ISC 2022 Mathematics - Specimen paper (Semester 2), Paper 1 with Answers
All 23 questions of the ISC 2022 Mathematics - Specimen paper (Semester 2), Paper 1, in printed order. Open a question to read it in full and see its answer.
- Q1(i) If $\int \frac{(\log x)^2}{x} \, dx = \frac{(\log x)^k}{k} + c$, then the value of $k$ is: 1 mark · MCQ
- Q1(ii) $\int_{0}^{2a} f(x) \, dx = \int_{0}^{a} f(x) \, dx + \int_{0}^{a} f(k-x) \, dx$, then the value of $k$ is: 1 mark · MCQ
- Q1(iii) The degree of the differential equation… 1 mark · MCQ
- Q1(iv) Given $\int e^x \left(\frac{x-1}{x^2}\right) \, dx = e^x f(x) + c$. Then $f(x)$ satisfying the equation is: 1 mark · MCQ
- Q1(v) Two cards are drawn out randomly from a pack of 52 cards one after the other, without replacement. The probability of first card… 1 mark · MCQ
- Q1(vi) If two balls are drawn from a bag containing 3 white, 4 black and 5 red balls. Then, the probability that the drawn balls are of… 1 mark · MCQ
- Q2(a) Evaluate: $\int \frac{x^3 - x^2 + x - 1}{x - 1} \, dx$ 2 marks · Short answer
- Q2(b) Evaluate: $\int \log_{10} x \, dx$ 2 marks · Short answer
- Q3(a) Solve the differential equation: $\operatorname{cosec} 3x \, dy - \operatorname{cosec} y \, dx = 0$ 2 marks · Short answer
- Q3(b) Solve the differential equation: $\frac{dy}{dx} = 2 - y$ 2 marks · Short answer
- Q4 Evaluate: $\int_{2}^{8} \frac{\sqrt{10-x}}{\sqrt{x} + \sqrt{10-x}} \, dx$ 2 marks · Short answer
- Q5(a) A bag contains 6 red and 5 blue balls and another bag contains 5 red and 8 blue balls. A ball is drawn from the first bag and… 4 marks · Short answer
- Q5(b) A bag contains 3 red and 4 white balls and another bag contains 2 red and 3 white balls. If one ball is drawn from the first bag… 4 marks · Short answer
- Q6 Evaluate: $\int \frac{1 + \sin x}{1 - \sin x} \, dx$ 4 marks · Short answer
- Q7 In a bolt factory, machines X, Y and Z manufacture 20%, 35% and 45% respectively of the total output. Of their output 8%, 6% and… 6 marks · Short answer
- Q8(a) Evaluate: $\int \frac{dx}{\sin x + \sin 2x}$ 6 marks · Short answer
- Q8(b) Evaluate: $\int_{0}^{\frac{\pi}{4}} \log(1 + \tan x) \, dx$ 6 marks · Short answer
- Q9 Choose the correct option for the following questions. The equation of the plane which is parallel to $2x - 3y + z = 0$ and which… 2 marks · Short answer
- Q10 Find the equation of the plane passing through the point $(1, 1, 1)$ and is perpendicular to the line… 2 marks · Short answer
- Q11 Using integration, find the area of the region bounded between the line $x = 4$ and the parabola $y^2 = 16x$. 4 marks · Short answer
- Q12 Choose the correct option for the following questions. If the regression line of $x$ on $y$ is $9x + 3y - 46 = 0$ and $y$ on $x$… 2 marks · Short answer
- Q13 For 5 observations of pairs $(x, y)$ of variables X and Y, the following results are obtained: $\sum x = 15$, $\sum y = 25$… 2 marks · Short answer
- Q14 A manufacturer wishes to produce two commodities A and B. The number of units of material, labour and equipment needed to produce… 4 marks · Short answer