ISC Mathematics 2020 Question Paper with Answers
All 40 questions of the ISC Mathematics 2020 Question Paper, in printed order. Open a question to read it in full and see its answer.
- Q1(i) Determine whether the binary operation $$ on $\mathbb{R}$ defined by $a b = |a - b|$ is commutative. Also, find the value of… 2 marks · Short answer
- Q1(ii) Prove that: $\tan^2(\sec^{-1} 2) + \cot^2(\operatorname{cosec}^{-1} 3) = 11$. 2 marks · Derivation
- Q1(iii) Without expanding at any stage, find the value of the determinant… 2 marks · Numerical
- Q1(iv) If… 2 marks · Numerical
- Q1(v) Find $\frac{dy}{dx}$ if $x^3 + y^3 = 3axy$. 2 marks · Short answer
- Q1(vi) The edge of a variable cube is increasing at the rate of $10\text{ cm/sec}$. How fast is the volume of the cube increasing when… 2 marks · Numerical
- Q1(vii) Evaluate: $\int_{4}^{5} |x - 5| \, dx$. 2 marks · Short answer
- Q1(viii) Form a differential equation of the family of the curves $y^2 = 4ax$. 2 marks · Short answer
- Q1(ix) A bag contains 5 white, 7 red and 4 black balls. If four balls are drawn one by one with replacement, what is the probability… 2 marks · Short answer
- Q1(x) Let $A$ and $B$ be two events such that $P(A) = \frac{1}{2}$, $P(B) = p$ and $P(A \cup B) = \frac{3}{5}$, find ‘$p$’ if $A$ and… 2 marks · Numerical
- Q2 If the function $f : \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \frac{3x + 4}{5x - 7}, x \ne \frac{7}{5}$ and… 4 marks · Derivation
- Q3(a) If $\cos^{-1}\frac{x}{2} + \cos^{-1}\frac{y}{3} = \theta$, then prove that $9x^2 - 12xy\cos\theta + 4y^2 = 36\sin^2\theta$. 4 marks · Derivation
- Q3(b) Evaluate: $\cos\left(2\cos^{-1} x + \sin^{-1} x\right)$ at $x = \frac{1}{5}$. 4 marks · Numerical
- Q4 Using properties of determinants, show that… 4 marks · Derivation
- Q5 Verify Rolle’s theorem for the function, $f(x) = -1 + \cos x$ in the interval $[0, 2\pi]$. 4 marks · Short answer
- Q6 If $y = e^{m\sin^{-1} x}$, prove that $(1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} = m^2 y$. 4 marks · Derivation
- Q7(a) The equation of tangent at $(2, 3)$ on the curve $y^2 = px^3 + q$ is $y = 4x - 7$. Find the values of ‘$p$’ and ‘$q$’. 4 marks · Short answer
- Q7(b) Using L’Hospital’s rule, evaluate: $\lim_{x \to 0} \frac{xe^x - \log(1 + x)}{x^2}$. 4 marks · Numerical
- Q8(a) Evaluate: $\int \frac{dx}{\sqrt{5x - 4x^2}}$. 4 marks · Short answer
- Q8(b) Evaluate: $\int \sin^3 x \cos^4 x \, dx$. 4 marks · Short answer
- Q9 Solve the differential equation $(1 + x^2)\frac{dy}{dx} = 4x^2 - 2xy$. 4 marks · Short answer
- Q10 Three persons A, B and C shoot to hit a target. Their probabilities of hitting the target are $\frac{5}{6}$, $\frac{4}{5}$ and… 4 marks · Short answer
- Q11 Solve the following system of linear equations using matrices: $x - 2y = 10, \quad 2x - y - z = 8, \quad -2y + z = 7$ 6 marks · Short answer
- Q12(a) Show that the radius of a closed right circular cylinder of given surface area and maximum volume is equal to half of its height. 6 marks · Derivation
- Q12(b) Prove that the area of right-angled triangle of given hypotenuse is maximum when the triangle is isosceles. 6 marks · Derivation
- Q13(a) Evaluate: $\int \tan^{-1}\sqrt{\frac{1-x}{1+x}} \, dx$. 6 marks · Short answer
- Q13(b) Evaluate: $\int \frac{2x + 7}{x^2 - x - 2} \, dx$. 6 marks · Short answer
- Q14 The probability that a bulb produced in a factory will fuse after 150 days of use is $0.05$. Find the probability that out of 5… 6 marks · Short answer
- Q15 [3×2] Write a vector of magnitude of 18 units in the direction of the vector $\hat{i} - 2\hat{j} - 2\hat{k}$. Find the angle… 6 marks · Short answer
- Q16(a) Prove that $\vec{a} \cdot [(\vec{b} + \vec{c}) \times (\vec{a} + 3\vec{b} + 4\vec{c})] = [\vec{a} \quad \vec{b} \quad \vec{c}]$. 4 marks · Derivation
- Q16(b) Using vectors, find the area of the triangle whose vertices are… 4 marks · Numerical
- Q17(a) Find the image of the point $(3, -2, 1)$ in the plane $3x - y + 4z = 2$. 4 marks · Short answer
- Q17(b) Determine the equation of the line passing through the point $(-1, 3, -2)$ and perpendicular to the lines… 4 marks · Short answer
- Q18 Draw a rough sketch of the curves $y^2 = x$ and $y^2 = 4 - 3x$ and find the area enclosed between them. 6 marks · Drawing
- Q19 [3×2] The selling price of a commodity is fixed at ₹ $60$ and its cost function is $C(x) = 35x + 250$. (i) Determine the profit… 6 marks · Short answer
- Q20(a) The correlation coefficient between $x$ and $y$ is $0.6$. If the variance of $x$ is $225$, the variance of $y$ is $400$, mean of… 4 marks · Short answer
- Q20(b) Find the regression coefficients $b_{yx}$, $b_{xy}$ and correlation coefficient ‘$r$’ for the following data… 4 marks · Numerical
- Q21(a) The marginal cost of the production of the commodity is $30 + 2x$, it is known that fixed costs are ₹ $200$, find: The total… 4 marks · Short answer
- Q21(b) The total cost function of a firm is given by $C(x) = \frac{1}{3}x^3 - 5x^2 + 30x - 15$ where the selling price per unit is given… 4 marks · Short answer
- Q22 A company uses three machines to manufacture two types of shirts, half sleeves and full sleeves. The number of hours required per… 6 marks · Short answer