Class 12 Mathematics (New Course) Question Paper 2082 (2025)

Which one is the relation between permutation and combination of \(n\) things taken \(r\) things at a time?
A) \(C(n, r) = r! \times P(n, r)\)    B) \(C(n, r) \times r! = P(n, r)\)
C) \(C(n, r) \times r = P(n, r)\)    D) \(C(n, r) = P(n, r) \times r\)

If the system of equations \(p_1 x + q_1 y = r_1\) and \(p_2 x + q_2 y = r_2\), what is the value of \(y\)?
A) \(\dfrac{\begin{vmatrix} p_1 & r_1 \\ p_2 & r_2 \end{vmatrix}}{\begin{vmatrix} p_1 & q_1 \\ p_2 & q_2 \end{vmatrix}}\)    B) \(\dfrac{\begin{vmatrix} p_1 & q_1 \\ p_2 & q_2 \end{vmatrix}}{\begin{vmatrix} r_1 & q_1 \\ r_2 & q_2 \end{vmatrix}}\)

C) \(\dfrac{\begin{vmatrix} r_1 & q_1 \\ r_2 & q_2 \end{vmatrix}}{\begin{vmatrix} p_1 & q_1 \\ p_2 & q_2 \end{vmatrix}}\)    D) \(\dfrac{\begin{vmatrix} r_1 & p_1 \\ r_2 & p_2 \end{vmatrix}}{\begin{vmatrix} p_1 & q_1 \\ p_2 & q_2 \end{vmatrix}}\)

Which one of the following is the value of \(\sec\dfrac{B}{2}\)?
A) \(\sqrt{\dfrac{ca}{s-b}}\)    B) \(\sqrt{\dfrac{bc}{s(s-a)}}\)    C) \(\sqrt{\dfrac{bc}{s(s-c)}}\)    D) \(\sqrt{\dfrac{ca}{s(s-b)}}\)

In which condition the line \(y = mx + c\) will be tangent to the circle \(x^2 + y^2 = a^2\)?
A) \(a = c\sqrt{1+m^2}\)    B) \(a = \pm c\sqrt{1+m^2}\)
C) \(c = \pm a\sqrt{1+m^2}\)    D) \(c = a\sqrt{1+m^2}\)

What is \(\vec{a} \times \vec{b}\) if \(\vec{a} = (0, 0, 1)\) and \(\vec{b} = (0, 1, 0)\)?
A) \((-1, 0, 0)\)    B) \((1, 0, 0)\)    C) \((0, 1, 0)\)    D) \((0, 0, -1)\)

If \(P(A) = \dfrac{1}{3}\), \(P(B) = \dfrac{2}{3}\) and \(P(A \cap B) = \dfrac{1}{5}\), which one of the following is \(P(B/A)\)?
A) \(\dfrac{3}{5}\)    B) \(\dfrac{2}{5}\)    C) \(\dfrac{2}{15}\)    D) \(\dfrac{1}{15}\)

Which one of the following is the slope of normal to the curve \(y = 3x^2 - x\) at \((-1, 1)\)?
A) \(-7\)    B) \(-5\)    C) \(\dfrac{1}{7}\)    D) \(\dfrac{1}{5}\)

What is the integral of \(\displaystyle\int \dfrac{dx}{9x^2 + 1}\)?
A) \(\dfrac{1}{3}\tan^{-1}(3x) + C\)    B) \(\dfrac{1}{27}\tan^{-1}(3x) + C\)
C) \(\dfrac{1}{27}\tan^{-1}\!\left(\dfrac{x}{3}\right) + C\)    D) \(\dfrac{1}{27}\tan^{-1}\!\left(\dfrac{3}{x}\right) + C\)

Which one of the following is the homogeneous differential equation?
A) \(\dfrac{dy}{dx} = \dfrac{x^2 + y^2}{x + y}\)    B) \(\dfrac{dy}{dx} = \dfrac{x^2 + y^2}{x - y}\)
C) \(\dfrac{dy}{dx} = \dfrac{x^2 + y^2}{x^2 - y^2}\)    D) \(\dfrac{dy}{dx} = \dfrac{x + y}{x^2 + y^2}\)

Which is the integrating factor of differential linear equation \(\cot x \dfrac{dy}{dx} = 1 - y\)?
A) \(\tan x\)    B) \(e^{\tan x}\)    C) \(e^{\sec^2 x}\)    D) \(\sec^2 x\)

