ISC Mathematics 2023 Question Paper with Answers
All 54 questions of the ISC Mathematics 2023 Question Paper, in printed order. Open a question to read it in full and see its answer.
- Q1(i) A relation $R$ on $\{1, 2, 3\}$ is given by $R = \{(1, 1), (2, 2), (1, 2), (3, 3), (2, 3)\}$. Then the relation $R$ is: 1 mark · MCQ
- Q1(ii) If $A$ is a square matrix of order 3, then $|2A|$ is equal to: 1 mark · MCQ
- Q1(iii) If the following function is continuous at $x = 2$ then the value of $k$ will be… 1 mark · MCQ
- Q1(iv) An edge of a variable cube is increasing at the rate of $10\text{ cm/sec}$. How fast will the volume of the cube increase if the… 1 mark · MCQ
- Q1(v) Let $f(x) = x^3$ be a function with domain $\{0, 1, 2, 3\}$. Then domain of $f^{-1}$ is: 1 mark · MCQ
- Q1(vi) For the curve $y^2 = 2x^3 - 7$, the slope of the normal at $(2, 3)$ is: 1 mark · MCQ
- Q1(vii) Evaluate: $\int \frac{x}{x^2 + 1}\,dx$ 1 mark · MCQ
- Q1(viii) The derivative of $\log x$ with respect to $\frac{1}{x}$ is: 1 mark · MCQ
- Q1(ix) The interval in which the function $f(x) = 5 + 36x - 3x^2$ increases will be: 1 mark · MCQ
- Q1(x) Evaluate: $\int_{-1}^1 x^{17}\cos^4 x\,dx$ 1 mark · MCQ
- Q1(xi) Solve the differential equation: $\frac{dy}{dx} = \operatorname{cosec} y$ 1 mark · Short answer
- Q1(xii) For what value of $k$ the matrix $\begin{bmatrix} 0 & k \\ -6 & 0 \end{bmatrix}$ is a skew symmetric matrix? 1 mark · One word
- Q1(xiii) Evaluate: $\int_0^1 |2x + 1|\,dx$ 1 mark · Numerical
- Q1(xiv) Evaluate: $\int \frac{1 + \cos x}{\sin^2 x}\,dx$ 1 mark · Short answer
- Q1(xv) A bag contains 19 tickets, numbered from 1 to 19. Two tickets are drawn randomly in succession with replacement. Find the… 1 mark · Numerical
- Q2(i) If $f(x) = [4 - (x - 7)^3]^{\frac{1}{5}}$ is a real invertible function, then find $f^{-1}(x)$. 2 marks · Short answer
- Q2(ii) Let $A = \mathbb{R} - \{2\}$ and $B = \mathbb{R} - \{1\}$. If $f : A \to B$ is a function defined by $f(x) = \frac{x - 1}{x - 2}$… 2 marks · Derivation
- Q3 Evaluate the following determinant without expanding. $\begin{vmatrix} 5 & 5 & 5 \\ a & b & c \\ b+c & c+a & a+b \end{vmatrix}$ 2 marks · Short answer
- Q4 The probability of the event $A$ occurring is $\frac{1}{3}$ and of the event $B$ occurring is $\frac{1}{2}$. If $A$ and $B$ are… 2 marks · Numerical
- Q5 Solve for $x$: $5\tan^{-1} x + 3\cot^{-1} x = 2\pi$ 2 marks · Short answer
- Q6(i) Evaluate: $\int \cos^{-1}(\sin x)\,dx$ 2 marks · Short answer
- Q6(ii) If $\int x^5 \cos(x^6)\,dx = k \sin(x^6) + C$, find the value of $k$. 2 marks · Short answer
- Q7 If $\tan^{-1}\left(\frac{x-1}{x+1}\right) + \tan^{-1}\left(\frac{2x-1}{2x+1}\right) = \tan^{-1}\left(\frac{23}{36}\right)$ then… 4 marks · Derivation
- Q8 If $y = e^{ax} \cos bx$, then prove that $\frac{d^2y}{dx^2} - 2a\frac{dy}{dx} + (a^2 + b^2)y = 0$ 4 marks · Derivation
- Q9(i) In a company, 15% of the employees are graduates and 85% of the employees are non-graduates. As per the annual report of the… 4 marks · Short answer
- Q9(ii) A problem in Mathematics is given to three students A, B and C. Their chances of solving the problem are $\frac{1}{2}$… 4 marks · Short answer
- Q10(i) Solve the differential equation: $(1 + y^2)\,dx = (\tan^{-1} y - x)\,dy$ 4 marks · Short answer
- Q10(ii) Solve the differential equation: $(x^2 - y^2)\,dx + 2xy\,dy = 0$ 4 marks · Short answer
- Q11 Use matrix method to solve the following system of equations: $\frac{2}{x} + \frac{3}{y} + \frac{10}{z} = 4$… 6 marks · Short answer
