Q.1 (i) The converse of contrapositive of \(\sim p \rightarrow q\) is _____.
Mathematics & Statistics Part 1
Every question from past board papers on Mathematics & Statistics Part 1, newest paper first. Open any question for its full worked answer.
Maharashtra HSC Mathematics and Statistics (Science) Board Paper March 2026 with Solutions
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Q.1 (ii) If \(A = \begin{bmatrix} 2 & -4 \\ 3 & 1 \end{bmatrix}\), then the adjoint of matrix \(A\) is _____.
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Q.1 (iii) If \(\tan^{-1}(2x) + \tan^{-1}(3x) = \dfrac{\pi}{4}\), then \(x =\) _____.
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Q.1 (iv) The angle between the line \(\bar{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k})\) and the plane \(\bar{r}\cdot(2\hat{i} - \hat{j} + \hat{k}) = 8\) is _____.
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Q.2 (i) Write the dual of \((p \vee q) \vee r \equiv p \vee (q \vee r)\)
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Q.2 (ii) Evaluate : \(\cos^{-1}\left(\dfrac{1}{2}\right) + 2\sin^{-1}\left(\dfrac{1}{2}\right)\)
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Q.3 Construct the switching circuit of the statement pattern \((\sim p \wedge q) \vee (p \wedge \sim r)\).
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Q.4 In \(\triangle ABC\), prove that \(a(b\cos C - c\cos B) = b^2 - c^2\).
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Q.5 Find the general solution of \(4\cos^2\theta = 3\).
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Q.6 Find \(k\), if the sum of the slopes of the lines represented by \(x^2 + kxy - 3y^2 = 0\) is twice their product.
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Q.7 Find the value of \(p\), for which the vectors \(\bar{a} = 3\hat{i} + 2\hat{j} + 9\hat{k}\) and \(\bar{b} = \hat{i} + p\hat{j} + 3\hat{k}\) are perpendicular to each other.
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Q.8 Find the vector equation of the line passing through the points \(A(1, 2, 3)\) and \(B(2, 3, 4)\).
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Q.15 Examine whether the statement pattern \((p \wedge q) \wedge (\sim p \vee \sim q)\) is a tautology or contradiction or contingency.
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Q.16 In \(\triangle ABC\), if \(A = 45^\circ,\ B = 60^\circ\) then find the ratio of its sides.
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Q.17 If two vertices of a triangle are \(A(3, 1, 4)\) and \(B(-4, 5, -3)\) and the centroid of the triangle is \(G(-1, 2, 1)\), then find the coordinates of the third vertex \(C\) of the triangle.
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Q.18 If \(D, E, F\) are the mid-points of the sides \(BC, CA, AB\) respectively of \(\triangle ABC\), then prove that \(\overline{AD} + \overline{BE} + \overline{CF} = \bar{0}\).
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Q.19 Show that the lines \(\bar{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(2\hat{i} - 2\hat{j} + \hat{k})\) and \(\bar{r} = (4\hat{i} - 3\hat{j} + 2\hat{k}) + \mu(\hat{i} - 2\hat{j} + 2\hat{k})\) intersect each other.
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Q.20 Find the cartesian equation of the plane \(\bar{r} = (\hat{i} - \hat{j}) + \lambda(\hat{i} + \hat{j} + \hat{k}) + \mu(\hat{i} - 2\hat{j} + 3\hat{k})\).
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Q.27 If \(A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\), prove that \(A\cdot(\operatorname{adj} A) = (\operatorname{adj} A)\cdot A = |A|\cdot I\)
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Q.28 Show that every homogeneous equation of degree two in \(x\) and \(y\) i.e. \(ax^2 + 2hxy + by^2 = 0\), represents a pair of lines passing through the origin, if \(h^2 - ab \ge 0\)
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Q.29 If \(A(\bar{a})\) and \(B(\bar{b})\) are any two points in space and \(R(\bar{r})\) be a point on the line segment \(AB\) dividing internally in the ratio \(m : n\) then prove that \(\bar{r} = \dfrac{m\bar{b} + n\bar{a}}{m + n}\).
