NCERT Solutions for Class 12 Maths Chapter 4 – Determinants Miscellaneous Exercise
Page No 141:
Question 1:
Prove that the determinant
is independent of θ.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6672/Chapter%204_html_m1f884975.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/editlive_ncert/77/2012_02_15_17_54_55/mathmlequation3065325959819493208.png)
Hence, Δ is independent of θ.
Question 2:
Without expanding the determinant, prove that
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6679/Chapter%204_html_m35d1ab09.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6679/Chapter%204_html_m3c295fc.gif)
Hence, the given result is proved.
Question 3:
Evaluate ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6684/Chapter%204_html_23fe4e04.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6684/Chapter%204_html_23fe4e04.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6684/Chapter%204_html_m2a6ea31.gif)
Expanding along C3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6684/Chapter%204_html_78843555.gif)
Question 4:
If a, b and c are real numbers, and
,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6688/Chapter%204_html_41cc95ea.gif)
Show that either a + b + c = 0 or a = b = c.
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6688/Chapter%204_html_m7b4eb4be.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6688/Chapter%204_html_de0ec94.gif)
Expanding along R1, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6688/Chapter%204_html_6069f2c1.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6688/Chapter%204_html_20447801.gif)
Hence, if Δ = 0, then either a + b + c = 0 or a = b = c.
Question 5:
Solve the equations ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6695/Chapter%204_html_6f4cea10.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6695/Chapter%204_html_6f4cea10.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6695/Chapter%204_html_m377141b6.gif)
Question 6:
Prove that ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6701/Chapter%204_html_5a72b0cd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6701/Chapter%204_html_5a72b0cd.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6701/Chapter%204_html_53bf5077.gif)
Expanding along R3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6701/Chapter%204_html_m629ccfa8.gif)
Hence, the given result is proved.
Question 7:
If ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6705/Chapter%204_html_2b41d46c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6705/Chapter%204_html_2b41d46c.gif)
Answer:
We know that
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6705/Chapter%204_html_1a7af877.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6705/Chapter%204_html_7fded7ff.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6705/Chapter%204_html_62c2672f.gif)
Page No 142:
Question 8:
Let
verify that
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6708/Chapter%204_html_61655144.gif)
(i) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6708/Chapter%204_html_2774581f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6708/Chapter%204_html_2774581f.gif)
(ii) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6708/Chapter%204_html_m372bb39f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6708/Chapter%204_html_m372bb39f.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6708/Chapter%204_html_m3f7303ea.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6708/Chapter%204_html_m6678b4af.gif)
(i)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6708/Chapter%204_html_5f871e24.gif)
We have,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6708/Chapter%204_html_1bb9104b.gif)
(ii)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6708/Chapter%204_html_44784180.gif)
Question 9:
Evaluate ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6713/Chapter%204_html_2207a667.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6713/Chapter%204_html_2207a667.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6713/Chapter%204_html_317181b7.gif)
Expanding along R1, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6713/Chapter%204_html_2d28f70b.gif)
Question 10:
Evaluate ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6716/Chapter%204_html_3177ffae.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6716/Chapter%204_html_3177ffae.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6716/Chapter%204_html_24808112.gif)
Expanding along C1, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6716/Chapter%204_html_m7a073c6a.gif)
Question 11:
Using properties of determinants, prove that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6717/Chapter%204_html_m2053ac31.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6717/Chapter%204_html_m13b66365.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6717/Chapter%204_html_3b1c868d.gif)
Expanding along R3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6717/Chapter%204_html_5c654038.gif)
Hence, the given result is proved.
Question 12:
Using properties of determinants, prove that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6719/Chapter%204_html_6e13f6e9.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6719/Chapter%204_html_37275d0f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6719/Chapter%204_html_m79fd7df8.gif)
Expanding along R3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6719/Chapter%204_html_77d332cc.gif)
Hence, the given result is proved.
Question 13:
Using properties of determinants, prove that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6723/Chapter%204_html_35583b6a.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6723/Chapter%204_html_m3a9c7a67.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6723/Chapter%204_html_m74d69704.gif)
Expanding along C1, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6723/Chapter%204_html_m356b29d8.gif)
Hence, the given result is proved.
Question 14:
Using properties of determinants, prove that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6726/Chapter%204_html_m14dcca89.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6726/Chapter%204_html_391432e3.gif)
Expanding along C1, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6726/Chapter%204_html_m7c6e807f.gif)
Hence, the given result is proved.
Question 15:
Using properties of determinants, prove that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6729/Chapter%204_html_m151f0f29.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6729/Chapter%204_html_m9108e5c.gif)
Hence, the given result is proved.
Question 16:
Solve the system of the following equations
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6733/Chapter%204_html_30a8d8b2.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6733/Chapter%204_html_m1eb2cd25.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6733/Chapter%204_html_m1eb2cd25.gif)
Then the given system of equations is as follows:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6733/Chapter%204_html_6303329e.gif)
This system can be written in the form of AX = B, where
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6733/Chapter%204_html_1b5998c5.gif)
Thus, A is non-singular. Therefore, its inverse exists.
Now,
A11 = 75, A12 = 110, A13 = 72
A21 = 150, A22 = −100, A23 = 0
A31 = 75, A32 = 30, A33 = − 24
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6733/Chapter%204_html_68fa224.gif)
Page No 143:
Question 17:
Choose the correct answer.
If a, b, c, are in A.P., then the determinant
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6734/Chapter%204_html_m5ab9cf8c.gif)
A. 0 B. 1 C. x D. 2x
Answer:
Answer: A
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6734/Chapter%204_html_m4df80470.gif)
Here, all the elements of the first row (R1) are zero.
Hence, we have Δ = 0.
The correct answer is A.
Question 18:
Choose the correct answer.
If x, y, z are nonzero real numbers, then the inverse of matrix
is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6736/Chapter%204_html_m2a170178.gif)
A.
B. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6736/Chapter%204_html_m1b38ff54.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6736/Chapter%204_html_m237b2b4b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6736/Chapter%204_html_m1b38ff54.gif)
C.
D. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6736/Chapter%204_html_m33e4252d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6736/Chapter%204_html_716c89bf.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6736/Chapter%204_html_m33e4252d.gif)
Answer:
Answer: A
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6736/Chapter%204_html_1a5715dc.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6736/Chapter%204_html_m451a05bd.gif)
The correct answer is A.
Question 19:
Choose the correct answer.
Let
, where 0 ≤ θ≤ 2π, then
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6741/Chapter%204_html_70daff3d.gif)
A. Det (A) = 0
B. Det (A) ∈ (2, ∞)
C. Det (A) ∈ (2, 4)
D. Det (A)∈ [2, 4]
Answer:
Answer: D
![](https://img-nm.mnimgs.com/img/study_content/content_ck_images/images/Selection_010(9).png)
0≤θ≤2π
⇒-1≤sinθ≤1
The correct answer is D.
![](https://img-nm.mnimgs.com/img/study_content/content_ck_images/images/Selection_012(8).png)