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Maharashtra HSC Class 12 Mathematics February 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
    The converse of contrapositive of ∼p→q\sim p \rightarrow q is _____.

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  2. Q.1 (ii)2 marks
    If A=[2−431]A = \begin{bmatrix} 2 & -4 \\ 3 & 1 \end{bmatrix}, then the adjoint of matrix AA is _____.

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  3. Q.1 (iii)2 marks
    If tan⁡−1(2x)+tan⁡−1(3x)=π4\tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}, then x=x = _____.

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  4. Q.1 (iv)2 marks
    The angle between the line rˉ=(i^+2j^+k^)+λ(i^+j^+k^)\bar{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k}) and the plane rˉ⋅(2i^−j^+k^)=8\bar{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 8 is _____.

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  5. Q.1 (v)2 marks
    If y=sec⁡(tan⁡−1x)y = \sec\left(\tan^{-1} x\right), then dydx\frac{dy}{dx} at x=1x = 1 is _____.

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  6. Q.1 (vi)2 marks
    The approximate value of the function f(x)=x3−3x+5f(x) = x^{3} - 3x + 5 at x=1.99x = 1.99 is _____.

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  7. Q.1 (vii)2 marks
    ∫121x2⋅e1xdx=\int_{1}^{2} \frac{1}{x^{2}} \cdot e^{\frac{1}{x}} dx = _____.

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  8. Q.1 (viii)2 marks
    If the p.d.f. of a continuous r.v. XX is f(x)=x+218f(x) = \frac{x + 2}{18}, for −2<x<4-2 < x < 4 =0= 0, otherwise then P(∣X∣<1)=P(|X| < 1) = _____

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  9. Q.2 (i)1 mark
    Write the dual of (p∨q)∨r≡p∨(q∨r)(p \vee q) \vee r \equiv p \vee (q \vee r)
  10. Q.2 (ii)1 mark
    Evaluate : cos⁡−1(12)+2sin⁡−1(12)\cos^{-1}\left(\frac{1}{2}\right) + 2\sin^{-1}\left(\frac{1}{2}\right)
  11. Q.2 (iii)1 mark
    Evaluate : ∫2x1+x2dx\int \frac{2x}{1 + x^{2}} dx
  12. Q.2 (iv)1 mark
    Write the degree of the differential equation (y′′′)2+3(y′′)3+3xy′+5y=0(y''')^{2} + 3(y'')^{3} + 3xy' + 5y = 0

Section B

2 marks each · attempt any 8

  1. Q.32 marks
    Construct the switching circuit of the statement pattern (∼p∧q)∨(p∧∼r)(\sim p \wedge q) \vee (p \wedge \sim r).
  2. Q.42 marks
    In ΔABC\Delta ABC, prove that a(bcos⁡C−ccos⁡B)=b2−c2a(b\cos C - c\cos B) = b^{2} - c^{2}.
  3. Q.52 marks
    Find the general solution of 4cos⁡2θ=34\cos^{2}\theta = 3.
  4. Q.62 marks
    Find kk, if the sum of the slopes of the lines represented by x2+kxy−3y2=0x^{2} + kxy - 3y^{2} = 0 is twice their product.
  5. Q.72 marks
    Find the value of pp, for which the vectors aˉ=3i^+2j^+9k^\bar{a} = 3\hat{i} + 2\hat{j} + 9\hat{k} and bˉ=i^+pj^+3k^\bar{b} = \hat{i} + p\hat{j} + 3\hat{k} are perpendicular to each other.
  6. Q.82 marks
    Find the vector equation of the line passing through the points A(1,2,3)A(1, 2, 3) and B(2,3,4)B(2, 3, 4).
  7. Q.92 marks
    Find dydx\frac{dy}{dx}, if x+y=a\sqrt{x} + \sqrt{y} = \sqrt{a}.
  8. Q.102 marks
    Find d2ydx2\frac{d^{2}y}{dx^{2}}, if y=x3+7x2−2x−9y = x^{3} + 7x^{2} - 2x - 9.
  9. Q.112 marks
    Test whether the function f(x)=x3+6x2+12x−7f(x) = x^{3} + 6x^{2} + 12x - 7 is increasing or decreasing for all x∈Rx \in R.
  10. Q.122 marks
    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?
  11. Q.132 marks
    Evaluate : ∫1+sin⁡2x⋅dx\int \sqrt{1 + \sin 2x} \cdot dx
  12. Q.142 marks
    Given that, X∼B(n,p)X \sim B(n, p), if n=10n = 10, E(X)=8E(X) = 8 then find Var(X)\text{Var}(X).

