The 20 one-mark questions from the March 2020 12th Standard Mathematics public exam, in paper order, with the correct option marked and a short explanation.
Q1
If \(u(x,y)=e^{x^2+y^2}\), then \(\frac{\partial u}{\partial x}\) is equal to :
- A. \(y^2u\)
- B. \(e^{x^2+y^2}\)
- C. \(2xu\)Correct
- D. \(x^2u\)
Explanation. Treating y as constant, \(\frac{\partial u}{\partial x}=e^{x^2+y^2}\cdot 2x=2xu\).
Q2
Subtraction is not a binary operation in :
- A. \(\mathbb{Q}\)
- B. \(\mathbb{R}\)
- C. \(\mathbb{Z}\)
- D. \(\mathbb{N}\)Correct
Explanation. In \(\mathbb{N}\), for example \(2-5=-3\notin\mathbb{N}\), so subtraction is not closed; it is closed in \(\mathbb{Z},\mathbb{Q},\mathbb{R}\).
Q3
The value of \(\int_0^{\pi}\sin^4 x\,dx\) is :
- A. \(\frac{3\pi}{2}\)
- B. \(\frac{3\pi}{10}\)
- C. \(\frac{3\pi}{8}\)Correct
- D. \(\frac{3\pi}{4}\)
Explanation. \(\int_0^{\pi}\sin^4x\,dx=2\int_0^{\pi/2}\sin^4x\,dx=2\cdot\frac{3}{4}\cdot\frac{1}{2}\cdot\frac{\pi}{2}=\frac{3\pi}{8}\).
Q4
A polynomial equation of degree n always has :
- A. exactly n rootsCorrect
- B. n distinct roots
- C. n real roots
- D. n imaginary roots
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.
Q5
If \(\rho(A)=\rho([A|B])\), then the system \(AX=B\) of linear equations is :
- A. inconsistent
- B. consistent and has a unique solution
- C. consistentCorrect
- D. consistent and has infinitely many solutions
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.
Q6
The vertex of the parabola \(x^2=8y-1\) is :
- A. \(\left(0,-\frac{1}{8}\right)\)
- B. \(\left(-\frac{1}{8},0\right)\)
- C. \(\left(\frac{1}{8},0\right)\)
- D. \(\left(0,\frac{1}{8}\right)\)Correct
Explanation. \(x^2=8\left(y-\frac{1}{8}\right)\), so the vertex is \(\left(0,\frac{1}{8}\right)\).
Q7
If \(\sin^{-1}x+\sin^{-1}y=\frac{2\pi}{3}\), then \(\cos^{-1}x+\cos^{-1}y\) is equal to :
- A. \(\pi\)
- B. \(\frac{2\pi}{3}\)
- C. \(\frac{\pi}{3}\)Correct
- D. \(\frac{\pi}{6}\)
Explanation. \(\cos^{-1}x+\cos^{-1}y=\left(\frac{\pi}{2}-\sin^{-1}x\right)+\left(\frac{\pi}{2}-\sin^{-1}y\right)=\pi-\frac{2\pi}{3}=\frac{\pi}{3}\).
Q8
The value of \(\sum_{n=1}^{13}\left(i^n+i^{n-1}\right)\) is :
- A. 0
- B. \(1+i\)Correct
- C. \(i\)
- D. 1
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 \(n=1\).)
Q9
\(\vec{r}=s\hat{i}+t\hat{j}\) is the equation of (s, t are parameters) :
- A. zox plane
- B. a straight line joining the points \(\hat{i}\) and \(\hat{j}\)
- C. xoy planeCorrect
- D. yoz plane
Explanation. Every point is \((s,t,0)\) with z = 0 and s, t free, so the equation represents the xoy plane.
Q10
The order of the differential equation of all circles with centre at (h, k) and radius 'a', where h, k and a are arbitrary constants, is :
- A. 1
- B. 2
- C. 3Correct
- D. 4
Explanation. \((x-h)^2+(y-k)^2=a^2\) has three arbitrary constants, so eliminating them gives a differential equation of order 3.
Q11
\(\arg(0)\) is :
- A. \(\infty\)
- B. 0
- C. \(\pi\)
- D. undefinedCorrect
Explanation. The complex number 0 has modulus 0 and no direction, so its argument is not defined.