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Q.30 Solve the L.P.P. graphically :
Minimize : \(z = 5x + 2y\),
Subject to, \(5x + y \ge 10,\ x + y \ge 6,\)
\(x \ge 0,\ y \ge 0\)
Maharashtra HSC Mathematics and Statistics (Science) Board Paper June 2026 with Solutions
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Q.1 (i) If \(A = \{1, 2, 3, 4, 5\}\) then which of the following is not true?
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Q.1 (ii) The value of \(\sin\left[\sin^{-1}\left(-\frac{1}{\sqrt{2}}\right) - 3\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\right] =\)
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Q.1 (iii) If \(\hat{p}, \hat{q}, \hat{r}\) are mutually perpendicular unit vectors and form a right handed triplet then the value of \(\hat{p}\cdot(\hat{q}\times\hat{r}) + \hat{q}\cdot(\hat{p}\times\hat{r}) + \hat{r}\cdot(\hat{p}\times\hat{q}) =\) _____
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Q.1 (iv) The Cartesian equation of the line passing through the points \(A(3, 4, -1)\) and \(B(2, -1, 3)\) is _____.
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Q.2 (i) Write the dual of \(p \wedge t\).
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Q.2 (ii) What is the distance of point \((3, -5, 6)\) from XZ plane?
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Q.3 Construct the switching circuit for the statement \((\sim p \wedge q) \vee (p \wedge \sim r)\).
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Q.4 Find the matrix of cofactors of matrix \(\begin{bmatrix} 5 & 2 \\ -1 & 6 \end{bmatrix}\).
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Q.5 Find the general solution of equation \(\cos 5\theta = \frac{1}{2}\).
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Q.6 Find the value of \(k\), if \(2x + y = 0\) is one of the lines represented by \(3x^2 + kxy + 2y^2 = 0\).
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Q.7 If \(\bar{a}, \bar{b}, \bar{c}\) are the position vectors of points \(A, B, C\) respectively and \(10\bar{a} = 7\bar{b} + 3\bar{c}\) then find the ratio in which the point \(C\) divides the line segment \(AB\).
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Q.8 Find the angle between the lines
\(\bar{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} - \hat{j} + \hat{k})\)
\(\bar{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \mu(\hat{i} + 2\hat{j} + \hat{k})\) -
Q.15 Using truth table, determine whether statement pattern \([(p \vee q) \wedge \sim p] \wedge \sim q\) is a tautology or a contradiction or a contingency.
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Q.16 Prove that: \(\sin^{-1}\left(\frac{3}{5}\right) + \cos^{-1}\left(\frac{12}{13}\right) = \sin^{-1}\left(\frac{56}{65}\right)\)
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Q.17 In \(\triangle ABC\), if \(a = 10\), \(b = 8\), \(c = 6\) then find the value of (i) \(\sin\left(\frac{C}{2}\right)\) (ii) \(\cos\left(\frac{B}{2}\right)\)
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Q.18 Are the four points \(A(1, 2, 1)\), \(B(2, -3, 4)\), \(C(3, 4, -5)\), \(D(2, 3, -2)\) coplanar? Justify your answer.
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Q.19 Find the distance between parallel lines \(\frac{x}{2} = \frac{y}{-1} = \frac{z}{2}\) and \(\frac{x+1}{2} = \frac{y-1}{-1} = \frac{z+1}{2}\)
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Q.20 Find the vector equation of the plane passing through the points \(A(2, 1, 1)\), \(B(0, 2, 3)\) and \(C(4, 5, 6)\).
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Q.27 Find the inverse of matrix \(\begin{bmatrix} -4 & -3 & -3 \\ 1 & 0 & 1 \\ 4 & 4 & 3 \end{bmatrix}\)
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Q.28 Prove that the homogeneous equation of degree two in \(x\) and \(y\), \(ax^2 + 2hxy + by^2 = 0\) represents a pair of lines passing through the origin if \(h^2 - ab \ge 0\).
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Q.29 Prove that, if \(\bar{a}, \bar{b}, \bar{c}\) are three non coplanar vectors then any vector \(\bar{r}\) in the space can be uniquely expressed as a linear combination of \(\bar{a}, \bar{b}, \bar{c}\).
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Q.30 Solve the linear programming problem (L.P.P.) by graphical method.
