The number of all possible positive integral value of $\alpha$ for which the roots of the quadratic equation $6x^2 - 11x + \alpha = 0$ are rational numbers is: (1) 5 (2) 3 (3) 4 (4) 2
If both the roots of the quadratic equation $x^2 - mx + 4 = 0$ are real and distinct and they lie in the interval $(1,5)$, then $m$ lies in the interval: Note: In the actual JEE paper interval was $[1,5]$ (1) $(-5,-4)$ (2) $(3,4)$ (3) $(5,6)$ (4) $(4,5)$
Let $z_0$ be a root of quadratic equation, $x^2 + x + 1 = 0$. If $z = 3 + 6iz_0^{81} - 3iz_0^{93}$, then $\arg(z)$ is equal to: (1) 0 (2) $\frac{\pi}{4}$ (3) $\frac{\pi}{6}$ (4) $\frac{\pi}{3}$
The number of natural numbers less than 7000 which can be formed by using the digits $0,1,3,7,9$ (repetition of digits allowed) is equal to: (1) 375 (2) 250 (3) 374 (4) 372
Let $a, b$ and $c$ be the $7^{\text{th}}, 11^{\text{th}}$ and $13^{\text{th}}$ terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then $\frac{a}{c}$ is equal to: (1) 2 (2) $\frac{7}{13}$ (3) $\frac{1}{2}$ (4) 4
Let $S$ be the set of all triangles in the $xy$-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in $S$ has area 50 sq. units, then the number of elements in the set $S$ is: (1) 36 (2) 32 (3) 9 (4) 18
Let the equations of two sides of a triangle be $3x - 2y + 6 = 0$ and $4x + 5y - 20 = 0$. If the orthocenter of this triangle is at $(1,1)$ then the equation of its third side is: (1) $122y + 26x + 1675 = 0$ (2) $26x - 122y - 1675 = 0$ (3) $26x + 61y + 1675 = 0$ (4) $122y - 26x - 1675 = 0$
Let $A(4,-4)$ and $B(9,6)$ be points on the parabola, $y^2 = 4x$. Let $C$ be chosen on the arc $AOB$ of the parabola, where $O$ is the origin, such that the area of $\triangle ACB$ is maximum. Then, the area (in sq. units) of $\triangle ACB$, is: (1) 32 (2) $31\frac{3}{4}$ (3) $30\frac{1}{2}$ (4) $31\frac{1}{4}$
A hyperbola has its centre at the origin, passes through the point $(4,2)$ and has transverse axis of length 4 along the $x$-axis. Then the eccentricity of the hyperbola is: (1) $\sqrt{3}$ (2) $\frac{3}{2}$ (3) $\frac{2}{\sqrt{3}}$ (4) 2
For each $x \in R$, let $[x]$ be the greatest integer less than or equal to $x$. Then $\lim_{x \rightarrow 0^-} \frac{x([x] + |x|)\sin[x]}{|x|}$ is equal to (1) 1 (2) 0 (3) $-\sin 1$ (4) $\sin 1$
A data consists of $n$ observations: $x_1, x_2, \ldots, x_n$. If $\sum_{i=1}^{n}(x_i + 1)^2 = 9n$ and $\sum_{i=1}^{n}(x_i - 1)^2 = 5n$, then the standard deviation of this data is (1) 5 (2) $\sqrt{7}$ (3) $\sqrt{5}$ (4) 2
If $A = \left[\begin{array}{ccc} e^t & e^{-t}\cos t & e^{-t}\sin t \\ e^t & -e^{-t}\cos t - e^{-t}\sin t & -e^{-t}\sin t + e^{-t}\cos t \\ e^t & 2e^{-t}\sin t & -2e^{-t}\cos t \end{array}\right]$, then $A$ is: (1) Invertible only if $t = \pi$ (2) Not invertible for any $t \in R$ (3) Invertible only if $t = \frac{\pi}{2}$ (4) Invertible for all $t \in R$
