Find the best tutors and institutes for Class 12 Tuition
Q2(i):
Evaluate the determinant: (i) $\begin{vmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{vmatrix}$
Solution :
Step 1: Theoretical Foundation of a $2 \times 2$ Determinant
For any square matrix of order $2$, defined as $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the determinant is calculated using the formula:
$\det(A) = |A| = (ad) - (bc)$
[Per the definition of the determinant of a $2 \times 2$ matrix].
Step 2: Assigning Variables from the Given Matrix
Given the determinant $\begin{vmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{vmatrix}$, we identify the elements as follows:
- $a = \cos\theta$
- $b = -\sin\theta$
- $c = \sin\theta$
- $d = \cos\theta$
Step 3: Execution of the Determinant Calculation
Substituting the variables into the formula $(ad - bc)$:
$\Delta = (\cos\theta)(\cos\theta) - (-\sin\theta)(\sin\theta)$
$\Delta = \cos^2\theta - (-\sin^2\theta)$
$\Delta = \cos^2\theta + \sin^2\theta$
Step 4: Simplification via Trigonometric Identity
Applying the fundamental Pythagorean identity of trigonometry:
$\cos^2\theta + \sin^2\theta = 1$
[Per the Fundamental Trigonometric Identity].
Step 5: Visual Representation
Final Solution: 1
More Questions from Class 12 Mathematics Determinants EXERCISE 4.1
- Q1: Evaluate the determinant: $\begin{vmatrix} 2 & 4 \\ -5 & -1 \end{vmatrix}$
- Q2(ii): Evaluate the determinant: (ii) $\begin{vmatrix} x^2 - x + 1 & x - 1 \\ x + 1 & x + 1 \end{vmatrix}$
- Q3: If A = $\begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}$, then show that $| 2A | = 4 | A |$
- Q4: If A = $\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{bmatrix}$, then show that $| 3 A | = 27 | A |$
- Q5(i): Evaluate the determinant: (i) $\begin{vmatrix} 3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{vmatrix}$
- Q5(ii): Evaluate the determinant: (ii) $\begin{vmatrix} 3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1 \end{vmatrix}$
- Q5(iii): Evaluate the determinant: (iii) $\begin{vmatrix} 0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0 \end{vmatrix}$
- Q5(iv): Evaluate the determinant: (iv) $\begin{vmatrix} 2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0 \end{vmatrix}$
- Q6: If A = $\begin{bmatrix} 1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9 \end{bmatrix}$, find $| A |$
- Q7(i): Find values of $x$, if (i) $\begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix} = \begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix}$
- Q7(ii): Find values of $x$, if (ii) $\begin{vmatrix} 2 & 3 \\ 4 & 5 \end{vmatrix} = \begin{vmatrix} x & 3 \\ 2x & 5 \end{vmatrix}$
- Q8: If $\begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} 6 & 2 \\ 18 & 6 \end{vmatrix}$, then $x$ is equal to
CBSE Solutions for Class 12 Mathematics Determinants
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Determinants
Super Tutor
Bilekahalli, Bangalore
6 yrs of
Exp
9 students
326 hrs of classes completed
500
per hour
Online Classes
I teach Chemistry to students of Class 11 and Class 12 and have been working in the teaching field for the past 7 years. Chemistry has always been a subject that I am passionate about, and I enjoy helping students understand concepts clearly and perform confidently in academics as well as competitive examinations. My goal is to make Chemistry simple, logical, and interesting for students through structured and concept-focused learning. I hold a Master of Science (M.Sc.) degree in Chemistry, which has provided me with strong subject knowledge and a deeper understanding of various chemical concepts. My academic background helps me explain topics in a clear and systematic way so that students can easily understand and apply them while solving numerical and theoretical questions. Over the years, I have worked with students from different learning backgrounds and understanding levels. This experience has taught me the importance of adapting teaching methods according to the student’s requirements. I focus on helping students strengthen their conceptual understanding, improve their analytical and problem-solving skills, and gain confidence in handling Chemistry effectively for both board exams and competitive exams. My teaching methodology is structured, student-centric, and exam-oriented. I follow a well-planned lesson structure where topics progress step by step from basic to advanced levels. Before starting a new topic, I assess the student’s current understanding level so that I can explain concepts in the most effective manner. I also use suitable real-life examples to simplify complex chemical concepts and make learning more relatable and easier to understand. I strongly believe that conceptual clarity is more important than rote learning. During my classes, I encourage students to ask questions freely and clear doubts without hesitation. If a student finds a concept difficult, I patiently re-explain it multiple times until complete understanding is achieved. I also conduct regular revision sessions, exercise practice, and detailed discussions of previous years’ board and competitive examination questions to strengthen preparation and retention. I conduct both online and offline classes depending on the convenience of students. During my sessions, I use PPT presentations and whiteboard explanations to make learning more organized, visual, and engaging. My classes are interactive and focused on building confidence, helping students solve problems independently, and preparing them effectively for examinations. As a teacher, my aim is to help students develop a strong foundation in Chemistry and remove the fear many students have towards the subject. I believe that with the right guidance, regular practice, and conceptual understanding, every student can excel in Chemistry. Helping students improve academically and achieve success in their exams continues to motivate me in my teaching journey.
As per my son's experience, he clears all the concepts and teaches very well, explaining repeatedly until it is fully understood.
Top Tutor
Agara Village, Bangalore
3 yrs of
Exp
33 hrs of classes completed
950
per hour
Online Classes
Have trained students for NEET and JEE MAINS and now my students are placed in reputated colleges.
I like the way she teach and elaborate aspects. She is the one simplifies complex subject into detailed and understandable thing.
Top Tutor
Sector 54, Chandigarh
15 yrs of
Exp
10 students
357 hrs of classes completed
750
per hour
Online Classes
i am physics teacher with 15 years of experience in teaching 11,12, competitive exams. i always try to explain topics more in practical form rather than theortical by giving examples from day to day life . topics like electrostatics , alternating current ,optics , semiconductors and logic gates are most imp and students usually got troubled there but with practical apporoach i make them go with happy and easy learning .
Puts in genuine effort Clear concepts Offer flexible timings Cover all types of questions in class Excellent teacher
Super Tutor
Amrapali Dream Valley, Noida
20 yrs of
Exp
5 students
228 hrs of classes completed
400
per hour
Online Classes
Student's Home
Tutor's home
My students have consistently scored 90+. My focus is always on clarity of fundamentals concepts.I can assure you that chemistry will become your favourite subject.I have taught students of best schools of Delhi and I know where some of school teachers lack which drives students away from chemistry.They start hating chemistry but my students love chemistry.Join for at least one month and experience it yourself.
My chemistry home tuition teacher is highly knowledgeable, patient, and dedicated. They explain every concept in a simple and easy-to-understand manner, making even difficult topics interesting. Their regular practice sessions, doubt-solving, and encouraging approach have greatly improved my understanding, confidence, and performance in chemistry. I truly appreciate their guidance and support.
Top Tutor
Indirapuram, Ghaziabad
5 yrs of
Exp
39 hrs of classes completed
Online Classes
I was a working professional .Left my job and started teaching as a tutor. I have been a national level debater . I am a famous teacher amongst my students .
