CBSE Class 12 · Mathematics

Vectors, 3D Geometry and Probability

19 practice questions, each with the answer and an explanation.

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If $l, m, n$ are the direction cosines of a straight line in three-dimensional space, what is the value of $l^{2} + m^{2} + n^{2}$ ?
Which formula gives the shortest distance $d$ between two skew lines $r = a_{1} + λ b_{1}$ and $r = a_{2} + μ b_{2}$ in three dimensions?
If $P (A) = 0.8$ , $P (B) = 0.5$ , and $P (B ∣ A) = 0.4$ , what is the value of $P (A \cap B)$ ?
Two events $E$ and $F$ with non-zero probabilities are statistically independent if and only if which equality holds?
Bag I contains 3 red and 4 black balls, while Bag II contains 5 red and 6 black balls. One bag is chosen at random with equal probability and a ball is drawn. If the drawn ball is red, what is the pro…
A fair coin is tossed $n = 10$ times. Let $X$ denote the number of heads obtained. What are the mean ( $μ$ ) and variance ( $σ^{2}$ ) of the random variable $X$ ?
In a linear programming problem, if the feasible region is bounded, where does the optimal (maximum or minimum) value of the objective function $Z = a x + b y$ always occur?
What is the inverse of the matrix $A = (2314)$ ?
Let $c$ be a critical point of function $f$ such that $f^{'} (c) = 0$ . If $f^{''} (c) < 0$ , what does the second derivative test conclude about $x = c$ ?
Which of the following is NOT a required hypothesis for Rolle's Theorem to hold on an interval $[a, b]$ ?
What is the derivative of $f (x) = tan^{- 1} (\frac{2 x}{1 - x ^{2}})$ with respect to $x$ for $∣ x ∣ < 1$ ?
A function $f (x, y)$ is homogeneous of degree $n$ if which relation is satisfied for all non-zero real scalars $t$ ?
If two non-zero vectors $a$ and $b$ are collinear (parallel or antiparallel), what is their vector cross product $a \times b$ ?
What is the cosine of the angle between two straight lines having direction ratios proportional to $(1, 2, 2)$ and $(2, - 2, 1)$ ?
For two general $n \times n$ matrices $A$ and $B$ , which matrix property does NOT hold in general?
Let $E_{1}, E_{2}, \dots, E_{n}$ be a partition of sample space $S$ . If $A$ is any event, what is the formula for $P (A)$ given by the Theorem on Total Probability?
For a system of linear equations $A X = B$ , if $∣ A ∣ = 0$ and $(adj A) B \neq = O$ , what is the nature of the solutions?
What is the perpendicular distance of the point $(2, 3, - 5)$ from the plane $x + 2 y - 2 z = 9$ ?
What is the value of the definite integral $\int_{0}^{1} \frac{1}{1 + x ^{2}} d x$ ?

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