Podcast
Questions and Answers
Given the matrix equation $AXB = C$, where $A = \begin{bmatrix} 2 & 1 \ 5 & 3 \end{bmatrix}$, $B = \begin{bmatrix} 5 & 3 \ 3 & 2 \end{bmatrix}$, and $C = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$, what is the matrix X?
Given the matrix equation $AXB = C$, where $A = \begin{bmatrix} 2 & 1 \ 5 & 3 \end{bmatrix}$, $B = \begin{bmatrix} 5 & 3 \ 3 & 2 \end{bmatrix}$, and $C = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$, what is the matrix X?
What is the consequence if the four given points with position vectors $6\hat{i} - 7\hat{j}$, $16\hat{i} - 19\hat{j} - 4\hat{k}$, $3\hat{i} - 6\hat{k}$, and $2\hat{i} - 5\hat{j} + 10\hat{k}$ are found to be coplanar?
What is the consequence if the four given points with position vectors $6\hat{i} - 7\hat{j}$, $16\hat{i} - 19\hat{j} - 4\hat{k}$, $3\hat{i} - 6\hat{k}$, and $2\hat{i} - 5\hat{j} + 10\hat{k}$ are found to be coplanar?
An urn contains 5 white and 8 black balls. Two successive drawings of 3 balls are made without replacement. What is the probability that the first draw yields 3 white balls and the second draw yields 3 black balls?
An urn contains 5 white and 8 black balls. Two successive drawings of 3 balls are made without replacement. What is the probability that the first draw yields 3 white balls and the second draw yields 3 black balls?
Based on the matrix calculation, what is the total amount of funds collected by school B?
Based on the matrix calculation, what is the total amount of funds collected by school B?
Signup and view all the answers
If the price of each item increases by 20%, what is the new price of 'Mats'?
If the price of each item increases by 20%, what is the new price of 'Mats'?
Signup and view all the answers
What is the value of the integral $\int (\sin^{-1} x)^3 dx$?
What is the value of the integral $\int (\sin^{-1} x)^3 dx$?
Signup and view all the answers
If $y(x)$ is a solution of the differential equation $\frac{dy}{dx} = 2y$ and $y(0) = 1$, what is the value of $y( \frac{1}{2} )$?
If $y(x)$ is a solution of the differential equation $\frac{dy}{dx} = 2y$ and $y(0) = 1$, what is the value of $y( \frac{1}{2} )$?
Signup and view all the answers
Which of these represents the original price of the items before any price changes?
Which of these represents the original price of the items before any price changes?
Signup and view all the answers
What matrix operation is used to find the total funds collected by each school?
What matrix operation is used to find the total funds collected by each school?
Signup and view all the answers
If we have the matrix P representing quantities of items by schools and original price matrix Q. What does the first element in the resulted matrix PQ represents?
If we have the matrix P representing quantities of items by schools and original price matrix Q. What does the first element in the resulted matrix PQ represents?
Signup and view all the answers
Given $P(B/A) = \frac{7}{15}$ and $P(A) = \frac{5}{143}$, what is the value of $P(A \cap B)$?
Given $P(B/A) = \frac{7}{15}$ and $P(A) = \frac{5}{143}$, what is the value of $P(A \cap B)$?
Signup and view all the answers
In the calculation of the integral, $I = \int (sin^{-1}x)^3 dx$, what substitution is made to begin solving?
In the calculation of the integral, $I = \int (sin^{-1}x)^3 dx$, what substitution is made to begin solving?
Signup and view all the answers
After applying integration by parts, what term is obtained when integrating $\int t^2 \sin t dt$?
After applying integration by parts, what term is obtained when integrating $\int t^2 \sin t dt$?
Signup and view all the answers
During the calculation of $I = \int (sin^{-1}x)^3 dx$, what is the result after one application of integration by parts?
During the calculation of $I = \int (sin^{-1}x)^3 dx$, what is the result after one application of integration by parts?
Signup and view all the answers
If $sin^{-1}x = t$, what is the expression for $\cos t$?
If $sin^{-1}x = t$, what is the expression for $\cos t$?
Signup and view all the answers
In the final solution for the integral $I = \int (sin^{-1}x)^3 dx$, what is the coefficient of the $(sin^{-1}x)( \sqrt{1-x^2} )$ term?
