ISC Mathematics 2027 Specimen Paper with Answers
All 48 questions of the ISC Mathematics 2027 Specimen Paper, in printed order. Open a question to read it in full and see its answer.
- Q1(i) A relation $R$ is defined on $\mathbb{Z}$ as $a R b$ if and only if $a^2 - 7ab + 6b^2 = 0$. Then $R$ is: 1 mark · MCQ
- Q1(ii) If $\theta = \sin^{-1}x + \cos^{-1}x - \tan^{-1}x$, $x \ge 0$, then the smallest interval in which $\theta$ lies is: 1 mark · MCQ
- Q1(iii) Let $A$ be the area of a triangle having vertices $(x_1, y_1)$, $(x_2, y_2)$ and $(x_3, y_3)$. Which of the following is correct? 1 mark · MCQ
- Q1(iv) A particle moves along the curve $6y = x^3 + 2$. At what point(s) on the curve, is the y-coordinate changing 8 times as fast as… 1 mark · MCQ
- Q1(v) Statement I: If a satellite's altitude function $h(t)$ has a positive derivative ($h'(t) > 0$), the satellite is moving away from… 1 mark · MCQ
- Q1(vi) For what value(s) of $k$ do the tangents of two curves $x = y^2$ and $xy = k$ cut at right angles? 1 mark · MCQ
- Q1(vii) Evaluate the nature of the point $(0,0)$ for the curve $y = x^4$, given that $f''(0) = 0$. 1 mark · MCQ
- Q1(viii) The expression for finding the area of the shaded region is: 1 mark · MCQ
- Q1(ix) Shown below is a solved anti-differentiation problem to obtain $f(x)$: $\frac{d}{dx}f(x) = \frac{1}{x(\log x)^2}$ such that… 1 mark · MCQ
- Q1(x) On solving $\int -\frac{3}{\sqrt{1-x^2}} dx$, Anil's answer was $\int -\frac{3}{\sqrt{1-x^2}} dx = 3 \cos^{-1} x + c$ and… 1 mark · MCQ
- Q1(xi) If $\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$, $\vec{b} = -\hat{i} + 2\hat{j} - 4\hat{k}$… 1 mark · MCQ
- Q1(xii) The point of intersection of the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{3-z}{-4}$ and… 1 mark · MCQ
- Q1(xiii) An objective function $Z = ax + by$ is maximum at points $(8,10)$ and $(7,12)$. If $a, b \ge 0$ and $ab = 72$, then the maximum… 1 mark · MCQ
- Q1(xiv) Statement I: Every Linear Programming Problem has at least one optimal solution. Statement II: If a Linear Programming Problem… 1 mark · MCQ
- Q1(xv) If $P(A) = m$, $P(B/\bar{A}) = 3m$, $P(B/A) = 6m$, then what will be $P(A/B)$? 1 mark · MCQ
- Q1(xvi) The curves given above can be considered as $f(x)$ and $f^{-1}(x)$. Observe the given graph: 1 mark · Assertion-reason
- Q1(xvii) $\int_{-1}^1 \frac{x^3}{\cos x} dx = 0$ 1 mark · Assertion-reason
- Q1(xviii) Using the properties of matrices, explain why the following concept is incorrect: Since $(a+b)(a-b) = a^2 - b^2$, therefore… 1 mark · Short answer
- Q1(xix) If the solution of the differential equation $\frac{dy}{dx} = \frac{ax+3}{2y+5}$ represents a circle, then find the value of $a$. 1 mark · Short answer
- Q1(xx) The probability distribution of random variable X is given below: X 1 2 3 4 5 P(X) m 3m a 5m b If $P(X \le 2) = 0.28$ and… 1 mark · Short answer
- Q2 A straight line $L_\theta$ has vector equation $\vec{r} = 5\hat{i} + \lambda(5\hat{i} + \sin\theta \hat{j} + \cos\theta \hat{k})$… 2 marks · Short answer
- Q3 In a school, for a period of 8 working days, it is equally likely that Tanishka is present or absent on any given day. What is… 2 marks · Short answer
- Q4 Solve the following differential equation: $\frac{dx}{dy} = \frac{x}{y} + \frac{f(x/y)}{f'(x/y)}$ 2 marks · Short answer
- Q5(i) Determine the values of constants $p$ and $q$ such that the function… 2 marks · Short answer
- Q5(ii) Observe the graph given below: State the value(s) of $x$ where the function has removable discontinuity. State the value(s) of… 2 marks · Short answer
