JEE Main 2019 (Online) 10th January Morning Slot

Let A be a point on the line \(\overrightarrow r = \left( {1 - 3\mu } \right)\widehat i + \left( {\mu - 1} \right)\widehat j + \left( {2 + 5\mu } \right)\widehat k\) and B(3, 2, 6) be a point in the space. Then the value of \(\mu\) for ...

The plane passing through the point (4, –1, 2) and parallel to the lines  \({{x + 2} \over 3} = {{y - 2} \over { - 1}} = {{z + 1} \over 2}\)  and  \({{x - 2} \over 1} = {{y - 3} \over 2} = {{z - 4} \over 3}\) also passes through the point -

The shortest distance between the point  \(\left( {{3 \over 2},0} \right)\)   and the curve y = \(\sqrt x\), (x > 0), is -

If the area enclosed between the curves y = kx2 and x = ky2, (k > 0), is 1 square unit. Then k is -

If the third term in the binomial expansion of \({\left( {1 + {x^{{{\log }_2}x}}} \right)^5}\) equals 2560, then a possible value of x is -

If  \({\sum\limits_{i = 1}^{20} {\left( {{{{}^{20}{C_{i - 1}}} \over {{}^{20}{C_i} + {}^{20}{C_{i - 1}}}}} \right)} ^3} = {k \over {21}}\)  then k is equal to

If a circle C passing through the point (4, 0) touches the circle x2 + y2 + 4x – 6y = 12 externally at the point (1, – 1), then the radius of C is :

Let z1 and z2 be any two non-zero complex numbers such that   \(3\left| {{z_1}} \right| = 4\left| {{z_2}} \right|.\)  If  \(z = {{3{z_1}} \over {2{z_2}}} + {{2{z_2}} \over {3{z_1}}}\)  then :

Let  \({\rm I} = \int\limits_a^b {\left( {{x^4} - 2{x^2}} \right)} dx.\)  If I is minimum then the ordered pair (a, b) is -

If  \({{dy} \over {dx}} + {3 \over {{{\cos }^2}x}}y = {1 \over {{{\cos }^2}x}},\,\,x \in \left( {{{ - \pi } \over 3},{\pi \over 3}} \right)\)  and  \(y\left( {{\pi \over 4}} \right) = {4 \over 3},\)  then  $$y\left( { - {\pi \over 4}} \r...

Let f : R \(\to\) R be a function such that f(x) = x3 + x2f'(1) + xf''(2) + f'''(3), x \(\in\) R. Then f(2) equals -

Consider a triangular plot ABC with sides AB = 7m, BC = 5m and CA = 6m. A vertical lamp-post at the mid point D of AC subtends an angle 30o at B. The height (in m) of the lamp-post is -

The equation of a tangent to the hyperbola 4x2 – 5y2 = 20 parallel to the line x – y = 2 is :

Let n \(\ge\) 2 be a natural number and \(0 < \theta < {\pi \over 2}.\) Then \(\int {{{{{\left( {{{\sin }^n}\theta - \sin \theta } \right)}^{1/n}}\cos \theta } \over {{{\sin }^{n + 1}}\theta }}} \,d\theta\) is equal to - (where C is a...

For each t \(\in\) R , let [t] be the greatest integer less than or equal to t
Then  $$\mathop {\lim }\limits_{x \to 1^ + } {{\left( {1 - \left| x \right| + \sin \left| {1 - x} \right|} \right)\sin \left( {{\pi \over 2}\left[ {1 - x} \ri...

Let  \(f\left( x \right) = \left\{ {\matrix{ {\max \left\{ {\left| x \right|,{x^2}} \right\}} & {\left| x \right| \le 2} \cr {8 - 2\left| x \right|} & {2 < \left| x \right| \le 4} \cr } } \right.\)
Let S be the set of points in ...

Consider the statement : "P(n) : n2 – n + 41 is prime". Then which one of the following is true ?

Let  d \(\in\) R, and 
$$A = \left[ {\matrix{
{ - 2} & {4 + d} & {\left( {\sin \theta } \right) - 2} \cr
1 & {\left( {\sin \theta } \right) + 2} & d \cr
5 & {\left( {2\sin \theta } \right) - d} & {\left( { - \sin \theta } \ri...

If the system of equations
x + y + z = 5
x + 2y + 3z = 9
x + 3y + az = \(\beta\)
has infinitely many solutions, then \(\beta\) \(-\) \(\alpha\) equals -

If the parabolas y2 = 4b(x – c) and y2 = 8ax have a common normal, then which on of the following is a valid choice for the ordered triad (a, b, c)?

An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards nu...

Consider the quadratic equation (c – 5)x2 – 2cx + (c – 4) = 0, c \(\ne\) 5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2, 3). Then t...

The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is -

In a class of 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of...

The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1, 3 and 8, then a ratio of other two observations is -

If 5, 5r, 5r2 are the lengths of the sides of a triangle, then r cannot be equal to :

A point P moves on the line 2x – 3y + 4 = 0. If Q(1, 4) and R (3, – 2) are fixed points, then the locus of the centroid of \(\Delta\)PQR is a line :

If the line 3x + 4y – 24 = 0 intersects the x-axis at the point A and the y-axis at the point B, then the incentre of the triangle OAB, where O is the origin, is :

The sum of all values of \(\theta\) \(\in\)\(\left( {0,{\pi \over 2}} \right)\) satisfying sin2 2\(\theta\) + cos4 2\(\theta\) = \({3 \over 4}\) is -

Let \(\overrightarrow a = 2\widehat i + {\lambda _1}\widehat j + 3\widehat k,\,\,\)   \(\overrightarrow b = 4\widehat i + \left( {3 - {\lambda _2}} \right)\widehat j + 6\widehat k,\)  and  $$\overrightarrow c = 3\widehat i + 6\widehat j ...