Class 12 Mathematics (New Course) Question Paper 2083 (2026)

Which one of the following represents \(\sum n^3\) ?
A) \(\dfrac{n(n+1)}{2}\)   B) \(\dfrac{n^2(n+1)}{4}\)
C) \(\dfrac{n^2(n+1)(n+2)}{4}\)   D) \(\dfrac{n^2(n+1)^2}{4}\)

Which one is the highest Common Factor (HCF) of \(n!\), \((n+1)!\), and \((n+2)!\) ?
A) \((n+2)!\)   B) \((n+1)!\)   C) \(n!\)   D) \(n\)

Which one of the following is equation of circle which touches \(x\)-axis and whose centre is at \((3, 4)\) ?
A) \(x^2 + y^2 + 6x + 8y + 9 = 0\)   B) \(x^2 + y^2 - 6x - 8y + 9 = 0\)
C) \(x^2 + y^2 + 6x + 8y + 16 = 0\)   D) \(x^2 + y^2 - 6x - 8y + 16 = 0\)

In any triangle ABC, \(a = 3\), \(b = 4\) and \(c = 5\), then which is equal to \(\sin\dfrac{A}{2}\) ?
A) \(\dfrac{3}{\sqrt{10}}\)   B) \(\dfrac{1}{\sqrt{10}}\)   C) \(\dfrac{1}{10}\)   D) \(\dfrac{3}{10}\)

If \(\vec{a}\) and \(\vec{b}\) be two vectors then the projection of the vector \(\vec{a}\) on \(\vec{b}\) is:
A) \(\dfrac{\vec{a} \cdot \vec{b}}{|\vec{b}|}\)   B) \(\dfrac{\vec{a} \cdot \vec{b}}{|\vec{a}|}\)   C) \(\vec{a} \cdot \vec{b}\)   D) \(\dfrac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\)

A and B are dependent events. \(P(A) = \dfrac{1}{2}\), \(P(B) = \dfrac{1}{3}\) and \(P(A \cap B) = \dfrac{1}{5}\), which one of the following is \(P(A/B)\) ?
A) \(\dfrac{1}{6}\)   B) \(\dfrac{2}{5}\)   C) \(\dfrac{3}{5}\)   D) \(\dfrac{19}{30}\)

Which one is the derivative of \(\log\!\left(\cosh\dfrac{x}{a}\right)\) ?
A) \(\dfrac{1}{a}\tanh\dfrac{x}{a}\)   B) \(\tanh\dfrac{x}{a}\)   C) \(\dfrac{1}{a}\text{sech}\dfrac{x}{a}\)   D) \(\dfrac{1}{a}\coth\dfrac{x}{a}\)

What is the value of \(\lim_{x \to 1}\left(\dfrac{x^3 - 1}{x^2 - 1}\right)\) ?
A) \(\dfrac{1}{2}\)   B) \(\dfrac{3}{2}\)   C) \(2\)   D) \(3\)

Which one is the solution of \(y\,dx - \dfrac{x}{2}\,dy = 0\) ?
A) \(y = c\sqrt{x}\)   B) \(x = c\sqrt{y}\)
C) \(y = cx\)   D) \(x = cy\)

The integrating factor for the differential equation \(\dfrac{dy}{dx} - \dfrac{y}{x} = x^2\) is
A) \(\log\dfrac{1}{x}\)   B) \(e^{-\frac{1}{x}}\)   C) \(\dfrac{1}{x}\)   D) \(-\dfrac{1}{x}\)

After forward row reduction in Gauss elimination method, if the augmented matrix of given system of three linear equations reduces to following form, then the system of equations has
\(\begin{bmatrix} a_1 & b_1 & c_1 & : & d_1 \\ 0 & b_2 & c_2 & : & d_2 \\ 0 & 0 & 0 & : & 0 \end{bmatrix}\)
A) Unique solutions   B) Infinitely many solutions
C) No solutions   D) Only one solution
OR
In a projectile, initial velocity is \(40\,\text{ms}^{-1}\) and angle of projection is \(30^\circ\), what is its time of flight ?
A) \(\dfrac{10}{g}\)   B) \(\dfrac{20}{g}\)   C) \(\dfrac{40}{g}\)   D) \(\dfrac{80}{g}\)

