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Maharashtra HSC Class 12 Mathematics July 2025 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
    The inverse of statement pattern (p∨q)→(p∧q)(p \vee q) \rightarrow (p \wedge q) is _____

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  2. Q.1 (ii)2 marks
    In ΔABC\Delta ABC, if a=2,b=3a = 2, b = 3 and sin⁡A=23\sin A = \frac{2}{3}, then ∠B=\angle B = _____

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  3. Q.1 (iii)2 marks
    If AB‾=2i^−4j^+7k^\overline{AB} = 2\hat{i} - 4\hat{j} + 7\hat{k} and initial point A≡(1,5,0)A \equiv (1, 5, 0) then terminal point BB is _____

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

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  5. Q.1 (v)2 marks
    If yy is a function of xx and log⁡(x+y)=xy\log(x + y) = xy then the value of (dydx)\left(\frac{dy}{dx}\right) at x=0x = 0 is _____

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  6. Q.1 (vi)2 marks
    If the displacement of a particle at time tt is given by S=2t3−5t2+4t−3S = 2t^{3} - 5t^{2} + 4t - 3, then its acceleration at time t=1t = 1 is _____

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  7. Q.1 (vii)2 marks
    The solution of the D.E. sec⁡2x⋅tan⁡y dx+sec⁡2y⋅tan⁡x dy=0\sec^{2} x \cdot \tan y \, dx + \sec^{2} y \cdot \tan x \, dy = 0 is _____

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  8. Q.1 (viii)2 marks
    If XX is waiting time in minutes for a bus and its p.d.f. is given by f(x)=15f(x) = \frac{1}{5}, for 0≤x≤50 \leq x \leq 5, =0= 0, otherwise then the probability that waiting time is between 1 and 3 is _____

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  9. Q.2 (i)1 mark
    Find the general solution of tan⁡θ=0\tan \theta = 0.
  10. Q.2 (ii)1 mark
    Find the magnitude of the vector aˉ=3i^+j^+7k^\bar{a} = 3\hat{i} + \hat{j} + 7\hat{k}
  11. Q.2 (iii)1 mark
    Find dydx\frac{dy}{dx}, if y=sin⁡(log⁡x)y = \sin(\log x).
  12. Q.2 (iv)1 mark
    Evaluate : ∫1x dx\int \frac{1}{\sqrt{x}} \, dx.

Section B

2 marks each · attempt any 8

  1. Q.32 marks
    Using truth table, prove that ∼p∧q≡(p∨q)∧∼p\sim p \wedge q \equiv (p \vee q) \wedge \sim p
  2. Q.42 marks
    Find the cofactors of the elements of the matrix [2−335]\begin{bmatrix} 2 & -3 \\ 3 & 5 \end{bmatrix}
  3. Q.52 marks
    Find the polar coordinates of the point whose cartesian coordinates are (−2,−2)(-\sqrt{2}, -\sqrt{2}).
  4. Q.62 marks
    In ΔABC\Delta ABC, if a=2,b=3,c=4a = 2, b = 3, c = 4, then prove that the triangle is obtuse angled.
  5. Q.72 marks
    If aˉ=3i^−j^+2k^,bˉ=2i^+j^−k^,\bar{a} = 3\hat{i} - \hat{j} + 2\hat{k}, \bar{b} = 2\hat{i} + \hat{j} - \hat{k}, then find ∣aˉ×bˉ∣\left|\bar{a} \times \bar{b}\right|
  6. Q.82 marks
    Find the cartesian equation of the plane passing through the point A(−1,2,3)A(-1, 2, 3), the direction ratios of whose normal are 0, 2, 5.
  7. Q.92 marks
    Find dydx\frac{dy}{dx}, if xx+yy=aax\sqrt{x} + y\sqrt{y} = a\sqrt{a}.
  8. Q.102 marks
    Show that the tangent to the curve y=x3−6x2+x+3y = x^{3} - 6x^{2} + x + 3 at the point (0, 3) is parallel to the line y=x+5y = x + 5.
  9. Q.112 marks
    Check whether the conditions of Rolle's theorem are satisfied by the function f(x)=x2−4x+3,x∈[1,3]f(x) = x^{2} - 4x + 3, x \in [1, 3]
  10. Q.122 marks
    Evaluate : ∫dxx+x−10\int \frac{dx}{x + x^{-10}}
  11. Q.132 marks
    Evaluate : ∫0−1e−x dx\int_{0}^{-1} e^{-x} \, dx
  12. Q.142 marks
    If X∼B(n,p)X \sim B(n, p) and E(X)=6,Var(X)=4.2,E(X) = 6, Var(X) = 4.2, then find nn and pp.

