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Maharashtra HSC Class 12 Mathematics June 2026 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
    If A={1,2,3,4,5}A = \{1, 2, 3, 4, 5\} then which of the following is not true?

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  2. Q.1 (ii)2 marks
    The value of sin⁡[sin⁡−1(−12)−3sin⁡−1(32)]=\sin\left[\sin^{-1}\left(-\frac{1}{\sqrt{2}}\right) - 3\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\right] =

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  3. Q.1 (iii)2 marks
    If p^,q^,r^\hat{p}, \hat{q}, \hat{r} are mutually perpendicular unit vectors and form a right handed triplet then the value of p^⋅(q^×r^)+q^⋅(p^×r^)+r^⋅(p^×q^)=\hat{p} \cdot (\hat{q} \times \hat{r}) + \hat{q} \cdot (\hat{p} \times \hat{r}) + \hat{r} \cdot (\hat{p} \times \hat{q}) = .......

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  4. Q.1 (iv)2 marks
    The Cartesian equation of the line passing through the points A(3,4,−1)A(3, 4, -1) and B(2,−1,3)B(2, -1, 3) is _____.

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  5. Q.1 (v)2 marks
    If y=log⁡xay = \log_{x} a then dydx=\frac{dy}{dx} = _____.

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  6. Q.1 (vi)2 marks
    If f(x)=sin⁡−1x1−x2f(x) = \frac{\sin^{-1} x}{\sqrt{1 - x^{2}}} and g(x)=esin⁡−1xg(x) = e^{\sin^{-1} x} then ∫f(x)⋅g(x)dx=\int f(x) \cdot g(x) dx = _____.

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  7. Q.1 (vii)2 marks
    The area bounded by the line y=2xy = 2x, XX - axis and the lines x=−1x = -1, x=4x = 4 is _____ sq. units.

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  8. Q.1 (viii)2 marks
    In a Binomial Distribution, E(X)=6E(X) = 6, Var(X)=4.2\text{Var}(X) = 4.2 then the value of nn is _____

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  9. Q.2 (i)1 mark
    Write the dual of p∧tp \land t
  10. Q.2 (ii)1 mark
    What is the distance of point (3,−5,6)(3, -5, 6) from XZ plane?
  11. Q.2 (iii)1 mark
    The displacement of a particle at time tt is given by S=3t2+4t+1S = 3t^{2} + 4t + 1. Find its velocity at time tt.
  12. Q.2 (iv)1 mark
    Write the order of differential equation (d2ydx2)52=dydx3\left(\frac{d^{2}y}{dx^{2}}\right)^{\frac{5}{2}} = \sqrt[3]{\frac{dy}{dx}}

Section B

2 marks each · attempt any 8

  1. Q.32 marks
    Construct the switching circuit for the statement (∼p∧q)∨(p∧∼r)(\sim p \land q) \vee (p \land \sim r).
  2. Q.42 marks
    Find the matrix of cofactors of matrix [52−16]\begin{bmatrix} 5 & 2 \\ -1 & 6 \end{bmatrix}.
  3. Q.52 marks
    Find the general solution of equation cos⁡5θ=12\cos 5\theta = \frac{1}{2}.
  4. Q.62 marks
    Find the value of kk, if 2x+y=02x + y = 0 is one of the lines represented by 3x2+kxy+2y2=03x^{2} + kxy + 2y^{2} = 0.
  5. Q.72 marks
    If aˉ,bˉ,cˉ\bar{a}, \bar{b}, \bar{c} are the position vectors of points AA, BB, CC respectively and 10aˉ=7bˉ+3cˉ10\bar{a} = 7\bar{b} + 3\bar{c} then find the ratio in which the point CC divides the line segment ABAB.
  6. Q.82 marks
    Find the angle between the lines rˉ=(i^+2j^+3k^)+λ(2i^−j^+k^)\bar{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} - \hat{j} + \hat{k}) rˉ=(i^+2j^+3k^)+μ(i^+2j^+k^)\bar{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \mu(\hat{i} + 2\hat{j} + \hat{k})
  7. Q.92 marks
    Test whether the function f(x)=x−1x,x∈R,x≠0f(x) = x - \frac{1}{x}, x \in R, x \neq 0 is increasing or decreasing.
  8. Q.102 marks
    Evaluate : ∫−11∣x∣ dx\int_{-1}^{1} |x| \, dx
  9. Q.112 marks
    Evaluate : ∫xlog⁡x dx\int x \log x \ dx
  10. Q.122 marks
    Find the area bounded by curve x2+y=0x^{2} + y = 0, XX - axis and lines x=1x = 1 and x=4x = 4.
  11. Q.132 marks
    The probability distribution of r.v. XX is as follows:
    X=xX = x01234
    P(X=x)P(X = x)0.1kk2k2k2k2kkk
    Find : (i) kk (ii) F(2)F(2)
  12. Q.142 marks
    Evaluate : ∫ex(1+x)cos⁡2(xex)dx\int \frac{e^{x}(1 + x)}{\cos^{2}(xe^{x})} dx

