UP Board Class 12 Mathematics 2026 Question Paper with Solution PDF

The value of \( \displaystyle \int_{0}^{\pi/2} \frac{dx}{1 + \sqrt{\tan x}} \) will be

• (A) 0
• (B) \(\frac{\pi}{2}\)
• (C) \(\frac{\pi}{4}\)
• (D) \(\frac{\pi}{8}\)

The degree of differential equation
\[ 9 \frac{d^2y}{dx^2} = \left\{ 1 + \left( \frac{dy}{dx} \right)^2 \right\}^{\frac{1}{3}} is \]

• (A) 1
• (B) 6
• (C) 3
• (D) 2

The value of expression \(\hat{i} \cdot \hat{i} - \hat{j} \cdot \hat{j} + \hat{k} \times \hat{k}\) is

• (A) 0
• (B) 1
• (C) 2
• (D) 3

The modulus function \( f : \mathbb{R} \to \mathbb{R}^+ \) given by \( f(x) = |x| \) is

• (A) one-one and onto
• (B) many-one and onto
• (C) one-one but not onto
• (D) neither one-one nor onto

A relation \( R = \{(a, b) : a = b - 1, b \geq 3\} \) is defined on set \( N \), then

• (A) \( (2, 4) \in R \)
• (B) \( (4, 5) \in R \)
• (C) \( (4, 6) \in R \)
• (D) \( (1, 3) \in R \)

Prove that the function \( f(x) = |x| \), is continuous at \( x = 0 \).

Find the degree of the differential equation
\[xy \frac{d^2y}{dx^2} + x \left( \frac{dy}{dx} \right)^2 - y \left( \frac{dy}{dx} \right) = 2\]

If \( P(A) = 0.12, P(B) = 0.15 \) and \( P(B/A) = 0.18 \), then find the value of \( P(A \cap B) \).

Find the angle between the vectors \(-2\hat{i} + \hat{j} + 3\hat{k}\) and \(3\hat{i} - 2\hat{j} + \hat{k}\).

If \( f : \mathbb{R} \to \mathbb{R} \) and \( g : \mathbb{R} \to \mathbb{R} \) be functions defined by \( f(x) = \cos x \) and \( g(x) = 3x^2 \) respectively, then prove that \( g \circ f \neq f \circ g \).

Find the general solution of differential equation \( ydx + (x - y^2)dy = 0 \).

Prove that (4, 4, 2), (3, 5, 2) and (-1, -1, 2) are vertices of a right angle triangle.

If three vectors \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) satisfying the condition \(\vec{a} + \vec{b} + \vec{c} = 0\). If \(|\vec{a}| = 3\),
\[|\vec{b}| = 4 and |\vec{c}| = 2, then find the value of \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}.\]

The radius of an air bubble is increasing at the rate of \(\frac{1}{2} \, cm/s\). At what rate is the volume of the bubble increasing while the radius is 1 cm?

Show that the function \( f(x) = 7x^2 - 3 \) is an increasing function when \( x > 0 \).

Find the unit vector perpendicular to each of the vectors (\( \vec{a} + \vec{b} \)) and (\( \vec{a} - \vec{b} \)) where \[\vec{a} = \hat{i} + \hat{j} + \hat{k}, \, \vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}.\]