List of practice Questions

A point on the plane passing through the points \((\sqrt{2},1,4), (0,-1,0)\) and \((0,0,1)\) is:

A box contains 5 white balls and 7 black balls. In how many ways can 3 balls be drawn from the box such that at least one black ball is included?}

Two dice are rolled. If \( A \) denotes the event that the same number shows on each die and \( B \) denotes the event that the sum of the numbers on both dice is greater than 7, then \( P(A \mid B) \) and \( P(B \mid A) \) respectively are:

The circumcentre of the triangle with vertices \( (-2,3), (2,-1), (4,0) \) is:

Let \[ \vec{a} = 2\hat{i} + \hat{j} - \hat{k} \quad \text{and} \quad \vec{b} = \hat{i} + 3\hat{j} - 5\hat{k} \] be two vectors, and \(\vec{r}\) be a vector along the vector \(3\vec{a} - 2\vec{b}\) such that \(|\vec{r}| = \sqrt{74}\). If the direction of \(\vec{r}\) is opposite to that of \(3\vec{a} - 2\vec{b}\), then \(\vec{r} =\)

If \(\vec{r}\) is the position vector of the centroid \(G\) of triangle \(ABC\) and \[ 2\hat{i} + \hat{j} - \hat{k} \quad \text{and} \quad 2\hat{i} + 4\hat{j} - 4\hat{k} \] are respectively the position vectors of its vertices \(A\) and \(B\), then \[ AG^2 + BG^2 + CG^2 =\ ? \]

If \[ \vec{a} = -2\hat{i} + 9\hat{j} - 6\hat{k}, \quad \vec{b} = t\hat{i} - 2\hat{j} + 6\hat{k} \] are vectors such that \(|\vec{a} + \vec{b}| = 25\), then the sum of the values of \(t =\ ?\)

If \[ \vec{a} = \hat{i} + 3\hat{j} + 13\hat{k}, \quad \vec{b} = 2\hat{i} - 4\hat{j} + 3\hat{k} \] are two vectors, then the component vector of \(\vec{a}\) perpendicular to \(\vec{b}\) is

Let \[ \vec{a} = 2\hat{i} - 3\hat{j} - 5\hat{k}, \quad \vec{b} = 3\hat{i} + 2\hat{j} - 5\hat{k} \] be two vectors and \(\vec{r}\) be a vector in the plane of \(\vec{a} \text{ and } \vec{b}\). If \(\vec{r}\) is orthogonal to \(5\hat{i} - 2\hat{j} + 3\hat{k}\) and the magnitude of \(\vec{r}\) is \(\sqrt{94}\), then \(|\vec{r} \cdot \vec{b}|\) is

If \(x_1, x_2, \ldots, x_n\) are 'n' observations and \(\bar{x}\) is their mean. If \[ \sum_{i=1}^{n}(x_i - \bar{x})^2 \text{ is almost zero, then a true statement among the following is} \]

If \( \cos^3 x \sin 4x = \sum_{r=0}^n a_r \sin rx \ \forall x \in \mathbb{R} \), then \( a_3 + a_5 : a_1 + a_7 = \)

In \(\triangle ABC\), if \(\angle A = 90^\circ\), then \((r_2 - r_1)(r_3 - r_1) = \)

The coefficient of \( x^3 \) in the expansion of \( (1 - 2x)^{\frac{1}{2}} (1 + 3x)^{\frac{1}{3}} \) is:

The number of arrangements of the letters of the word ARRANGEMENT in which two E’s do not occur adjacently is:

The number of elements in the set \( \left\{ x \in \mathbb{N} : \binom{20 - 2x}{x - 3} \in \mathbb{N} \right\} \) is

\(\frac{\cot A}{1 - \tan A} + \frac{\tan A}{1 - \cot A} =\)

If \( \theta = \frac{\pi}{9} \), then \( 1 + 27 \tan^2 \theta - 33 \tan^4 \theta + \tan^6 \theta = \)

If \( c \) and \( d \) are the roots of \( x^2 + ax + b = 0 \), then a root of the equation} \[ x^2 + (4c + a)x + (b + 2ac + 4c^2) = 0 \] is:

A matrix whose elements \( a_{ij} \) are defined by \( a_{ij} = \frac{1}{3}|i - 5j|, \; i, j = 1, 2, 3 \) is

If \( \alpha \) is the modulus of \( z_1 = 4 + 3i \), then a point that does not lie in the region represented by } \[ |z - z_1| \leq \alpha \] is

If \( a \) and \( b \) are the roots of the equation \( y^2 + y + 1 = 0 \), then the value of \( a^4 + b^4 + a^{-1}b^{-1} \) is

If \( f:\mathbb{R} \to \mathbb{R} \) is defined by \( f(x+y) = f(x) + f(y) \ \forall x, y \in \mathbb{R} \) and \( f(1) = 7 \), then \(\sum\limits_{r=1}^n f(r) =\)

Let \( A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \), where \( \theta \in \mathbb{R} \). Then \( A^n \), for any positive integer \( n \), is equal to:

A person of height 1.6 m is walking away from a lamp post of height 4 m along a straight line on the horizontal ground. The lamp post and the person are always vertical. If the speed of the person is 60 cm/sec, then find the speed of tip point of its shadow with respect to the person in cm/sec.

Arrange the following stations of Trance Siberian Railway from East to West direction.
A. Chita
B. Vladivostak
C. Moscow
D. Omsk
Choose the correct answer from the options given below :