Section C

3 marks each · attempt any 8

  1. Q.153 marks
    In ΔABC\Delta ABC, prove that a2=b2+c2−2bccos⁡Aa^{2} = b^{2} + c^{2} - 2bc \cos A
  2. Q.163 marks
    Find the combined equation of pair of lines passing through (2, 3) and perpendicular to the lines 3x+2y−1=03x + 2y - 1 = 0 and x−3y+2=0x - 3y + 2 = 0.
  3. Q.173 marks
    Show that the acute angle θ\theta between the lines represented by ax2+2hxy+by2=0ax^{2} + 2hxy + by^{2} = 0 is given by, tan⁡θ=∣2h2−aba+b∣\tan \theta = \left|\frac{2\sqrt{h^{2} - ab}}{a + b}\right|
  4. Q.183 marks
    Using vector method, prove that the medians of a triangle are concurrent.
  5. Q.193 marks
    Find the vector equation of the plane passing through the point A(−1,2,−5)A(-1, 2, -5) and parallel to the vectors 4i^−j^+3k^4\hat{i} - \hat{j} + 3\hat{k} and i^+j^−k^\hat{i} + \hat{j} - \hat{k}.
  6. Q.203 marks
    Find the shortest distance between the lines, x−12=y−23=z−34\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} and x−23=y−44=z−55\frac{x - 2}{3} = \frac{y - 4}{4} = \frac{z - 5}{5}.
  7. Q.213 marks
    Water is being poured at the rate of 27 m3^{3}/sec into a cylindrical vessel of base radius 3 m. Find the rate at which the water level is rising.
  8. Q.223 marks
    Evaluate : ∫sin⁡(x+a)cos⁡(x−b) dx\int \frac{\sin(x + a)}{\cos(x - b)} \, dx
  9. Q.233 marks
    Solve the D.E. dydx+yx=x3−3\frac{dy}{dx} + \frac{y}{x} = x^{3} - 3
  10. Q.243 marks
    Obtain the differential equation by eliminating the arbitrary constants from y=c1cos⁡(log⁡x)+c2sin⁡(log⁡x)y = c_{1} \cos(\log x) + c_{2} \sin(\log x).
  11. Q.253 marks
    The probability distribution of XX is as follows :
    xx01234
    P[X=x]P[X=x]0.1kk2k2k2k2kkk
    Find (i) kk, (ii) P(X<2)P(X < 2), (iii) P[1≤X<4]P[1 \leq X < 4]
  12. Q.263 marks
    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?

Section D

4 marks each · attempt any 5

  1. Q.274 marks
    Give an alternative equivalent simple circuit for the following switching circuit :
  2. Q.284 marks
    The sum of three numbers is 2. If twice of the second number is added to the sum of first and third number we get 0. Adding five times the first number to twice the sum of second and third number we get 7. Find the numbers using matrix method.
  3. Q.294 marks
    Using properties of scalar triple product, prove that [aˉ+bˉbˉ+cˉcˉ+aˉ]=2[aˉ  bˉ  cˉ][\bar{a} + \bar{b} \quad \bar{b} + \bar{c} \quad \bar{c} + \bar{a}] = 2 [\bar{a} \; \bar{b} \; \bar{c}]
  4. Q.304 marks
    Solve the following L.P.P. using graphical method : Maximize, z=9x+13yz = 9x + 13y Subject to, 2x+3y≤182x + 3y \leq 18, 2x+y≤102x + y \leq 10 x≥0,y≥0x \geq 0, y \geq 0
  5. Q.314 marks
    If x=f(t)x = f(t) and y=g(t)y = g(t) are differentiable functions of tt, so that yy is a function of xx and dxdt≠0\frac{dx}{dt} \neq 0, then prove that dydx=(dydt)(dxdt)\frac{dy}{dx} = \frac{\left(\frac{dy}{dt}\right)}{\left(\frac{dx}{dt}\right)} Hence find dydx\frac{dy}{dx}, if y=at2y = at^{2} and x=2atx = 2at.
  6. Q.324 marks
    Prove that : ∫a2−x2⋅dx=x2⋅a2−x2+a22⋅sin⁡−1(xa)+c\int \sqrt{a^{2} - x^{2}} \cdot dx = \frac{x}{2} \cdot \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} \cdot \sin^{-1}\left(\frac{x}{a}\right) + c
  7. Q.334 marks
    Evaluate : ∫012dx(1−2x2)⋅1−x2\int_{0}^{\frac{1}{2}} \frac{dx}{(1 - 2x^{2}) \cdot \sqrt{1 - x^{2}}}
  8. Q.344 marks
    Find the area of the region lying between the parabolas y2=4xy^{2} = 4x and x2=4yx^{2} = 4y.