The number of values of $\theta \in (0, \pi)$ for which the system of linear equations  

$x + 3y + 7z = 0$  

$-x + 4y + 7z = 0$  

$(\sin 3\theta)x + (\cos 2\theta)y + 2z = 0$  

has a non-trivial solution, is -

1

Let $f : (-1, 1) \to \mathbb{R}$ be a function defined by $f(x) = \max \{-|x|, -\sqrt{1 - x^2}\}$. If $K$ be the set of…

Topic: JEE Main 2019 (10 January Evening Shift)

2

Let $f$ be a differentiable function such that $f'(x) = 7 - \dfrac{3}{4}\,\dfrac{f(x)}{x}$, for $x>0$, and $f(1)\neq 4…

Topic: JEE Main 2019 (10 January Evening Shift)

3

If $\displaystyle \int_{0}^{x} f(t)\,dt \;=\; x^{2} \;+\; \int_{x}^{1} t^{2} f(t)\,dt$, then $f'\!\left(\tfrac{1}{2}\r…

Topic: JEE Main 2019 (10 January Evening Shift)

1

The curve amongst the family of curves represented by the differential equation $(x^2 - y^2)dx + 2xy\,dy = 0$ which…

Topic: JEE Main 2019 (10 January Evening Shift)

2

Let $z = \left(\dfrac{\sqrt{3}}{2} + \dfrac{i}{2}\right)^5 + \left(\dfrac{\sqrt{3}}{2} - \dfrac{i}{2}\right)^5.$ If $…

Topic: JEE Main 2019 (10 January Evening Shift)

3

The positive value of $\lambda$ for which the coefficient of $x^2$ in the expression $x^2 \left( \sqrt{x} + \dfrac{\l…

Topic: JEE Main 2019 (10 January Evening Shift)

4

The value of $\lambda$ such that the sum of the squares of the roots of the quadratic equation $x^2 + (3 - \lambda)x …

Topic: JEE Main 2019 (10 January Evening Shift)

5

Let $A = \begin{bmatrix} 2 & b & 1 \\ b & b^2 + 1 & b \\ 1 & b & 2 \end{bmatrix}$ where $b > 0$. Then the minimum va…

Topic: JEE Main 2019 (10 January Evening Shift)

6

If $\vec{\alpha} = (\lambda - 2)\vec{a} + \vec{b}$ and $\vec{\beta} = (4\lambda - 2)\vec{a} + 3\vec{b}$ be two given ve…

Topic: JEE Main 2019 (10 January Evening Shift)

7

Let $S = \{(x, y) \in \mathbb{R}^2 : \dfrac{y^2}{1 + r} - \dfrac{x^2}{1 - r} = 1 ; \, r \neq \pm 1 \}$. Then $S$ repr…

Topic: JEE Main 2019 (10 January Evening Shift)

8

The value of $\displaystyle \int_{-\pi/2}^{\pi/2} \dfrac{dx}{[x] + [\sin x] + 4}$, where $[t]$ denotes the greatest i…

Topic: JEE Main 2019 (10 January Evening Shift)

9

Two vertices of a triangle are $(0,2)$ and $(4,3)$. If its orthocenter is at the origin, then its third vertex lies in …

Topic: JEE Main 2019 (10 January Evening Shift)

10

Let $a_1,a_2,\dots,a_{10}$ be in G.P. with $a_i>0$ for $i=1,2,\dots,10$ and $S$ be the set of pairs $(r,k)$, $r,k\in\ma…

Topic: JEE Main 2019 (10 January Evening Shift)