ISC Mathematics 2023 Specimen Paper with Answers
All 46 questions of the ISC Mathematics 2023 Specimen Paper, in printed order. Open a question to read it in full and see its answer.
- Q1(i) Which of the following is NOT an equivalence relation on $\mathbb{Z}$? 1 mark · MCQ
- Q1(ii) Let $A$ be the set of all $50$ cards numbered from $1$ to $50$. Let $f: A \to \mathbb{N}$ be a function defined by $f(x) =$ card… 1 mark · MCQ
- Q1(iii) The value of $\tan^{-1} \sqrt{3} - \sec^{-1}(-2)$ is equal to 1 mark · MCQ
- Q1(iv) If $A = \begin{pmatrix} 1 & b \\\\ 0 & 1 \end{pmatrix}$, then $A^n$ ($n \in \mathbb{N}$) is equal to 1 mark · MCQ
- Q1(v) If $A$ is a $3 \times 3$ matrix such that… 1 mark · MCQ
- Q1(vi) If… 1 mark · MCQ
- Q1(vii) If $\sin^{-1} x + \sin^{-1} y = \frac{\pi}{2}$, then $\frac{dy}{dx}$ is equal to 1 mark · MCQ
- Q1(viii) If $f(x) = \frac{4-x^2}{4x-x^3}$ then the function is: 1 mark · MCQ
- Q1(ix) The degree of the differential equation… 1 mark · MCQ
- Q1(x) If $A$ and $B$ are two events such that $P(A) > 0$ and $P(B) \neq 1$, then $P(\bar{A}/\bar{B})$ is 1 mark · MCQ
- Q1(xi) Write the smallest equivalence relation on the set $A = \\{a, b, c\\}$ 1 mark · Short answer
- Q1(xii) If $\begin{pmatrix} 3 & 5 \\\\ 7 & 9 \end{pmatrix} = X + Y$, where $X$ is skew-symmetric matrix and $Y$ is symmetric matrix. Find… 1 mark · Short answer
- Q1(xiii) If $\int \log 2x \\, dx = x \log 2x - k + c$ where $k$ is a function of $x$, then find $k$. 1 mark · Short answer
- Q1(xiv) $50$ tickets in a box are numbered $00, 01, 02, \dots, 49$. One ticket is drawn randomly from the box. Find the probability of… 1 mark · Short answer
- Q1(xv) A speaks truth in $60\\%$ of cases and B speaks truth in $90\\%$ of the cases. In what percentage of cases are they likely to… 1 mark · Short answer
- Q2(a) If $y = \sqrt{\sin x + y}$, then find $\frac{dy}{dx}$. 2 marks · Short answer
- Q2(b) Find the point on the curve $y^2 = 4x + 8$ for which the abscissa and ordinate changes at the same rate. 2 marks · Short answer
- Q3 Let $f: \mathbb{R} \to \mathbb{R}$ defined as $f(x) = 2x - 3$. Find $f^{-1}(x)$ domain and range of $f^{-1}(x)$ 2 marks · Short answer
- Q4 The function $f$ is defined for all $x \in \mathbb{R}$. The line with equation $y = 6x - 1$ is the tangent to the graph of $f$ at… 2 marks · Short answer
- Q5(i) Evaluate: $\int [\sin(\log x) + \cos(\log x)] \\, dx$ 2 marks · Short answer
- Q5(ii) Evaluate: $\int_0^{\frac{\pi}{2}} \frac{a \sin x + b \cos x}{\sin x + \cos x} \\, dx$ 2 marks · Short answer
- Q6 Solve the differential equation: $(1 + y^2)(1 + \log x) \\, dx + x \\, dy = 0$ 2 marks · Short answer
- Q7 If $\cos^{-1} \frac{x}{a} + \cos^{-1} \frac{y}{b} = \alpha$, prove that… 4 marks · Derivation
- Q8 If $y = (x + \sqrt{1 + x^2})^n$, then prove that $(1 + x^2) \frac{d^2y}{dx^2} + x \frac{dy}{dx} = n^2 y$. 4 marks · Derivation
- Q9(i) Evaluate: $\int_{-1}^2 |x^3 - x| \\, dx$ 4 marks · Short answer
- Q9(ii) Evaluate: $\int \sqrt{\csc x - 1} \\, dx$ 4 marks · Short answer
