ISC Mathematics 2018 Specimen Paper with Answers
All 40 questions of the ISC Mathematics 2018 Specimen Paper, in printed order. Open a question to read it in full and see its answer.
- Q1(i) A binary operation $$ defined on $\mathbb{Q} - \{1\}$ is given by $a b = a + b - ab$. Find the identity element. 2 marks · Short answer
- Q1(ii) Without expanding at any stage, find the value of the determinant… 2 marks · Short answer
- Q1(iii) Solve: $\sin^{-1}(\cos(\sin^{-1} x)) = \frac{\pi}{3}$ 2 marks · Short answer
- Q1(iv) Find the value of $k$ if $M = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$ and $M^2 - kM - I_2 = 0$. 2 marks · Short answer
- Q1(v) Evaluate: $\int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{3}{2}} x}{\sin^{\frac{3}{2}} x + \cos^{\frac{3}{2}} x} \, dx$ 2 marks · Short answer
- Q1(vi) Find $\frac{dy}{dx}$, if $x = at^2$ and $y = 2at$. 2 marks · Short answer
- Q1(vii) Find the differential equation of the family of curves $y = Ae^x + Be^{-x}$, where $A$ and $B$ are arbitrary constants. 2 marks · Short answer
- Q1(viii) Find the intervals in which the function $f(x)$ is strictly increasing where, $f(x) = 10 - 6x - 2x^2$. 2 marks · Short answer
- Q1(ix) A family has two children. What is the probability that both children are boys, given that at least one of them is a boy? 2 marks · Short answer
- Q1(x) Given that the events $A$ and $B$ are such that $P(A) = \frac{1}{2}$, $P(A \cup B) = \frac{3}{5}$ and $P(B) = k$. Find $k$ if… 2 marks · Short answer
- Q2 Let $R^+$ be the set of all positive real numbers and $f: R^+ \to [4, \infty): f(x) = x^2 + 4$. Show that inverse of $f$ exists… 4 marks · Derivation
- Q3 Using properties of determinants, prove… 4 marks · Derivation
- Q4 Prove that $\tan^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{4} - \frac{1}{2} \cos^{-1}\left(\frac{4}{5}\right)$. 4 marks · Derivation
- Q5(a) Prove that the function $f(x) = |x - 1|, x \in \mathbb{R}$, is continuous at $x = 1$ but not differentiable. 4 marks · Derivation
- Q5(b) Verify Rolle’s Theorem for the following function: $f(x) = e^x \sin x, x \in [0, \pi]$ 4 marks · Short answer
- Q6 If $y = e^{a \cos^{-1} x}$, where $-1 \leq x \leq 1$, then show that: $(1 - x^2) y_2 - xy_1 - a^2y = 0$ 4 marks · Derivation
- Q7(a) Evaluate: $\int \frac{6x+7}{\sqrt{(x-5)(x-4)}} \, dx$ 4 marks · Short answer
- Q7(b) Evaluate: $\int_{1}^{3} (x^2 + x) \, dx$, expressing as a limit of sum. 4 marks · Short answer
- Q8(a) Find the equations of the normals to the curve $y = x^3 + 2x + 6$ which are parallel to the line $x + 14y + 4 = 0$. 4 marks · Short answer
- Q8(b) A circular disc of radius 3 cm. is heated. Due to expansion its radius increases at the rate of $0.05\text{ cm/s}$. Find the rate… 4 marks · Short answer
- Q9 Solve the following differential equation: $x \frac{dy}{dx} + 2y = x^2 \log x$ 4 marks · Short answer
- Q10 Let $X$ denote the number of hours you study during a randomly selected school day. The probability that $X$ can take the values… 4 marks · Short answer
- Q11(a) Evaluate… 6 marks · Long answer
- Q11(b) Using elementary transformations, find the inverse of the matrix… 6 marks · Long answer
- Q12(a) Show that the altitude of a right circular cone of maximum volume that can be inscribed in a sphere of radius $r$ is… 6 marks · Derivation
- Q12(b) An open topped box is to be made by removing equal squares from each corner of a $3\text{ m}$ by $8\text{ m}$ rectangle sheet of… 6 marks · Long answer
- Q13 Evaluate: $\int \frac{3x+5}{x^3 - x^2 - x + 1} \, dx$ 6 marks · Long answer
- Q14 A, B and C throw a die one after the other in the same order till one of them gets a '6' and wins the game. Find their respective… 6 marks · Long answer
- Q15 [3×2] Find the area of the parallelogram whose adjacent sides are given by the vectors… 6 marks · Short answer
- Q16(a) Show that… 4 marks · Derivation
- Q16(b) Show that: $\vec{a} \cdot (\vec{b} + \vec{c}) \times (\vec{a} + 2\vec{b} + 3\vec{c}) = [\vec{a} \; \vec{b} \; \vec{c}]$ 4 marks · Derivation
- Q17(a) Find the shortest distance between the lines $\frac{x-8}{3} = \frac{y+9}{-16} = \frac{z-10}{7}$ and… 4 marks · Short answer
- Q17(b) Find the cartesian equation of the plane passing through the intersection of the planes… 4 marks · Short answer
- Q18 Using integration, find the area of the following region… 6 marks · Long answer
- Q19 [3×2] Find the cost of increasing from 100 to 200 units if the marginal cost in Rupees per unit is given by the function… 6 marks · Short answer
- Q20(a) Given that the observations are $(9,-4), (10, -3), (11,-1), (13,1), (14,3), (15,5), (16,8)$, find the two lines of regression… 4 marks · Short answer
- Q20(b) Find the regression coefficient $b_{yx}$ and $b_{xy}$ and the two lines of regression for the following data: X 2 6 4 7 5 Y 8 8 5… 4 marks · Short answer
- Q21(a) If the demand function is given by $x = \frac{600-p}{8}$, where the price is ₹ $p$ per unit and the manufacturer produces $x$… 4 marks · Short answer
- Q21(b) The fixed cost of new product is ₹ $35000$ and the variable cost per unit is ₹ $500$. If the demand function: $p = 5000 - 100x$… 4 marks · Short answer
- Q22 A toy company manufactures two types of dolls A and B. Market test and available resources have indicated that the combined… 6 marks · Long answer