NCERT Solutions for Class 12 Maths Chapter 4 – Determinants Ex 4.1
Page No 108:
Question 1:
Evaluate the determinants in Exercises 1 and 2.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6411/Chapter%204_html_54bbaa6a.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6411/Chapter%204_html_54bbaa6a.gif)
Question 2:
Evaluate the determinants in Exercises 1 and 2.
(i)
(ii) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6412/Chapter%204_html_m4d7be874.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6412/Chapter%204_html_m362f25b6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6412/Chapter%204_html_m4d7be874.gif)
Answer:
(i)
= (cos θ)(cos θ) − (−sin θ)(sin θ) = cos2 θ+ sin2 θ = 1
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6412/Chapter%204_html_m362f25b6.gif)
(ii) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6412/Chapter%204_html_m4d7be874.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6412/Chapter%204_html_m4d7be874.gif)
= (x2 − x + 1)(x + 1) − (x − 1)(x + 1)
= x3 − x2 + x + x2 − x + 1 − (x2 − 1)
= x3 + 1 − x2 + 1
= x3 − x2 + 2
Question 3:
If
, then show that![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6413/Chapter%204_html_76def806.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6413/Chapter%204_html_m7a2d4992.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6413/Chapter%204_html_76def806.gif)
Answer:
The given matrix is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6413/Chapter%204_html_m7a2d4992.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6413/Chapter%204_html_5a46c228.gif)
Question 4:
If
, then show that![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6416/Chapter%204_html_46c04212.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6416/Chapter%204_html_m1261167e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6416/Chapter%204_html_46c04212.gif)
Answer:
The given matrix is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6416/Chapter%204_html_m1261167e.gif)
It can be observed that in the first column, two entries are zero. Thus, we expand along the first column (C1) for easier calculation.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6416/Chapter%204_html_275ad411.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6416/Chapter%204_html_6b5dffe6.gif)
From equations (i) and (ii), we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6416/Chapter%204_html_171d0b0c.gif)
Hence, the given result is proved.
Question 5:
Evaluate the determinants
(i)
(iii) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6418/Chapter%204_html_m665735e9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6418/Chapter%204_html_570bb78c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6418/Chapter%204_html_m665735e9.gif)
(ii)
(iv) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6418/Chapter%204_html_1edf2581.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6418/Chapter%204_html_m621a4e2e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6418/Chapter%204_html_1edf2581.gif)
Answer:
(i) Let
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6418/Chapter%204_html_m35290334.gif)
It can be observed that in the second row, two entries are zero. Thus, we expand along the second row for easier calculation.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6418/Chapter%204_html_m1094e20c.gif)
(ii) Let
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6418/Chapter%204_html_m347cad35.gif)
By expanding along the first row, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6418/Chapter%204_html_m380503db.gif)
(iii) Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6418/Chapter%204_html_799a7cf2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6418/Chapter%204_html_799a7cf2.gif)
By expanding along the first row, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6418/Chapter%204_html_162bb8c6.gif)
(iv) Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6418/Chapter%204_html_m229d49f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6418/Chapter%204_html_m229d49f.gif)
By expanding along the first column, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6418/Chapter%204_html_45cc20ca.gif)
Page No 109:
Question 6:
If
, find
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6419/Chapter%204_html_m685cfbea.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6419/Chapter%204_html_m6c09cbfe.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6419/Chapter%204_html_m195cd5a2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6419/Chapter%204_html_m195cd5a2.gif)
By expanding along the first row, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6419/Chapter%204_html_7a717b7.gif)
Question 7:
Find values of x, if
(i)
2451=2x46x(ii)
2345=x32x5
Answer:
(i) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6427/Chapter%204_html_m5b1bf1d8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6427/Chapter%204_html_m5b1bf1d8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6427/Chapter%204_html_m4c59b137.gif)
(ii) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6427/Chapter%204_html_m3b7a5670.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6427/Chapter%204_html_m3b7a5670.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6427/Chapter%204_html_m3820bd31.gif)
Question 8:
If
, then x is equal to
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6432/Chapter%204_html_2ce2e5e0.gif)
(A) 6 (B) ±6 (C) −6 (D) 0
Answer:
Answer: B
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6432/Chapter%204_html_2ce2e5e0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6432/Chapter%204_html_245292a.gif)
Hence, the correct answer is B.