Karnataka 2nd PUC Mathematics 2026 Question Paper with Solution PDF : Available Here

If a relation \(R\) in the set \(\{1,2,3\}\) be defined by \(R=\{(1,1),(2,2)\}\), then \(R\) is

• (A) Symmetric but not transitive
• (B) Transitive but not symmetric
• (C) Symmetric and transitive
• (D) Neither symmetric nor transitive

The domain of \(\tan^{-1}x\) is

• (A) \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\)
• (B) \((0,\pi)\)
• (C) \([-1,1]\)
• (D) \((-\infty,\infty)\)

A matrix has 13 elements. The number of possible different orders it can have is

• (A) 1
• (B) 2
• (C) 3
• (D) 4

For the matrix \(A=\begin{bmatrix}5 & 0
0 & 5\end{bmatrix}\), the value of \(|adj\,A|\) is

• (A) 25
• (B) 5
• (C) 0
• (D) 1

The derivative of \(\sin^{-1}x\) exists in the interval

• (A) \([-1,1]\)
• (B) \((-1,1)\)
• (C) \(\mathbb{R}\)
• (D) \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\)

If \(x-y=\pi\), then \(\dfrac{dy}{dx}\) is

• (A) \(\pi\)
• (B) \(-\pi\)
• (C) \(1\)
• (D) \(-1\)

The minimum value of \(f(x)=|x|,\; x\in \mathbb{R}\) is

• (A) 0
• (B) 1
• (C) 2
• (D) Does not exist

Statement I: The function \(f(x)=x^2\) is decreasing in the interval \((0,\infty)\).
Statement II: Any function \(y=f(x)\) is decreasing if \(\dfrac{dy}{dx}<0\).
Which of the following is correct?

• (A) Both the Statements I and II are true
• (B) Both the Statements I and II are false
• (C) Statement I is true and Statement II is false
• (D) Statement I is false and Statement II is true

The antiderivative of \(\dfrac{1}{x\sqrt{x^2-1}},\; x>1\) with respect to \(x\) is

• (A) \(\sin^{-1}x + C\)
• (B) \(\cos^{-1}x + C\)
• (C) \(\sec^{-1}x + C\)
• (D) \(\cot^{-1}x + C\)

The value of \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^{7}x \, dx \]
is

• (A) 1
• (B) 0
• (C) \(-1\)
• (D) 7

If \(\vec{a}\) is a nonzero vector of magnitude \(a\) and \(\lambda\) is a nonzero scalar, then \(\lambda \vec{a}\) is a unit vector if

• (A) \(\lambda = 1\)
• (B) \(\lambda = -1\)
• (C) \(a = |\lambda|\)
• (D) \(a = \dfrac{1}{|\lambda|}\)

The position vector of the midpoint of the line joining the points \(P(2,3,4)\) and \(Q(4,1,-2)\) is

• (A) \(3\hat{i} + 2\hat{j} + \hat{k}\)
• (B) \(3\hat{i} + 2\hat{j} - \hat{k}\)
• (C) \(\hat{i} - \hat{j} - 3\hat{k}\)
• (D) \(-\hat{i} + \hat{j} + 3\hat{k}\)

The direction ratios of the \(x\)-axis are

• (A) \((0,k,0)\)
• (B) \((0,0,k)\)
• (C) \((k,0,0)\)
• (D) \((k,k,k)\)

The probability of obtaining an even prime number on each die when a pair of dice is rolled is

• (A) \(\frac{1}{36}\)
• (B) \(\frac{1}{6}\)
• (C) \(\frac{1}{18}\)
• (D) \(\frac{1}{4}\)

If \(A\) and \(B\) are independent events with \(P(A)=0.3\) and \(P(B)=0.4\), then \(P(A \cap B)\) is

• (A) \(1.2\)
• (B) \(0.12\)
• (C) \(0.7\)
• (D) \(\frac{3}{4}\)

The left hand derivative of \(|x|\) with respect to \(x\) at \(x=0\) is

The point of inflection of the function \(f(x)=x^3\) in the interval \([-1,1]\) is

If \(m\) and \(n\) are respectively the order and degree of the differential equation \(2x^2\dfrac{d^2y}{dx^2}-3\dfrac{dy}{dx}+y=0\), then \(m+n=\)

The value of \(\hat{i}\cdot\hat{i}+\hat{j}\cdot\hat{j}\) is

If \(F\) is an event of a sample space \(S\), then \(P(S|F)=\)