In the final solution for the integral $I = \int (sin^{-1}x)^3 dx$, what is the coefficient of the $(sin^{-1}x)( \sqrt{1-x^2} )$ term?
Signup and view all the answers
What is the first step after substituting $t = sin^{-1}x$ in the integral $I = \int (sin^{-1}x)^3 dx$?
What is the first step after substituting $t = sin^{-1}x$ in the integral $I = \int (sin^{-1}x)^3 dx$?
Signup and view all the answers
In the expression $x(sin^{-1}x)^3 + 3(sin^{-1}x)^2\sqrt{1 - x^2} - 6(sin^{-1}x)x - 6\sqrt{1-x^2}+ C$ for $\int (sin^{-1}x)^3 dx$, how many times is integration by parts applied?
In the expression $x(sin^{-1}x)^3 + 3(sin^{-1}x)^2\sqrt{1 - x^2} - 6(sin^{-1}x)x - 6\sqrt{1-x^2}+ C$ for $\int (sin^{-1}x)^3 dx$, how many times is integration by parts applied?
Signup and view all the answers
Given the equation $(1 + y)(2 + \sin x) = 4$, what is the expression for $y$ in terms of $x$?
Given the equation $(1 + y)(2 + \sin x) = 4$, what is the expression for $y$ in terms of $x$?
Signup and view all the answers
Using the derived expression for $y$, what is the value of $y$ when $x = \frac{\pi}{2}$?
Using the derived expression for $y$, what is the value of $y$ when $x = \frac{\pi}{2}$?
Signup and view all the answers
What is the integrating factor when solving the linear differential equation $\frac{dy}{dx} + \frac{2}{x} y = x$?
What is the integrating factor when solving the linear differential equation $\frac{dy}{dx} + \frac{2}{x} y = x$?
Signup and view all the answers
In the given differential equation $\frac{dy}{dx} + \frac{2}{x} y = x$, what is the term that corresponds to $Q$ in the standard form $\frac{dy}{dx} + Py = Q$?
In the given differential equation $\frac{dy}{dx} + \frac{2}{x} y = x$, what is the term that corresponds to $Q$ in the standard form $\frac{dy}{dx} + Py = Q$?
Signup and view all the answers
Which of the following equations represents a linear differential equation?
Which of the following equations represents a linear differential equation?
Signup and view all the answers
Given the equation $(1+y)(2+\sin x)=4$, and that $y = \frac{2 - \sin x}{2 + \sin x}$, find the value of $x$ where $y=1$.
Given the equation $(1+y)(2+\sin x)=4$, and that $y = \frac{2 - \sin x}{2 + \sin x}$, find the value of $x$ where $y=1$.
Signup and view all the answers
What does the given differential equation $\frac{dy}{dx} + \frac{2}{x} y = x$ represent?
What does the given differential equation $\frac{dy}{dx} + \frac{2}{x} y = x$ represent?
Signup and view all the answers
In the equation $(1+y)(2+\sin x) = 4$, which of the following values is a valid range for the value of $y$?
In the equation $(1+y)(2+\sin x) = 4$, which of the following values is a valid range for the value of $y$?
Signup and view all the answers
If $u = \sin^{-1}(\frac{2x}{1+x^2})$ and $x = \tan\theta$, what is the simplified expression for u in terms of $\theta$?
If $u = \sin^{-1}(\frac{2x}{1+x^2})$ and $x = \tan\theta$, what is the simplified expression for u in terms of $\theta$?
Signup and view all the answers
Given $v = \cos^{-1}(\frac{1-x^2}{1+x^2})$ and $x = \tan\theta$, what is the simplified expression for $v$ in terms of $\theta$?
Given $v = \cos^{-1}(\frac{1-x^2}{1+x^2})$ and $x = \tan\theta$, what is the simplified expression for $v$ in terms of $\theta$?
Signup and view all the answers
If $x = \tan \theta$ and $0 < x < 1$, what is the range of $\theta$?
If $x = \tan \theta$ and $0 < x < 1$, what is the range of $\theta$?
Signup and view all the answers
Given $u = 2\tan^{-1}x$, what is $\frac{du}{dx}$?
Given $u = 2\tan^{-1}x$, what is $\frac{du}{dx}$?
Signup and view all the answers
What is the value of $\frac{du/dx}{dv/dx}$?
What is the value of $\frac{du/dx}{dv/dx}$?
