ISC Mathematics 2021 Specimen Paper with Answers
All 46 questions of the ISC Mathematics 2021 Specimen Paper, in printed order. Open a question to read it in full and see its answer.
- Q1(i) Let $R$ be the relation in the set $N$, given by $R = \{(x, y) : x = y + 3 \wedge y > 5\}$. Choose the correct answer from the… 1 mark · MCQ
- Q1(ii) If $A = \{1, 2, 3\}$, $B = \{x, y\}$, then the number of functions from $A$ to $B$ is: 1 mark · MCQ
- Q1(iii) Simplified value of $\sin\left[\frac{\pi}{2} - \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]$ is: 1 mark · MCQ
- Q1(iv) Which of the following statements is correct: 1 mark · MCQ
- Q1(v) If $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, then $A^2$ is equal to: 1 mark · MCQ
- Q1(vi) For what value of $k$ inverse does not exist for the matrix $\begin{bmatrix} 1 & 2 \\ k & 6 \end{bmatrix}$: 1 mark · MCQ
- Q1(vii) Identify from the given options the slope of the normal to the curve $x^2 + 3y + y^2 = 5$ at $(1, 1)$: 1 mark · MCQ
- Q1(viii) The function $f(x) = x^3 - 3x$ is strictly decreasing on: 1 mark · MCQ
- Q1(ix) The function $f(x) = \sin x + \cos x$, $x \in (0, \pi/2)$; Maximum value of $f(x)$ is: 1 mark · MCQ
- Q1(x) If $P(A) = \frac{4}{5}$ and $P(A \cap B) = \frac{7}{10}$, then the value of $P(B/A)$ is: 1 mark · MCQ
- Q1(xi) Write total number of functions from set A to set B, where $A = \{1, 2, 3\}$, $B = \{p, q, r, s\}$. 1 mark · Short answer
- Q1(xii) If A is square matrix of order 3, with $|A| = 4$, then write the value of $|-2A|$. 1 mark · Short answer
- Q1(xiii) Find the sum of order and degree of the differential equation… 1 mark · Short answer
- Q1(xiv) $A$ and $B$ appeared for an interview for two vacancies. The probability of $A$’s selection is $\frac{4}{5}$ and $B$’s selection… 1 mark · Short answer
- Q1(xv) If $A$ and $B$ are independent events such that $P(A) = 1/4$ and $P(B) = 2/3$. Find $P(A \cap B)$. 1 mark · Short answer
- Q2(a) Differentiate the function with reference to $x$: $f(x) = \cos^{-1}\left(\sqrt{\frac{1 - \cos x}{2}}\right)$ 2 marks · Short answer
- Q2(b) Find $\frac{dy}{dx}$, if $x = at^2$ and $y = 2at$. 2 marks · Short answer
- Q3 Show that the function $f : N \to N$, defined by $f(x) = 5x - 3, \forall x \in N$, is one-one function but not onto function. 2 marks · Derivation
- Q4 Using L’Hospital’s Rule, evaluate: $\lim_{x \to 0} \frac{8^x - 4^x}{4x}$ 2 marks · Short answer
- Q5(a) Evaluate: $\int \frac{\sec^2 x}{\operatorname{cosec}^2 x} dx$ 2 marks · Short answer
- Q5(b) Evaluate: $\int_0^{\pi/2} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} dx$ 2 marks · Short answer
- Q6 Solve $\frac{dy}{dx} = 1 - xy + y - x$ 2 marks · Short answer
- Q7 Prove that $\tan^{-1}\frac{1}{2} = \frac{\pi}{4} - \frac{1}{2} \cos^{-1}\left(\frac{4}{5}\right)$. 4 marks · Derivation
- Q8 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
- Q9(a) Evaluate: $\int_{-6}^3 |x + 3| dx$ 4 marks · Short answer
- Q9(b) Evaluate: $\int_0^\pi \frac{x \tan x}{\sec x + \tan x} dx$ 4 marks · Short answer
- Q10(a) 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 · Short answer
- Q10(b) A word consists of 10 different alphabets, in which there are 4 consonants and 6 vowels. Four alphabets are chosen at random… 4 marks · Short answer
- Q11(a) Using matrices, solve the following system of equations: $2x - 3y + 5z = 11$, $3x + 2y - 4z = -5$ and $x + y - 2z = -3$ 6 marks · Short answer
- Q11(b) Solve the matrix equation… 6 marks · Short answer
- Q12(a) Evaluate: $\int \tan^{-1}\sqrt{x} dx$ 6 marks · Short answer
- Q12(b) Evaluate: $\int_{-\pi/2}^{\pi/2} \frac{\cos x}{1 + e^x} dx$ 6 marks · Short answer
- Q13 A cone is inscribed in a sphere of radius 12 cm. If the volume of the cone is maximum, find its height. 6 marks · Short answer
- Q14 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 · Short answer
- Q15 [5×1] In sub-parts (i) and (ii) choose the correct options and in sub-parts (iii) to (v), answer the questions as instructed. If… 5 marks · Short answer
- Q16(a) If $\vec{a}$ and $\vec{b}$ are perpendicular vectors, $|\vec{a} + \vec{b}| = 13$ and $|\vec{a}| = 5$, find the value of… 2 marks · Numerical
- Q16(b) Find $\lambda$ if the scalar projection of $\vec{a} = \lambda\hat{i} + \hat{j} + 4\hat{k}$ on… 2 marks · Short answer
- Q17(a) Find the distance of the point $A(3, 4, 4)$ from the point, where the line joining points $P(3, -4, -5)$ and $Q(2, -3, 1)$… 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 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
- Q19 [5×1] In sub-parts (i) and (ii) choose the correct options and in sub-parts (iii) to (v), answer the questions as instructed. The… 5 marks · Short answer
- Q20(a) A company produces a commodity with ₹ 24,000 as fixed cost. The variable cost estimated to be 25% of the total revenue received… 2 marks · Short answer
- Q20(b) The total cost function for a production is given by $C(x) = \frac{3}{4}x^2 - 7x + 27$. Find the number of units produced for… 2 marks · Short answer
- Q21 Find the equation of the regression line of $y$ on $x$, if the observations $(x, y)$ are as follows… 4 marks · Short answer
- Q22(a) A carpenter has 90, 80 and 50 running feet respectively of teak wood, plywood and rosewood which is used to produce product A and… 4 marks · Short answer
- Q22(b) A farm is engaged in breeding hens. The hens are fed products A and B grown in the farm which contains nutrients P, Q and R. One… 4 marks · Short answer