JEE Main 2022 (Online) 26th July Morning Shift

The length of the perpendicular from the point \((1,-2,5)\) on the line passing through \((1,2,4)\) and parallel to the line \(x+y-z=0=x-2 y+3 z-5\) is :

Let \(\mathrm{Q}\) and \(\mathrm{R}\) be two points on the line \(\frac{x+1}{2}=\frac{y+2}{3}=\frac{z-1}{2}\) at a distance \(\sqrt{26}\) from the point \(P(4,2,7)\). Then the square of the area of the triangle \(P Q R\) is ___________.

Let the function \(f(x)=2 x^{2}-\log _{\mathrm{e}} x, x>0\), be decreasing in \((0, \mathrm{a})\) and increasing in \((\mathrm{a}, 4)\). A tangent to the parabola \(y^{2}=4 a x\) at a point \(\mathrm{P}\) on it passes through the point $$(8...

The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = ya is \({{364} \over 3}\), is equal to :

If the coefficients of \(x\) and \(x^{2}\) in the expansion of \((1+x)^{\mathrm{p}}(1-x)^{\mathrm{q}}, \mathrm{p}, \mathrm{q} \leq 15\), are \(-3\) and \(-5\) respectively, then the coefficient of \(x^{3}\) is equal to _____________.

Let O be the origin and A be the point \({z_1} = 1 + 2i\). If B is the point \({z_2}\), \({\mathop{\rm Re}\nolimits} ({z_2}) < 0\), such that OAB is a right angled isosceles triangle with OB as hypotenuse, then which of the following is NOT...

If \(a = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {{{2n} \over {{n^2} + {k^2}}}}\) and \(f(x) = \sqrt {{{1 - \cos x} \over {1 + \cos x}}}\), \(x \in (0,1)\), then :

If \(\mathrm{n}(2 \mathrm{n}+1) \int_{0}^{1}\left(1-x^{\mathrm{n}}\right)^{2 \mathrm{n}} \mathrm{d} x=1177 \int_{0}^{1}\left(1-x^{\mathrm{n}}\right)^{2 \mathrm{n}+1} \mathrm{~d} x\), then \(\mathrm{n} \in \mathbf{N}\) is equal to __________...

If \({{dy} \over {dx}} + 2y\tan x = \sin x,\,0 < x < {\pi \over 2}\) and \(y\left( {{\pi \over 3}} \right) = 0\), then the maximum value of \(y(x)\) is :

Let a curve \(y=y(x)\) pass through the point \((3,3)\) and the area of the region under this curve, above the \(x\)-axis and between the abscissae 3 and \(x(>3)\) be \(\left(\frac{y}{x}\right)^{3}\). If this curve also passes through the p...

Let the tangent drawn to the parabola \(y^{2}=24 x\) at the point \((\alpha, \beta)\) is perpendicular to the line \(2 x+2 y=5\). Then the normal to the hyperbola \(\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1\) at the point $$(\alpha...

\(\tan \left(2 \tan ^{-1} \frac{1}{5}+\sec ^{-1} \frac{\sqrt{5}}{2}+2 \tan ^{-1} \frac{1}{8}\right)\) is equal to :

Let f : R \(\to\) R be a continuous function such that \(f(3x) - f(x) = x\). If \(f(8) = 7\), then \(f(14)\) is equal to :

If the function $$f(x) = \left\{ {\matrix{
{{{{{\log }_e}(1 - x + {x^2}) + {{\log }_e}(1 + x + {x^2})} \over {\sec x - \cos x}}} & , & {x \in \left( {{{ - \pi } \over 2},{\pi \over 2}} \right) - \{ 0\} } \cr
k & , & {x = 0} \cr

...

If \(f(x) = \left\{ {\matrix{ {x + a} & , & {x \le 0} \cr {|x - 4|} & , & {x > 0} \cr } } \right.\) and \(g(x) = \left\{ {\matrix{ {x + 1} & , & {x < 0} \cr {{{(x - 4)}^2} + b} & , & {x \ge 0} \cr } } \right.\) are c...

