The system of linear equations

$ x + \lambda y - z = 0 $

$ \lambda x - y - z = 0 $

$ x + y - \lambda z = 0 $

has a non-trivial solution for:

1

The integral $ \displaystyle \int \frac{2x^{12} + 5x^{9}}{(x^{5} + x^{2} + 1)^{3}}, dx $ is equal to:

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2

The area (in sq. units) of the region $ {(x,y) : y^{2} \ge 2x \ \text{and} \ x^{2} + y^{2} \le 4x,\ x \ge 0,\ y \ge 0} …

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3

If a curve $y = f(x)$ passes through the point $(1,-1)$ and satisfies the differential equation $ y(1+xy),dx = x,dy $, …

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4

Let two fair six-faced dice $A$ and $B$ be thrown simultaneously. If $E_{1}$ is the event that die $A$ shows up four, $…

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5

If the standard deviation of the numbers $2, 3, a,$ and $11$ is $3.5$, then which of the following is true?

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6

A value of $\theta$ for which $ \displaystyle \frac{2 + 3i \sin \theta}{1 - 2i,} \cdot \frac{1}{\sin \theta} $ is purel…

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7

Let $ p = \displaystyle \lim_{x \to 0^{+}} \left(1 + \tan^{2}\sqrt{x}\right)^{\tfrac{1}{x}} $ then $\log p$ is equal to:

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8

For $x \in \mathbb{R}$, $f(x) = |\log 2 - \sin x|$ and $g(x) = f(f(x))$, then:

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1

A wire of length $2$ units is cut into two parts which are bent respectively to form a square of side $= x$ units and a…

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2

The eccentricity of the hyperbola whose length of the latus rectum is equal to $8$ and the length of its conjugate axis…

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3

$A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}$ and $A \,\text{adj}\, A = A\,A^{T}$, then $5a + b$ is equal to:

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4

If one of the diameters of the circle, given by the equation, $ x^{2} + y^{2} - 4x + 6y - 12 = 0 $, is a chord of a cir…

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5

Two sides of a rhombus are along the lines, $x - y + 1 = 0$ and $7x - y - 5 = 0$. If its diagonals intersect at $(-1, -…

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6

If the $2^{\text{nd}}, 5^{\text{th}}$ and $9^{\text{th}}$ terms of a non-constant A.P. are in G.P., then the common rat…

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7

If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged…

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8

The sum of all real values of $x$ satisfying the equation $(x^{2} - 5x + 5)^{x^{2} + 4x - 60} = 1$ is :

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9

If $f(x) + 2f\left(\dfrac{1}{x}\right) = 3x,; x \ne 0,$ and $S = {x \in \mathbb{R} : f(x) = f(-x)}$, then $S$ :

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