Note: In question No. 17 e) write \(\dfrac{d^2y}{dx^2}\) in place of \(\dfrac{d^2y}{2x^2}\).

a) State the principle of mathematical induction. [1]
b) Write the general term of \((x + a)^n\). [1]
c) When does given linear equations become inconsistent ? [1]
d) State Demoivre's theorem for positive integral power. [1]
e) Write any one property of combination. [1]

a) How many word can be formed from the letter of the word "DAUGHTER" so that vowels always come together ? [2]
b) Find the cube roots of : \(-\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}\,i\) [3]

a) Find the solution of triangle given by \(a = 2\), \(b = \sqrt{6}\) and \(A = 45^\circ\) [2]
b) Two vectors \(\vec{a} = 2\vec{i} - 3\vec{j} + 4\vec{k}\) and \(\vec{b} = -\vec{i} + 5\vec{j} + 2\vec{k}\) are given. Is the area of parallelogram formed by \(\vec{a}\) and \(\vec{b}\) a whole number ? Find it by calculation. [3]

a) Find the equation of tangent to the curve \(y^2 = 4ax\) at \((t_1,\, t_2)\). [2]
b) Find the equation of hyperbola with focus at \((-7,\, 0)\) and eccentricity \(= \dfrac{7}{4}\) [3]

The height in inches (X) and weight in kg (Y) of 7 students are given below:

a) If a particle moves in a path given by \(S = f(t)\), write the velocity. [1]
b) Write the derivative of \(\sinh^{-1}x\) with respect to \(x\). [1]
c) Write the integral of \(\int \cosh x\, dx\). [1]
d) Write the integral of \(\displaystyle\int \dfrac{dx}{x^2 - a^2}\). [1]
e) Write the order of \(\dfrac{d^2y}{dx^2} + y\dfrac{dy}{dx} = 5x\) [1]

a) Using L. Hospital's rule, find the limit of \(\lim_{x \to 0}\left(\dfrac{x - \sin x}{x^2}\right)\) [2]
b) Solve: \(\sqrt[3]{\dfrac{dy}{dx} + \dfrac{1}{x}y} = x\). [3]

Use simplex method to solve maximize \(Z = 8x + 36y\) subject to constraints
\(2x + 6y \leq 30,\quad x + 6y \leq 18,\quad x,\, y \geq 0\) [5]
OR
a) A mass of 10 kg is acted on by a constant force which in 5 seconds, produced a velocity of 20m per second. Find the force if the mass was initially rest. [2]
b) A particle of mass 5 kg slides down ache inclination of the plane. If smooth inclined plane to the horizontal is \(30^\circ\). Find the acceleration of the particle and reaction between the particle and the plane. [3]

a) A Committee of 5 members is to be formed from 6 boys and 4 girls. In how many ways can this be done so as to include at most three girls. [3]
b) Prove that \(1 + 3 + 5 + \ldots + (2n-1) = n^2\) using principle of mathematical induction. [3]
c) The total cost of 2 pen and 3 book is Rs. 1,100 and the total cost of 4 pen and 1 book in Rs. 700. Find the cost of a pen and a book using Cramer's rule. [2]

a) Prove that : \((c + a - b)\!\left(\cot\dfrac{C}{2} + \cot\dfrac{A}{2}\right) = 2b\cot\dfrac{B}{2}\) [3]
b) Find vertex, focus, and equation of directrix of conic \(9x^2 + 4y^2 = 36\). [3]
c) Find the angle between \(\vec{a} = \vec{i} + 2\vec{j} - \vec{k}\) and \(\vec{b} = \vec{i} - \vec{j} + \vec{k}\) [2]

a) A container is made up of a hollow inverted right circular cone whose height is 24 cm and the radius at the top is 16 cm. Water is flowing in the container at the rate of 8 cm³/sec and find rate of change of depth of water when depth is 12 cm. [3]
b) Integrate : \(\displaystyle\int \dfrac{y^2 + 1}{(y^2 + 49)(y^2 + 4)}\, dy\) [3]
c) Solve : \(x\,dy + y\,dx = \sqrt{x^2 + a^2}\, dx\) [2]