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Maharashtra HSC Class 12 Mathematics July 2024 question paper

Maximum marks 80 · Time 3 hours

Try each question first, then open its model answer. Where the paper offers a choice, both questions are shown with OR between them.

Section A

2 marks each; 1 mark each

  1. Q.1 (i)2 marks
    cos⁡[tan⁡−1(13)+tan⁡−1(12)]=\cos\left[\tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1}\left(\frac{1}{2}\right)\right] = _____.

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  2. Q.1 (ii)2 marks
    If θ\theta is the angle between two vectors aˉ\bar{a} and bˉ\bar{b} and ∣aˉ⋅bˉ∣=∣aˉ×bˉ∣\left|\bar{a} \cdot \bar{b}\right| = \left|\bar{a} \times \bar{b}\right| then θ\theta is equal to _____.

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  3. Q.1 (iii)2 marks
    The angle between the lines rˉ=(i^+2j^−3k^)+λ(3i^+2j^+6k^)\bar{r} = (\hat{i} + 2\hat{j} - 3\hat{k}) + \lambda(3\hat{i} + 2\hat{j} + 6\hat{k}) and rˉ=(5i^−2j^+7k^)+μ(i^+2j^+2k^)\bar{r} = (5\hat{i} - 2\hat{j} + 7\hat{k}) + \mu(\hat{i} + 2\hat{j} + 2\hat{k}) is _____.

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  4. Q.1 (iv)2 marks
    The perpendicular distance of the plane rˉ⋅(2i^+3j^−k^)=5\bar{r} \cdot (2\hat{i} + 3\hat{j} - \hat{k}) = 5 from the origin is _____.

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  5. Q.1 (v)2 marks
    If x=exyx = e^{\frac{x}{y}} then dydx=\frac{dy}{dx} = _____.

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  6. Q.1 (vi)2 marks
    y=c2+cxy = c^{2} + \frac{c}{x} is solution of _____.

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  7. Q.1 (vii)2 marks
    Given that X∼B(n,p)X \sim B(n, p). If n=10n = 10 and p=0.4p = 0.4 then E(X)E(X) and Var(X)Var(X) respectively are _____.

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  8. Q.1 (viii)2 marks
    The approximate value of tan⁡(44∘30′)\tan(44^{\circ}30'), given that 1∘=0.0175c1^{\circ} = 0.0175^{c}, is _____

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  9. Q.2 (i)1 mark
    Find the combined equation of the pair of lines 2x+y=02x + y = 0 and 3x−y=03x - y = 0
  10. Q.2 (ii)1 mark
    Find the value of sin⁡−1(sin⁡5π3)\sin^{-1}\left(\sin\frac{5\pi}{3}\right)
  11. Q.2 (iii)1 mark
    Evaluate : ∫5x3xdx\int \frac{5^{x}}{3^{x}} dx
  12. Q.2 (iv)1 mark
    Write the integrating factor (I.F.) of the differential equation dydx+y=e−x\frac{dy}{dx} + y = e^{-x}.

Section B

2 marks each · attempt any 8

  1. Q.32 marks
    If the statements pp, qq are true statements and rr, ss are false statements, then determine the truth value of the statement pattern : (q∧r)∨(∼p∧s)(q \wedge r) \vee (\sim p \wedge s)
  2. Q.42 marks
    Find the inverse of matrix AA by elementary row transformations, where A=[2−3−12]A = \begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}
  3. Q.52 marks
    Find the polar co-ordinates of the point whose Cartesian co-ordinates are (1,−3)\left(1, -\sqrt{3}\right).
  4. Q.62 marks
    Find the acute angle between the lines represented by xy+y2=0xy + y^{2} = 0.
  5. Q.72 marks
    Using the truth table, show that the statement pattern p→(q→p)p \rightarrow (q \rightarrow p) is a tautology.
  6. Q.82 marks
    If tan⁡−1(2x)+tan⁡−1(3x)=π4\tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4} then find the value of xx, where 0<3x<10 < 3x < 1
  7. Q.92 marks
    Find the points on the curve given by y=x3−6x2+x+3y = x^{3} - 6x^{2} + x + 3 where the tangents are parallel to the line y=x+5y = x + 5.
  8. Q.102 marks
    Evaluate : ∫ex(1+x)dxsin⁡2(xex)\int \frac{e^{x}(1 + x) dx}{\sin^{2}(xe^{x})}
  9. Q.112 marks
    The displacement of a particle at a time tt is given by s=2t3−5t2+4t−3s = 2t^{3} - 5t^{2} + 4t - 3. Find the time when acceleration is 14 ft/sec214 \, \text{ft/sec}^{2}.
  10. Q.122 marks
    Evaluate : ∫0π/41+sin⁡2x dx\int_{0}^{\pi/4} \sqrt{1 + \sin 2x} \, dx
  11. Q.132 marks
    The probability distribution of XX is as follows :
    X=xX = x01234
    P(X=x)P(X = x)0.1kk2k2k2k2kkk
    Find (a) kk (b) P(X<2)P(X < 2)
  12. Q.142 marks
    Find the particular solution of : rdrdθ+cos⁡θ=5r\frac{dr}{d\theta} + \cos\theta = 5 at r=2r = \sqrt{2} and θ=0\theta = 0