- Q6(i) Parag is working on a school project on right-angled triangles. He draws a right-angled triangle PQR with $\angle R = 90^\circ$… 2 marks · Short answer
- Q6(ii) The equation given below has equal roots: $ax^2 + \sin^{-1}(x^2 - 2x + 2) + \cos^{-1}(x^2 - 2x + 2) = 0$ What is the value of… 2 marks · Short answer
- Q7(i) Find the vector projection of $\vec{B} = 6\hat{i} + 3\hat{j} + 2\hat{k}$ on $\vec{A} = \hat{i} - 2\hat{j} - 2\hat{k}$ and the… 2 marks · Short answer
- Q7(ii) If the position vectors of three points A, B, C are respectively $\hat{i} + \hat{j} + \hat{k}$, $2\hat{i} + 3\hat{j} - 4\hat{k}$… 2 marks · Short answer
- Q8 Three landmarks of Dehradun are joined by straight roads. Clock tower of Dehradun is considered as the 'origin'. IMA (I) is 3 km… 2 marks · Short answer
- Q9 The feasible region determined by some constraints is represented by the shaded region in the graph given below: Formulate the… 3 marks · Short answer
- Q10 If $x = 3z + 1$ and $y = f(x)$, then prove that $9\frac{d^2y}{dx^2} = \frac{d^2y}{dz^2}$. 3 marks · Short answer
- Q11 Given the function $f(x) = \log\left[\frac{\sqrt{1+x^2}+x}{\sqrt{1+x^2}-x}\right] + \tan^{-1}\left(\frac{2x}{1-x^2}\right)$: Find… 3 marks · Short answer
- Q12(i) A vector $\vec{n}$ of magnitude 8 units is inclined to x-axis at $45^\circ$, y-axis at $60^\circ$ and at an acute angle with… 3 marks · Short answer
- Q12(ii) The planes $\pi_1$ and $\pi_2$ have equations $2x + 6y - 2z = 5$ and $3x + 9y + pz = -\frac{51}{2}$ respectively. Verify that the… 3 marks · Short answer
- Q13(i) If $x, y, z$ are distinct non zero real numbers, prove that… 3 marks · Short answer
- Q13(ii) A school is organising its Annual Day function and plans to decorate every chair with a ribbon and every table with a cover… 3 marks · Short answer
- Q14 Let $\int_1^5 3f(x) dx = 12$. Show that $\int_5^1 f(x) dx = -4$. Find the value of… 3 marks · Short answer
- Q15(i) The diagram below shows the graph of $f(x) = 2x\sqrt{a^2 - x^2}$, for $-1 \le x \le a$, where $a > 1$. The line L is the tangent… 3 marks · Short answer
- Q15(ii) A line with equation $y = -3x + 9$ intersects the axes at the points P and Q. A parabola of the form $y = ax^2 + c$, where… 3 marks · Short answer
- Q16 The graph of $f(x) = x - 6\sqrt{x} + 1$ is shown below. State the natural domain of $f(x)$. Does $f(x)$ have an inverse function?… 5 marks · Short answer
- Q17(i) An engineering team is designing a section of a new roller coaster track. The vertical profile of the track for a specific… 5 marks · Case based
- Q17(ii) A large industrial water tank is shaped like an inverted right circular cone with a semi-vertical angle of $\tan^{-1}(0.5)$… 5 marks · Case based
- Q18(i) Evaluate the following integrals: Evaluate: $\int \ln x dx$. Hence, evaluate: $\int \frac{\ln[\ln(\frac{1+x}{1-x})]}{1-x^2} dx$. 5 marks · Short answer
- Q18(ii) Evaluate the following integrals: Evaluate: $\int \tan x dx$. Hence, evaluate… 5 marks · Short answer
- Q19(i) In a game show, a contestant is shown three closed doors: Door A, Door B and Door C. Behind one door is a laptop while the other… 5 marks · Short answer
- Q19(ii) In a large organisation, only 1 out of every 1,500 emails contains a harmful attachment. An automated filtering system is used to… 5 marks · Short answer
- Q20 A ball is thrown vertically downwards from the top of a cliff, and its position is tracked from when it is first thrown until it… 5 marks · Short answer