Mathematics Class 12 Sample Paper

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?

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?

If the price of each item increases by 20%, what is the new price of 'Mats'?

What is the value of the integral $\int (\sin^{-1} x)^3 dx$?

If $y(x)$ is a solution of the differential equation $\frac{dy}{dx} = 2y$ and $y(0) = 1$, what is the value of $y( \frac{1}{2} )$?

Which of these represents the original price of the items before any price changes?

What matrix operation is used to find the total funds collected by each school?

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?

Given $P(B/A) = \frac{7}{15}$ and $P(A) = \frac{5}{143}$, what is the value of $P(A \cap B)$?

In the calculation of the integral, $I = \int (sin^{-1}x)^3 dx$, what substitution is made to begin solving?

After applying integration by parts, what term is obtained when integrating $\int t^2 \sin t dt$?

During the calculation of $I = \int (sin^{-1}x)^3 dx$, what is the result after one application of integration by parts?

If $sin^{-1}x = t$, what is the expression for $\cos t$?

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?

What is the first step after substituting $t = sin^{-1}x$ in the integral $I = \int (sin^{-1}x)^3 dx$?

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?

Given the equation $(1 + y)(2 + \sin x) = 4$, what is the expression for $y$ in terms of $x$?

Using the derived expression for $y$, what is the value of $y$ when $x = \frac{\pi}{2}$?

What is the integrating factor when solving the linear differential equation $\frac{dy}{dx} + \frac{2}{x} y = x$?

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 $\frac{dy}{dx} + Py = Q$?

Which of the following equations represents a linear differential equation?

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 $y=1$.

What does the given differential equation $\frac{dy}{dx} + \frac{2}{x} y = x$ represent?

In the equation $(1+y)(2+\sin x) = 4$, which of the following values is a valid range for the value of $y$?

If $u = \sin^{-1}(\frac{2x}{1+x^2})$ and $x = \tan\theta$, what is the simplified expression for u 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 $\theta$?

If $x = \tan \theta$ and $0 < x < 1$, what is the range of $\theta$?

Given $u = 2\tan^{-1}x$, what is $\frac{du}{dx}$?

What is the value of $\frac{du/dx}{dv/dx}$?

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 = 2\tan^{-1}x$ and $v= 2\tan^{-1}x$, what is the relationship between $\frac{du}{dx}$ and $\frac{dv}{dx}$?

What is the first step in simplifying the integral ∫ √2 (2 − 3/2x − x²) dx?

After completing the square, what expression is obtained inside the square root?

What trigonometric substitution is suggested by the form of the integral after completing the square?

What is the result of the simplified integral with the term involving arcsin?

What is the value of ‘I’ after applying the limits of integration and substituting the value from previous result?

In the integral I = ∫1/(sin x + sec x) dx, what is the first step to simplify the integrand?

After rewriting in terms of sine and cosine, what is the simplified form of the integrand?

In the second approach, the integrand is split into two integrals. What decomposition is used?

What substitution is made for the first integral after splitting?

What substitution is made for the second integral after splitting?

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 second integral substitution easier?

After the substitutions, the first integral simplifies to an integral of which form?

After the substitutions, the second integral simplifies to an integral of which form?

What is the final expression of the given integral after solving?