If the system of linear equations $x - 4y + 7z = g$; $3y - 5z = h$; $-2x + 5y - 9z = k$ is consistent, then: (1) $g + h + 2k = 0$ (2) $g + 2h + k = 0$ (3) $2g + h + k = 0$ (4) $g + h + k = 0$
Let $f:[0,1] \rightarrow R$ be such that $f(xy) = f(x) \cdot f(y)$, for all $x,y \in [0,1]$, and $f(0) \neq 0$. If $y = y(x)$ satisfies the differential equation, $\frac{dy}{dx} = f(x)$ with $y(0) = 1$ then $y\left(\frac{1}{4}\right) + y\left(\frac{3}{4}\right)$ is equal to: (1) 5 (2) 2 (3) 3 (4) 4
Let $A = \{x \in R : x$ is not a positive integer$\}$. Define a function $f: A \rightarrow R$ as $f(x) = \frac{2x}{x-1}$, then $f$ is: (1) Injective but not surjective (2) Not injective (3) Surjective but not injective (4) Neither injective nor surjective
Let $f$ be a differentiable function from $R$ to $R$ such that $|f(x) - f(y)| \leq 2|x-y|^{3/2}$, for all $x,y \in R$. If $f(0) = 1$ then $\int_0^1 f^2(x)\,dx$ is equal to (1) 0 (2) 1 (3) 2 (4) $\frac{1}{2}$
If $x = 3\tan t$ and $y = 3\sec t$, then the value of $\frac{d^2y}{dx^2}$ at $t = \frac{\pi}{4}$, is: (1) $\frac{1}{6}$ (2) $\frac{1}{6\sqrt{2}}$ (3) $\frac{1}{3\sqrt{2}}$ (4) $\frac{3}{2\sqrt{2}}$
If $f(x) = \int \frac{\left(5x^8 + 7x^6\right)}{\left(x^2 + 1 + 2x^7\right)^2}\,dx,\,(x \geq 0)$, and $f(0) = 0$, then the value of $f(1)$ is (1) $\frac{-1}{4}$ (2) $\frac{1}{2}$ (3) $\frac{1}{4}$ (4) $-\frac{1}{2}$
If $\int_0^{\pi/3} \frac{\tan\theta}{\sqrt{2k\sec\theta}}\,d\theta = 1 - \frac{1}{\sqrt{2}},\,(k > 0)$, then the value of $k$ is (1) $\frac{1}{2}$ (2) 1 (3) 2 (4) 4
The area of the region $A = \{(x,y): 0 \leq y \leq x|x| + 1$ and $-1 \leq x \leq 1\}$ in sq. units, is (1) $\frac{4}{3}$ (2) 2 (3) $\frac{1}{3}$ (4) $\frac{2}{3}$
Let $\vec{a} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \sqrt{2}\hat{\mathrm{k}},\, \vec{b} = b_1\hat{\mathrm{i}} + b_2\hat{\mathrm{j}} + \sqrt{2}\hat{\mathrm{k}}$ and $\vec{c} = 5\hat{\mathrm{i}} + \hat{\mathrm{j}} + \sqrt{2}\hat{\mathrm{k}}$ be three vectors such that the projection vector of $\vec{b}$ on $\vec{a}$ is $|\vec{a}|$. If $\vec{a} + \vec{b}$ is perpendicular to $\vec{c}$, then $|\vec{b}|$ is equal to: (1) $\sqrt{22}$ (2) $\sqrt{32}$ (3) 6 (4) 4
If the lines $x = ay + b,\, z = cy + d$ and $x = a'z + b',\, y = c'z + d'$ are perpendicular, then (1) $cc' + a + a' = 0$ (2) $aa' + c + c' = 0$ (3) $bb' + cc' + 1 = 0$ (4) $ab' + bc' + 1 = 0$
An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red, is: (1) $\frac{21}{49}$ (2) $\frac{26}{49}$ (3) $\frac{32}{49}$ (4) $\frac{27}{49}$