore, we have:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6501/Chapter%204_html_m20edecc0.gif

When 5 − k = −7, k = 5 + 7 = 12.

When 5 − k = 7, k = 5 − 7 = −2.

Hence, k = 12, −2.

The correct Answer is D.

Also Read : Exercise-4.4-Chapter-4-Determinants-class-12-ncert-solutions-Maths

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