Q.1 (v) If \(y = \sec(\tan^{-1} x)\), then \(\dfrac{dy}{dx}\) at \(x = 1\) is _____.
Mathematics & Statistics Part 2
Every question from past board papers on Mathematics & Statistics Part 2, newest paper first. Open any question for its full worked answer.
Maharashtra HSC Mathematics and Statistics (Science) Board Paper March 2026 with Solutions
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Q.1 (vi) The approximate value of the function \(f(x) = x^3 - 3x + 5\) at \(x = 1.99\) is _____.
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Q.1 (vii) \(\displaystyle\int_1^2 \frac{1}{x^2}\cdot e^{\frac{1}{x}}\,dx =\) _____.
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Q.1 (viii) If the p.d.f. of a continuous r.v. \(X\) is
\(f(x) = \dfrac{x + 2}{18}\), for \(-2 \lt x \lt 4\)
\(\quad = 0\), otherwise
then \(P(|X| \lt 1) =\) _____ -
Q.2 (iii) Evaluate : \(\displaystyle\int \frac{2x}{1 + x^2}dx\)
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Q.2 (iv) Write the degree of the differential equation \((y''')^2 + 3(y'')^3 + 3xy' + 5y = 0\)
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Q.9 Find \(\dfrac{dy}{dx}\), if \(\sqrt{x} + \sqrt{y} = \sqrt{a}\).
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Q.10 Find \(\dfrac{d^2y}{dx^2}\), if \(y = x^3 + 7x^2 - 2x - 9\).
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Q.11 Test whether the function \(f(x) = x^3 + 6x^2 + 12x - 7\) is increasing or decreasing for all \(x \in R\).
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Q.12 A stone is dropped into a quiet lake and waves in the form of circles are generated. Radius of the circular wave increases at the rate of 3 cm/sec. How fast the area enclosed is increasing when the radius is 8 cm?
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Q.13 Evaluate : \(\displaystyle\int\sqrt{1 + \sin 2x}\cdot dx\)
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Q.14 Given that, \(X \sim B(n, p)\), if \(n = 10,\ E(X) = 8\) then find \(\text{Var}(X)\).
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Q.21 If \(y = f(u)\) is a differentiable function of \(u\) and \(u = g(x)\) is a differentiable function of \(x\) then prove that \(y\) is a differentiable function of \(x\) and \(\dfrac{dy}{dx} = \dfrac{dy}{du}\times\dfrac{du}{dx}\).
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Q.22 Verify LMVT for the function \(f(x) = \log x\), on \([1, e]\).
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Q.23 Evaluate : \(\displaystyle\int\frac{\sin x}{\sin 3x}dx\)
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Q.24 Solve the D.E. \(3e^x\tan y\,dx + (1 + e^x)\sec^2 y\,dy = 0\).
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Q.25 Find \(E(X)\) and \(V(X)\), where X is the number obtained on uppermost face, when a fair die is thrown.
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Q.26 A pair of dice is thrown 4 times. If getting a doublet is considered as success, find the probability of two successes.
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Q.31 Evaluate : \(\displaystyle\int\frac{3x^2 + 4x - 5}{(x^2 - 1)(x + 2)}\cdot dx\)
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Q.32 Prove that : \(\displaystyle\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\cdot dx\)
Hence, find \(\displaystyle\int_{\pi/6}^{\pi/3}\sin^2 x\cdot dx\) -
Q.33 Find the area enclosed between the circle \(x^2 + y^2 = 1\) and the line \(x + y = 1\) lying in the first quadrant.
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Q.34 Solve the differential equation \(x^2\dfrac{dy}{dx} = x^2 + xy + y^2\).
Maharashtra HSC Mathematics and Statistics (Science) Board Paper June 2026 with Solutions
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Q.1 (v) If \(y = \log_x a\) then \(\frac{dy}{dx} =\) _____.
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Q.1 (vi) If \(f(x) = \frac{\sin^{-1}x}{\sqrt{1-x^2}}\) and \(g(x) = e^{\sin^{-1}x}\) then \(\int f(x)\cdot g(x)\,dx =\) _____.
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Q.1 (vii) The area bounded by the line \(y = 2x\), X-axis and the lines \(x = -1\), \(x = 4\) is _____ sq. units.
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Q.1 (viii) In a Binomial Distribution, \(E(X) = 6\), \(\mathrm{Var}(X) = 4.2\) then the value of \(n\) is _____
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Q.2 (iii) The displacement of a particle at time \(t\) is given by \(S = 3t^2 + 4t + 1\). Find its velocity at time \(t\).
