ISC Mathematics 2017 Question Paper with Answers
All 38 questions of the ISC Mathematics 2017 Question Paper, in printed order. Open a question to read it in full and see its answer.
- Q1(i) If the matrix $\begin{pmatrix} 6 & -x^2 \\ 2x - 15 & 10 \end{pmatrix}$ is symmetric, find the value of $x$. 3 marks · Numerical
- Q1(ii) If $y - 2x - k = 0$ touches the conic $3x^2 - 5y^2 = 15$, find the value of $k$. 3 marks · Numerical
- Q1(iii) Prove that $\frac{1}{2}\cos^{-1}\left(\frac{1-x}{1+x}\right) = \tan^{-1}\sqrt{x}$. 3 marks · Derivation
- Q1(iv) Using L’Hospital’s Rule, evaluate: $\lim_{x \to \pi/2} \left(x \tan x - \frac{\pi}{2} \sec x\right)$ 3 marks · Short answer
- Q1(v) Evaluate: $\int \frac{1}{x^2}\sin^2\left(\frac{1}{x}\right) dx$ 3 marks · Short answer
- Q1(vi) Evaluate: $\int_{0}^{\pi/4} \log(1 + \tan\theta) \, d\theta$ 3 marks · Short answer
- Q1(vii) By using the data $\bar{x} = 25$, $\bar{y} = 30$, $b_{yx} = 1.6$ and $b_{xy} = 0.4$, find: The regression equation $y$ on $x$… 3 marks · Short answer
- Q1(viii) A problem is given to three students whose chances of solving it are $\frac{1}{4}, \frac{1}{5}$ and $\frac{1}{3}$ respectively… 3 marks · Short answer
- Q1(ix) If $a + ib = \frac{x+iy}{x-iy}$, prove that $a^2 + b^2 = 1$ and $\frac{b}{a} = \frac{2xy}{x^2-y^2}$. 3 marks · Derivation
- Q1(x) Solve: $\frac{dy}{dx} = 1 - xy + y - x$ 3 marks · Short answer
- Q2(a) Using properties of determinants, prove that… 5 marks · Derivation
- Q2(b) Given that: $A = \begin{pmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{pmatrix}$ and… 5 marks · Short answer
- Q3(a) Solve the equation for $x$: $\sin^{-1} x + \sin^{-1}(1 - x) = \cos^{-1} x, \quad x \neq 0$ 5 marks · Short answer
- Q3(b) If $A$, $B$ and $C$ are the elements of Boolean algebra, simplify the expression $(A' + B')(A + C') + B'(B + C)$. Draw the… 5 marks · Short answer
- Q4(a) Verify Lagrange’s mean value theorem for the function: $f(x) = x(1 - \log x)$ and find the value of ‘c’ in the interval $[1, 2]$. 5 marks · Numerical
- Q4(b) Find the coordinates of the centre, foci and equation of directrix of the hyperbola $x^2 - 3y^2 - 4x = 8$. 5 marks · Short answer
- Q5(a) If $y = \cos(\sin x)$, show that: $\frac{d^2y}{dx^2} + \tan x \frac{dy}{dx} + y \cos^2 x = 0$ 5 marks · Derivation
- Q5(b) Show that the surface area of a closed cuboid with square base and given volume is minimum when it is a cube. 5 marks · Derivation
- Q6(a) Evaluate: $\int \frac{\sin 2x}{(1 + \sin x)(2 + \sin x)} \, dx$ 5 marks · Short answer
- Q6(b) Draw a rough sketch of the curve $y^2 = 4x$ and find the area of the region enclosed by the curve and the line $y = x$. 5 marks · Short answer
- Q7(a) Calculate the Spearman’s rank correlation coefficient for the following data and interpret the result: $X$ 35 54 80 95 73 73 35… 5 marks · Numerical
- Q7(b) Find the line of best fit for the following data, treating $x$ as dependent variable (Regression equation $x$ on $y$): $X$ 14 12… 5 marks · Short answer
- Q8(a) In a class of 60 students, 30 opted for Mathematics, 32 opted for Biology and 24 opted for both Mathematics and Biology. If one… 5 marks · Short answer
- Q8(b) Bag A contains 1 white, 2 blue and 3 red balls. Bag B contains 3 white, 3 blue and 2 red balls. Bag C contains 2 white, 3 blue… 5 marks · Short answer
- Q9(a) Prove that locus of $z$ is circle and find its centre and radius if $\frac{z-i}{z-1}$ is purely imaginary. 5 marks · Derivation
- Q9(b) Solve: $(x^2 - yx^2) \, dy + (y^2 + xy^2) \, dx = 0$ 5 marks · Short answer
- Q10(a) If $\vec{a}, \vec{b}, \vec{c}$ are three mutually perpendicular vectors of equal magnitude, prove that… 5 marks · Short answer
- Q10(b) Find the value of $\lambda$ for which the four points with position vectors $6\hat{i} - 7\hat{j}$… 5 marks · Numerical
- Q11(a) Show that the lines $\frac{x-4}{1} = \frac{y+3}{-4} = \frac{z+1}{7}$ and $\frac{x-1}{2} = \frac{y+1}{-3} = \frac{z+10}{8}$… 5 marks · Short answer
- Q11(b) Find the equation of the plane passing through the point $(1, -2, 1)$ and perpendicular to the line joining the points… 5 marks · Short answer
- Q12(a) A fair die is rolled. If face 1 turns up, a ball is drawn from Bag A. If face 2 or 3 turns up, a ball is drawn from Bag B. If… 5 marks · Short answer
- Q12(b) An urn contains 25 balls of which 10 balls are red and the remaining green. A ball is drawn at random from the urn, the colour is… 5 marks · Short answer
- Q13(a) A machine costs ₹ 60,000 and its effective life is estimated to be 25 years. A sinking fund is to be created for replacing the… 5 marks · Short answer
- Q13(b) A farmer has a supply of chemical fertilizer of type A which contains 10% nitrogen and 6% phosphoric acid and of type B which… 5 marks · Short answer
- Q14(a) The demand for a certain product is represented by the equation $p = 500 + 25x - \frac{x^2}{3}$ in rupees where $x$ is the number… 5 marks · Short answer
- Q14(b) A bill of ₹ 60,000 payable 10 months after date was discounted for ₹ 57,300 on 30th June, 2007. If the rate of interest was… 5 marks · Short answer
- Q15(a) The price relatives and weights of a set of commodities are given below: Commodity A B C D Price relatives 125 120 127 119… 5 marks · Short answer
- Q15(b) Construct 3 yearly moving averages from the following data and show on a graph against the original data: Year 2000 2001 2002… 5 marks · Short answer