Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m$, then $m$ is ______.
Previous 10 Questions — JEE Main 2026 (22 January Morning Shift)
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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 fir…
Topic: JEE Main 2026 (22 January Morning Shift)
Let the solution curve of the differential equation$xdy - ydx = \sqrt{x^2 + y^2},dx,; x>0,; y(1)=0$be $y = y(x)$. Th…
Topic: JEE Main 2026 (22 January Morning Shift)
The number of solutions of $\tan^{-1}(4x) + \tan^{-1}(6x) = \frac{\pi}{6}$, where $-\frac{1}{2\sqrt{6}} < x < \frac{1}{…
Topic: JEE Main 2026 (22 January Morning Shift)
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…
Topic: JEE Main 2026 (22 January Morning Shift)
The coefficient of $x^{48}$ in $(1 + x)^2(1 + x^2 + 3(1 + x)^3 + \cdots + 100(1 + x)^{100})$ is equal to
Topic: JEE Main 2026 (22 January Morning Shift)
The value of $\displaystyle \int_{-\pi/2}^{\pi/2} \frac{1}{[x] + 4},dx$, where $[\cdot]$ denotes the greatest integer f…
Topic: JEE Main 2026 (22 January Morning Shift)
If the domain of the function
$f(x) = \sin^{-1}\left(\frac{5 - x}{3 + 2x}\right) + \log\left(\frac{1}{10 - x}\right)$
…
Topic: JEE Main 2026 (22 January Morning Shift)
If $A = \begin{bmatrix} 2 & 3 \ 3 & 5 \end{bmatrix}$, then the determinant of the matrix
$(A^{2025} - 3A^{2024} + A^{2…
Topic: JEE Main 2026 (22 January Morning Shift)
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…
Topic: JEE Main 2026 (22 January Morning Shift)
If the image of the point $P(1, 2, a)$ in the line
$\frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2}$
is $Q(5, b, c)$, …
Topic: JEE Main 2026 (22 January Morning Shift)
Next 10 Questions — JEE Main 2026 (22 January Morning Shift)
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22. Let A be a 3 × 3 matrix such that A + A^T = O. If
$$
A \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}
= \begin{bmatrix…
Topic: JEE Main 2026 (22 January Morning Shift)
If $\displaystyle \int (\sin x)^{-\frac{11}{2}} (\cos x)^{\frac{5}{2}} dx = -\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}} - \f…
Topic: JEE Main 2026 (22 January Morning Shift)
If $\displaystyle \frac{\cos^2 48^\circ - \sin^2 12^\circ}{\sin^2 24^\circ - \sin^2 6^\circ} = \frac{\alpha + \beta\sqr…
Topic: JEE Main 2026 (22 January Morning Shift)
If $\dfrac{\cos^2 48^\circ - \sin^2 12^\circ}{\sin^2 24^\circ - \sin^2 6^\circ} = \dfrac{\alpha + \beta\sqrt{5}}{2}$, w…
Topic: JEE Main 2026 (22 January Morning Shift)
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…
Topic: JEE Main 2026 (22 January Morning Shift)
