List of top Mathematics Questions asked in UP Board XII

Find the area of a triangle \( \triangle ABC \) whose vertices are A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1).
A car is started from a point P at time t = 0 and is stopped at the point Q. The distance x metre covered by the car in t second is given by \( x = t^2(2 - \frac{t}{3}) \). Find the time required by the car to reach at point Q and also find the distance between P and Q.
Find the differential equation of the family of curves denoted by \(y = a \sin(x+b)\), where a and b are arbitrary constants.
A person has a contract of construction. The probability of being a strike is 0.65. The probabilities of completing the construction work on time in both conditions are 0.80 and 0.32 whether the strike is not happened and it is happened respectively. Then find the probability of completing the construction work in due time.
A box is formed by a 3 m x 8 m rectangular steel-sheet on cutting the squares of length x m from its each corner to form the box without cover. Then find the maximum volume of the box so formed.
The modulus of two vectors \(\vec{a}\) and \(\vec{b}\) are \(\sqrt{3}\) and 4 respectively, and \(\vec{a} \cdot \vec{b} = 6\). Then find the angle between the vectors \(\vec{a}\) and \(\vec{b}\).
Find the area of a parallelogram whose adjacent sides are the vectors \( \vec{a} = \hat{i} - \hat{j} + 3\hat{k} \) and \( \vec{b} = 2\hat{i} - 7\hat{j} + \hat{k} \).
If the ordered pairs (2x - 3, 5) and (x, y - 1) are equal, then find the numbers x and y.
Prove that \( 0 \leq P(E) \leq 1 \), where P(E) is the probability of the event E.
Obtain the differential equation of the family of curves \(y = \frac{2ce^{2x}}{1+ce^{2x}}\).
Find the direction cosine of Z-axis.
If the function f: N \(\rightarrow\) N, is defined by f(x) = x - 1 for all x > 2 and f(1) = f(2) = 1, then f is
Obtain the projection of the vector \( \vec{a} = 2\hat{i} + 3\hat{j} + 5\hat{k} \) on the vector \( \vec{b} = \hat{i} + 3\hat{j} + \hat{k} \).
If y = 5x\(^2\) + 4, then at the point with x-coordinate 2, the slope is
Prove that \( \sin^{-1} x = \cos^{-1} \sqrt{1 - x^2} \).
Find the general solution of the differential equation \(y - x \frac{dy}{dx} = x + y \frac{dy}{dx}\).
Find the area of the region bounded by the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
Solve by matrix method the following system of equations:
\(2x + y + z = 1\)
\(x - 2y - z = \frac{3}{2}\)
\(3y - 5z = 9\)
Prove that the semi-vertical angle of the cone of given slant height and maximum volume is tan\(^{-1}\) \(\sqrt{2}\).
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.
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 \(\vec{r}=(4\hat{i}+5\hat{j}+6\hat{k})+\mu(2\hat{i}+3\hat{j}+\hat{k})\).
Find the area of the parallelogram whose diagonals are \( \vec{a} = 3\hat{i} + \hat{j} - 2\hat{k} \) and \( \vec{b} = \hat{i} - 3\hat{j} + 4\hat{k} \).
Find the general solution of the differential equation \(x\frac{dy}{dx} + 2y = x^2\) (\(x \neq 0\)).
Maximize \(Z = x + 2y\) by graphical method under the constraints \(x+y \le 1, -x+y \le 0, x \ge 0, y \ge 0\).
If \( \theta \) be the angle between two unit vectors \( \hat{a} \) and \( \hat{b} \), prove that \( \sin\frac{\theta}{2} = \frac{1}{2} |\hat{a} - \hat{b}| \).