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Q.2 (iv) Write the order of differential equation \(\left(\frac{d^2y}{dx^2}\right)^{\frac{5}{2}} = \sqrt[3]{\frac{dy}{dx}}\)
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Q.9 Test whether the function \(f(x) = x - \frac{1}{x},\ x \in R,\ x \ne 0\) is increasing or decreasing.
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Q.10 Evaluate: \(\int_{-1}^{1} |x|\,dx\)
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Q.11 Evaluate: \(\int x\log x\,dx\)
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Q.12 Find the area bounded by curve \(x^2 + y = 0\), X-axis and lines \(x = 1\) and \(x = 4\).
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Q.13 The probability distribution of r.v. \(X\) is as follows:
\(X = x\) 0 1 2 3 4 \(P(X = x)\) 0.1 \(k\) \(2k\) \(2k\) \(k\) Find: (i) \(k\) (ii) \(F(2)\)
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Q.14 Evaluate: \(\int\frac{e^x(1 + x)}{\cos^2(xe^x)}\,dx\)
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Q.21 If \(y = \log\left[\frac{x + \sqrt{x^2 + a^2}}{\sqrt{x^2 + a^2} - x}\right]\), find \(\frac{dy}{dx}\).
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Q.22 A wire of length 36 cm is bent to form a rectangle. Find its dimensions, if the area of the rectangle is maximum.
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Q.23 Verify Rolle's theorem for the function \(f(x) = x^2 - 5x + 9,\ x \in [1, 4]\)
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Q.24 Solve the differential equation \(x\frac{dy}{dx} - \sqrt{x^2 + y^2} = y\)
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Q.25 The following is the p.d.f. of continuous r.v.:
\(f(x) = \frac{x}{8};\ 0 \lt x \lt 4\)
\(= 0\), otherwise
Find (i) expression for c.d.f. of \(X\). (ii) \(F(x)\) at 0.5 and 5 -
Q.26 If the p.m.f. of r.v. \(X\) be \(P(X = x) = {}^4C_x\left(\frac{5}{9}\right)^x\left(\frac{4}{9}\right)^{4-x}\) for \(x = 0, 1, 2, 3, 4\); \(= 0\) otherwise, then find \(E(X)\) and \(\mathrm{Var}(X)\).
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Q.31 If \(x = f(t)\) and \(y = g(t)\) are differentiable functions of \(t\), so that \(y\) is differentiable function of \(x\) and \(\frac{dx}{dt} \ne 0\) then prove that \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\).
Hence find \(\frac{dy}{dx}\), if \(x = \sin t,\ y = \cos t\). -
Q.32 Evaluate: \(\int\frac{2}{3 + 2\sin^2 x + 5\cos^2 x}\,dx\)
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Q.33 Prove that, \(\int_{-a}^{a} f(x)\,dx = 2\int_{0}^{a} f(x)\,dx\), if \(f(x)\) is even
\(= 0\), if \(f(x)\) is odd -
Q.34 A body cools according to Newton's law from 100°C to 60°C in 20 minutes. The temperature of the surrounding being 20°C. How long will it take to cool down to 40°C?
Maharashtra HSC Mathematics and Statistics (Science) Board Paper March 2025 with Solutions
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Q.1 (v) Let \(f(1) = 3\), \(f'(1) = -\dfrac{1}{3}\), \(g(1) = -4\) and \(g'(1) = -\dfrac{8}{3}\). The derivative of \(\sqrt{[f(x)]^2 + [g(x)]^2}\) w.r.t. \(x\) at \(x = 1\) is _____.
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Q.1 (vi) If the mean and variance of a binomial distribution are 18 and 12 respectively, then \(n\) is equal to _____.
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Q.1 (vii) The value of \(\int x^x(1 + \log x)\,dx\) is equal to _____.
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Q.1 (viii) The area bounded by the line \(y = x\), X-axis and the lines \(x = -1\) and \(x = 4\) is equal to _____. (in square units)
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Q.2 (iii) Write the order of the differential equation \(\sqrt{1 + \left(\dfrac{dy}{dx}\right)^2} = \left(\dfrac{d^2y}{dx^2}\right)^{\frac32}\)
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Q.2 (iv) Write the condition for the function \(f(x)\), to be strictly increasing, for all \(x \in R\).
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Q.9 Divide the number 20 into two parts such that sum of their squares is minimum.
