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then the determinant of the matrix \( \left(A^{2025} - 3A^{2024} + A^{2023}\right) \) is

If the image of the point \( P(1, 2, a) \) in the line \( \frac{x-6}{3} = \frac{y-7}{2} = \frac{7 - z}{2} \) is \( Q(5, b, c) \), then \( a^{2} + b^{2} + c^{2} \) is equal to:

If the sum of the first four terms of an A.P. is \( 6 \) and the sum of its first six terms is \( 4 \), then the sum of its first twelve terms is

The number of solutions of \( \tan^{-1}4x + \tan^{-1}6x = \frac{\pi}{6} \), where \( -\frac{1}{2\sqrt{6}} < x < \frac{1}{2\sqrt{6}} \), is equal to:

The coefficient of \( x^{48} \) in \( (1 + x) + 2(1 + x)^{2} + 3(1 + x)^{3} + \ldots + 100(1 + x)^{100} \) is:

Let \( f(x) = x^{2025} - x^{2000}, \ x \in [0,1] \) and the minimum value of the function \( f(x) \) in the interval \( [0,1] \) be \( (80)^{80}(n)^{-81} \). Then \( n \) is equal to:

Two distinct numbers \( a \) and \( b \) are selected at random from \( 1, 2, 3, \ldots, 50 \). The probability, that their product \( ab \) is divisible by \( 3 \), is:

The value of \( \int_{-\pi/2}^{\pi/2}\frac{1}{[x]+4}\,dx \), where \( [ \, \cdot \, ] \) denotes the greatest integer function, is:

Let the solution curve of the differential equation \( x\,dy-y\,dx=\sqrt{x^{2}+y^{2}}\,dx ,\ x>0 ,\ y(1)=0 \), be \( y=y(x) \). Then \( y(3) \) is equal to:

Let \( f:[1,\infty)\to \)(\(\mathbb{R}\)) be a differentiable function. If \( 6\int_{1}^{x} f(t)\,dt=3x f(x)+x^{3}-4 \) for all \( x\ge 1 \), then the value of \( f(2)-f(3) \) is:

The number of distinct real solutions of the equation \( x\lvert x + 4\rvert + 3\lvert x + 2\rvert + 10 = 0 \) is:

Let the line \( x = -1 \) divide the area of the region \( \{(x,y) : 1 + x^2 \le y \le 3 - x\} \) in the ratio \( m : n \), \( \gcd(m,n) = 1 \). Then \( m + n \) is equal to:

Let $\overrightarrow{AB} = 2\hat{i} + 4\hat{j} - 5\hat{k}$ and $\overrightarrow{AD} = \hat{i} + 2\hat{j} + \lambda\hat{k}, \lambda \in \mathbb{R}$. Let the projection of the vector $\vec{v} = \hat{i} + \hat{j} + \hat{k}$ on the diagonal $\overrightarrow{AC}$ of the parallelogram $ABCD$ be of length one unit. If $\alpha, \beta$, where $\alpha > \beta$, be the roots of the equation $\lambda^2x^2 - 6\lambda x + 5 = 0$, then $2\alpha - \beta$ is equal to:

If a random variable \( x \) has the probability distribution

Then \( P(3 < x \le 6) \) is equal to:

If the domain of the function \( f(x) = \sin^{-1}\!\left(\dfrac{5 - x}{3 + 2x}\right) + \dfrac{1}{\log\_{e}(10 - x)} \) is \( (-\infty, \alpha] \cup [\beta, \gamma) - \{\delta\} \), then \( 6(\alpha + \beta + \gamma + \delta) \) is equal to:

If the chord joining the points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ on the parabola $y^2 = 12x$ subtends a right angle at the vertex of the parabola, then $x_1x_2 - y_1y_2$ is equal to:

Let the set of all values of \( r \), for which the circles \( (x + 1)^2 + (y + 4)^2 = r^2 \) and \( x^2 + y^2 - 4x - 2y - 4 = 0 \) intersect at two distinct points be the interval \( (\alpha, \beta) \). Then \( \alpha \beta \) is equal to:

Let \( P(\alpha, \beta, \gamma) \) be the point on the line \( \frac{x-1}{2} = \frac{y+1}{-3} = z \) at a distance \( 4\sqrt{14} \) from the point \( (1, -1, 0) \) and nearer to the origin. Then the shortest distance, between the lines \( \frac{x - \alpha}{1} = \frac{y - \beta}{2} = \frac{z - \gamma}{3} \) and \( \frac{x + 5}{2} = \frac{y - 10}{1} = \frac{z - 3}{1} \), is equal to:

If the line \( \alpha x + 2y = 1 \), where \( \alpha \in \mathbb{R} \), does not meet the hyperbola \( x^2 - 9y^2 = 9 \), then a possible value of \( \alpha \) is:

Let \( ABC \) be a triangle. Consider four points \( p_1, p_2, p_3, p_4 \) on the side \( AB \), five points \( p_5, p_6, p_7, p_8, p_9 \) on the side \( BC \), and four points \( p_{10}, p_{11}, p_{12}, p_{13} \) on the side \( AC \). None of these points is a vertex of the triangle \( ABC \). Then the total number of pentagons, that can be formed by taking all the vertices from the points \( p_1, p_2, \ldots, p_{13} \), is __________.

Let \( \alpha = \frac{-1 + i\sqrt{3}}{2} \) and \( \beta = \frac{-1 - i\sqrt{3}}{2} \), \( i = \sqrt{-1} \). If \( (7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10} \), then \( m \) is ______________.

Let \( A \) be a \( 3 \times 3 \) matrix such that \( A + A^{T} = O \). If  and \( \det(\operatorname{adj}(2\operatorname{adj}(A + I))) = (2)^{\alpha} \cdot (3)^{\beta} \cdot (11)^{\gamma} \); \( \alpha, \beta, \gamma \) are non-negative integers, then \( \alpha + \beta + \gamma \) is equal to ________.

If $\int (\sin x)^{-\frac{11}{2}} (\cos x)^{-\frac{5}{2}} dx = -\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}} - \frac{p_2}{q_2}(\cot x)^{\frac{5}{2}} - \frac{p_3}{q_3}(\cot x)^{\frac{1}{2}} + \frac{p_4}{q_4}(\cot x)^{-\frac{3}{2}} + C$, where $p_i$ and $q_i$ are positive integers with $\gcd(p_i, q_i)=1$ for $i=1,2,3,4$ and $C$ is the constant of integration, then $\frac{15p_1p_2p_3p_4}{q_1q_2q_3q_4}$ is equal to _________.

If \( \frac{\cos^2 48^\circ-\sin^2 12^\circ}{\sin^2 24^\circ-\sin^2 6^\circ}=\frac{\alpha+\beta\sqrt{5}}{2} \), where \( \alpha, \beta \in \)(\(\mathbb{R}\)) , then \( \alpha+\beta \) is equal to ______________.