Minimize: \(z = 8x + 10y\)
Subject to, \(2x + y \ge 7\), \(2x + 3y \ge 15\), \(x \ge 0\), \(y \ge 2\)
Maharashtra HSC Mathematics and Statistics (Science) Board Paper March 2025 with Solutions
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Q.1 (i) If \(A = \{1, 2, 3, 4, 5\}\) then which of the following is not true?
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Q.1 (ii) In \(\triangle ABC\), \((a + b)\cdot\cos C + (b + c)\cos A + (c + a)\cdot\cos B\) is equal to _____.
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Q.1 (iii) If \(|\bar a| = 5\), \(|\bar b| = 13\) and \(|\bar a \times \bar b| = 25\) then \(|\bar a \cdot \bar b|\) is equal to _____.
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Q.1 (iv) The vector equation of the line passing through the point having position vector \(4\hat i - \hat j + 2\hat k\) and parallel to vector \(-2\hat i - \hat j + \hat k\) is given by _____.
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Q.2 (i) Write the negation of the statement: ‘\(\exists n \in N\) such that \(n + 8 \gt 11\)’
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Q.2 (ii) Write unit vector in the opposite direction to \(\bar u = 8\hat i + 3\hat j - \hat k\).
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Q.3 Using truth table, prove that the statement patterns \(p \leftrightarrow q\) and \((p \wedge q) \vee (\sim p \wedge \sim q)\) are logically equivalent.
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Q.4 Find the adjoint of the matrix \(\begin{bmatrix} 2 & -2 \\ 4 & 3 \end{bmatrix}\).
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Q.5 Find the general solution of \(\tan^2\theta = 1\).
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Q.6 Find the co-ordinates of the points of intersection of the lines represented by \(x^2 - y^2 - 2x + 1 = 0\).
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Q.7 A line makes angles of measure 45° and 60° with the positive directions of the Y and Z axes respectively. Find the angle made by the line with the positive direction of the X-axis.
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Q.8 Find the vector equation of the plane passing through the point having position vector \(2\hat i + 3\hat j + 4\hat k\) and perpendicular to the vector \(2\hat i + \hat j - 2\hat k\).
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Q.15 Express the following switching circuit in the symbolic form of logic. Construct the switching table and interpret it:
[Figure: Switches \(S_1\) and \(S_2\) are connected in parallel; this parallel combination is connected in series with switches \(S_1'\) and \(S_2'\), a battery and a lamp L.]
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Q.16 Prove that: \(2\tan^{-1}\left(\dfrac13\right) + \cos^{-1}\left(\dfrac35\right) = \dfrac{\pi}{2}\).
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Q.17 In \(\triangle ABC\) if \(a = 13\), \(b = 14\), \(c = 15\) then find the values of (i) \(\sec A\) (ii) \(\operatorname{cosec}\dfrac{A}{2}\).
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Q.18 A line passes through the points \((6, -7, -1)\) and \((2, -3, 1)\). Find the direction ratios and the direction cosines of the line. Show that the line does not pass through the origin.
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Q.19 Find the cartesian and vector equations of the line passing through \(A(1, 2, 3)\) and having direction ratios 2, 3, 7.
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Q.20 Find the vector equation of the plane passing through points \(A(1, 1, 2)\), \(B(0, 2, 3)\) and \(C(4, 5, 6)\).
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Q.27 Find the inverse of \(\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}\) by elementary row transformations.
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Q.28 Prove that homogeneous equation of degree two in \(x\) and \(y\), \(ax^2 + 2hxy + by^2 = 0\) represents a pair of lines passing through the origin if \(h^2 - ab \ge 0\). Hence show that equation \(x^2 + y^2 = 0\) does not represent a pair of lines.
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Q.29 Let \(\bar a\) and \(\bar b\) be non-collinear vectors. If vector \(\bar r\) is coplanar with \(\bar a\) and \(\bar b\) then show that there exist unique scalars \(t_1\) and \(t_2\) such that \(\bar r = t_1\bar a + t_2\bar b\). For \(\bar r = 2\hat i + 7\hat j + 9\hat k\), \(\bar a = \hat i + 2\hat j\), \(\bar b = \hat j + 3\hat k\), find \(t_1\), \(t_2\).
