Sample QuestionsMODEL PAPER 2025 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
A linear programming problem (LPP) along eith the graph of its constraints is shown below.The correspondig objective function is :Z = 18x + 10y which has to be minimized. The smallest value of the objective function Z is 134 and is obtained at the corner point (3,8).
The optimal solution of the above linear programming problem_______________.
For the linear programming problem (LPP), the objective function is Z= 4x+3y and the feasible region determined by a set of constraints is shown in the graph:
Which of the following statements is true?
For any two events A and B,if $P(\bar{A})=\frac{1}{2}, P(\bar{B})=\frac{2}{3}$ and $P(A \cap B)=\frac{1}{4}$, then $P(\bar{A} / \bar{B})$ equals:
View full solution →If the points $\left(x_1, y_1\right),\left(x_2, y_2\right)$ and $\left(x_1+x_2, y_1+y_2\right)$ are collinear, then $x_1 y_2$ is equal to
View full solution →If for a square matrix A, A.(adjA) $=\left[\begin{array}{ccc}2025 & 0 & 0 \\ 0 & 2025 & 0 \\ 0 & 0 & 2025\end{array}\right]$ then the value of $|A|+|a d j A |$ is equal to.
View full solution →Assertion (A): Consider the function defined as $f(x)=|x|+|x-1|, x \in R$. Then $f(x)$ is not differentiable at x = 0 x = 1
Reason (R): Suppose f be defined and continuous on $(a, b)$ and $c \in(a, b)$, then $f(x)$ is not differentiable at $x=c$ if $\lim _{h \rightarrow 0^{-}} \frac{f(c+h)-f(c)}{h} \neq \lim _{h \rightarrow 0^{+}} \frac{f(c+h)-f(c)}{h}$.
View full solution →Assertion (A): The function $f: R-\left\{(2 n+1) \frac{\pi}{2}: n \in Z\right\} \rightarrow(-\infty,-1] \cup[1, \infty)$ defined by $f(x)=\sec x$ is not one - one function in its domain.
Reason (R): The line y = 2 meets the graph of the function at more than one point.
View full solution →A person standing at $0\ (0,0,0)$ is watching an aeroplane which is at the coordinate point $4\ (4,0,3)$. At the same time he saw a bird at the coordinate point $B\ (0,0,1)$. Find the angles which $\overrightarrow{B A}$ makes with the $x,y$ and $z$ axes.
View full solution →Differentiate the following function with respect to $x:x :(\cos x)^x ;\left(\right.$ where $\left.x \in\left(0, \frac{\pi}{2}\right)\right)$
View full solution →Find the derivative of $\tan^{-1}x$ with respect to $\log.x; ($where $x \in (1,\infty)).$
View full solution →The two co-initial adjacent sides of a parallelogram are $2 \hat{\imath}-4 \hat{\jmath}-5 \hat{k}$ and $2 \hat{\imath}+2 \hat{\jmath}+3 \hat{k}$ Find its diagonals and use them to find the area of the parallelogram.
View full solution →If vectors $\vec{a}=2 \hat{ i }+2 \hat{\jmath}+3 \hat{ k }, \vec{b}=-\hat{ i }+2 \hat{\jmath}+\hat{ k }$ and $\vec{c}=3 \hat{ i }+\hat{ j }$ are such that $\vec{b}+\lambda \vec{c}$ is perpendicular to $\vec{a}$, then find the value of $\lambda$.
View full solution →The probability that it rains today is 0.4. If it rains today, the probability that it will rain tomorrow is 0.8. If it does not rain today, the probability that it will rain tomorrow is 0.7. If
P1: denotes the probability that it does not rain today.
P2: denotes the probability that it will not rain tomorrow, if it rains today.
P3: denotes the probability that it will rain tomorrow, if it does not rain today.
P4: denotes the probability that it will not rain tomorrow, if it does not rain today.
(i) Find the value of $P_1 \times P_4-P_2 \times P_3$
(ii) Calculate the probability of raining tomorrow.
View full solution →A random variable $X$ can take all non negative integral values and the probability that $X$ takes the value is proportional to $5^{-r}$. Find $P(X<3)$
View full solution →Consider the following Linear Programming Problem:
Minimise Z = x + 2y
Subject to $2 x+y \geq 3, x+2 y \geq 6, x, y \geq 0$
Show graphically that the minimum of Z occurs at more than two points
View full solution →Evaluate : $\int_0^1 x(1-x)^n d x$; $($ where $n \in N)$
View full solution →Find the vector and the cartesian equation of the line that passes through through (-1, 2, 7) and is perpendicular to the lines $\vec{r}=2 \hat{\imath}+\hat{\jmath}-3 \hat{ k }+\lambda(\hat{\imath}+2 \hat{\jmath}+5 \hat{ k })$ and $\vec{r}=3 \hat{\imath}+3 \hat{\jmath}-7 \hat{ k }+\mu(3 \hat{\imath}-2 \hat{\jmath}+5 \hat{ k })$.
