List of top Questions asked in UP Board XII

Let A and B are independent events and P(A)=0.3 and P(B)=0.4 ; then find P(B/A).
Find the area of a triangle whose vertices are A(2, 2, 2), B(2, 1, 3) and C(3, 2, 1).
Let function \( f:N \to Y \) is defined as \( f(x)=4x+3 \) where \( Y=\{y \in N : y=4x+3 \text{ for } x \in N\} \). Prove that f is invertible, also find the inverse of the function f.
Find the minimum value of the objective function \( Z = -50x + 20y \) by graphical method under the following constraints :
\( 2x - y \geq -5 \)
\( 3x + y \geq 3 \)
\( 2x - 3y \leq 12 \)
\( x \geq 0, y \geq 0 \)
Prove that \( \sin^{-1} x = \tan^{-1} [x / \sqrt{1-x^2}] \).
Find the unit vector along the vector \( \vec{a} = 2\hat{i} + 3\hat{j} + \hat{k} \).
Find the direction-cosines of the sum of the vectors \( \vec{a} = 3\hat{i} + 4\hat{j} - 3\hat{k} \) and \( \vec{b} = -2\hat{i} - 3\hat{j} + \hat{k} \).
If the functions \( f:\mathbb{R} \to \mathbb{R} \) and \( g:\mathbb{R} \to \mathbb{R} \) are defined as \( f(x)=\cos x \) and \( g(x)=3x^2 \) respectively then find gof.
If two dice are thrown together then find the probability of getting the sum eight.
If the position vectors of the points A and B are \(\hat{i}+\hat{j}+\hat{k}\) and \(2\hat{i}+5\hat{j}\) respectively, then find the unit vector along the straight line AB.
Find the differential equation representing the family of curves \(y=2mx\).
Prove that \(\int_{\pi/6}^{\pi/3} \frac{dx}{1+\sqrt{\tan x}} = \frac{\pi}{12}\).
If the function \( f : \mathbb{R} \to \mathbb{R} \) is defined as \( f(x) = 3x \) then f is
If \( X = \{ a, b, c \} \) and \( Y = \{ 1, 2, 3 \} \) and the mapping f is given by \( f(a)=2, f(b)=3, f(c)=1 \), then
If \( \int \log x \, dx = x \log x + k(x) + c \) then
The number of arbitrary constants in a general solution of a differential equation of fifth order is
Find the area enclosed by the curve \( y = x^2 \) and the line \( y = 16 \).
If \( E_1 \) and \( E_2 \) are mutually exclusive events, then prove that \( P(E_1) + P(E_2) = P(E_1 \cup E_2) + P(E_1 \cap E_2) \).
Solve the following system of equations by matrix method:
\(x + y + 2z = 1\)
\(3x + 2y + z = 7\)
\(2x + y + 3z = 2\)
Solve : \( \frac{dy}{dx} = \frac{x + 2y - 3}{2x + y - 3} \).
Find the area of a triangle \( \triangle ABC \) whose vertices are A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1).
If the function \(f: [0, \frac{\pi}{2}] \rightarrow \mathbb{R}\) is given by \(f(x) = \sin x\) and function \(g: [0, \frac{\pi}{2}] \rightarrow \mathbb{R}\) is given by \(g(x) = \cos x\), then prove that f and g are one-one but \(f + g\) is not one-one.
Minimize Z = 3x + 2y by graphical method under the following constraints: \(x + 2y \leq 10\), \(3x + y \leq 15\), \(x \geq 0\), \(y \geq 0\).
Differentiate the function \(x^{\cos x}\) with respect to x.
Find the differential equation of the family of curves denoted by \(y = a \sin(x+b)\), where a and b are arbitrary constants.