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Q.30 Solve the linear programming problem graphically.
Maximize: \(z = 3x + 5y\)
Subject to: \(x + 4y \le 24\), \(3x + y \le 21\), \(x + y \le 9\), \(x \ge 0, y \ge 0\)Also find the maximum value of \(z\).
Maharashtra HSC Mathematics and Statistics (Science) Board Paper July 2025 with Solutions
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Q.1 (i) The inverse of statement pattern \((p \vee q) \to (p \wedge q)\) is _____
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Q.1 (ii) In \(\triangle ABC\), if \(a = 2,\ b = 3\) and \(\sin A = \frac{2}{3}\), then \(\angle B =\) _____
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Q.1 (iii) If \(\overline{AB} = 2\hat{i} - 4\hat{j} + 7\hat{k}\) and initial point \(A \equiv (1, 5, 0)\) then terminal point \(B\) is _____
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Q.1 (iv) The angle between the lines \(\bar{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} - 2\hat{j} + \hat{k})\) and \(\bar{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \mu(\hat{i} + 2\hat{j} + 2\hat{k})\) is _____
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Q.2 (i) Find the general solution of \(\tan\theta = 0\).
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Q.2 (ii) Find the magnitude of the vector \(\bar{a} = 3\hat{i} + \hat{j} + 7\hat{k}\)
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Q.3 Using truth table, prove that \(\sim p \wedge q \equiv (p \vee q) \wedge \sim p\)
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Q.4 Find the cofactors of the elements of the matrix \(\begin{bmatrix} 2 & -3 \\ 3 & 5 \end{bmatrix}\)
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Q.5 Find the polar coordinates of the point whose cartesian coordinates are \((-\sqrt{2}, -\sqrt{2})\).
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Q.6 In \(\triangle ABC\), if \(a = 2,\ b = 3,\ c = 4\), then prove that the triangle is obtuse angled.
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Q.7 If \(\bar{a} = 3\hat{i} - \hat{j} + 2\hat{k},\ \bar{b} = 2\hat{i} + \hat{j} - \hat{k}\), then find \(|\bar{a} \times \bar{b}|\)
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Q.8 Find the cartesian equation of the plane passing through the point \(A(-1, 2, 3)\), the direction ratios of whose normal are 0, 2, 5.
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Q.15 In \(\triangle ABC\), prove that \(a^2 = b^2 + c^2 - 2bc\cos A\)
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Q.16 Find the combined equation of pair of lines passing through (2, 3) and perpendicular to the lines \(3x + 2y - 1 = 0\) and \(x - 3y + 2 = 0\).
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Q.17 Show that the acute angle \(\theta\) between the lines represented by \(ax^2 + 2hxy + by^2 = 0\) is given by, \(\tan\theta = \left|\frac{2\sqrt{h^2 - ab}}{a + b}\right|\)
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Q.18 Using vector method, prove that the medians of a triangle are concurrent.
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Q.19 Find the vector equation of the plane passing through the point \(A(-1, 2, -5)\) and parallel to the vectors \(4\hat{i} - \hat{j} + 3\hat{k}\) and \(\hat{i} + \hat{j} - \hat{k}\).
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Q.20 Find the shortest distance between the lines, \(\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}\) and \(\frac{x - 2}{3} = \frac{y - 4}{4} = \frac{z - 5}{5}\).
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Q.27 Give an alternative equivalent simple circuit for the following switching circuit :
[Figure: Switch \(S_1\) is connected in series with a parallel combination of two switches \(S_1'\) (upper branch) and \(S_2\) (lower branch); the circuit is completed through a battery and a lamp L.]
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Q.28 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.
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Q.29 Using properties of scalar triple product, prove that \([\bar{a} + \bar{b}\ \ \bar{b} + \bar{c}\ \ \bar{c} + \bar{a}] = 2[\bar{a}\ \bar{b}\ \bar{c}]\)
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Q.30 Solve the following L.P.P. using graphical method :
Maximize, \(z = 9x + 13y\)
Subject to, \(2x + 3y \le 18\),
\(2x + y \le 10\)
\(x \ge 0,\ y \ge 0\)
Maharashtra HSC Mathematics and Statistics (Science) Board Paper March 2024 with Solutions
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Q.1 (i) The dual of statement \(t \vee (p \vee q)\) is _____.
