UP Board Class 12 Mathematics Question Paper 2024 (Code 324 EX) Available- Download Here with Solution PDF

UP Board Class 12 Mathematics Questions with Solutions

Question 1:

(a) If orders of matrices A and B are \( p \times q \) and \( q \times r \) respectively, then order of AB is:

• (A) \( p \times r \)
• (B) \( r \times p \)
• (C) \( q \times p \)
• (D) None of these

(b) At which point is the slope of the line \( y = x + 1 \) equal to the slope of the curve \( y^2 = 4x \)?

• (A) \( (1,2) \)
• (B) \( (2,1) \)
• (C) \( (1,-2) \)
• (D) \( (-1,2) \)

(c) The value of the integral \( \int \sqrt{1 + \sin 2x} \,dx \) is:

• (A) \( \sin x + \cos x + c \)
• (B) \( \sin x - \cos x + c \)
• (C) \( \cos x - \sin x + c \)
• (D) \( \cos x + \sin x + c \)

(d) If vectors \( \mathbf{5i - \lambda j + 2k} \) and \( \mathbf{2i + 3j + 4k} \) are perpendicular, find \( \lambda \):

• (A) \( 3 \)
• (B) \( 4 \)
• (C) \( 6 \)
• (D) \( 0 \)

(e) The degree of the differential equation \( d^2y/dx^2 = \sqrt{x} + \left(\frac{dy}{dx}\right)^2 \) is:

• (A) \( 4 \)
• (B) \( 3 \)
• (C) \( 1 \)
• (D) \( 2 \)

(a) Show that the function

is not continuous at \( x = 0 \).

(b) Find the principal value of \( \sin^{-1} \left( \sin \frac{7\pi}{4} \right) \).

(c) If \( y = Ae^x + B \) where \( A, B \) are constants, then show that \( \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0 \).

(d) Solve the differential equation \[ \frac{dy}{dx} = \frac{1 + x^2}{1 + y^2} \]

(e) The probability of \( A \) winning the race is \( \frac{1}{3} \) and that of \( B \) is \( \frac{1}{4} \). In this race, find the probability that neither \( A \) nor \( B \) can win the race.

(a) \( R \) is a relation on a set of natural numbers \( N \) defined by \[ R = \{(a, b): a, b \in N and a = b^2 \} \]
Is \( (a, b) \in R \), \( (b, c) \in R \Rightarrow (a, c) \in R \) true? Justify it by one example.

(b) If \( y = \frac{x^2 + 3x + 4}{e^x \cos x} \), find \( \frac{dy}{dx} \).

(c) Solve the differential equation \[ \frac{dy}{dx} = e^x \sin x. \]

(d) A die is thrown once. The number on the die is a multiple of 3 is denoted by \( E \), and the number on the die is even is denoted by \( F \). Are \( E \) and \( F \) independent events?

(a) Differentiate \( y = (\cos x)^{\sin x} \).

(b) Find the probability of 53 Sundays in a leap year.

(c) Find the vector equation of the line which passes through the point \( (5,2,-4) \) and is parallel to the vector \( 3\hat{i} + 2\hat{j} - 8\hat{k} \).

(d) Find the value of:
\[ \mathbf{a} \times (\mathbf{b} + \mathbf{c}) + \mathbf{b} \times (\mathbf{c} + \mathbf{a}) + \mathbf{c} \times (\mathbf{a} + \mathbf{b}) \]

(a) If \( R \) is the relation “less than” from \( A = \{1,2,3,4,5\} \) to \( B = \{1,4,5\} \), find the set of ordered pairs corresponding to \( R \). Also define this relation from \( B \) to \( A \).

(b). Find the values of \( x, y, z \) if the matrix \( A \) satisfies the equation \( A^T A = I \), where

(c) Differentiate \( y = x^x + (\cos x)^{\tan x} \) with respect to \( x \).

(d) Find the minimum value of \( z = x + 3y \) under the following constraints:

\( x + y \leq 8 \)
\( 3x + 5y \geq 15 \)
\( x \geq 0, y \geq 0 \)

(e) Find the shortest distance between the lines
\[ \mathbf{r} = \hat{i} + \hat{j} + \lambda (2\hat{i} - \hat{j} + \hat{k}) \]
\[ \mathbf{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu (3\hat{i} - 5\hat{j} + 2\hat{k}) \]

(a) Find two positive numbers whose sum is 15 and the sum of their squares is minimum.

(b) Find the area of the circle \( x^2 + y^2 = a^2 \).

(c) Find the perpendicular unit vectors on the vectors \[ \mathbf{a} = 2\hat{i} - \hat{j} + \hat{k} \quad and \quad \mathbf{b} = 3\hat{i} + 4\hat{j} - \hat{k} \]
and find the sine of the angle between them.

(d) If  prove that , where \( n \in \mathbb{N} \).

(e) The probabilities of solving a question by \( A \) and \( B \) independently are \( \frac{1}{2} \) and \( \frac{1}{3} \) respectively. If both of them try to solve it independently, find the probability that:

[(i)] none of them solved it.
[(ii)] at least one of them solved it.

(a) If matrix , then find \( A^{-1} \).

(b) Solve the system of equations by matrix method:
\[ 2x + 3y + 3z = 5 \] \[ x - 2y + z = -4 \] \[ 3x - y - 2z = 3 \]

(a) Solve the differential equation:
\[ (\tan^{-1} y - x) \, dy = (1 + y^2) \, dx \]

(b) Solve the differential equation:
\[ \frac{dy}{dx} + \frac{y}{x} = x^2, \quad y = 1 when x = 1 \]

(a) Evaluate:
\[ I = \int_0^{\pi} \frac{x \, dx}{a^2 \cos^2 x + b^2 \sin^2 x} \]

(b) Prove that:
\[ \int_0^{\pi/2} \sin 2x \tan^{-1} (\sin x) \,dx = \left(\frac{\pi}{2} - 1\right) \]