The 20 one-mark questions from the May 2022 12th Standard Mathematics public exam, in paper order, with the correct option marked and a short explanation.
Q1
If \(f(x)=\begin{cases}2x & 0\le x\le a\\ 0 & \text{otherwise}\end{cases}\) is a probability density function of a random variable, then the value of a is :
- A. 3
- B. 1Correct
- C. 4
- D. 2
Explanation. Total probability must be 1: \(\int_0^a 2x\,dx=a^2=1\), so \(a=1\) (a > 0).
Q2
Which one of the following is not true in the case of discrete random variable X ?
- A. \(\lim_{x\to\infty}F(x)=F(\infty)=1\)
- B. \(0\le F(x)\le 1\) for all \(x\in\mathbb{R}\)
- C. \(F(x)\) is real valued decreasing function.Correct
- D. \(\lim_{x\to-\infty}F(x)=F(-\infty)=0\)
Explanation. A cumulative distribution function F(x) is non-decreasing, not decreasing; the other three properties are true.
Q3
If \(f(x)=\frac{x}{x+1}\), then its differential is :
- A. \(\frac{1}{x+1}\,dx\)
- B. \(\frac{-1}{(x+1)^2}\,dx\)
- C. \(\frac{-1}{x+1}\,dx\)
- D. \(\frac{1}{(x+1)^2}\,dx\)Correct
Explanation. \(f'(x)=\frac{(x+1)-x}{(x+1)^2}=\frac{1}{(x+1)^2}\), so \(df=\frac{1}{(x+1)^2}\,dx\).
Q4
The value of \(\int_0^1 x(1-x)^{99}\,dx\) is :
- A. \(\frac{1}{10010}\)
- B. \(\frac{1}{11000}\)
- C. \(\frac{1}{10001}\)
- D. \(\frac{1}{10100}\)Correct
Explanation. Using \(\int_0^a f(x)dx=\int_0^a f(a-x)dx\): \(\int_0^1(1-x)x^{99}dx=\frac{1}{100}-\frac{1}{101}=\frac{1}{10100}\).
Q5
The principal value of \(\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)\) is :
- A. \(\frac{\pi}{2}\)
- B. \(\frac{\pi}{3}\)
- C. \(\frac{5\pi}{6}\)
- D. \(\frac{\pi}{6}\)Correct
Explanation. \(\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\) and \(\frac{\pi}{6}\in[0,\pi]\), so the principal value is \(\frac{\pi}{6}\).
Q6
If \(A=\begin{bmatrix}2&3\\5&-2\end{bmatrix}\) be such that \(\lambda A^{-1}=A\), then \(\lambda\) is :
- A. 19Correct
- B. 17
- C. 21
- D. 14
Explanation. \(\lambda A^{-1}=A\Rightarrow A^2=\lambda I\). Here \(A^2=\begin{bmatrix}19&0\\0&19\end{bmatrix}=19I\), so \(\lambda=19\).
Q7
If \(\alpha,\beta\) and \(\gamma\) are the zeros of \(x^3+px^2+qx+r\), then \(\sum\frac{1}{\alpha}\) is :
- A. \(\frac{q}{r}\)
- B. \(-\frac{q}{r}\)Correct
- C. \(-\frac{q}{p}\)
- D. \(-\frac{p}{r}\)
Explanation. \(\sum\frac{1}{\alpha}=\frac{\alpha\beta+\beta\gamma+\gamma\alpha}{\alpha\beta\gamma}=\frac{q}{-r}=-\frac{q}{r}\).
Q8
If \((1+i)(1+2i)(1+3i)\cdots(1+ni)=x+iy\) then the value \(2\cdot5\cdot10\cdots(1+n^2)\) is :
- A. \(x^2+y^2\)Correct
- B. 1
- C. \(1+n^2\)
- D. \(i\)
Explanation. Taking modulus squared on both sides: \(|1+i|^2|1+2i|^2\cdots|1+ni|^2=2\cdot5\cdot10\cdots(1+n^2)=x^2+y^2\).
Q9
The minimum value of the function \(|3-x|+9\) is :
- A. 6
- B. 0
- C. 9Correct
- D. 3
Explanation. \(|3-x|\ge 0\) with equality at \(x=3\), so the minimum value is \(0+9=9\).
Q10
The value of \(\sum_{n=1}^{12} i^n\) is :
- A. 0Correct
- B. 1
- C. -1
- D. \(i\)
Explanation. Any four consecutive powers of \(i\) add to 0 \((i-1-i+1=0)\); 12 terms form three such groups, so the sum is 0.
