List of top Mathematics Questions asked in JEE Main

Let
\(\begin{array}{l} f\left(x\right)=3^{\left(x^2-2\right)^3+4},x\in \mathbb{R}.\end{array}\)
Then which of the following statements are true?
P : x = 0 is a point of local minima of f
Q : x = √2 is a point of inflection of f
R : f ′ is increasing for x > √2

Let \(\vec{a} = 3\hat{i} + \hat{j}\) and \(\vec{b} = \hat{i} + 2\hat{j} + \hat{k}\).
Let \(\vec{c}\) be a vector satisfying \(\vec{a} \times (\vec{b} \times \vec{c}) = \vec{b} + \lambda \vec{c}\)
If \(\vec{b}\) and \(\vec{c}\) are non-parallel, then the value of \(λ\) is

The angle of elevation of the top of a tower from a point A due north of it is α and from a point B at a distance of 9 units due west of A is \(\cos^{-1}\left(\frac{3}{\sqrt{13}}\right)\)
If the distance of the point B from the tower is 15 units, then cot α is equal to :

Let\( A(1, 1), B(-4, 3) \)and \(C(-2, -5)\) be vertices of a triangle ABC, P be a point on side BC, and \(Δ1\) and \(Δ2\) be the areas of triangles APB and ABC, respectively. If \(Δ1 : Δ2 = 4 : 7\), then the area enclosed by the lines AP, AC and the x-axis is

Let a vertical tower AB of height 2h stands on a horizontal ground. Let from a point P on the ground a man can see up to height h of the tower with an angle of elevation 2α.
When from P, he moves a distance d in the direction of \(\overrightarrow{AP}\).
he can see the top B of the tower with an angle of elevation α. if \(d = \sqrt7\) h, then tan α is equal to

Let \(S = \{θ ∈ (0, 2π) : 7 cos^2θ – 3 sin^2θ – 2 cos^22θ = 2\}\).
Then, the sum of roots of all the equations \(x^2 – 2 (tan^2θ + cot^2θ) x + 6 sin^2θ = 0, θ ∈ S\), is _______.

Let \(\hat a\) and \(\hat b\) be two unit vectors such that the angle between them is \(\frac{\pi}{4}\). If θ is the angle between the vectors (\(\hat a\)+\(\hat b\)) and (\(\hat a\)+2\(\hat b\)+2(\(\hat a\)×\(\hat b\))), then the value of 164 cos²θ is equal to :

Let \(A = \begin{bmatrix} 0 & -2 \\[0.3em] 2 & 0 \end{bmatrix}\). If \(M\) and \(N\) are two matrices given by \(M = \displaystyle\sum_{k=1}^{10} A^{2k}\) and \(N = \displaystyle\sum_{k=1}^{10} A^{2k-1}\)
then \(MN^2 \) is :

Let \(ƒ(x)\) be a polynomial function such that \(ƒ(x) + ƒ^′(x) + ƒ^{′′}(x) = x^5 + 64\). Then, the value of \(\lim\limits_{x \to 1}\frac {f(x)}{x−1}\) is equal to :

Let \(C_r\) denote the binomial coefficient of \(x^r \)in the expansion of \((1 + x)^{10}\). If for \(α, β∈R\), \(C_1 + 3⋅2 C_2 + 5⋅3 C_3 + …\) upto 10 terms =\(\frac {α×211}{2^β−1}(C_0+\frac {C_1}{2}+\frac {C_2}{3}…..\)upto 10 terms) then the value of \(α + β\) is equal to _______.

If
\(\begin{array}{l}\sum_{K=1}^{10}K^2\left(^{10}C_K\right)^2 = 22000L,\end{array}\)then L is equal to _____.

Let \(\vec a=a_1\hat i+a_2\hat j+a_3\hat k\), \(a_i>0,\ i=1, 2, 3\) be a vector which makes equal angles with the coordinate axes OX, OY and OZ. Also, let the projection of \(\vec a\) on the vector \(3\hat i+4\hat j\) be 7. Let \(\vec b\) be a vector obtained by rotating \(\vec a\) with 90°. If \(\vec a,\vec b\) and x-axis are co-planar, then projection of a vector \(\vec b\) on \(3\hat i+4\hat j\) is equal to :

Let the lines
\(L_1:\vec r=λ(\hat i+2\hat j+3 \hat k), \ λ∈R\)
\(L_2:\vec r=(\hat i+3\hat j+\hat k)+μ(\hat i+\hat j+\hat 5k), \ μ∈R\)
intersect at the point \(S\). If a plane \(ax + by – z + d = 0\) passes through \(S\) and is parallel to both the lines \(L_1\) and \(L_2\) then the value of \(a + b + d\) is equal to _______.

A vector \(\vec{a}\)
is parallel to the line of intersection of the plane determined by the vectors
\(\hat{i},\hat{i}+\hat{j} \)and the plane determined by the vectors
\(\hat{i}−\hat{j},\hat{i}+\hat{k}\). The obtuse angle between
\(\vec{a}\) and the vector \(\vec{b}=\hat{i}−2\hat{j}+2\hat{k}\)
is

\(\begin{array}{l}\text{Let}~\vec{a} = \alpha \hat{i} + \hat{j} + \beta \hat{k}~\text{and}~ \vec{b} = 3 \hat{i} + 5\hat{j} + 4 \hat{k}~ \text{be two vectors, such that }\vec{a} \times \vec{b} = -\hat{i} + 9\hat{j} + 12 \hat{k}.\end{array}\)
\(\begin{array}{l}~\text{Then the projection of } \vec{b}-2\vec{a} ~\text{on}~ \vec{b}+ \vec{a}~\text {is equal to}\end{array}\)

The letters of the work ‘MANKIND’ are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word ‘MANKIND’ is ______.

If
\(\sum\limits_{k=1}^{31}\) \((^{31}C_k) (^{31}C_{k-1})\) \(-\sum\limits_{k=1}^{30}\) \((^{30}C_k) (^{30}C_{k-1})\) \(= \frac{α (60!)} {(30!) (31!)}\)
where \(α ∈ R\), then the value of 16α is equal to

If
\((^{40}C_0) + (^{41}C_1) + (^{42}C_2) + ...... + (^{60}C_{20}) \frac{m}{n} ^{60}C_{20}\)
m and n are coprime, then m + n is equal to _____.

The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is _____.

The number of ways to distribute 30 identical candies among four children C₁, C₂, C₃ and C₄ so that C₂ receives atleast 4 and atmost 7 candies, C₃ receives atleast 2 and atmost 6 candies, is equal to:

Numbers are to be formed between 1000 and 3000, which are divisible by 4, using the digits 1, 2, 3, 4, 5 and 6 without repetition of digits. Then the total number of such numbers is ______________.