Let $A$ be a $3 \times 3$ real matrix such that $A^2(A-2 I)-4(A-I)=O$, where $I$ and $O$ are the identity and null matrices, respectively. If $A^5=\alpha A^2+\beta A+\gamma I$, where $\alpha, \beta$, and $\gamma$ are real constants, then $\alpha+\beta+\gamma$ is equal to :
Previous 10 Questions — JEE Main 2025 (2 April Evening Shift)
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If the system of equations
$ \begin{aligned} & 2 x+\lambda y+3 z=5 \\ & 3 x+2 y-z=7 \\ & 4 x+5 y+\mu z=9 \end{aligned} …
Topic: JEE Main 2025 (2 April Evening Shift)
Let the point P of the focal chord PQ of the parabola $y^2=16 x$ be $(1,-4)$. If the focus of the parabola divides the …
Topic: JEE Main 2025 (2 April Evening Shift)
If the mean and the variance of $6,4, a, 8, b, 12,10,13$ are 9 and 9.25 respectively, then $a+b+a b$ is equal to :
Topic: JEE Main 2025 (2 April Evening Shift)
Let $f:[1, \infty) \rightarrow[2, \infty)$ be a differentiable function. If $10 \int_1^1 f(\mathrm{t}) \mathrm{dt}=5 x …
Topic: JEE Main 2025 (2 April Evening Shift)
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If $\displaystyle \lim_{x \to 0} \frac{\cos(2x) + a\cos(4x) - b}{x^4}$ is finite, then $(a + b)$ is equal to:
Topic: JEE Main 2025 (2 April Evening Shift)
If the image of the point $P(1, 0, 3)$ in the line joining the points $A(4, 7, 1)$ and $B(3, 5, 3)$ is $Q(\alpha, \beta…
Topic: JEE Main 2025 (2 April Evening Shift)
If $\theta \in \left[-\dfrac{7\pi}{6}, \dfrac{4\pi}{3}\right]$, then the number of solutions of
$\sqrt{3}\csc^2\theta -…
Topic: JEE Main 2025 (2 April Evening Shift)
Let $A = {1, 2, 3, \ldots, 100}$ and $R$ be a relation on $A$ such that
$R = {(a, b) : a = 2b + 1}$.
Let $(a_1, a_2), (…
Topic: JEE Main 2025 (2 April Evening Shift)
If the domain of the function $f(x) = \dfrac{1}{\sqrt{10 + 3x - x^2}} + \dfrac{1}{\sqrt{x + |x|}}$ is $(a, b)$, then $(…
Topic: JEE Main 2025 (2 April Evening Shift)
If the domain of the function $f(x) = \dfrac{1}{\sqrt{10 + 3x - x^2}} + \dfrac{1}{\sqrt{x + |x|}}$ is $(a, b)$, then $(…
Topic: JEE Main 2025 (2 April Evening Shift)
The line $L_1$ is parallel to the vector $\vec{a} = -3\hat{i} + 2\hat{j} + 4\hat{k}$ and passes through the point $(7, …
Topic: JEE Main 2025 (2 April Evening Shift)
The number of terms of an A.P. is even.
The sum of all the odd terms is $24$, the sum of all the even terms is $30$, an…
Topic: JEE Main 2025 (2 April Evening Shift)
Let $(a, b)$ be the point of intersection of the curve $x^2 = 2y$ and the straight line $y - 2x - 6 = 0$ in the second …
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If the length of the minor axis of an ellipse is equal to one-fourth of the distance between the foci, then the eccentr…
Topic: JEE Main 2025 (2 April Evening Shift)