Section C

3 marks each · attempt any 8

  1. Q.153 marks
    In ΔABC\Delta ABC, if acos⁡A=bcos⁡Ba \cos A = b \cos B then prove that the triangle is either a right angled or an isosceles triangle.
  2. Q.163 marks
    Are the four points A(1,−1,1)A(1, -1, 1), B(−1,1,1)B(-1, 1, 1), C(1,1,1)C(1, 1, 1) and D(2,−3,4)D(2, -3, 4) co-planar? Justify your answer.
  3. Q.173 marks
    Find the difference between the slopes of the lines given by (tan⁡2θ+cos⁡2θ)x2−2xytan⁡θ+(sin⁡2θ)y2=0(\tan^{2}\theta + \cos^{2}\theta)x^{2} - 2xy\tan\theta + (\sin^{2}\theta)y^{2} = 0
  4. Q.183 marks
    Find the vector equation of the line passing through the point (i^+2j^+3k^)(\hat{i} + 2\hat{j} + 3\hat{k}) and perpendicular to the vectors i^+j^+k^\hat{i} + \hat{j} + \hat{k} and 2i^−j^+k^2\hat{i} - \hat{j} + \hat{k}.
  5. Q.193 marks
    Let aˉ\bar{a} and bˉ\bar{b} be non-collinear vectors. If vector rˉ\bar{r} is co-planar with aˉ\bar{a} and bˉ\bar{b} then prove that there exists unique scalars t1t_{1} and t2t_{2} such that rˉ=t1aˉ+t2bˉ\bar{r} = t_{1}\bar{a} + t_{2}\bar{b}. Hence find t1t_{1} and t2t_{2} for rˉ=i^+j^\bar{r} = \hat{i} + \hat{j}, aˉ=2i^−j^\bar{a} = 2\hat{i} - \hat{j}, bˉ=i^−2j^\bar{b} = \hat{i} - 2\hat{j}.
  6. Q.203 marks
    Find the equation of the plane passing through the intersection of the planes x+2y+3z+4=0x + 2y + 3z + 4 = 0 and 4x+3y+2z+1=04x + 3y + 2z + 1 = 0 and the origin.
  7. Q.213 marks
    Find dydx\frac{dy}{dx} if y=tan⁡−1(3−x3+x)y = \tan^{-1}\left(\sqrt{\frac{3 - x}{3 + x}}\right)
  8. Q.223 marks
    Find the approximate value of f(x)=x3+5x2−2x+3f(x) = x^{3} + 5x^{2} - 2x + 3 at x=1.98x = 1.98.
  9. Q.233 marks
    Evaluate : ∫sin⁡(x+a)cos⁡(x−b)dx\int \frac{\sin(x + a)}{\cos(x - b)} dx
  10. Q.243 marks
    Solve the differential equation dr+(2rcot⁡θ+sin⁡2θ)dθ=0dr + (2r\cot\theta + \sin 2\theta) d\theta = 0
  11. Q.253 marks
    Let X∼B(10,0.2)X \sim B(10, 0.2). Find (a) P(X=1)P(X = 1) (b) P(X≥1)P(X \geq 1)
  12. Q.263 marks
    Find the expected value, variance and standard deviation of r.v. XX whose p.m.f. is given as :
    X=xX = x123
    P(X)P(X)15\frac{1}{5}25\frac{2}{5}25\frac{2}{5}

Section D

4 marks each · attempt any 5

  1. Q.274 marks
    Give an alternative arrangement for the following circuit, so that new circuit has minimum switches :
  2. Q.284 marks
    If A=[2−11−12−11−12]A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix} then find A−1A^{-1} by Adjoint method.
  3. Q.294 marks
    In ΔABC\Delta ABC, DD and EE are points on BCBC and ACAC respectively such that BD=2DCBD = 2DC and AE=3ECAE = 3EC. Let PP be the point of intersection of ADAD and BEBE. Find ratio BPPE\frac{BP}{PE} using the vector method.
  4. Q.304 marks
    A firm manufactures two products AA and BB on which profit earned per unit are ₹3 and ₹4 respectively. Each product is processed on two machines M1M_{1} and M2M_{2}. The product A requires one minute of processing time on M1M_{1} and two minutes on M2M_{2}, while product BB requires one minute on M1M_{1} and one minute on M2M_{2}. Machine M1M_{1} is available for use not more than 450 minutes, while M2M_{2} is available for 600 minutes during any working day. Find the number of units of products AA and BB to be manufactured to get maximum profit.
  5. Q.314 marks
    If y=f(u)y = f(u) is a differentiable function of uu and u=g(x)u = g(x) is differentiable function of xx such that the composite function y=f[g(x)]y = f[g(x)] is a differentiable function of xx then prove that : dydx=dydu×dudx\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} Hence find ddx(1sin⁡x)\frac{d}{dx}\left(\frac{1}{\sqrt{\sin x}}\right).
  6. Q.324 marks
    Evaluate : ∫x2sin⁡3x dx\int x^{2} \sin 3x \, dx.
  7. Q.334 marks
    Prove that : ∫−aaf(x)dx=2∫0af(x) dx\int_{-a}^{a} f(x) dx = 2\int_{0}^{a} f(x) \, dx if ff is an even function =0= 0 , if ff is an odd function Hence find the value of ∫−11tan⁡−1x dx\int_{-1}^{1} \tan^{-1} x \, dx.
  8. Q.344 marks
    Find the area of the ellipse : x2a2+y2b2=1\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 Hence write area of x225+y216=1\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1