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Q.10 Evaluate: \(\int x^9\cdot\sec^2(x^{10})\,dx\)
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Q.11 Evaluate: \(\int \dfrac{1}{25 - 9x^2}\,dx\)
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Q.12 Evaluate: \(\displaystyle\int_{-\pi/4}^{\pi/4} \dfrac{1}{1 - \sin x}\,dx\)
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Q.13 Find the area of the region bounded by the parabola \(y^2 = 16x\) and its latus rectum.
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Q.14 Suppose that \(X\) is waiting time in minutes for a bus and its p.d.f. is given by:
\(f(x) = \dfrac15\), for \(0 \le x \le 5\)
\(= 0\), otherwise.Find the probability that: (i) waiting time is between 1 to 3 minutes. (ii) waiting time is more than 4 minutes.
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Q.21 Find the \(n^{\text{th}}\) order derivative of \(\log x\).
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Q.22 The displacement of a particle at time \(t\) is given by \(s = 2t^3 - 5t^2 + 4t - 3\). Find the velocity and displacement at the time when the acceleration is 14 ft/sec\(^2\).
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Q.23 Find the equations of tangent and normal to the curve \(y = 2x^3 - x^2 + 2\) at point \(\left(\dfrac12, 2\right)\).
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Q.24 Three coins are tossed simultaneously, \(X\) is the number of heads. Find the expected value and variance of \(X\).
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Q.25 Solve the differential equation: \(x\dfrac{dy}{dx} = x\cdot\tan\left(\dfrac yx\right) + y\).
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Q.26 Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Find the probability that: (i) all the five cards are spades. (ii) none is spade.
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Q.31 If \(x = f(t)\) and \(y = g(t)\) are differentiable functions of \(t\) so that \(y\) is a function of \(x\) and if \(\dfrac{dx}{dt} \ne 0\) then prove that \(\dfrac{dy}{dx} = \dfrac{\;\dfrac{dy}{dt}\;}{\dfrac{dx}{dt}}\).
Hence find the derivative of \(7^x\) w.r.t. \(x^7\).
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Q.32 Evaluate: \(\displaystyle\int e^{\sin^{-1}x}\left(\dfrac{x + \sqrt{1 - x^2}}{\sqrt{1 - x^2}}\right)dx\)
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Q.33 Prove that: \(\displaystyle\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx\)
Hence evaluate: \(\displaystyle\int_0^3 \dfrac{\sqrt x}{\sqrt x + \sqrt{3 - x}}\,dx\)
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Q.34 If a body cools from 80°C to 50°C at room temperature of 25°C in 30 minutes, find the temperature of the body after 1 hour.
Maharashtra HSC Mathematics and Statistics (Science) Board Paper July 2025 with Solutions
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Q.1 (v) If \(y\) is a function of \(x\) and \(\log(x + y) = xy\) then the value of \(\left(\frac{dy}{dx}\right)\) at \(x = 0\) is _____
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Q.1 (vi) If the displacement of a particle at time \(t\) is given by \(S = 2t^3 - 5t^2 + 4t - 3\), then its acceleration at time \(t = 1\) is _____
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Q.1 (vii) The solution of the D.E. \(\sec^2 x \cdot \tan y\,dx + \sec^2 y \cdot \tan x\,dy = 0\) is _____
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Q.1 (viii) If \(X\) is waiting time in minutes for a bus and its p.d.f. is given by \(f(x) = \frac{1}{5}\), for \(0 \le x \le 5\), \(= 0\), otherwise, then the probability that waiting time is between 1 and 3 is _____
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Q.2 (iii) Find \(\frac{dy}{dx}\), if \(y = \sin(\log x)\).
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Q.2 (iv) Evaluate : \(\int \frac{1}{\sqrt{x}}\,dx\).
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Q.9 Find \(\frac{dy}{dx}\), if \(x\sqrt{x} + y\sqrt{y} = a\sqrt{a}\).
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Q.10 Show that the tangent to the curve \(y = x^3 - 6x^2 + x + 3\) at the point \((0, 3)\) is parallel to the line \(y = x + 5\).
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Q.11 Check whether the conditions of Rolle's theorem are satisfied by the function \(f(x) = x^2 - 4x + 3,\ x \in [1, 3]\)
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Q.12 Evaluate : \(\int \frac{dx}{x + x^{-10}}\)
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Q.13 Evaluate : \(\int_0^{-1} e^{-x}\,dx\)
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Q.14 If \(X \sim B(n, p)\) and \(E(X) = 6,\ Var(X) = 4.2\), then find \(n\) and \(p\).
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Q.21 Water is being poured at the rate of 27 m\(^3\)/sec into a cylindrical vessel of base radius 3 m. Find the rate at which the water level is rising.