Section C

3 marks each · attempt any 8

  1. Q.153 marks
    Examine whether the statement pattern (p∧q)∧(∼p∨∼q)(p \wedge q) \wedge (\sim p \vee \sim q) is a tautology or contradiction or contingency.
  2. Q.163 marks
    In ΔABC\Delta ABC, if A=45∘A = 45^{\circ}, B=60∘B = 60^{\circ} then find the ratio of its sides.
  3. Q.173 marks
    If two vertices of a triangle are A(3,1,4)A(3, 1, 4) and B(−4,5,−3)B(-4, 5, -3) and the centroid of the triangle is G(−1,2,1)G(-1, 2, 1), then find the coordinates of the third vertex CC of the triangle.
  4. Q.183 marks
    If DD, EE, FF are the mid-points of the sides BCBC, CACA, ABAB respectively of ΔABC\Delta ABC, then prove that AD‾+BE‾+CF‾=0ˉ\overline{AD} + \overline{BE} + \overline{CF} = \bar{0}.
  5. Q.193 marks
    Show that the lines rˉ=(i^+j^−k^)+λ(2i^−2j^+k^)\bar{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(2\hat{i} - 2\hat{j} + \hat{k}) and rˉ=(4i^−3j^+2k^)+μ(i^−2j^+2k^)\bar{r} = (4\hat{i} - 3\hat{j} + 2\hat{k}) + \mu(\hat{i} - 2\hat{j} + 2\hat{k}) intersect each other.
  6. Q.203 marks
    Find the cartesian equation of the plane rˉ=(i^−j^)+λ(i^+j^+k^)+μ(i^−2j^+3k^)\bar{r} = (\hat{i} - \hat{j}) + \lambda(\hat{i} + \hat{j} + \hat{k}) + \mu(\hat{i} - 2\hat{j} + 3\hat{k}).
  7. Q.213 marks
    If y=f(u)y = f(u) is a differentiable function of uu and u=g(x)u = g(x) is a differentiable function of xx then prove that yy is a differentiable function of xx and dydx=dydu×dudx\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}.
  8. Q.223 marks
    Verify LMVT for the function f(x)=log⁡xf(x) = \log x, on [1,e][1, e].
  9. Q.233 marks
    Evaluate : ∫sin⁡xsin⁡3xdx\int \frac{\sin x}{\sin 3x} dx
  10. Q.243 marks
    Solve the D.E. 3extan⁡y dx+(1+ex)sec⁡2y dy=03e^{x} \tan y \, dx + (1 + e^{x})\sec^{2} y \, dy = 0.
  11. Q.253 marks
    Find E(X)E(X) and V(X)V(X), where XX is the number obtained on uppermost face, when a fair die is thrown.
  12. Q.263 marks
    A pair of dice is thrown 4 times. If getting a doublet is considered as success, find the probability of two successes.

Section D

4 marks each · attempt any 5

  1. Q.274 marks
    If A=[1234]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, prove that A⋅(adj A)=(adj A)⋅A=∣A∣⋅IA \cdot (\text{adj } A) = (\text{adj } A) \cdot A = |A| \cdot I
  2. Q.284 marks
    Show that every homogeneous equation of degree two in xx and yy i.e. 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
    If A(aˉ)A(\bar{a}) and B(bˉ)B(\bar{b}) are any two points in space and R(rˉ)R(\bar{r}) be a point on the line segment ABAB dividing internally in the ratio m:nm : n then prove that rˉ=mbˉ+naˉm+n\bar{r} = \frac{m\bar{b} + n\bar{a}}{m + n}.
  4. Q.304 marks
    Solve the L.P.P. graphically : Minimize : z=5x+2yz = 5x + 2y, Subject to, 5x+y≥105x + y \geq 10, x+y≥6x + y \geq 6, x≥0x \geq 0, y≥0y \geq 0
  5. Q.314 marks
    Evaluate : ∫3x2+4x−5(x2−1)(x+2)⋅dx\int \frac{3x^{2} + 4x - 5}{(x^{2} - 1)(x + 2)} \cdot dx
  6. Q.324 marks
    Prove that : ∫abf(x)dx=∫abf(a+b−x)⋅dx\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a + b - x) \cdot dx Hence, find ∫π6π3sin⁡2x⋅dx\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sin^{2} x \cdot dx
  7. Q.334 marks
    Find the area enclosed between the circle x2+y2=1x^{2} + y^{2} = 1 and the line x+y=1x + y = 1 lying in the first quadrant.
  8. Q.344 marks
    Solve the differential equation x2⋅dydx=x2+xy+y2x^{2} \cdot \frac{dy}{dx} = x^{2} + xy + y^{2}.