JEE Main 2017 (Offline)

The distance of the point (1, 3, – 7) from the plane passing through the point (1, –1, – 1), having normal
perpendicular to both the lines
\({{x - 1} \over 1} = {{y + 2} \over { - 2}} = {{z - 4} \over 3}\)
and
$${{x - 2} \over 2} = {{y + 1}...

If the image of the point P(1, –2, 3) in the plane, 2x + 3y – 4z + 22 = 0 measured parallel to the line,
\({x \over 1} = {y \over 4} = {z \over 5}\) is Q, then PQ is equal to:

The normal to the curve y(x – 2)(x – 3) = x + 6 at the point where the curve intersects the y-axis passes
through the point :

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the
maximum area (in sq. m) of the flower-bed, is :

The area (in sq. units) of the region
\(\left\{ {\left( {x,y} \right):x \ge 0,x + y \le 3,{x^2} \le 4y\,and\,y \le 1 + \sqrt x } \right\}\) is

The value of \(\left( {{}^{21}{C_1} - {}^{10}{C_1}} \right) + \left( {{}^{21}{C_2} - {}^{10}{C_2}} \right) + \left( {{}^{21}{C_3} - {}^{10}{C_3}} \right)\)
\(\left( {{}^{21}{C_4} - {}^{10}{C_4}} \right)\)$$ + .... + \left( {{}^{21}{C_{10}} ...

The radius of a circle, having minimum area, which touches the curve y = 4 – x2 and the lines, y = |x| is :

Let \(\omega\) be a complex number such that 2\(\omega\) + 1 = z where z = \(\sqrt {-3}\). If
$$\left| {\matrix{
1 & 1 & 1 \cr
1 & { - {\omega ^2} - 1} & {{\omega ^2}} \cr
1 & {{\omega ^2}} & {{\omega ^7}} \cr

} } \right...

The integral \(\int\limits_{{\pi \over 4}}^{{{3\pi } \over 4}} {{{dx} \over {1 + \cos x}}}\) is equal to

If \(\left( {2 + \sin x} \right){{dy} \over {dx}} + \left( {y + 1} \right)\cos x = 0\) and y(0) = 1,
then \(y\left( {{\pi \over 2}} \right)\) is equal to :

If for \(x \in \left( {0,{1 \over 4}} \right)\), the derivatives of
\({\tan ^{ - 1}}\left( {{{6x\sqrt x } \over {1 - 9{x^3}}}} \right)\) is \(\sqrt x .g\left( x \right)\), then \(g\left( x \right)\) equals

The eccentricity of an ellipse whose centre is at the origin is \({1 \over 2}\). If one of its directrices is x = – 4, then the
equation of the normal to it at \(\left( {1,{3 \over 2}} \right)\) is :

Let \(a\), b, c \(\in R\). If \(f\)(x) = ax2 + bx + c is such that
\(a\) + b + c = 3 and \(f\)(x + y) = \(f\)(x) + \(f\)(y) + xy, \(\forall x,y \in R,\)
then \(\sum\limits_{n = 1}^{10} {f(n)}\) is equal to

The function \(f:R \to \left[ { - {1 \over 2},{1 \over 2}} \right]\) defined as
\(f\left( x \right) = {x \over {1 + {x^2}}}\), is

Let a vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P be a point
on the ground such that AP = 2AB. If \(\angle\)BPC = \(\beta\), then tan\(\beta\) is equal to:

A hyperbola passes through the point P\(\left( {\sqrt 2 ,\sqrt 3 } \right)\) and has foci at \(\left( { \pm 2,0} \right)\). Then the tangent to this hyperbola at P also passes through the point :

Let \({I_n} = \int {{{\tan }^n}x\,dx} ,\,\left( {n > 1} \right).\)
If \({I_4} + {I_6}\) = \(a{\tan ^5}x + b{x^5} + C\), where C is a constant of integration,
then the ordered pair \(\left( {a,b} \right)\) is equal to

\(\mathop {\lim }\limits_{x \to {\pi \over 2}} {{\cot x - \cos x} \over {{{\left( {\pi - 2x} \right)}^3}}}\) equals

The following statement
\(\left( {p \to q} \right) \to \left[ {\left( { \sim p \to q} \right) \to q} \right]\) is :

If S is the set of distinct values of 'b' for which the following system of linear equations
x + y + z = 1
x + ay + z = 1
ax + by + z = 0
has no solution, then S is :

If \(A = \left[ {\matrix{ 2 & { - 3} \cr { - 4} & 1 \cr } } \right]\),
then adj(3A2 + 12A) is equal to

A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are
ladies and 4 are men. Assume X and Y have no common friends. Then the total number of ways in which X
and Y together can throw a party i...

For three events A, B and C, P(Exactly one of A or B occurs) = P(Exactly one of B or C occurs) = P
(Exactly one of C or A occurs) = \({1 \over 4}\)
and P(All the three events occur simultaneously) = \({1 \over {16}}\).
Then the
probabilit...

If two different numbers are taken from the set {0, 1, 2, 3, ........, 10}; then the probability that their sum as
well as absolute difference are both multiple of 4, is :

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with
replacement, then the variance of the number of green balls drawn is :

If for a positive integer n, the quadratic equation
\(x\left( {x + 1} \right) + \left( {x + 1} \right)\left( {x + 2} \right)\)\(+ .... + \left( {x + \overline {n - 1} } \right)\left( {x + n} \right)\)\(= 10n\)
has two consecutive integral...

For any three positive real numbers a, b and c,
9(25\({a^2}\) + b2) + 25(c2 - 3\(a\)c) = 15b(3\(a\) + c).
Then

Let k be an integer such that the triangle with vertices (k, – 3k), (5, k) and (–k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point :

If \(5\left( {{{\tan }^2}x - {{\cos }^2}x} \right) = 2\cos 2x + 9\),
then the value of \(\cos 4x\) is :

Let \(\overrightarrow a = 2\widehat i + \widehat j -2 \widehat k\) and \(\overrightarrow b = \widehat i + \widehat j\).
Let \(\overrightarrow c\) be a vector such that \(\left| {\overrightarrow c - \overrightarrow a } \right| = 3\),
$$\...