The 20 one-mark questions from the Supplementary 2021 12th Standard Mathematics public exam, in paper order, with the correct option marked and a short explanation.
Q1
The inverse of \(\begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix}\) is :
- A. \(\begin{bmatrix} 3 & -1 \\ -5 & -3 \end{bmatrix}\)
- B. \(\begin{bmatrix} 2 & -1 \\ -5 & 3 \end{bmatrix}\)Correct
- C. \(\begin{bmatrix} -3 & 5 \\ 1 & -2 \end{bmatrix}\)
- D. \(\begin{bmatrix} -2 & 5 \\ 1 & -3 \end{bmatrix}\)
Explanation. \(|A| = 6-5 = 1\), so \(A^{-1} = \frac{1}{1}\begin{bmatrix} 2 & -1 \\ -5 & 3 \end{bmatrix}\).
Q2
The centre of the hyperbola \(\frac{(x-1)^2}{16} - \frac{(y+1)^2}{25} = 1\) is :
- A. \(\left(\frac{1}{2}, -\frac{1}{2}\right)\)
- B. \((-1, 1)\)
- C. \((1, -1)\)Correct
- D. \((0, 0)\)
Explanation. Comparing with \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\), the centre is \((h, k) = (1, -1)\).
Q3
The order and degree of the differential equation \(\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^{\frac{1}{3}} + x^{\frac{1}{4}} = 0\) are :
- A. 2, 6
- B. 2, 3Correct
- C. 2, 4
- D. 3, 3
Explanation. The highest derivative is \(\frac{d^2y}{dx^2}\), so order 2. Writing \(\left(\frac{dy}{dx}\right)^{\frac{1}{3}} = -\left(\frac{d^2y}{dx^2} + x^{\frac{1}{4}}\right)\) and cubing gives \(\left(\frac{d^2y}{dx^2} + x^{\frac{1}{4}}\right)^3 = -\frac{dy}{dx}\), so the degree is 3.
Q4
A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six sided die and 1, 2, 3, 4 of a four sided die is rolled and the sum is determined. If the random variable X denote the sum, then the number of elements in the inverse image of 7 is :
- A. 3
- B. 1
- C. 4Correct
- D. 2
Explanation. \(X^{-1}(7) = \{(3,4), (4,3), (5,2), (6,1)\}\), which has 4 elements.
Q5
If \(|z| = 1\), then the value of \(\frac{1+z}{1+\bar{z}}\) is :
- A. \(\frac{1}{z}\)
- B. \(z\)Correct
- C. 1
- D. \(\bar{z}\)
Explanation. Since \(z\bar{z} = |z|^2 = 1\), \(\bar{z} = \frac{1}{z}\). Then \(\frac{1+z}{1+\frac{1}{z}} = \frac{z(1+z)}{z+1} = z\).
Q6
The value of \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^2 x \cos x \, dx\) is :
- A. 0
- B. \(\frac{3}{2}\)
- C. \(\frac{2}{3}\)Correct
- D. \(\frac{1}{2}\)
Explanation. Put \(t = \sin x\): the integral is \(\left[\frac{\sin^3 x}{3}\right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}} = \frac{1}{3} - \left(-\frac{1}{3}\right) = \frac{2}{3}\).
Q7
The function \(f(x) = x^2\), in the interval \([0, \infty)\) is :
- A. cannot be determined
- B. increasing functionCorrect
- C. increasing and decreasing function
- D. decreasing function
Explanation. \(f'(x) = 2x \ge 0\) on \([0, \infty)\) (zero only at \(x = 0\)), so \(f\) is increasing there.
Q8
The volume of the parallelepiped with its edges represented by the vectors \(\hat{i}+\hat{j},\ \hat{i}+2\hat{j},\ \hat{i}+\hat{j}+\pi\hat{k}\) is :
- A. \(\pi\)Correct
- B. \(\frac{\pi}{2}\)
- C. \(\frac{\pi}{4}\)
- D. \(\frac{\pi}{3}\)
Explanation. Volume \(= \left|\begin{vmatrix} 1 & 1 & 0 \\ 1 & 2 & 0 \\ 1 & 1 & \pi \end{vmatrix}\right| = \pi(2-1) = \pi\).
Q9
In the set \(\mathbb{R}\) of real numbers '\(*\)' is defined as follows. Which one of the following is not a binary operation on \(\mathbb{R}\) ?
- A. \(a * b = a\)
- B. \(a * b = \min(a, b)\)
- C. \(a * b = a^b\)Correct
- D. \(a * b = \max(a, b)\)
Explanation. \(a^b\) need not be a real number, e.g. \((-1)^{\frac{1}{2}}\) is not real, so \(a * b = a^b\) is not closed on \(\mathbb{R}\). The other three always give real numbers.
