Mathematics
If x, y, z are in GP, then which of the following is/are correct?
1. ln(3x), ln(3y), ln(3z) are in AP
2. xyz + ln(x), xyz + ln(y), xyz + ln(z) are in HP
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The quadratic equation 3x2 - (k2 + 5k)x + 3k2 - 5k = 0 has real roots of equal magnitude and opposite sign. Which one of the following is correct?
$\frac{1}{b+c}, \frac{1}{c+a},\frac{1}{a+b}$ are in HP, then which of the following is/are correct?
1. a, b, c are in AP
2. (b + c)2, (c + a)2, (a + b)2 are in GP. Select the correct answer using the code given below.
$\rm A=\begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix}$ where a ∈ ℕ, then is A100 - A50 - 2A25 equal to?
where I is the identity matrix.
Suppose 20 distinct points are placed randomly on a circle. Which of the following statements is/are correct?
1. The number of straight lines that can be drawn by joining any two of these points is 380.
2. The number of triangles that can be drawn by joining any three of these points is 1140.
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What is the value of following?
$\rm cot \left[sin^{-1} \frac{3}{5}+cot^{-1}\frac{3}{2} \right]$
Consider the following statements in respect of the roots of the equation x3 - 8 = 0 :
1. The roots are non-collinear.
2. The roots lie on a circle of unit radius.
Which of the above statements is/are correct?
Consider the following statements in respect of sets:
1. The union over the intersection of sets is distributive.
2. The complement of the union of two sets is equal to the intersection of their complements.
3. If the difference between the two sets is equal to the empty set, then the two sets must be equal.
Which of the above statements are correct?
Consider the following statements in respect of relations and functions:
1. All relations are functions but all functions are not relations.
2. A relation from A to B is a subset of Cartesian product A × B.
3. A relation in A is a subset of Cartesian product A × A.
Which of the above statements are correct?
What is the value of the following determinant?
$\begin{vmatrix} \cos \rm C & \tan \rm A & 0\\ \sin \rm B & 0 & -\tan \rm A\\ 0 & \sin \rm B & \cos \rm C \end{vmatrix}$
If $x = \frac{a}{b-c}$, $y = \frac{b}{c - a}$, $z = \frac{c}{a - b}$ then what is the value of the following?
$\begin{vmatrix} 1 & -x & x\\ 1 & 1 & -y\\ 1 & z & 1 \end{vmatrix}$
Consider the following in respect of the matrix $\rm A = \begin{bmatrix} 1 & 1 & 1\\ 1 & 1 & 1\\ 1 & 1 & 1 \end{bmatrix}$
1. Inverse of A does not exist
2. A3 = A
3. 3A = A2
Which of the above are correct?
Consider the following statements:
1. The third vertex has at least one irrational coordinate.
2. The area is irrational.
Which of the above statements is/are correct?
Consider the following statements in respect of such parabolas:
1. One of the parabolas passes through the origin (0, 0).
2. The focus of one of the parabolas lies at (-2, 0).
Which of the above statements is/are correct?
Consider the following statements:
1. A-line in space can have infinitely many direction ratios.
2. It is possible for certain lines that the sum of the squares of direction cosines can be equal to the sum of its direction cosines.
Which of the above statements is/are correct?
Let $\rm \vec{a}, \vec{b}$ and $\rm\vec{c}$ be unit vectors such that $\rm\vec{a} \times \vec{b}$ is perpendicular to $\vec{c}$. If θ is the angle between $\rm\vec{a}$ and $\rm\vec{b}$, then which of the following is/are correct?
1. $\rm\vec{a} \times \vec{b} = sin ~\theta~ \vec{c} $
2. $\rm\vec {a} \cdot (\vec{b}\times \vec{c})=0$
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Let $\rm\vec {a}, \vec{b}$ and $ \rm \vec {c}$ be three vectors such that $\rm\vec {a}, \vec{b}$ and $ \rm \vec {c}$ are co-planar. Which of the following is/are correct?
1. $\rm(\vec{a}\times \vec{b})\times \vec{c}$ is co-planar with $\rm\vec {a}$ and $\rm\vec {b}$
2. $\rm(\vec{a}\times \vec{b})\times \vec{c}$ is perpendicular to $\rm\vec {a}$ and $\rm\vec {b}$
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Let $\rm f(x) = \left\{\begin{matrix} 1+\frac{x}{2k}, & 0 < x < 2\\\ kx, & 2 \le x < 4 \end {matrix}\right.$
If $\displaystyle\lim_{x\rightarrow 2}$ f(x) exists, then what is the value of k?
Consider the following statements in respect of f(x) = |x| - 1
1. f(x) is continuous at x = 1.
2. f(x) is differentiable at x = 0.
Which of the above statements is/are correct?
Consider the following statements in respect of the function $\rm f(x) = sin \left(\frac{1}{x^2}\right)$, x ≠ 0:
1. It is continuous at x = 0, if f(0) = 0.
2. It is continuous at $x = \frac{2}{\sqrt{\pi}}$.
Which of the above statements is/are correct?
Consider the following statements in respect of the function f(x) = x2 + 1 in the interval [1, 2]:
1. The maximum value of the function is 5.
2. The minimum value of the function is 2.
Which of the above statements is/are correct?
Consider the following statements in respect of the function f(x) = x + $\rm \frac{1}{x}$:
1. The local maximum value of f(x) is less than its local minimum value.
2. The local maximum value of f(x) occurs at x = 1.
Which of the above statements is/are correct?
Consider the following relations for two events E and F :
1. P(E ∩ F) ≥ P(E) + P(F) - 1
2. P(E ∪ F) = P(E) + P(F) + P(E ∩ F)
3. P(E ∪ F) ≤ P(E) + P(F)
Which of the above relations is/are correct?
Consider the following statements:
1. The regression line of y on x is $\rm y = \frac{3}{4}x+2$
2. The regression line of x on y is $\rm x = \frac{3}{4}y+\frac{1}{4}$
Which of the above statements is/are correct?
Consider the following statements:
1. The coefficient of correlations r is $\rm \frac{3}{4}$.
2. The means of x and y are 3 and 4 respectively.
Which of the above statements is/are correct?
For the set of numbers x, x, x + 2, x + 3, x + 10 where x is a natural number, which of the following is/are correct?
1. Mean > Mode
2. Median > Mean
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