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Q.22 Evaluate : \(\int \frac{\sin(x + a)}{\cos(x - b)}\,dx\)
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Q.23 Solve the D.E. \(\frac{dy}{dx} + \frac{y}{x} = x^3 - 3\)
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Q.24 Obtain the differential equation by eliminating the arbitrary constants from \(y = c_1\cos(\log x) + c_2\sin(\log x)\).
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Q.25 The probability distribution of \(X\) is as follows :
\(x\) 0 1 2 3 4 \(P[X = x]\) 0.1 \(k\) \(2k\) \(2k\) \(k\) Find (i) \(k\), (ii) \(P(X \lt 2)\), (iii) \(P[1 \le X \lt 4]\)
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Q.26 In a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four correct answers just by guessing?
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Q.31 If \(x = f(t)\) and \(y = g(t)\) are differentiable functions of \(t\), so that \(y\) is a function of \(x\) and \(\frac{dx}{dt} \ne 0\), then prove that \(\frac{dy}{dx} = \frac{\left(\frac{dy}{dt}\right)}{\left(\frac{dx}{dt}\right)}\)
Hence find \(\frac{dy}{dx}\), if \(y = at^2\) and \(x = 2at\). -
Q.32 Prove that : \(\int \sqrt{a^2 - x^2}\,dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + c\)
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Q.33 Evaluate : \(\int_0^{\frac{1}{2}} \frac{dx}{(1 - 2x^2)\cdot\sqrt{1 - x^2}}\)
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Q.34 Find the area of the region lying between the parabolas \(y^2 = 4x\) and \(x^2 = 4y\).
Maharashtra HSC Mathematics and Statistics (Science) Board Paper March 2024 with Solutions
Full paper →-
Q.1 (v) The slope of the tangent to the curve \(x = \sin\theta\) and \(y = \cos 2\theta\) at \(\theta = \frac{\pi}{6}\) is _____.
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Q.1 (vi) If \(\int_{-\pi/4}^{\pi/4} x^3 \cdot \sin^4 x \, dx = k\) then \(k =\) _____.
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Q.1 (vii) The integrating factor of linear differential equation \(x\frac{dy}{dx} + 2y = x^2 \log x\) is _____.
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Q.1 (viii) If the mean and variance of a binomial distribution are 18 and 12 respectively, then the value of \(n\) is _____.
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Q.2 (iii) Evaluate : \(\int \frac{1}{x^2 + 25}\,dx\)
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Q.2 (iv) A particle is moving along X-axis. Its acceleration at time \(t\) is proportional to its velocity at that time. Find the differential equation of the motion of the particle.
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Q.9 Find \(\frac{dy}{dx}\), if \(y = (\log x)^x\).
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Q.10 Evaluate : \(\int \log x \, dx\).
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Q.11 Evaluate : \(\int_0^{\pi/2} \cos^2 x \, dx\)
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Q.12 Find the area of the region bounded by the curve \(y = x^2\), and the lines \(x = 1\), \(x = 2\) and \(y = 0\).
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Q.13 Solve : \(1 + \frac{dy}{dx} = \operatorname{cosec}(x + y)\) ; put \(x + y = u\).
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Q.14 If two coins are tossed simultaneously, write the probability distribution of the number of heads.
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Q.21 If \(y = \sin^{-1} x\), then show that : \((1 - x^2)\frac{d^2y}{dx^2} - x \cdot \frac{dy}{dx} = 0\).
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Q.22 Find the approximate value of \(\tan^{-1}(1.002)\).
[Given : \(\pi = 3.1416\)] -
Q.23 Prove that : \(\int \frac{1}{a^2 - x^2}\,dx = \frac{1}{2a}\log\left(\frac{a + x}{a - x}\right) + c\).
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Q.24 Solve the differential equation : \(x \cdot \frac{dy}{dx} - y + x \cdot \sin\left(\frac{y}{x}\right) = 0\).
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Q.25 Find \(k\), if
\(f(x) = kx^2(1 - x)\), for \(0 \lt x \lt 1\),
\(= 0\) otherwise
is the p.d.f. of random variable \(X\). -
Q.26 A die is thrown 6 times, if ‘getting an odd number’ is success, find the probability of 5 successes.
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Q.31 If \(x = f(t)\) and \(y = g(t)\) are differentiable functions of \(t\), so that \(y\) is function of \(x\) and \(\frac{dx}{dt} \ne 0\) then prove that \(\frac{dy}{dx} = \frac{dy/dt}{dx/dt}\).
