List of top Mathematics Questions asked in JEE Main

Let \( \overrightarrow a \) , \( \overrightarrow b \) and \( \overrightarrow c \) be three unit vectors such that
\( \overrightarrow a + \vec b + \overrightarrow c = \overrightarrow 0 \). If \( \lambda = \overrightarrow a .\vec b + \vec b.\overrightarrow c + \overrightarrow c .\overrightarrow a \) and
\( \overrightarrow d = \overrightarrow a \times \vec b + \vec b \times \overrightarrow c + \overrightarrow c \times \overrightarrow a \), then the ordered pair, \( \left( {\lambda ,\overrightarrow d } \right) \) is equal to :

The number of ordered pairs (r, k) for which
6.³⁵C_r = (k² - 3). ³⁶C_{r + 1}, where k is an integer, is :

If the system of linear equations,
x + y + z = 6
x + 2y + 3z = 10
3x + 2y + \( \lambda \)z = \( \mu \)
has more than two solutions, then \( \mu \) - \( \lambda \)² is equal to ______.

Let \( {a_1} \) , \( {a_2} \) , \( {a_3} \) ,....... be a G.P. such that
\( {a_1} \) < 0, \( {a_1} \) + \( {a_2} \) = 4 and \( {a_3} \) + \( {a_4} \) = 16.
If \( \sum\limits_{i = 1}^9 {{a_i}} = 4\lambda \), then \( \lambda \) is equal to:

Let y = y(x) be the solution curve of the differential equation,

\( \left( {{y^2} - x} \right){{dy} \over {dx}} = 1 \), satisfying y(0) = 1. This curve intersects the x-axis at a point whose abscissa is :

If the function ƒ defined on \( \left( { - {1 \over 3},{1 \over 3}} \right) \) by

f(x) = \( \left\{ {\matrix{ {{1 \over x}{{\log }_e}\left( {{{1 + 3x} \over {1 - 2x}}} \right),} & {when\,x \ne 0} \cr {k,} & {when\,x = 0} \cr } } \right. \)

is continuous, then k is equal to_______.

The area (in sq. units) of the region
{(x, y) \( \in \) R² | 4x² \( \le \) y \( \le \) 8x + 12} is :

Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point (2, 1, 6). Then the image of R in the plane P is :

If the sum of the coefficients of all even powers of x in the product
(1 + x + x² + ....+ x²ⁿ)(1 - x + x² - x³ + ...... + x²ⁿ) is 61, then n is equal to _______.

If y = mx + 4 is a tangent to both the parabolas, y² = 4x and x² = 2by, then b is equal to :

Let the function, ƒ:[-7, 0]\( \to \)R be continuous on [-7,0] and differentiable on (-7, 0). If ƒ(-7) = - 3 and ƒ'(x) \( \le \) 2, for all x \( \in \) (-7,0), then for all such functions ƒ, ƒ(-1) + ƒ(0) lies in the interval:

If the system of linear equations
2x + 2ay + az = 0
2x + 3by + bz = 0
2x + 4cy + cz = 0,
where a, b, c \( \in \) R are non-zero distinct; has a non-zero solution, then:

The logical statement (p \( \Rightarrow \) q) \( \Lambda \) ( q \( \Rightarrow \) ~p) is equivalent to :

A vector \( \overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k\left( {\alpha ,\beta \in R} \right) \) lies in the plane of the vectors, \( \overrightarrow b = \widehat i + \widehat j \) and \( \overrightarrow c = \widehat i - \widehat j + 4\widehat k \). If \( \overrightarrow a \) bisects the angle between \( \overrightarrow b \) and \( \overrightarrow c \), then:

If g(x) = x² + x - 1 and
(goƒ) (x) = 4x² - 10x + 5, then ƒ\( \left( {{5 \over 4}} \right) \) is equal to:

Let \( \alpha \) and \( \beta \) be two real roots of the equation
(k + 1)tan²x - \( \sqrt 2 \) . \( \lambda \)tanx = (1 - k), where k(\( \ne \) - 1) and \( \lambda \) are real numbers. if tan² (\( \alpha \) + \( \beta \)) = 50, then a value of \( \lambda \) is:

The greatest positive integer k, for which 49^k + 1 is a factor of the sum
49¹²⁵ + 49¹²⁴ + ..... + 49² + 49 + 1, is:

The area of the region, enclosed by the circle x² + y² = 2 which is not common to the region bounded by the parabola y² = x and the straight line y = x, is:

If \( {\mathop{\rm Re}\nolimits} \left( {{{z - 1} \over {2z + i}}} \right) = 1 \), where z = x + iy, then the point (x, y) lies on a :

Let x^k + y^k = a^k, (a, k > 0 ) and \( {{dy} \over {dx}} + {\left( {{y \over x}} \right)^{{1 \over 3}}} = 0 \), then k is:

If ƒ(a + b + 1 - x) = ƒ(x), for all x, where a and b are fixed positive real numbers, then

\( {1 \over {a + b}}\int_a^b {x\left( {f(x) + f(x + 1)} \right)} dx \) is equal to:

Let S be the set of points where the function, ƒ(x) = |2-|x-3||, x \( \in \) R is not differentiable. Then \( \sum\limits_{x \in S} {f(f(x))} \) is equal to_____.

Let \( \alpha \) be a root of the equation x² + x + 1 = 0 and the
matrix A = \( {1 \over {\sqrt 3 }}\left[ {\matrix{ 1 & 1 & 1 \cr 1 & \alpha & {{\alpha ^2}} \cr 1 & {{\alpha ^2}} & {{\alpha ^4}} \cr } } \right] \)

then the matrix A³¹ is equal to

\( \mathop {\lim }\limits_{x \to 2} {{{3^x} + {3^{3 - x}} - 12} \over {{3^{ - x/2}} - {3^{1 - x}}}} \) is equal to_______.

If the variance of the first n natural numbers is 10 and the variance of the first m even natural numbers is 16, then m + n is equal to_____.