Two simultaneous equations are given as \(3x + 4y = 13\) and \(x - 2y = 1\). What is the equation after eliminating \(x\)?
A) \(10y = 10\)    B) \(10y = 16\)    C) \(y = 10\)    D) \(2y = 10\)
OR
What is the maximum height attained by a particle in a projectile motion if initial velocity and angle of inclination are \(40\) m/sec and \(30^\circ\)? \([g = 10 \text{ ms}^{-2}]\)
A) \(20\) m    B) \(40\) m    C) \(80\) m    D) \(160\) m

a) Write the expansion of \(\log_e(1-x)\); \(\lvert x \rvert < 1\). [1]
b) Write the total number of permutations of a set having \(n\) elements. [1]
c) State De-Moivre's theorem. [1]
d) Write the sum of cubes of first \(n\) natural numbers. [1]
e) Write the augmented matrix of the system of equation \(3x + 2y - 1 = 0\) and \(4x + y = 3\). [1]

a) A committee is to be chosen from 'a' boys and 6 girls and is to consist 2 boys and 3 girls. If 120 committees are formed, what is the number of boys represented by 'a'? [2]
b) The square roots of any complex number are \((\sqrt{3} + i)\) and \((-\sqrt{3} - i)\). Write the complex number in polar form. [3]

a) In any triangle PQR, if \(p\sin^3\dfrac{R}{2} + r\sin^3\dfrac{P}{2} = \dfrac{q}{2}\), prove that the sides are in A.P. [3]
b) If \(\vec{a} = 4\vec{i} - 3\vec{j} + 2\vec{k}\) and \(\vec{b} = 3\vec{i} - 2\vec{j} + 4\vec{k}\) are two vectors, find the projection of \(\vec{b}\) on \(\vec{a}\). [2]

a) Find the eccentricity of conic \(3x^2 - 4y^2 - 6x = 0\). [2]
b) Find the eccentricity of ellipse whose major axis is four times its minor axis and passes through the point \((4, 2)\). [3]

Consider the following data for supply (X) and the price (Y) of a commodity for last six years.

a) Write the derivative of \(\text{cosech}^{-1}(x)\). [1]
b) Define L-Hospital's rule. [1]
c) Write the condition where the curve \(y = f(x)\) has tangent parallel to y-axis. [1]
d) Write the integral of \(\displaystyle\int \dfrac{1}{x^2 - a^2}\, dx\). [1]
e) Write the standard form of first order linear differential equation. [1]

a) Find the derivative of \(\coth^{-1}(\sin x)\). [2]
b) Integrate: \(\displaystyle\int \dfrac{dx}{x^3 - x^2 - 2x}\). [3]

Using simplex method, maximize \(P(x, y) = 10x + 3y\) subject to constraint \(6x + 7y \leq 42\), \(x + 3y \leq 42\), \(x + 3y \leq 9\), \(x \geq 0\), \(y \geq 0\). [5]
OR
Two forces A and B acting parallel to the length and base of an inclined plane respectively, would each of them singly support a weight 'R' on the plane, prove that \(\dfrac{1}{A^2} = \dfrac{1}{B^2} + \dfrac{1}{R^2}\). [5]

a) If the middle term in the expansion \(\left(\dfrac{a}{2} + 2\right)^8\) is 1120, find the value of \(a\). [2]
b) Using mathematical induction, prove that \(1 + 7 + 13 + 19 + \cdots + (6n-5) = n(3n-2)\). [3]
c) Solve the following linear equations by using matrix method
\(7x - 2y = 18\), \(3x + 7z = 33\), \(x + y + z = 12\). [3]

a) The scalar product of two vectors and cross product of two vectors are interrelated. Explain. [3]
b) If the cosines of two angles of a triangle are proportional to the opposite sides, show that it is an isosceles triangle. [2]
c) Establish the condition that the line \(ax + by + c = 0\) may be normal to the parabola \(x^2 = 4ay\). [3]

a) Find the rate of change of volume of a sphere with respect to its surface area when radius is 7 cm. [2]
b) Integrate: \(\displaystyle\int \dfrac{dx}{5 - 3\cos x}\). [3]
c) Solve: \((1 + x^2)\,dy - (1 + y^2)\,dx = 0\). [3]