Let $f, g : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(n + 1) = f(n) + f(1) \forall n \in \mathbb{N}$ and $g$ be any arbitrary function. Which of the following statements is $NOT$ true?
Problem 2 of 30
Let the lines $(2 - i)z = (2 + i)\overline{z}$ and $(2 + i)z + (i - 2)\overline{z} - 4i = 0$, (here $i^2 = -1$) be normal to a circle $C$. If the line $iz + \overline{z} + 1 + i = 0$ is tangent to this circle $C$, then its radius is:
Problem 3 of 30
The integer $'k'$, for which the inequality $x^2 - 2(3k - 1)x + 8k^2 - 7 > 0$ is valid for every $x$ in $\mathbb{R}$, is:
If Rolle's theorem holds for the function $f(x) = x^3 - ax^2 + bx - 4$, $x \in [1, 2]$ with $f'\left(\dfrac{4}{3}\right) = 0$, then ordered pair $(a, b)$ is equal to:
Problem 6 of 30
$\lim\limits_{n \rightarrow \infty}\left(1 + \dfrac{1 + \dfrac{1}{2} + ... + \dfrac{1}{n}}{n^2}\right)^n$ is equal to:
The value of $\displaystyle \int\limits_{-1}^{1} x^2e^{\lfloor x^3 \rfloor} dx$, where $\lfloor t \rfloor$ denotes the greatest integer $\le t$, is:
Problem 9 of 30
If a curve passes through the origin and the slope of the tangent to it at any point $(x, y)$ is $\dfrac{x^2 - 4x + y + 8}{x - 2}$, then this curve also passes through the point:
Problem 10 of 30
The image of the point $(3, 5)$ in the line $x - y + 1 = 0$, lies on:
Problem 11 of 30
A tangent is drawn to the parabola $y^2 = 6x$ which is perpendicular to the line $2x + y = 1$. Which of the following points does $NOT$ lie on it?
Problem 12 of 30
If the curves, $\dfrac{x^2}{a} + \dfrac{y^2}{b} = 1$ and $\dfrac{x^2}{c} + \dfrac{y^2}{d} = 1$ intersect each other at an angle of $90^\circ$, then which of the following relations is $TRUE$?
Problem 13 of 30
Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations $l + m + n = 0$ and $l^2 + m^2 - n^2 = 0$. Then the value of $\sin^{4}\alpha + \cos^{4}\alpha$ is:
Problem 14 of 30
The equation of the line through the point $(0, 1, 2)$ and perpendicular to the line $\dfrac{x - 1}{2} = \dfrac{y + 1}{3} = \dfrac{z-1}{-2}$ is:
Problem 15 of 30
When a missile is fired from a ship, the probability that it is intercepted is $\dfrac{1}{3}$ and the probability that the missile hits the target, given that it is not intercepted, is $\dfrac{3}{4}$. If three missiles are fired independently from the ship, then the probability that all three hit the target, is:
Problem 16 of 30
The coefficients $a, b$ and $c$ of the quadratic equation $ax^2 + bx + c = 0$ are obtained by throwing a dice three times. The probability that this equation has equal roots is:
Problem 17 of 30
All possible values of $\theta \in [0, 2\pi]$ for which $\sin 2\theta + \tan 2\theta > 0$ lie in:
Problem 18 of 30
The total number of positive integral solutions $(x, y, z)$ such that $xyz = 24$ is:
Problem 19 of 30
A man is observing, from the top of a tower, a boat speeding towards the tower from a certain point $A$, with uniform speed. At the point, angle of depression of the boat with the man's eye is $30^\circ$ (Ignore the man's height). After sailing for $20$ seconds, towards the base of the tower (which is at the level of water), the boat has reached a point $B$, where the angle of depression is $45^\circ$. Then the time taken (in seconds) by the boat to reach the base of the tower is:
Problem 20 of 30
The statement $A \rightarrow (B \rightarrow A)$ is equivalent to:
Problem 21 of 30
If $A = \begin{bmatrix} 0 & -\tan\left(\dfrac{\theta}{2}\right) \\ \tan\left(\dfrac{\theta}{2}\right) & 0\end{bmatrix}$ and $(I_2 + A)(I_2 - A)^{-1} = \begin{bmatrix} a & -b \\ b & a\end{bmatrix}$,
then $13(a^2 + b^2)$ is equal to
Problem 22 of 30
Let $A = \begin{bmatrix} x & y & z \\ y & z & x \\ z & x & y\end{bmatrix}$, where $x, y$ and $z$ are real numbers such that $x + y + z \gt 0$ and $xyz = 2$. If $A^2 = I_3$, then the value of $x^3 + y^3 + z^3$ is ______.
Problem 23 of 30
If the system of equations
$\phantom{0000} kx + y + 2z = 1$
$\phantom{0000} 3x - y - 2z = 2$
$\phantom{0000} -2x - 2y - 4z = 3$
has infinitely many solutions, then $k$ is equal to ______.
Problem 24 of 30
The total number of numbers, lying between $100$ and $1000$ that can be formed with the digits $1, 2, 3, 4, 5$, if the repetition of digits is not allowed and the numbers are divisible by either $3$ or $5$, is ______.
Problem 25 of 30
Let $A_1, A_2, A_3, ...$ be squares such that for each $n \geqslant 1$, the length of the side of $A_n$ equals the length of diagonal of $A_{n+1}$. If the length of $A_1$ is $12$ cm, then the smallest value of $n$ for which area of $A_n$ is less than one, is ______.
Problem 26 of 30
The number of points, at which the function $f(x) = |2x + 1| - 3|x + 2| + |x^2 + x - 2|$, $x \in \mathbb{R}$ is not differentiable, is ______.
Problem 27 of 30
Let $f(x)$ be a polynomial of degree $6$ in $x$, in which the coefficient of $x^6$ is unity and it has extrema at $x = -1$ and $x = 1$. If $\lim\limits_{x \rightarrow 0} \dfrac{f(x)}{x^3} = 1$, then $5\cdot f(2)$ is equal to ______.
Problem 28 of 30
The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area $A$.
Then $A^4$ is equal to ______.
Problem 29 of 30
The locus of the point of intersection of the lines $\sqrt{3}kx + ky - 4\sqrt{3} = 0$ and $\sqrt{3}x - y - 4(\sqrt{3})k = 0$ is a conic, whose eccentricity is ______.
Problem 30 of 30
Let $\overrightarrow{a} = \hat{i} + 2\hat{j} - \hat{k}$, $\overrightarrow{b} = \hat{i} - \hat{j}$ and $\overrightarrow{c} = \hat{i} - \hat{j} - \hat{k}$ be three given vectors. If $\overrightarrow{r}$ is a vector such that $\overrightarrow{r} \times \overrightarrow{a} = \overrightarrow{c} \times \overrightarrow{a}$ and $\overrightarrow{r} \cdot \overrightarrow{b} = 0$, then $\overrightarrow{r} \cdot \overrightarrow{a}$ is equal to ______.