12th Standard Mathematics March 2020: Online Practice Test

Explanation. Treating y as constant, \(\frac{\partial u}{\partial x}=e^{x^2+y^2}\cdot 2x=2xu\).

Explanation. In \(\mathbb{N}\), for example \(2-5=-3\notin\mathbb{N}\), so subtraction is not closed; it is closed in \(\mathbb{Z},\mathbb{Q},\mathbb{R}\).

Explanation. \(\int_0^{\pi}\sin^4x\,dx=2\int_0^{\pi/2}\sin^4x\,dx=2\cdot\frac{3}{4}\cdot\frac{1}{2}\cdot\frac{\pi}{2}=\frac{3\pi}{8}\).

Explanation. By the fundamental theorem of algebra, a polynomial equation of degree n has exactly n roots (counting multiplicity); they need not be distinct or real.

Explanation. By the Rouche-Capelli theorem, equal ranks mean the system is consistent; whether the solution is unique depends on comparing the rank with the number of unknowns, which is not given.

Explanation. \(x^2=8\left(y-\frac{1}{8}\right)\), so the vertex is \(\left(0,\frac{1}{8}\right)\).

Explanation. \(\cos^{-1}x+\cos^{-1}y=\left(\frac{\pi}{2}-\sin^{-1}x\right)+\left(\frac{\pi}{2}-\sin^{-1}y\right)=\pi-\frac{2\pi}{3}=\frac{\pi}{3}\).

Explanation. Any 4 consecutive powers of \(i\) add to 0. \(\sum_{n=1}^{13}i^n=i^{13}=i\) and \(\sum_{n=1}^{13}i^{n-1}=i^{12}=1\), so the sum is \(1+i\). (The paper prints the index as \(i=1\); it means \(n=1\).)

Explanation. Every point is \((s,t,0)\) with z = 0 and s, t free, so the equation represents the xoy plane.

Explanation. \((x-h)^2+(y-k)^2=a^2\) has three arbitrary constants, so eliminating them gives a differential equation of order 3.

Explanation. The complex number 0 has modulus 0 and no direction, so its argument is not defined.

Explanation. \(\tan^{-1}\frac{1}{4}+\tan^{-1}\frac{2}{9}=\tan^{-1}\frac{17/36}{17/18}=\tan^{-1}\frac{1}{2}\). Since \(\cos\left(2\tan^{-1}\frac{1}{2}\right)=\frac{1-1/4}{1+1/4}=\frac{3}{5}\), option (2) is also equal to it; the paper has two correct options.

Explanation. Velocity \(v=s'(t)=6t-2=0\) gives \(t=\frac{1}{3}\).

Explanation. \(P=2\left(x+\frac{100}{x}\right)\); \(P'=0\) gives \(x=10\), a square of side 10 m, so \(P=40\) m.

Explanation. \(\sigma=\sqrt{npq}=\sqrt{25\times0.8\times0.2}=\sqrt{4}=2\).

Explanation. For a circle the coefficients of \(x^2\) and \(y^2\) are equal, so \(b=3\): \(x^2+y^2+4x-6y+3=0\). Radius \(=\sqrt{4+9-3}=\sqrt{10}\).

Explanation. Write the second plane as \(x+2y+3z+\frac{7}{2}=0\). Distance \(=\frac{|7-7/2|}{\sqrt{14}}=\frac{7}{2\sqrt{14}}=\frac{\sqrt{7}}{2\sqrt{2}}\).

Explanation. \((AB)^{-1}=B^{-1}A^{-1}\), so \(B^{-1}=(AB)^{-1}A\). Here \(A=(A^{-1})^{-1}=\begin{bmatrix}3 & 1\\ 2 & 1\end{bmatrix}\), giving \(B^{-1}=\begin{bmatrix}2 & -5\\ -3 & 8\end{bmatrix}\).

Explanation. \(\int\frac{dx}{\sqrt{4-9x^2}}=\frac{1}{3}\sin^{-1}\frac{3x}{2}\); from 0 to \(\frac{2}{3}\) this is \(\frac{1}{3}\cdot\frac{\pi}{2}=\frac{\pi}{6}\).

Explanation. Multiplying by \(\frac{dy}{dx}\) gives \(\left(\frac{dy}{dx}\right)^2+1=0\): the highest derivative is of first order and its power is 2, so order 1, degree 2.