Signup and view all the answers
If $u = \sin^{-1}(\frac{2x}{1+x^2})$, and $v = \cos^{-1}(\frac{1-x^2}{1+x^2})$, and $x = \tan \theta$, what is the relationship between $u$ and $v$?
If $u = \sin^{-1}(\frac{2x}{1+x^2})$, and $v = \cos^{-1}(\frac{1-x^2}{1+x^2})$, and $x = \tan \theta$, what is the relationship between $u$ and $v$?
Signup and view all the answers
If $ u = 2\tan^{-1}x$ and $v= 2\tan^{-1}x$, what is the relationship between $\frac{du}{dx}$ and $\frac{dv}{dx}$?
If $ u = 2\tan^{-1}x$ and $v= 2\tan^{-1}x$, what is the relationship between $\frac{du}{dx}$ and $\frac{dv}{dx}$?
Signup and view all the answers
What is the first step in simplifying the integral ∫ √2 (2 − 3/2x − x²) dx?
What is the first step in simplifying the integral ∫ √2 (2 − 3/2x − x²) dx?
Signup and view all the answers
After completing the square, what expression is obtained inside the square root?
After completing the square, what expression is obtained inside the square root?
Signup and view all the answers
What trigonometric substitution is suggested by the form of the integral after completing the square?
What trigonometric substitution is suggested by the form of the integral after completing the square?
Signup and view all the answers
What is the result of the simplified integral with the term involving arcsin?
What is the result of the simplified integral with the term involving arcsin?
Signup and view all the answers
What is the value of ‘I’ after applying the limits of integration and substituting the value from previous result?
What is the value of ‘I’ after applying the limits of integration and substituting the value from previous result?
Signup and view all the answers
In the integral I = ∫1/(sin x + sec x) dx, what is the first step to simplify the integrand?
In the integral I = ∫1/(sin x + sec x) dx, what is the first step to simplify the integrand?
Signup and view all the answers
After rewriting in terms of sine and cosine, what is the simplified form of the integrand?
After rewriting in terms of sine and cosine, what is the simplified form of the integrand?
Signup and view all the answers
In the second approach, the integrand is split into two integrals. What decomposition is used?
In the second approach, the integrand is split into two integrals. What decomposition is used?
Signup and view all the answers
What substitution is made for the first integral after splitting?
What substitution is made for the first integral after splitting?
Signup and view all the answers
What substitution is made for the second integral after splitting?
What substitution is made for the second integral after splitting?
Signup and view all the answers
What identity is used to transform 2 + 2 sin x cos x to make the first integral substitution easier?
What identity is used to transform 2 + 2 sin x cos x to make the first integral substitution easier?
Signup and view all the answers
What identity is used to transform 2 + 2 sin x cos x to make the second integral substitution easier?
What identity is used to transform 2 + 2 sin x cos x to make the second integral substitution easier?
Signup and view all the answers
After the substitutions, the first integral simplifies to an integral of which form?
After the substitutions, the first integral simplifies to an integral of which form?
Signup and view all the answers
After the substitutions, the second integral simplifies to an integral of which form?
After the substitutions, the second integral simplifies to an integral of which form?
Signup and view all the answers
What is the final expression of the given integral after solving?
What is the final expression of the given integral after solving?
Signup and view all the answers
Study Notes
- This document appears to be a mathematics sample paper for class 12.
- The paper covers various topics in mathematics, including multiple choice questions (MCQs), short answer questions (SAQs), long answer questions (LAQs) and integrated units of assessment.
- There are internal choices provided in some questions.
- The exam duration is 3 hours, and the total marks are 80.
- The paper is divided into five sections: A, B, C, D, and E. Each section is compulsory.
- Section A has 18 multiple choice questions (MCQs) and two assertion-reason questions.
- Section B has 5 very short answer questions (VSAQs).
- Section C has 6 short answer questions (SAQs).
- Section D has 4 long answer questions (LAQs).
- Section E has 3 integrated assessment questions (source-based, case-based, passage-based).
- The topics covered are likely to include calculus, linear algebra, and other relevant topics in mathematics for class 12.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Related Documents
Description
Prepare for your Class 12 mathematics exam with this comprehensive sample paper. It includes multiple choice questions, short answer questions, long answer questions, and integrated assessments. Covering vital topics such as calculus and linear algebra, this paper is designed to help you familiarize yourself with the exam structure and question types.