Q10
The position of a particle 's' moving at any time t is given by \(s(t) = 5t^2 - 2t - 8\). The time at which the particle is at rest, is :
- A. 1
- B. 0
- C. 3
- D. \(\frac{1}{3}\)Correct
Explanation. At rest, \(v = s'(t) = 0\). As printed, \(10t - 2 = 0\) gives \(t = \frac{1}{5}\), which is not an option. The paper has a misprint: the textbook question is \(s(t) = 3t^2 - 2t - 8\), for which \(6t - 2 = 0\) gives \(t = \frac{1}{3}\) (intended answer).
Q11
If the function \(f(x) = \frac{1}{12}\) for \(a < x < b\), represents a probability density function of a continuous random variable X, then which of the following cannot be the values of a and b ?
- A. 7 and 19
- B. 0 and 12
- C. 16 and 24Correct
- D. 5 and 17
Explanation. \(\int_a^b \frac{1}{12}\,dx = 1\) gives \(b - a = 12\). For 16 and 24, \(b - a = 8\), so these cannot be the values.
Q12
If P(x, y) be any point on \(16x^2 + 25y^2 = 400\) with foci \(F_1(3, 0)\) and \(F_2(-3, 0)\), then \(PF_1 + PF_2\) is :
- A. 10Correct
- B. 8
- C. 12
- D. 6
Explanation. \(\frac{x^2}{25} + \frac{y^2}{16} = 1\) is an ellipse with \(a = 5\). For any point on it, \(PF_1 + PF_2 = 2a = 10\).
Q13
If the planes \(\vec{r} \cdot (2\hat{i} - \lambda\hat{j} + \hat{k}) = 3\) and \(\vec{r} \cdot (4\hat{i} + \hat{j} - \mu\hat{k}) = 5\) are parallel, then the values of \(\lambda\) and \(\mu\) are respectively :
- A. \(-\frac{1}{2}, -2\)Correct
- B. \(\frac{1}{2}, -2\)
- C. \(\frac{1}{2}, 2\)
- D. \(-\frac{1}{2}, 2\)
Explanation. Normals must be parallel: \(\frac{2}{4} = \frac{-\lambda}{1} = \frac{1}{-\mu}\), so \(\lambda = -\frac{1}{2}\) and \(\mu = -2\).
Q14
A zero of \(x^3 + 64\) is :
- A. \(4i\)
- B. 0
- C. \(-4\)Correct
- D. 4
Explanation. \((-4)^3 + 64 = -64 + 64 = 0\), so \(-4\) is a zero.
Q15
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, \(\frac{dy}{y} = -P\,dx\), so \(\log y = -\int P\,dx + \log c\), i.e. \(y = ce^{-\int P\,dx}\).
Q16
\(\int_0^{\frac{\pi}{2}} \sin^7 x \, dx =\)
- A. \(\frac{\pi}{2}\)
- B. \(\int_0^{\frac{\pi}{2}} \cos^7 x \, dx\)Correct
- C. 0
- D. 1
Explanation. Using \(\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx\) with \(a = \frac{\pi}{2}\), \(\sin^7 x\) becomes \(\cos^7 x\). (Both equal \(\frac{16}{35}\).)
Q17
The value of \(\sin^{-1}\left(\frac{1}{2}\right) + \cos^{-1}\left(\frac{1}{2}\right)\) is :
- A. 0
- B. \(\frac{\pi}{2}\)Correct
- C. \(\frac{\pi}{3}\)
- D. \(\pi\)
Explanation. \(\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}\) for \(|x| \le 1\); here \(\frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{2}\).
Q18
If A, B and C are invertible matrices of some order, then which one of the following is not true ?
- A. \(\det A^{-1} = (\det A)^{-1}\)
- B. \(\text{adj } A = |A|A^{-1}\)
- C. \((ABC)^{-1} = C^{-1}B^{-1}A^{-1}\)
- D. \(\text{adj}(AB) = (\text{adj } A)(\text{adj } B)\)Correct
Explanation. The correct rule is \(\text{adj}(AB) = (\text{adj } B)(\text{adj } A)\) (reversal law), so option (4) is not true; the other three are standard results.
Q19
The value of the complex number \((i^{25})^3\) is equal to :
- A. 1
- B. \(i\)
- C. \(-i\)Correct
- D. \(-1\)
Explanation. \(i^{25} = i^{24} \cdot i = i\), so \((i^{25})^3 = i^3 = -i\).
Q20
If we measure the side of a cube to be 4 cm with an error of 0.1 cm, then the error in calculation of the volume is (in cubic cm) :
- A. 2
- B. 0.4
- C. 4.8Correct
- D. 0.45
Explanation. \(V = x^3\), so \(dV = 3x^2\,dx = 3(16)(0.1) = 4.8\) cubic cm.
