UP Board Class 12 Mathematics Question Paper 2025 (Code 324 JA) Available- Download Here with Solution PDF

The relation R = { (a,b): b = a + 2 } defined in the set A = { 1, 2, 3, 4, 5 } is

• (A) not reflexive and symmetric, but transitive
• (B) not reflexive and transitive, but symmetric
• (C) not symmetric and transitive, but reflexive
• (D) not reflexive, not symmetric and not also transitive

If the orders of the matrices A and B are \(m \times n\) and \(n \times p\) respectively, then the order of AB will be

• (A) \(m \times p\)
• (B) \(p \times m\)
• (C) \(m \times n\)
• (D) \(n \times p\)

The degree of the differential equation \(7\left(\frac{d^3y}{dx^3}\right)^2 + 5\left(\frac{d^2y}{dx^2}\right)^3 + x\frac{dy}{dx} + y = 0\) will be

• (A) 3
• (B) 2
• (C) 6
• (D) 5

The direction cosines of the vector \(\hat{i} + \hat{j} - 2\hat{k}\) are

• (A) (1, 1, -2)
• (B) \( \left( \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, -\frac{2}{\sqrt{6}} \right) \)
• (C) \( \left( \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}} \right) \)
• (D) \( \left( -\frac{1}{\sqrt{6}}, -\frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}} \right) \)

If R* be the set of all non-zero real numbers, then the mapping \(f: R^* \to R^*\) defined by \(f(x) = \frac{1}{x}\) is

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

Find the principal value of \(\cos^{-1}\left(-\frac{1}{\sqrt{2}}\right)\).

Test whether the function \(f: \mathbb{R} \to \mathbb{R}\) defined by \( f(x) = \begin{cases} x+5, & if x \le 1
x-5, & if x > 1 \end{cases} \) is continuous at \(x=1\).

Find the value of \(\int x^3 e^{x^4} dx\).

If \(P(B) = \frac{9}{13}\) and \(P(A \cap B) = \frac{4}{13}\), find \(P(A|B)\).

Let \(\vec{a} = \hat{i} + 2\hat{j}\) and \(\vec{b} = 2\hat{i} + \hat{j}\). Is \(|\vec{a}| = |\vec{b}|\)? Are the vectors \(\vec{a}\) and \(\vec{b}\) equal?

Find the value of \(\frac{dy}{dx}\) if \(y = \tan^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right)\), where \(-\frac{1}{\sqrt{3}} < x < \frac{1}{\sqrt{3}}\).

Find the value of \(\int \frac{1}{x^2 - a^2} dx\).

If \(R_1\) and \(R_2\) are equivalence relations in the set A, prove that \(R_1 \cap R_2\) is also an equivalence relation in A.

If two vectors \(\vec{a}\) and \(\vec{b}\) are such that \(|\vec{a}| = 2\), \(|\vec{b}| = 3\) and \(\vec{a} \cdot \vec{b} = 4\), find \(|\vec{a} - \vec{b}|\).

The Cartesian equation of a line is \(\frac{x+3}{2} = \frac{y-5}{4} = \frac{z+6}{2}\). Find its vector equation.

A die is thrown once. If the event 'the number obtained on the die is a multiple of 3' is represented by E and 'the number obtained on the die is even' is represented by F, tell whether the events E and F are independent.

If \(A' = \begin{bmatrix} -2 & 3
1 & 2 \end{bmatrix}\) and \(B = \begin{bmatrix} -1 & 0
1 & 2 \end{bmatrix}\), find \((A+2B)'\).

Show that the function \(f(x) = \log \sin x\) is increasing in the interval \((0, \frac{\pi}{2})\) and decreasing in \((\frac{\pi}{2}, \pi)\).

Find the value of the determinant \( \begin{vmatrix} x+y+2z & x & y
z & y+z+2x & y
z & x & z+x+2y \end{vmatrix} \).

If \(y = x^{x^{x^{\dots to infinity}}}\), prove that \(x \frac{dy}{dx} = \frac{y^2}{1 - y \log x}\).

Evaluate: \(\int \sqrt{3 - 2x - x^2} dx\).

Prove that \(\cot^{-1}\left(\frac{\sqrt{1+\sin x} + \sqrt{1-\sin x}}{\sqrt{1+\sin x} - \sqrt{1-\sin x}}\right) = \frac{x}{2}\), \(x \in (0, \pi/4)\).

For the differential equation \(xy\frac{dy}{dx} = (x+2)(y+2)\), find the curve passing through the point \((1, -1)\).

A person has taken the contract of a construction work. The probability of strike is 0.65. The probabilities of the construction work being completed on time in the circumstances of no strike and strike are respectively 0.80 and 0.32. Find the probability of the construction work being completed on time.

Minimize Z = 200x + 500y by graphical method subject to the following constraints:
x + 2y \(\ge\) 10, 3x + 4y \(\le\) 24, x \(\ge\) 0, y \(\ge\) 0.

Find the shortest distance between the lines whose vector equations are the following:
\(\vec{r} = (1-t)\hat{i} + (t-2)\hat{j} + (3-2t)\hat{k}\) and \(\vec{r} = (s+1)\hat{i} + (2s-1)\hat{j} - (2s+1)\hat{k}\).

For two vectors \(\vec{a}\) and \(\vec{b}\) prove that \(|\vec{a} + \vec{b}| \le |\vec{a}| + |\vec{b}|\).

Solve: \(ydx - (x + 2y^2)dy = 0\).

If \(A = \begin{bmatrix} 2 & -3 & 5
3 & 2 & -4
1 & 1 & -2 \end{bmatrix}\), find \(A^{-1}\). Using \(A^{-1}\) solve the following system of equations:
2x - 3y + 5z = 11
3x + 2y - 4z = -5
x + y - 2z = -3.

If \(A = \begin{bmatrix} 0 & -\tan(\alpha/2)
\tan(\alpha/2) & 0 \end{bmatrix}\) and I is the identity matrix of order 2, prove that \(I+A = (I-A)\begin{bmatrix} \cos\alpha & -\sin\alpha
\sin\alpha & \cos\alpha \end{bmatrix}\).

Prove that the semi-vertical angle of a right circular cone of given surface and maximum volume is \(\sin^{-1}(1/3)\).

Prove that \(\int_0^{\pi/2} \log(\cos x) dx = -\frac{\pi}{2}\log 2\).

Find the area of the region bounded by the ellipse \(\frac{x^2}{9^2} + \frac{y^2}{4^2} = 1\).

Solve: \((\tan^{-1}y - x)dy = (1+y^2)dx\).