Hence find \(\frac{dy}{dx}\), if \(x = at^2\), \(y = 2at\). -
Q.32 A box with a square base is to have an open top. The surface area of box is 147 sq.cm. What should be its dimensions in order that the volume is largest?
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Q.33 Evaluate : \(\int \frac{5e^x}{(e^x + 1)(e^{2x} + 9)}\,dx\)
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Q.34 Prove that :
\(\int_0^{2a} f(x)\,dx = \int_0^a f(x)\,dx + \int_0^a f(2a - x)\,dx\)
Hence show that :
\(\int_0^{\pi} \sin x\,dx = 2\int_0^{\pi/2} \sin x\,dx\)
Maharashtra HSC Mathematics and Statistics (Science) Board Paper July 2024 with Solutions
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Q.1 (v) If \(x=e^{\frac{x}{y}}\) then \(\dfrac{dy}{dx}=\) ____.
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Q.1 (vi) \(y=c^2+\dfrac{c}{x}\) is solution of ____.
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Q.1 (vii) Given that \(X\sim B(n,p)\). If \(n=10\) and \(p=0.4\) then \(E(X)\) and \(Var(X)\) respectively are ____.
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Q.1 (viii) The approximate value of \(\tan(44^\circ30')\), given that \(1^\circ=0.0175^c\), is ____
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Q.2 (iii) Evaluate : \(\displaystyle\int\dfrac{5^x}{3^x}\,dx\)
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Q.2 (iv) Write the integrating factor (I.F.) of the differential equation \(\dfrac{dy}{dx}+y=e^{-x}\).
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Q.9 Find the points on the curve given by \(y=x^3-6x^2+x+3\) where the tangents are parallel to the line \(y=x+5\).
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Q.10 Evaluate : \(\displaystyle\int\dfrac{e^x(1+x)\,dx}{\sin^2(xe^x)}\)
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Q.11 The displacement of a particle at a time \(t\) is given by \(s=2t^3-5t^2+4t-3\). Find the time when acceleration is 14 ft/sec2.
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Q.12 Evaluate : \(\displaystyle\int_0^{\pi/4}\sqrt{1+\sin2x}\,dx\)
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Q.13 The probability distribution of \(X\) is as follows :
\(X=x\) 0 1 2 3 4 \(P(X=x)\) 0.1 \(k\) \(2k\) \(2k\) \(k\) Find (a) \(k\) (b) \(P(X\lt 2)\)
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Q.14 Find the particular solution of : \(r\dfrac{dr}{d\theta}+\cos\theta=5\) at \(r=\sqrt2\) and \(\theta=0\)
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Q.21 Find \(\dfrac{dy}{dx}\) if \(y=\tan^{-1}\left(\sqrt{\dfrac{3-x}{3+x}}\right)\)
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Q.22 Find the approximate value of \(f(x)=x^3+5x^2-2x+3\) at \(x=1.98\).
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Q.23 Evaluate : \(\displaystyle\int\dfrac{\sin(x+a)}{\cos(x-b)}\,dx\)
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Q.24 Solve the differential equation \(dr+(2r\cot\theta+\sin2\theta)\,d\theta=0\)
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Q.25 Let \(X\sim B(10,0.2)\). Find (a) \(P(X=1)\) (b) \(P(X\ge1)\)
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Q.26 Find the expected value, variance and standard deviation of r.v. \(X\) whose p.m.f. is given as :
\(X=x\) 1 2 3 \(P(X)\) \(\frac15\) \(\frac25\) \(\frac25\) -
Q.31 If \(y=f(u)\) is a differentiable function of \(u\) and \(u=g(x)\) is differentiable function of \(x\) such that the composite function \(y=f[g(x)]\) is a differentiable function of \(x\) then prove that : \(\dfrac{dy}{dx}=\dfrac{dy}{du}\times\dfrac{du}{dx}\). Hence find \(\dfrac{d}{dx}\left(\dfrac{1}{\sqrt{\sin x}}\right)\).
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Q.32 Evaluate : \(\displaystyle\int x^2\sin3x\,dx\)
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Q.33 Prove that :
\(\displaystyle\int_{-a}^{a}f(x)\,dx=2\int_0^af(x)\,dx\), if \(f\) is an even function
\(=0\), if \(f\) is an odd function.
Hence find the value of \(\displaystyle\int_{-1}^{1}\tan^{-1}x\,dx\). -
Q.34 Find the area of the ellipse : \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\). Hence write area of \(\dfrac{x^2}{25}+\dfrac{y^2}{16}=1\)