Let \(f(x) = \left\{ {\matrix{ {{x^3} - {x^2} + 10x - 7,} & {x \le 1} \cr { - 2x + {{\log }_2}({b^2} - 4),} & {x > 1} \cr } } \right.\).
Then the set of all values of b, for which f(x) has maximum value at x = 1, is :

The statement \((\sim(\mathrm{p} \Leftrightarrow \,\sim \mathrm{q})) \wedge \mathrm{q}\) is :

If the system of linear equations.
\(8x + y + 4z = - 2\)
\(x + y + z = 0\)
\(\lambda x - 3y = \mu\)
has infinitely many solutions, then the distance of the point \(\left( {\lambda ,\mu , - {1 \over 2}} \right)\) from the plane $$8x + y + ...

Let A be a 2 \(\times\) 2 matrix with det (A) = \(-\) 1 and det ((A + I) (Adj (A) + I)) = 4. Then the sum of the diagonal elements of A can be :

The number of 5-digit natural numbers, such that the product of their digits is 36 , is __________.

The mean and variance of a binomial distribution are \(\alpha\) and \(\frac{\alpha}{3}\) respectively. If \(\mathrm{P}(X=1)=\frac{4}{243}\), then \(\mathrm{P}(X=4\) or 5\()\) is equal to :

Let \(\mathrm{E}_{1}, \mathrm{E}_{2}, \mathrm{E}_{3}\) be three mutually exclusive events such that \(\mathrm{P}\left(\mathrm{E}_{1}\right)=\frac{2+3 \mathrm{p}}{6}, \mathrm{P}\left(\mathrm{E}_{2}\right)=\frac{2-\mathrm{p}}{8}\) and $$\math...

If for some \(\mathrm{p}, \mathrm{q}, \mathrm{r} \in \mathbf{R}\), not all have same sign, one of the roots of the equation $$\left(\mathrm{p}^{2}+\mathrm{q}^{2}\right) x^{2}-2 \mathrm{q}(\mathrm{p}+\mathrm{r}) x+\mathrm{q}^{2}+\mathrm{r}^{...

The number of distinct real roots of the equation \(x^{5}\left(x^{3}-x^{2}-x+1\right)+x\left(3 x^{3}-4 x^{2}-2 x+4\right)-1=0\) is ______________.

Consider two G.Ps. 2, 22, 23, ..... and 4, 42, 43, .... of 60 and n terms respectively. If the geometric mean of all the 60 + n terms is \({(2)^{{{225} \over 8}}}\), then \(\sum\limits_{k = 1}^n {k(n - k)}\) is equal to :

The series of positive multiples of 3 is divided into sets : \(\{3\},\{6,9,12\},\{15,18,21,24,27\}, \ldots\) Then the sum of the elements in the \(11^{\text {th }}\) set is equal to ____________.

A point \(P\) moves so that the sum of squares of its distances from the points \((1,2)\) and \((-2,1)\) is 14. Let \(f(x, y)=0\) be the locus of \(\mathrm{P}\), which intersects the \(x\)-axis at the points \(\mathrm{A}\), \(\mathrm{B}\) a...

The equations of the sides \(\mathrm{AB}, \mathrm{BC}\) and \(\mathrm{CA}\) of a triangle \(\mathrm{ABC}\) are \(2 x+y=0, x+\mathrm{p} y=15 \mathrm{a}\) and \(x-y=3\) respectively. If its orthocentre is \((2, a),-\frac{1}{2}<\mathrm{a}<2\),...

Let \(S=\left\{\theta \in[0,2 \pi]: 8^{2 \sin ^{2} \theta}+8^{2 \cos ^{2} \theta}=16\right\} .\) Then $$n(s) + \sum\limits_{\theta \in S}^{} {\left( {\sec \left( {{\pi \over 4} + 2\theta } \right)\cos ec\left( {{\pi \over 4} + 2\theta } ...

Let \(\overrightarrow{\mathrm{a}}=\alpha \hat{i}+\hat{j}-\hat{k}\) and \(\overrightarrow{\mathrm{b}}=2 \hat{i}+\hat{j}-\alpha \hat{k}, \alpha>0\). If the projection of \(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\) on th...