Q11
If the vectors \(2\hat{i}-\hat{j}+3\hat{k},\ 3\hat{i}+2\hat{j}+\hat{k},\ \hat{i}+m\hat{j}+4\hat{k}\) are coplanar, then the value of m is :
- A. 2
- B. 3
- C. -2
- D. -3Correct
Explanation. Coplanar means the scalar triple product is 0: \(\begin{vmatrix}2&-1&3\\3&2&1\\1&m&4\end{vmatrix}=2(8-m)+11+3(3m-2)=21+7m=0\), so \(m=-3\).
Q12
The general equation of a circle with centre \((-3,-4)\) and radius 3 units is :
- A. \(x^2+y^2-6x+8y-16=0\)
- B. \(x^2+y^2-6x-8y+16=0\)
- C. \(x^2+y^2+6x-8y+16=0\)
- D. \(x^2+y^2+6x+8y+16=0\)Correct
Explanation. \((x+3)^2+(y+4)^2=9\) gives \(x^2+y^2+6x+8y+16=0\).
Q13
The solution of \(\frac{dy}{dx}+p(x)y=0\) is :
- A. \(x=ce^{-\int p\,dy}\)
- B. \(y=ce^{\int p\,dx}\)
- C. \(x=ce^{\int p\,dy}\)
- D. \(y=ce^{-\int p\,dx}\)Correct
Explanation. Separating variables: \(\frac{dy}{y}=-p\,dx\Rightarrow \log y=-\int p\,dx+\log c\), so \(y=ce^{-\int p\,dx}\).
Q14
The value of \(\int_0^\infty e^{-3x}x^2\,dx\) is :
- A. \(\frac{4}{27}\)
- B. \(\frac{7}{27}\)
- C. \(\frac{2}{27}\)Correct
- D. \(\frac{5}{27}\)
Explanation. Using \(\int_0^\infty e^{-ax}x^n\,dx=\frac{n!}{a^{n+1}}\): \(\frac{2!}{3^3}=\frac{2}{27}\).
Q15
The point of inflection of the curve \(y=(x-1)^3\) is :
- A. \((1,0)\)Correct
- B. \((0,0)\)
- C. \((1,1)\)
- D. \((0,1)\)
Explanation. \(y''=6(x-1)\) is 0 at \(x=1\) and changes sign there; \(y(1)=0\), so the point is \((1,0)\).
Q16
The angle between the lines \(\frac{x-4}{2}=\frac{y}{1}=\frac{z+1}{-2}\) and \(\frac{x-1}{4}=\frac{y+1}{-4}=\frac{z-2}{2}\) is :
- A. \(\frac{\pi}{2}\)Correct
- B. \(\frac{\pi}{4}\)
- C. \(\frac{2\pi}{3}\)
- D. \(\frac{\pi}{3}\)
Explanation. Direction ratios \((2,1,-2)\) and \((4,-4,2)\): dot product \(8-4-4=0\), so the lines are perpendicular.
Q17
Which one of the following is a binary operation on N ?
- A. MultiplicationCorrect
- B. Division
- C. Subtraction
- D. All the above
Explanation. The product of two natural numbers is a natural number, but \(2-3\) and \(2\div3\) are not in N.
Q18
Which one of the following is incorrect ?
- A. If A is a square matrix of order n, and \(\lambda\) is a scalar, then \(\text{Adj}(\lambda A)=\lambda^n(\text{Adj}\,A)\).Correct
- B. Adjoint of a symmetric matrix is also a symmetric matrix.
- C. \(A(\text{Adj}\,A)=(\text{Adj}\,A)A=|A|I\).
- D. Adjoint of a diagonal matrix is also a diagonal matrix.
Explanation. Each cofactor of \(\lambda A\) is an \((n-1)\)-order minor, so \(\text{Adj}(\lambda A)=\lambda^{n-1}\,\text{Adj}\,A\), not \(\lambda^n\).
Q19
If \(\sin x\) is the integrating factor of the linear differential equation \(\frac{dy}{dx}+Py=Q\), then P is :
- A. \(\tan x\)
- B. \(\log\sin x\)
- C. \(\cot x\)Correct
- D. \(\cos x\)
Explanation. \(e^{\int P\,dx}=\sin x\Rightarrow\int P\,dx=\log\sin x\Rightarrow P=\cot x\).
Q20
The length of the latus rectum of the parabola \(x^2=24y\) is :
- A. 8
- B. 24Correct
- C. 6
- D. 12
Explanation. Comparing with \(x^2=4ay\), \(4a=24\), so the latus rectum length is \(4a=24\).
