ISC Mathematics 2018 Question Paper with Answers
All 40 questions of the ISC Mathematics 2018 Question Paper, in printed order. Open a question to read it in full and see its answer.
- Q1(i) The binary operation $* : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is defined as $a * b = 2a + b$. Find $(2 * 3) * 4$. 2 marks · Short answer
- Q1(ii) If $A = \begin{pmatrix} 5 & a \\ b & 0 \end{pmatrix}$ and $A$ is a symmetric matrix, show that $a = b$. 2 marks · Derivation
- Q1(iii) Solve: $3\tan^{-1}x + \cot^{-1}x = \pi$ 2 marks · Short answer
- Q1(iv) Without expanding at any stage, find the value of: $\begin{vmatrix} a & b & c \\ a+2x & b+2y & c+2z \\ x & y & z \end{vmatrix}$ 2 marks · Numerical
- Q1(v) Find the value of constant $k$ so that the function $f(x)$ defined as… 2 marks · Numerical
- Q1(vi) Find the approximate change in the volume $V$ of a cube of side $x$ metres caused by decreasing the side by $1\%$. 2 marks · Short answer
- Q1(vii) Evaluate: $\int \frac{x^3 + 5x^2 + 4x + 1}{x^2} \, dx$. 2 marks · Short answer
- Q1(viii) Find the differential equation of the family of concentric circles $x^2 + y^2 = a^2$. 2 marks · Short answer
- Q1(ix) If $A$ and $B$ are events such that $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$ and $P(A \cap B) = \frac{1}{4}$, then find… 2 marks · Short answer
- Q1(x) In a race, the probabilities of $A$ and $B$ winning the race are $\frac{1}{3}$ and $\frac{1}{6}$ respectively. Find the… 2 marks · Short answer
- Q2 If the function $f(x) = \sqrt{2x-3}$ is invertible then find its inverse. Hence prove that $(f \circ f^{-1})(x) = x$. 4 marks · Derivation
- Q3 If $\tan^{-1}a + \tan^{-1}b + \tan^{-1}c = \pi$, prove that $a + b + c = abc$. 4 marks · Derivation
- Q4 Use properties of determinants to solve for $x$: $\begin{vmatrix} x+a & b & c \\ c & x+b & a \\ a & b & x+c \end{vmatrix} = 0$… 4 marks · Short answer
- Q5(a) Show that the function $f(x) = \begin{cases} x^2, & x \le 1 \\ \frac{1}{x}, & x > 1 \end{cases}$ is continuous at $x = 1$ but not… 4 marks · Derivation
- Q5(b) Verify Rolle’s theorem for the following function: $f(x) = e^{-x} \sin x$ on $[0, \pi]$. 4 marks · Short answer
- Q6 If $x = \tan\left(\frac{1}{a} \log y\right)$, prove that $(1 + x^2) \frac{d^2y}{dx^2} + (2x - a) \frac{dy}{dx} = 0$. 4 marks · Derivation
- Q7 Evaluate: $\int \tan^{-1}\sqrt{x} \, dx$. 4 marks · Short answer
- Q8(a) Find the points on the curve $y = 4x^3 - 3x + 5$ at which the equation of the tangent is parallel to the x-axis. 4 marks · Short answer
- Q8(b) Water is dripping out from a conical funnel of semi-vertical angle $\frac{\pi}{4}$ at the uniform rate of… 4 marks · Short answer
- Q9(a) Solve: $\sin x \frac{dy}{dx} - y = \sin x \tan\frac{x}{2}$. 4 marks · Short answer
- Q9(b) The population of a town grows at the rate of $10\%$ per year. Using differential equation, find how long will it take for the… 4 marks · Short answer
- Q10(a) Using matrices, solve the following system of equations: $2x - 3y + 5z = 11$ $3x + 2y - 4z = -5$ $x + y - 2z = -3$ 6 marks · Short answer
- Q10(b) Using elementary transformation, find the inverse of the matrix… 6 marks · Short answer
- Q11 $A$ speaks truth in $60\%$ of the cases, while $B$ in $40\%$ of the cases. In what percent of cases are they likely to contradict… 4 marks · Short answer
- Q12 A cone is inscribed in a sphere of radius $12\text{ cm}$. If the volume of the cone is maximum, find its height. 6 marks · Short answer
- Q13(a) Evaluate: $\int \frac{x-1}{\sqrt{x^2-x}} \, dx$. 6 marks · Short answer
- Q13(b) Evaluate: $\int_{0}^{\pi/2} \frac{\cos^2 x}{1 + \sin x \cos x} \, dx$. 6 marks · Short answer
- Q14 From a lot of 6 items containing 2 defective items, a sample of 4 items are drawn at random. Let the random variable $X$ denote… 6 marks · Short answer
- Q15 [3×2] Find $\lambda$ if the scalar projection of $\vec{a} = \lambda\hat{\imath} + \hat{\jmath} + 4\hat{k}$ on… 6 marks · Short answer
- Q16(a) If $A, B, C$ are three non-collinear points with position vectors $\vec{a}, \vec{b}, \vec{c}$, respectively, then show that the… 4 marks · Derivation
- Q16(b) Show that the four points $A, B, C$ and $D$ with position vectors $4\hat{\imath} + 5\hat{\jmath} + \hat{k}$… 4 marks · Derivation
- Q17(a) Draw a rough sketch of the curve and find the area of the region bounded by curve $y^2 = 8x$ and the line $x = 2$. 4 marks · Drawing
- Q17(b) Sketch the graph of $y = |x + 4|$. Using integration, find the area of the region bounded by the curve $y = |x + 4|$ and $x = -6$… 4 marks · Drawing
- Q18 Find the image of a point having position vector $3\hat{\imath} - 2\hat{\jmath} + \hat{k}$ in the plane… 6 marks · Short answer
- Q19 [3×2] Given the total cost function for $x$ units of a commodity as: $C(x) = \frac{1}{3}x^3 + 3x^2 - 16x + 2$. Find: (i) Marginal… 6 marks · Short answer
- Q20(a) Find the line of regression of $y$ on $x$ from the following table. $x$ 1 2 3 4 5 $y$ 7 6 5 4 3 Hence, estimate the value of $y$… 4 marks · Short answer
- Q20(b) From the given data: Variable $x$ $y$ Mean 6 8 Standard Deviation 4 6 and correlation coefficient: $\frac{2}{3}$. Find… 4 marks · Short answer
- Q21(a) A product can be manufactured at a total cost $C(x) = \frac{x^2}{100} + 100x + 40$, where $x$ is the number of units produced… 4 marks · Short answer
- Q21(b) A manufacturer’s marginal cost function is $\frac{500}{\sqrt{2x+25}}$. Find the cost involved to increase production from 100… 4 marks · Short answer
- Q22 A manufacturing company makes two types of teaching aids A and B of Mathematics for Class X. Each type of A requires 9 labour… 6 marks · Short answer