Maharashtra HSC Mathematics and Statistics (Science) Board Paper July 2024 with Solutions

Q.1 (i) \(\cos\left[\tan^{-1}\left(\dfrac{1}{3}\right)+\tan^{-1}\left(\dfrac{1}{2}\right)\right]=\) ____.

Q.1 (ii) If \(\theta\) is the angle between two vectors \(\bar a\) and \(\bar b\) and \(|\bar a\cdot\bar b|=|\bar a\times\bar b|\) then \(\theta\) is equal to ____.

Q.1 (iii) The angle between the lines \(\bar r=(\hat i+2\hat j-3\hat k)+\lambda(3\hat i+2\hat j+6\hat k)\) and \(\bar r=(5\hat i-2\hat j+7\hat k)+\mu(\hat i+2\hat j+2\hat k)\) is ____.

Q.1 (iv) The perpendicular distance of the plane \(\bar r\cdot(2\hat i+3\hat j-\hat k)=5\) from the origin is ____.

Q.1 (v) If \(x=e^{\frac{x}{y}}\) then \(\dfrac{dy}{dx}=\) ____.

Q.1 (vi) \(y=c^2+\dfrac{c}{x}\) is solution of ____.

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 ____.

Q.1 (viii) The approximate value of \(\tan(44^\circ30')\), given that \(1^\circ=0.0175^c\), is ____

Q.2 (i) Find the combined equation of the pair of lines \(2x+y=0\) and \(3x-y=0\)

Q.2 (ii) Find the value of \(\sin^{-1}\left(\sin\dfrac{5\pi}{3}\right)\)

Q.2 (iii) Evaluate : \(\displaystyle\int\dfrac{5^x}{3^x}\,dx\)

Q.2 (iv) Write the integrating factor (I.F.) of the differential equation \(\dfrac{dy}{dx}+y=e^{-x}\).

Q.3 If the statements \(p, q\) are true statements and \(r, s\) are false statements, then determine the truth value of the statement pattern : \((q\wedge r)\vee(\sim p\wedge s)\)

Q.4 Find the inverse of matrix \(A\) by elementary row transformations, where \(A=\begin{bmatrix}2&-3\\-1&2\end{bmatrix}\)

Q.5 Find the polar co-ordinates of the point whose Cartesian co-ordinates are \((1,-\sqrt3)\).

Q.6 Find the acute angle between the lines represented by \(xy+y^2=0\).

Q.7 Using the truth table, show that the statement pattern \(p\to(q\to p)\) is a tautology.

Q.8 If \(\tan^{-1}(2x)+\tan^{-1}(3x)=\dfrac{\pi}{4}\) then find the value of \(x\), where \(0\lt 3x\lt 1\)

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 \(y=x+5\).

Q.10 Evaluate : \(\displaystyle\int\dfrac{e^x(1+x)\,dx}{\sin^2(xe^x)}\)

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.

Q.12 Evaluate : \(\displaystyle\int_0^{\pi/4}\sqrt{1+\sin2x}\,dx\)

Q.13 The probability distribution of \(X\) is as follows :

Find (a) \(k\) (b) \(P(X\lt 2)\)

Q.14 Find the particular solution of : \(r\dfrac{dr}{d\theta}+\cos\theta=5\) at \(r=\sqrt2\) and \(\theta=0\)

Q.15 In \(\triangle ABC\), if \(a\cos A=b\cos B\) then prove that the triangle is either a right angled or an isosceles triangle.

Q.16 Are the four points \(A(1,-1,1)\), \(B(-1,1,1)\), \(C(1,1,1)\) and \(D(2,-3,4)\) co-planar? Justify your answer.

Q.17 Find the difference between the slopes of the lines given by \((\tan^2\theta+\cos^2\theta)x^2-2xy\tan\theta+(\sin^2\theta)y^2=0\)

Q.18 Find the vector equation of the line passing through the point \((\hat i+2\hat j+3\hat k)\) and perpendicular to the vectors \(\hat i+\hat j+\hat k\) and \(2\hat i-\hat j+\hat k\).

Q.19 Let \(\bar a\) and \(\bar b\) be non-collinear vectors. If vector \(\bar r\) is co-planar with \(\bar a\) and \(\bar b\) then prove that there exists unique scalars \(t_1\) and \(t_2\) such that \(\bar r=t_1\bar a+t_2\bar b\). Hence find \(t_1\) and \(t_2\) for \(\bar r=\hat i+\hat j\), \(\bar a=2\hat i-\hat j\), \(\bar b=\hat i-2\hat j\).

Q.20 Find the equation of the plane passing through the intersection of the planes \(x+2y+3z+4=0\) and \(4x+3y+2z+1=0\) and the origin.

Q.21 Find \(\dfrac{dy}{dx}\) if \(y=\tan^{-1}\left(\sqrt{\dfrac{3-x}{3+x}}\right)\)

Q.22 Find the approximate value of \(f(x)=x^3+5x^2-2x+3\) at \(x=1.98\).

Q.23 Evaluate : \(\displaystyle\int\dfrac{\sin(x+a)}{\cos(x-b)}\,dx\)

Q.24 Solve the differential equation \(dr+(2r\cot\theta+\sin2\theta)\,d\theta=0\)

Q.25 Let \(X\sim B(10,0.2)\). Find (a) \(P(X=1)\) (b) \(P(X\ge1)\)

Q.26 Find the expected value, variance and standard deviation of r.v. \(X\) whose p.m.f. is given as :

Q.27 Give an alternative arrangement for the following circuit, so that new circuit has minimum switches :

[Figure: Three branches connected in parallel between the two terminals, in series with a battery and a lamp L. Branch 1: switches \(S_1\) and \(S_2'\) in series. Branch 2: switches \(S_1'\) and \(S_2\) in series. Branch 3: switches \(S_1'\) and \(S_2'\) in series.]

Q.28 If \(A=\begin{bmatrix}2&-1&1\\-1&2&-1\\1&-1&2\end{bmatrix}\) then find \(A^{-1}\) by Adjoint method.

Q.29 In \(\triangle ABC\), \(D\) and \(E\) are points on \(BC\) and \(AC\) respectively such that \(BD=2DC\) and \(AE=3EC\). Let \(P\) be the point of intersection of \(AD\) and \(BE\). Find ratio \(\dfrac{BP}{PE}\) using the vector method.

Q.30 A firm manufactures two products \(A\) and \(B\) on which profit earned per unit are ₹3 and ₹4 respectively. Each product is processed on two machines \(M_1\) and \(M_2\). The product A requires one minute of processing time on \(M_1\) and two minutes on \(M_2\), while product \(B\) requires one minute on \(M_1\) and one minute on \(M_2\). Machine \(M_1\) is available for use not more than 450 minutes, while \(M_2\) is available for 600 minutes during any working day. Find the number of units of products \(A\) and \(B\) to be manufactured to get maximum profit.

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 \(\dfrac{d}{dx}\left(\dfrac{1}{\sqrt{\sin x}}\right)\).

Q.32 Evaluate : \(\displaystyle\int x^2\sin3x\,dx\)

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 \(\dfrac{x^2}{25}+\dfrac{y^2}{16}=1\)