ISC Mathematics 2019 Question Paper with Answers
All 40 questions of the ISC Mathematics 2019 Question Paper, in printed order. Open a question to read it in full and see its answer.
- Q1(i) If $f : \mathbb{R} \to \mathbb{R}$, $f(x) = x^3$ and $g : \mathbb{R} \to \mathbb{R}$, $g(x) = 2x^2 + 1$, and $\mathbb{R}$ is the… 2 marks · Short answer
- Q1(ii) Solve: $\sin(2\tan^{-1}x) = 1$ 2 marks · Short answer
- Q1(iii) Using determinants, find the values of $k$, if the area of triangle with vertices $(-2, 0)$, $(0, 4)$ and $(0, k)$ is $4$ square… 2 marks · Short answer
- Q1(iv) Show that $(A + A')$ is symmetric matrix, if $A = \begin{pmatrix} 2 & 4 \\\\ 3 & 5 \end{pmatrix}$. 2 marks · Derivation
- Q1(v) $f(x) = \frac{x^2 - 9}{x - 3}$ is not defined at $x = 3$. What value should be assigned to $f(3)$ for continuity of $f(x)$ at… 2 marks · Short answer
- Q1(vi) Prove that the function $f(x) = x^3 - 6x^2 + 12x + 5$ is increasing on $\mathbb{R}$. 2 marks · Derivation
- Q1(vii) Evaluate: $\int \frac{\sec^2 x}{\operatorname{cosec}^2 x}\\,dx$ 2 marks · Short answer
- Q1(viii) Using L'Hospital's Rule, evaluate: $\lim_{x \to 0} \frac{8^x - 4^x}{4x}$ 2 marks · Short answer
- Q1(ix) Two balls are drawn from an urn containing 3 white, 5 red and 2 black balls, one by one without replacement. What is the… 2 marks · Short answer
- Q1(x) If events $A$ and $B$ are independent, such that $P(A) = \frac{3}{5}$, $P(B) = \frac{2}{3}$, find $P(A \cup B)$. 2 marks · Short answer
- Q2 If $f : A \to A$ and $A = \mathbb{R} - \left\\{\frac{8}{5}\right\\}$, show that the function $f(x) = \frac{8x + 3}{5x - 8}$ is… 4 marks · Short answer
- Q3(a) Solve for $x$: $\tan^{-1}\left(\frac{x - 1}{x - 2}\right) + \tan^{-1}\left(\frac{x + 1}{x + 2}\right) = \frac{\pi}{4}$ 4 marks · Short answer
- Q3(b) If $\sec^{-1} x = \operatorname{cosec}^{-1} y$, show that $\frac{1}{x^2} + \frac{1}{y^2} = 1$ 4 marks · Derivation
- Q4 Using properties of determinants prove that… 4 marks · Derivation
- Q5(a) Show that the function $f(x) = |x - 4|$, $x \in \mathbb{R}$ is continuous, but not differentiable at $x = 4$. 4 marks · Derivation
- Q5(b) Verify the Lagrange's mean value theorem for the function: $f(x) = x + \frac{1}{x}$ in the interval $[1, 3]$ 4 marks · Short answer
- Q6 If $y = e^{\sin^{-1} x}$ and $z = e^{-\cos^{-1} x}$, prove that $\frac{dy}{dz} = e^{\frac{\pi}{2}}$ 4 marks · Derivation
- Q7 A 13 m long ladder is leaning against a wall, touching the wall at a certain height from the ground level. The bottom of the… 4 marks · Numerical
- Q8(a) Evaluate: $\int \frac{x(1 + x^2)}{1 + x^4}\\,dx$ 4 marks · Short answer
- Q8(b) Evaluate: $\int_{-6}^3 |x + 3|\\,dx$ 4 marks · Short answer
- Q9 Solve the differential equation: $\frac{dy}{dx} = \frac{x + y + 2}{2(x + y) - 1}$ 4 marks · Short answer
- Q10 Bag A contains 4 white balls and 3 black balls, while Bag B contains 3 white balls and 5 black balls. Two balls are drawn from… 4 marks · Numerical
- Q11 Solve the following system of linear equations using matrix method: $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 9$… 6 marks · Long answer
- Q12(a) The volume of a closed rectangular metal box with a square base is $4096\text{ cm}^3$. The cost of polishing the outer surface of… 6 marks · Numerical
- Q12(b) Find the point on the straight line $2x + 3y = 6$, which is closest to the origin. 6 marks · Numerical
- Q13 Evaluate: $\int_0^\pi \frac{x\tan x}{\sec x + \tan x}\\,dx$ 6 marks · Short answer
- Q14(a) Given three identical Boxes A, B and C, Box A contains 2 gold and 1 silver coin, Box B contains 1 gold and 2 silver coins and Box… 6 marks · Numerical
- Q14(b) Determine the binomial distribution where mean is 9 and standard deviation is $\frac{3}{2}$. Also, find the probability of… 6 marks · Numerical
- Q15 [3×2] If $\vec{a}$ and $\vec{b}$ are perpendicular vectors, $|\vec{a} + \vec{b}| = 13$ and $|\vec{a}| = 5$, find the value of… 6 marks · Short answer
- Q16(a) If $\vec{a} = -\hat{i} - 2\hat{j} + 3\hat{k}$, $\vec{b} = 2\hat{i} + 3\hat{j} - 5\hat{k}$, prove that $\vec{a}$ and… 4 marks · Derivation
- Q16(b) If $\vec{a}$ and $\vec{b}$ are non-collinear vectors, find the value of $x$ such that the vectors… 4 marks · Numerical
- Q17(a) Find the equation of the plane passing through the intersection of the planes $2x + 2y - 3z - 7 = 0$ and $2x + 5y + 3z - 9 = 0$… 4 marks · Short answer
- Q17(b) Find the equation of the lines passing through the point $(2, 1, 3)$ and perpendicular to the lines… 4 marks · Short answer
- Q18 Draw a rough sketch and find the area bounded by the curve $x^2 = y$ and $x + y = 2$. 6 marks · Drawing
- Q19 [3×2] A company produces a commodity with ₹ 24,000 as fixed cost. The variable cost estimated to be 25% of the total revenue… 6 marks · Short answer
- Q20(a) The following results were obtained with respect to two variables $x$ and $y$: $\sum x = 15$, $\sum y = 25$, $\sum xy = 83$… 4 marks · Short answer
- Q20(b) Find the equation of the regression line of $y$ on $x$, if the observations $(x, y)$ are as follows… 4 marks · Short answer
- Q21(a) The cost function of a product is given by $C(x) = \frac{x^3}{3} - 45x^2 - 900x + 36$ where $x$ is the number of units produced… 4 marks · Numerical
- Q21(b) The marginal cost function of $x$ units of a product is given by $\text{MC} = 3x^2 - 10x + 3$. The cost of producing one unit is… 4 marks · Short answer
- Q22 A carpenter has 90, 80 and 50 running feet respectively of teak wood, plywood and rosewood which is used to produce product A and… 6 marks · Drawing