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UP Board XII
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Mathematics
List of top Mathematics Questions asked in UP Board XII
A die was thrown twice and the sum of the numbers which appeared was found to be 6. Find the conditional probability that the number 4 appears at least once.
UP Board XII - 2025
UP Board XII
Mathematics
Probability
Find the shortest distance between the lines whose vector equations are \(\vec{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda(\hat{i}-3\hat{j}+2\hat{k})\) and \(\vec{r}=(4\hat{i}+5\hat{j}+6\hat{k})+\mu(2\hat{i}+3\hat{j}+\hat{k})\).
UP Board XII - 2025
UP Board XII
Mathematics
3D Geometry
Find the general solution of the differential equation \(x\frac{dy}{dx} + 2y = x^2\) (\(x \neq 0\)).
UP Board XII - 2025
UP Board XII
Mathematics
Differential Equations
Evaluate : \(\int \frac{dx}{\sqrt{x^2 - a^2}}\).
UP Board XII - 2025
UP Board XII
Mathematics
Calculus
Let a relation R be defined in the set \(\mathbb{N} \times \mathbb{N}\) as follows: \((a, b) R (c, d)\) if and only if \(a + d = b + c\). Prove that R is an equivalence relation.
UP Board XII - 2025
UP Board XII
Mathematics
Relations and functions
Find the projection of the vector \(\vec{a} = 2\hat{i} + 3\hat{j} + 2\hat{k}\) on the vector \(\vec{b} = \hat{i} + 2\hat{j} + \hat{k}\).
UP Board XII - 2025
UP Board XII
Mathematics
Vector Algebra
In a leap year, selected at random, find the probability that there are 53 Tuesdays.
UP Board XII - 2025
UP Board XII
Mathematics
Probability
Differentiate \(x^{\sin x}\) with respect to \(x\), while \(x>0\).
UP Board XII - 2025
UP Board XII
Mathematics
Calculus
Evaluate: \( \int \text{cosec}\,x(\text{cosec}\,x + \cot x) \,dx \).
UP Board XII - 2025
UP Board XII
Mathematics
Calculus
Find the principal value of \( \text{cosec}^{-1}(-\sqrt{2}) \).
UP Board XII - 2025
UP Board XII
Mathematics
Trigonometric Identities
Find the direction-cosines of the y-axis.
UP Board XII - 2025
UP Board XII
Mathematics
3D Geometry
Test whether the function defined by \( f(x) = x^2 - \sin(x) + 5 \) is continuous at \( x = \pi \).
UP Board XII - 2025
UP Board XII
Mathematics
Calculus
If the coordinates of the points P and Q are respectively (2, 3, 0) and (–1, –2, –4), the vector \( \vec{PQ} \) will be
UP Board XII - 2025
UP Board XII
Mathematics
Vector Algebra
Let [x] represents the greatest integer which is less than or equal to x. Then the function \(f: \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = [x]\) will be
UP Board XII - 2025
UP Board XII
Mathematics
Relations and functions
The order of the differential equation \( \left( \frac{d^3y}{dx^3} \right)^2 + \log x \left( \frac{d^2y}{dx^2} \right)^3 + 5y = \cos x \) will be
UP Board XII - 2025
UP Board XII
Mathematics
Differential Equations
Solve: \((\tan^{-1}y - x)dy = (1+y^2)dx\).
UP Board XII - 2025
UP Board XII
Mathematics
Differential Equations
Find the area of the region bounded by the ellipse \(\frac{x^2}{9^2} + \frac{y^2}{4^2} = 1\).
UP Board XII - 2025
UP Board XII
Mathematics
Calculus
For two vectors \(\vec{a}\) and \(\vec{b}\) prove that \(|\vec{a} + \vec{b}| \le |\vec{a}| + |\vec{b}|\).
UP Board XII - 2025
UP Board XII
Mathematics
Vector Algebra
Prove that the semi-vertical angle of a right circular cone of given surface and maximum volume is \(\sin^{-1}(1/3)\).
UP Board XII - 2025
UP Board XII
Mathematics
Calculus
Solve: \(ydx - (x + 2y^2)dy = 0\).
UP Board XII - 2025
UP Board XII
Mathematics
Differential Equations
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.
UP Board XII - 2025
UP Board XII
Mathematics
Calculus
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.
UP Board XII - 2025
UP Board XII
Mathematics
Probability
Show that the function \(f(x) = \log \sin x\) is increasing in the interval \((0, \frac{\pi}{2})\) and decreasing in \((\frac{\pi}{2}, \pi)\).
UP Board XII - 2025
UP Board XII
Mathematics
Calculus
Evaluate: \(\int \sqrt{3 - 2x - x^2} dx\).
UP Board XII - 2025
UP Board XII
Mathematics
Calculus
If \(y = x^{x^{x^{\dots \text{to infinity}}}}\), prove that \(x \frac{dy}{dx} = \frac{y^2}{1 - y \log x}\).
UP Board XII - 2025
UP Board XII
Mathematics
Calculus
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