View full solution →Draw the rough sketch of the curve $y=20 \cos 2 x ;\left(\right.$ where $\left.\frac{\pi}{6} \leq x \leq \frac{\pi}{3}\right)$
Using integration, find the area of the region bounded by the curve $y = 20 \cos2x$ from the ordinates
$x=\frac{\pi}{6}$ to $x=\frac{\pi}{3}$ and the $x-$axis.
View full solution →Find the image of the point (1,2,1) with respect to the line$\frac{x-3}{1}=\frac{y+1}{2}=\frac{z-1}{3}$ Also find the equation of the line joining the given point and its image.
View full solution →If $(x-a)^2+(y-b)^2=c^2$, for some $c>0$, prove that $\frac{\left[1+\left(\frac{d y}{d x}\right)^2\right]^{\frac{3}{2}}}{\frac{d^2 y}{d x^2}}$ is a constant independent of $a$ and $b$.
View full solution →If $f: R \rightarrow R$ is defined by $f(x)=|x|^3$, show that $f^{\prime \prime}(x)$ exists for all real $x$ and find it.
View full solution →Find the shortest distance between the lines $I_1$ and $l_2$
$\vec{r}=(-\hat{\imath}-\hat{\jmath}-\hat{k})+\lambda(7 \hat{\imath}-6 \hat{\jmath}+\hat{k})$ and $\vec{r}=(3 \hat{\imath}+5 \hat{\jmath}+7 \hat{k})+\mu(\hat{\imath}-2 \hat{\jmath}+\hat{k})$where $\lambda$ and $\mu$ are parameters.
View full solution →Arka bought two cages of birds: Cage $-I $ contains $5$ parrots and $1$ owl and Cage $-II$ contains $6$ parrots. One day Arka forgot to lock both cages and two birds flew from Cage $-I$ to Cage $-II$ $($simultaneously$)$. Then two birds flew back from cage $-II$ to cage $-I \ ($simultaneously$)$.
Assume that all the birds have equal chances of flying.
On the basis of the above information, answer the following questions: $-$
$(i)$ When two birds flew from Cage $-I$ to Cage $-II$ and two birds flew back from Cage $-II$ to Cage $-I$ then find the probability that the owl is still in Cage $-I.$
$(ii)$ When two birds flew from Cage $-I$ to Cage $-II$ and two birds flew back from Cage $-II$ to Cage-I, the owl is still seen in Cage $-I,$ what is the probability that one parrot and the owl flew from Cage $-I$ to Cage $-II$?
View full solution →Ramesh, the owner of a sweet selling shop, purchased some rectangular card board sheets of dimension 25 cm by 40 cm to make container packets without top. Let x cm be the length of the side of the square to be cut out from each corner to give that sheet the shape of the container by folding up the flaps.
Based on the above information answer the following questions.
(i) Express the volume (V) of each container as function of x only.
(ii) Find $\frac{d V}{d x}$
(iii) (a) for what value of x, the volume of each container is maximum ?
OR
(iii) (b) Check whether V has a point of inflection at x $x=\frac{65}{6}$ or not ?
View full solution →An organization conducted bike race under $2$ different categories$-$boys and girls. In all, there were $250$ participants. Among all of them finally three from Category $1$ and two from Category $2$ were selected for the final race. Ravi forms two sets $B$ and $G$ with these participants for his college project.
Let $B =\left\{b_1, b_2, b_3\right\}, G =\left\{g_1, g_2\right\}$ where $B$ represents the set of boys selected and $G$ the set of girls who were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions.
On the basis of the above information, answer the following questions:
$(i)$ Ravi wishes to form all the relations possible from $B$ to $G$. How many such relations are possible?
$(ii)$ Among these relations, how many are functions from $B$ to $G$ ?
$OR$
$(iii) (b)$ If the track of the final race $($for the biker $b_1)$ follows the curve
$x^2=4 y ;($ where $0 \leq x \leq 20 \sqrt{2} \ 0 \leq y \leq 200)$,then state whether the track represents a one$-$one and onto function or not. $($Justify$).$
View full solution →