Section C

3 marks each · attempt any 8

  1. Q.153 marks
    Using truth table, determine whether statement pattern [(p∨q)∧∼p]∧∼q[(p \vee q) \land \sim p] \land \sim q is a tautology or a contradiction or a contingency.
  2. Q.163 marks
    Prove that : sin⁡−1(35)+cos⁡−1(1213)=sin⁡−1(5665)\sin^{-1}\left(\frac{3}{5}\right) + \cos^{-1}\left(\frac{12}{13}\right) = \sin^{-1}\left(\frac{56}{65}\right)
  3. Q.173 marks
    In ΔABC\Delta ABC, if a=10a = 10, b=8b = 8, c=6c = 6 then find the value of (i) sin⁡(C2)\sin\left(\frac{C}{2}\right) (ii) cos⁡(B2)\cos\left(\frac{B}{2}\right)
  4. Q.183 marks
    Are the four points A(1,2,1)A(1, 2, 1), B(2,−3,4)B(2, -3, 4), C(3,4,−5)C(3, 4, -5), D(2,3,−2)D(2, 3, -2) coplanar? Justify your answer.
  5. Q.193 marks
    Find the distance between parallel lines x2=y−1=z2\frac{x}{2} = \frac{y}{-1} = \frac{z}{2} and x+12=y−1−1=z+12\frac{x + 1}{2} = \frac{y - 1}{-1} = \frac{z + 1}{2}
  6. Q.203 marks
    Find the vector equation of the plane passing through the points A(2,1,1)A(2, 1, 1), B(0,2,3)B(0, 2, 3) and C(4,5,6)C(4, 5, 6).
  7. Q.213 marks
    If y=log⁡[x+x2+a2x2+a2−x]y = \log\left[\frac{x + \sqrt{x^{2} + a^{2}}}{\sqrt{x^{2} + a^{2}} - x}\right], find dydx\frac{dy}{dx}.
  8. Q.223 marks
    A wire of length 36 cm is bent to form a rectangle. Find its dimensions, if the area of the rectangle is maximum.
  9. Q.233 marks
    Verify Rolle's theorem for the function f(x)=x2−5x+9f(x) = x^{2} - 5x + 9, x∈[1,4]x \in [1, 4]
  10. Q.243 marks
    Solve the differential equation xdydx−x2+y2=yx \frac{dy}{dx} - \sqrt{x^{2} + y^{2}} = y
  11. Q.253 marks
    The following is the p.d.f. of continuous r.v. f(x)=x8f(x) = \frac{x}{8}; 0<x<40 < x < 4 =0= 0, Otherwise Find (i) expression for c.d.f of XX. (ii) F(x)F(x) at 0.5 and 5
  12. Q.263 marks
    If the p.m.f of r.v. XX be P(X=x)=4Cx(59)x(49)4−xP(X = x) = {}^{4}C_{x}\left(\frac{5}{9}\right)^{x}\left(\frac{4}{9}\right)^{4 - x} for x=0,1,2,3,4x = 0, 1, 2, 3, 4 =0= 0 ; otherwise then find E(X)E(X) and Var(X)\text{Var}(X).

Section D

4 marks each · attempt any 5

  1. Q.274 marks
    Find the inverse of matrix [−4−3−3101443]\begin{bmatrix} -4 & -3 & -3 \\ 1 & 0 & 1 \\ 4 & 4 & 3 \end{bmatrix}
  2. Q.284 marks
    Prove that the homogeneous equation of degree two in xx and yy, ax2+2hxy+by2=0ax^{2} + 2hxy + by^{2} = 0 represents a pair of lines passing through the origin if h2−ab≥0h^{2} - ab \geq 0
  3. Q.294 marks
    Prove that, if aˉ,bˉ,cˉ\bar{a}, \bar{b}, \bar{c} are three non coplanar vectors then any vector rˉ\bar{r} in the space can be uniquely expressed as a linear combination of aˉ,bˉ,cˉ\bar{a}, \bar{b}, \bar{c}.
  4. Q.304 marks
    Solve the linear programming problem (L.P.P.) by graphical method. Minimize : z=8x+10yz = 8x + 10y Subject to, 2x+y≥72x + y \geq 7 2x+3y≥152x + 3y \geq 15 x≥0x \geq 0 y≥2y \geq 2
  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 differentiable function of xx and dxdt≠0\frac{dx}{dt} \neq 0 then prove that dydx=dydtdxdt\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}. Hence find dydx\frac{dy}{dx}, if x=sin⁡tx = \sin t y=cos⁡ty = \cos t.
  6. Q.324 marks
    Evaluate : ∫23+2sin⁡2x+5cos⁡2xdx\int \frac{2}{3 + 2\sin^{2} x + 5\cos^{2} x} 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 f(x)f(x) is even =0= 0, if f(x)f(x) is odd
  8. Q.344 marks
    A body cools according to Newton's law from 100ºC to 60ºC in 20 minutes. The temperature of the surrounding being 20ºC. How long will it take to cool down to 40ºC?