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Q.1 (ii) The principle solutions of the equation \(\cos\theta = \frac{1}{2}\) are _____.
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Q.1 (iii) If \(\alpha, \beta, \gamma\) are direction angles of a line and \(\alpha = 60^\circ\), \(\beta = 45^\circ\), then \(\gamma =\) _____.
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Q.1 (iv) The perpendicular distance of the plane \(\bar{r} \cdot (3\hat{i} + 4\hat{j} + 12\hat{k}) = 78\), from the origin is _____.
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Q.2 (i) Write the compound statement ‘Nagpur is in Maharashtra and Chennai is in Tamilnadu’ symbolically.
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Q.2 (ii) If the vectors \(2\hat{i} - 3\hat{j} + 4\hat{k}\) and \(p\hat{i} + 6\hat{j} - 8\hat{k}\) are collinear, then find the value of \(p\).
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Q.3 Construct the truth table for the statement pattern : \([(p \to q) \wedge q] \to p\)
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Q.4 Check whether the matrix \(\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}\) is invertible or not.
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Q.5 In \(\Delta ABC\), if \(a = 18\), \(b = 24\) and \(c = 30\) then find the value of \(\sin\left(\frac{A}{2}\right)\).
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Q.6 Find \(k\), if the sum of the slopes of the lines represented by \(x^2 + kxy - 3y^2 = 0\) is twice their product.
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Q.7 If \(\bar{a}, \bar{b}, \bar{c}\) are the position vectors of the points \(A, B, C\) respectively and \(5\bar{a} - 3\bar{b} - 2\bar{c} = \bar{0}\), then find the ratio in which the point \(C\) divides the line segment \(BA\).
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Q.8 Find the vector equation of the line passing through the point having position vector \(4\hat{i} - \hat{j} + 2\hat{k}\) and parallel to the vector \(-2\hat{i} - \hat{j} + \hat{k}\).
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Q.15 Express the following switching circuit in the symbolic form of logic. Construct the switching table :
[Figure: Switches \(S_1\) and \(S_2\) are connected in parallel; this parallel combination is connected in series with switches \(S_1'\) and \(S_2'\), a lamp L and a battery.]
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Q.16 Prove that : \(\tan^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(\frac{1}{3}\right) = \frac{\pi}{4}\)
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Q.17 In \(\Delta ABC\), prove that : \(\frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} = \frac{a^2 + b^2 + c^2}{2abc}\).
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Q.18 Prove by vector method, the angle subtended on a semicircle is a right angle.
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Q.19 Find the shortest distance between the lines \(\bar{r} = (4\hat{i} - \hat{j}) + \lambda(\hat{i} + 2\hat{j} - 3\hat{k})\) and \(\bar{r} = (\hat{i} - \hat{j} - 2\hat{k}) + \mu(\hat{i} + 4\hat{j} - 5\hat{k})\)
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Q.20 Find the angle between the line \(\bar{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k})\) and the plane \(\bar{r} \cdot (2\hat{i} + \hat{j} + \hat{k}) = 8\).
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Q.27 Solve the following system of equations by the method of reduction : \(x + y + z = 6\), \(y + 3z = 11\), \(x + z = 2y\).
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Q.28 Prove that the acute angle \(\theta\) between the lines represented by the equation \(ax^2 + 2hxy + by^2 = 0\) is \(\tan\theta = \left|\frac{2\sqrt{h^2 - ab}}{a + b}\right|\).
Hence find the condition that the lines are coincident. -
Q.29 Find the volume of the parallelopiped whose vertices are \(A(3, 2, -1)\), \(B(-2, 2, -3)\), \(C(3, 5, -2)\) and \(D(-2, 5, 4)\).
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Q.30 Solve the following L.P.P. by graphical method :
Maximize : \(z = 10x + 25y\)
Subject to : \(0 \le x \le 3\),
\(0 \le y \le 3\),
\(x + y \le 5\).
Also find the maximum value of \(z\).