Q12
\(\tan^{-1}\left(\frac{1}{4}\right)+\tan^{-1}\left(\frac{2}{9}\right)\) is :
- A. \(\tan^{-1}\left(\frac{1}{2}\right)\)Correct
- B. \(\frac{1}{2}\cos^{-1}\left(\frac{3}{5}\right)\)
- C. \(\frac{1}{2}\sin^{-1}\left(\frac{3}{5}\right)\)
- D. \(\frac{1}{2}\tan^{-1}\left(\frac{3}{5}\right)\)
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.
Q13
The position of a particle moving along a horizontal line of any time t is given by \(s(t)=3t^2-2t-8\). The time at which the particle is at rest, is :
- A. \(t=3\)
- B. \(t=0\)
- C. \(t=\frac{1}{3}\)Correct
- D. \(t=1\)
Explanation. Velocity \(v=s'(t)=6t-2=0\) gives \(t=\frac{1}{3}\).
Q14
The least possible perimeter (in meter) of a rectangle of area \(100\ m^2\) is :
- A. 50
- B. 10
- C. 20
- D. 40Correct
Explanation. \(P=2\left(x+\frac{100}{x}\right)\); \(P'=0\) gives \(x=10\), a square of side 10 m, so \(P=40\) m.
Q15
A random variable X has binomial distribution with n = 25 and p = 0.8, then the standard deviation of X is :
- A. 2Correct
- B. 6
- C. 4
- D. 3
Explanation. \(\sigma=\sqrt{npq}=\sqrt{25\times0.8\times0.2}=\sqrt{4}=2\).
Q16
The radius of the circle \(3x^2+by^2+4bx-6by+b^2=0\) is :
- A. \(\sqrt{11}\)
- B. 1
- C. 3
- D. \(\sqrt{10}\)Correct
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 \(=\sqrt{4+9-3}=\sqrt{10}\).
Q17
The distance between the planes \(x+2y+3z+7=0\) and \(2x+4y+6z+7=0\) is :
- A. \(\frac{7}{2\sqrt{2}}\)
- B. \(\frac{\sqrt{7}}{2\sqrt{2}}\)Correct
- C. \(\frac{7}{2}\)
- D. \(\frac{\sqrt{7}}{2}\)
Explanation. Write the second plane as \(x+2y+3z+\frac{7}{2}=0\). Distance \(=\frac{|7-7/2|}{\sqrt{14}}=\frac{7}{2\sqrt{14}}=\frac{\sqrt{7}}{2\sqrt{2}}\).
Q18
If \((AB)^{-1}=\begin{bmatrix}12 & -17\\ -19 & 27\end{bmatrix}\) and \(A^{-1}=\begin{bmatrix}1 & -1\\ -2 & 3\end{bmatrix}\), then \(B^{-1}=\)
- A. \(\begin{bmatrix}8 & -5\\ -3 & 2\end{bmatrix}\)
- B. \(\begin{bmatrix}2 & -5\\ -3 & 8\end{bmatrix}\)Correct
- C. \(\begin{bmatrix}8 & 5\\ 3 & 2\end{bmatrix}\)
- D. \(\begin{bmatrix}3 & 1\\ 2 & 1\end{bmatrix}\)
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 \(B^{-1}=\begin{bmatrix}2 & -5\\ -3 & 8\end{bmatrix}\).
Q19
The value of \(\int_0^{2/3}\frac{dx}{\sqrt{4-9x^2}}\) is :
- A. \(\pi\)
- B. \(\frac{\pi}{6}\)Correct
- C. \(\frac{\pi}{2}\)
- D. \(\frac{\pi}{4}\)
Explanation. \(\int\frac{dx}{\sqrt{4-9x^2}}=\frac{1}{3}\sin^{-1}\frac{3x}{2}\); from 0 to \(\frac{2}{3}\) this is \(\frac{1}{3}\cdot\frac{\pi}{2}=\frac{\pi}{6}\).
Q20
The order and degree of the differential equation \(\frac{dx}{dy}+\frac{dy}{dx}=0\) are :
- A. 2, degree not defined
- B. 1, 2Correct
- C. 2, 1
- D. 2, 2
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.
