JEE (Advanced) 2017 Paper-1

Which of the following is(are) NOT the square of a 3×3 matrix with real entries?

Let a, b, x and y be real numbers such that a - b = 1 and $$y \neq 0$$. If the complex number z = x + iy satisfies $$IM\left(\frac{az + b}{z+1}\right) = y$$, then which of the following is(are) possible value(s) of x?

Let X and Y be two events such that $$P(X) = \frac{1}{3}, P(X \mid Y) = \frac{1}{2}$$ and $$P(Y \mid X) = \frac{2}{5}$$. Then

For how many values of P, the circle $$x^2 + y^2 + 2x + 4y - p = 0$$ and the coordinate axes have exactly three common points?

Let $$R \rightarrow R$$ be a differentiable function such that $$f(0) = 0, f\left(\frac{\pi}{2}\right) = 3$$ and $$f'(0) = 1$$. If $$ g(x) = \int_{x}^{\frac{\pi}{2}}\left[f'(t) \cosec t - \cos t \cosec t f(t)\right] dt$$ for $$x \in \left(0, \frac{\pi}{2}\right]$$, then $$\lim_{x \rightarrow 0}g(x) =$$

For a real number $$\alpha$$, if the system $$\begin{bmatrix}1 & \alpha & \alpha^2 \\\alpha & 1 & \alpha\\\alpha^2 & \alpha & 1 \end{bmatrix}\begin{bmatrix}x\\y\\z \end{bmatrix} = \begin{bmatrix}1\\-1\\1 \end{bmatrix}$$ of linear equations, has infinitely many solutions, then $$1+ \alpha + \alpha^2 =$$

Words of length 10 are formed using the letters A, B, C, D, E, F, G,H, I, J. Let 𝑥 be the number of such words where no letter is repeated; and let 𝑦 be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, $$\frac{y}{9x} =$$

The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?

For $$a = \sqrt{2}$$, if a tangent is drawn to a suitable conic (Column 1) at the point of contact (-1, 1), then which of the following options is the only CORRECT combination for obtaining equation?

If a tangent to a suitable conic (Column 1) is found to be y = x + 8 and its point of contact is (8, 16), then which of the following options is the only CORRECT combination?