- Q10(i) A student answers a multiple choice question with $5$ alternatives, of which exactly one is correct. The probability that he… 4 marks · Short answer
- Q10(ii) A candidate takes three tests in succession and the probability of passing the first test is $\frac{1}{2}$. The probability of… 4 marks · Short answer
- Q11 Find $A^{-1}$, if $A = \begin{pmatrix} 1 & 2 & 0 \\\\ -2 & -1 & -2 \\\\ 0 & -1 & 1 \end{pmatrix}$. Using $A^{-1}$, solve the… 6 marks · Long answer
- Q12(i) Find the particular solution of the differential equation given below: $(1 + x^2) \\, dy = (\tan^{-1} x - y) \\, dx$, given that… 6 marks · Long answer
- Q12(ii) Evaluate: $\int \frac{1}{\sin^4 x + \cos^4 x} \\, dx$ 6 marks · Long answer
- Q13(i) Find the maximum volume of the cylinder which can be inscribed in a sphere of radius $3\sqrt{3}\text{ cm}$. (find the answer in… 6 marks · Long answer
- Q13(ii) Find the coordinates of a point on the curve $y = x^2 + 7x + 2$ which is closest to the straight line $y = 3x - 3$. 6 marks · Long answer
- Q14 A biased four-sided die with faces labelled 1, 2, 3 and 4 is rolled and recorded. Let $X$ be the result obtained when the die is… 6 marks · Long answer
- Q15 In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed. If… 5 marks · Short answer
- Q16(i) Find the area of the triangle whose adjacent sides are $\hat{i} + 4\hat{j} - \hat{k}$ and $\hat{i} + \hat{j} + 2\hat{k}$. 2 marks · Short answer
- Q16(ii) For any two non-zero vectors $\vec{a}$ and $\vec{b}$, if $|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$, then find the… 2 marks · Short answer
- Q17(i) Show that $\frac{4-x}{-1} = \frac{y+3}{-4} = \frac{z+1}{7}$ and $\frac{x-1}{2} = \frac{y+1}{-3} = \frac{z+10}{8}$ intersect each… 4 marks · Derivation
- Q17(ii) Find the equation of the plane passing through the points $(0, 4, -3)$ and $(6, -4, 3)$ if the sum of their intercepts on three… 4 marks · Short answer
- Q18 If the area is bounded by the parabola $y^2 = 16x$ and the line $y = 4mx$ is $\frac{1}{12}\text{ sq units}$, then using… 4 marks · Numerical
- Q19 In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed. Which… 5 marks · Short answer
- Q20(i) The total revenue function $R(x) = x + 2x^3 - 3\cdot 5x^2$. Find the point where marginal revenue curve cuts the co-ordinate axis. 2 marks · Short answer
- Q20(ii) For the revenue function $R(x) = \frac{bx}{a+x}$, show that the marginal revenue function is increasing for all $b < 0$ and… 2 marks · Derivation
- Q21 The line of regression of marks in Maths ($X$) and marks in English ($Y$) for a class of $50$ students is $3Y - 5X + 180 = 0$… 4 marks · Short answer
- Q22(i) Solve the following linear programming problem graphically and interpret your result of $Z = 2x - 5y$ subject to the constraints… 4 marks · Short answer
- Q22(ii) The standard weight of a special purpose brick is $5\text{ kg}$ and it must contain two basic ingredients $B_1$ and $B_2$. $B_1$… 4 marks · Short answer