- Q12(i) Prove that the semi-vertical angle of the right circular cone of given volume and least curved area is $\cot^{-1}\sqrt{2}$. 6 marks · Derivation
- Q12(ii) A running track of $440\text{ m}$ is to be laid out enclosing a football field. The football field is in the shape of a rectangle… 6 marks · Numerical
- Q13(i) Evaluate: $\int \frac{3e^{2x} - 2e^x}{e^{2x} + 2e^x - 8}\,dx$ 6 marks · Short answer
- Q13(ii) Evaluate: $\int \frac{2}{(1 - x)(1 + x^2)}\,dx$ 6 marks · Short answer
- Q14 A box contains 30 fruits, out of which 10 are rotten. Two fruits are selected at random one by one without replacement from the… 6 marks · Short answer
- Q15(i) If $|\vec{a}| = 3$, $|\vec{b}| = \frac{\sqrt{2}}{3}$ and $\vec{a} \times \vec{b}$ is a unit vector then the angle between… 1 mark · MCQ
- Q15(ii) The distance of the point $2\hat{i} + \hat{j} - \hat{k}$ from the plane $\vec{r} \cdot (\hat{i} - 2\hat{j} + 4\hat{k}) = 9$ will… 1 mark · MCQ
- Q15(iii) Find the area of the parallelogram whose diagonals are $\hat{i} - 3\hat{j} + \hat{k}$ and $\hat{i} + \hat{j} + \hat{k}$. 1 mark · Numerical
- Q15(iv) Write the equation of the plane passing through the point $(2, 4, 6)$ and making equal intercepts on the coordinate axes. 1 mark · Short answer
- Q15(v) If the two vectors $3\hat{i} + a\hat{j} + \hat{k}$ and $2\hat{i} - \hat{j} + 8\hat{k}$ are perpendicular to each other, then find… 1 mark · Numerical
- Q16(i) If $A(1, 2, -3)$ and $B(-1, -2, 1)$ are the end points of a vector $\vec{AB}$ then find the unit vector in the direction of… 2 marks · Short answer
- Q16(ii) If $\hat{a}$ is unit vector and $(2\vec{x} - 3\hat{a}) \cdot (2\vec{x} + 3\hat{a}) = 91$, find the value of $|\vec{x}|$. 2 marks · Numerical
- Q17(i) Find the equation of the plane passing through the point $(1, 1, -1)$ and perpendicular to the planes $x + 2y + 3z = 7$ and… 4 marks · Short answer
- Q17(ii) A line passes through the point $(2, -1, 3)$ and is perpendicular to the lines… 4 marks · Short answer
- Q18 Find the area of the region bounded by the curve $x^2 = 4y$ and the line $x = 4y - 2$. 4 marks · Numerical
- Q19(i) If the demand function is given by $p = 1500 - 2x - x^2$ then find the marginal revenue when $x = 10$: 1 mark · MCQ
- Q19(ii) If the two regression coefficients are $0.8$ and $0.2$, then the value of coefficient of correlation $r$ will be: 1 mark · MCQ
- Q19(iii) Out of the two regression lines $x + 2y - 5 = 0$ and $2x + 3y = 8$, find the line of regression of $y$ on $x$. 1 mark · Short answer
- Q19(iv) The cost function is $C(x) = 3x^2 - 6x + 5$. Find the average cost when $x = 2$. 1 mark · Numerical
- Q19(v) The fixed cost of a product is ₹ 30,000 and its variable cost per unit is ₹ 800. If the demand function is $p(x) = 4500 - 100x$… 1 mark · Numerical
- Q20(i) The total cost function for $x$ units is given by $C(x) = \sqrt{6x + 5} + 2500$. Show that the marginal cost decreases as the… 2 marks · Derivation
- Q20(ii) The average revenue function is given by $AR = 25 - \frac{x}{4}$. Find total revenue function and marginal revenue function. 2 marks · Short answer
- Q21 Solve the following Linear Programming Problem graphically: Maximise $Z = 5x + 2y$ subject to: $x - 2y \le 2$ $3x + 2y \le 12$… 4 marks · Short answer
- Q22(i) The following table shows the Mean, the Standard Deviation and the coefficient of correlation of two variables $x$ and $y$… 4 marks · Short answer
- Q22(ii) An analyst analysed 102 trips of a travel company. He studied the relation between travel expenses ($y$) and the duration ($x$)… 4 marks · Short answer