Let $\vec{c}$ and $\vec{d}$ be vectors such that $|\vec{c} + \vec{d}| = \sqrt{29}$ and $\vec{c} \times (2\hat{i} + 3\hat{j} + 4\hat{k}) = (2\hat{i} + 3\hat{j} + 4\hat{k}) \times \vec{d}$. If $\lambda_1, \lambda_2 (\lambda_1 > \lambda_2)$ are the possible values of $(\vec{c} + \vec{d}) \cdot (-7\hat{i} + 2\hat{j} + 3\hat{k})$, then the equation
$Kx^2 + (K - 5K + \lambda_1)xy + \left(3K + \frac{\lambda_2}{2}\right)y^2 - 8x + 12y + \lambda_2 = 0$
represents a circle, for $K$ equal to:
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Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive terms such that $a_1 a_2 a_3 a_4 = 64$ and $a_1 + a_2 + a_…
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Let $\vec{a} = -\hat{i} + 2\hat{j} + 2\hat{k}$, $\vec{b} = 8\hat{i} + 7\hat{j} - 3\hat{k}$ and $\vec{c}$ be a vector su…
Topic: JEE Main 2026 (21 January Morning Shift)
Let a point $A$ lie between the parallel lines $L_1$ and $L_2$ such that its distances from $L_1$ and $L_2$ are $6$ and…
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The number of relations, defined on the set ${a,b,c,d}$, which are both reflexive and symmetric, is equal to:
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The area of the region, inside the ellipse $x^2 + 4y^2 = 4$ and outside the region bounded by the curves $y = |x| - 1$ …
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If the domain of the function
$ f(x) = \cos^{-1}\left(\frac{2x-5}{11-3x}\right) + \sin^{-1}(2x^2 - 3x + 1) $
is the int…
Topic: JEE Main 2026 (21 January Morning Shift)
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Let $y = y(x)$ be the solution curve of the differential equation $(1 + x^2)dy + (y - \tan^{-1}x)dx = 0$, $y(0) = 1$. T…
Topic: JEE Main 2026 (21 January Morning Shift)
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The value of $\int_{-\pi/6}^{\pi/6} \left( \frac{\pi + 4x}{1 - \sin(x + \pi/6)} \right) dx$ is equal to:
Topic: JEE Main 2026 (21 January Morning Shift)
Let the mean and variance of $7$ observations $2, 4, 10, x, 12, 14, y$, $x > y$, be $8$ and $16$ respectively. Two numb…
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Let $(\alpha, \beta, \gamma)$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5,4,2)$ on th…
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If the coefficient of $x$ in the expansion of $(ax^2 + bx + c)(1 - 2x)^{26}$ is $-56$ and the coefficients of $x^2$ and…
Topic: JEE Main 2026 (21 January Morning Shift)
If $x^2 + x + 1 = 0$, then the value of
$\left(x + \frac{1}{x}\right)^4 + \left(x^2 + \frac{1}{x^2}\right)^4 + \left(x^…
Topic: JEE Main 2026 (21 January Morning Shift)
