If $f(x)=\left\{\begin{array}{l}k x+5 \text {, when } x \leq 2 \\ x+1, \text { when } x>2\end{array}\right.$ is continuous at $x=2$ then $k= ?$
View full solution →Question types
Model Paper 10 question types
45 questions across 6 question groups — pick any mix to generate a MATHS paper with step-by-step answer keys.
45
Questions
6
Question groups
5
Question types
01
M.C.Q (1 Marks)
18 Q→02Assertion (A) & Reason (B) MCQ
2 Q→032 Marks Questions
7 Q→043 Marks Question
9 Q→055 Marks Questions
6 Q→06Case study (4 Marks)
3 Q→Sample Questions
Model Paper 10 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $P(A)=\frac{3}{10}, P(B)=\frac{2}{5}$ and $P(A \cup B)=\frac{3}{5}$, then $P(B / A)+P(A / B)$ equals
View full solution →the order of the single matrix obtained from
$\left[\begin{array}{cc}1 & -1 \\ 0 & 2 \\ 2 & 3\end{array}\right]_{3 \times 2}\left\{\left[\begin{array}{ccc}-1 & 0 & 2 \\ 2 & 0 & 1\end{array}\right]_{2 \times 3}-\left[\begin{array}{lll}0 & 1 & 23 \\ 1 & 0 & 21\end{array}\right]_{2 \times 3}\right\}$ is
View full solution →$\left[\begin{array}{cc}1 & -1 \\ 0 & 2 \\ 2 & 3\end{array}\right]_{3 \times 2}\left\{\left[\begin{array}{ccc}-1 & 0 & 2 \\ 2 & 0 & 1\end{array}\right]_{2 \times 3}-\left[\begin{array}{lll}0 & 1 & 23 \\ 1 & 0 & 21\end{array}\right]_{2 \times 3}\right\}$ is
If $y=\log \left(\sin e^x\right)$, then $\frac{d y}{d x}$ is:
View full solution →If the vectors $4 \hat{i}+11 \hat{j}+m \hat{k}, 7 \hat{i}+2 \hat{j}+6 \hat{k}$ and $\hat{i}+5 \hat{j}+4 \hat{k}$ are coplanar, then $m =$
View full solution →Assertion (A): A relation $R =\{( a , b )$ : $| a - b |<3\}$ defined on the set $A =\{1,2,3,4\}$ is reflexive.
Reason (R): A relation $R$ on the set $A$ is said to be reflexive if for $(a, b) \in R$ and $(b, c) \in R$, we have $(a, c) \in R$.
View full solution →Reason (R): A relation $R$ on the set $A$ is said to be reflexive if for $(a, b) \in R$ and $(b, c) \in R$, we have $(a, c) \in R$.
Assertion (A): The rate of change of area of a circle with respect to its radius $r$ when $r=6 cm$ is $12 \pi cm^2 / cm$.
Reason (R): Rate of change of area of a circle with respect to its radius r is $\frac{d A}{d r}$, where A is the area of the circle.
View full solution →Reason (R): Rate of change of area of a circle with respect to its radius r is $\frac{d A}{d r}$, where A is the area of the circle.
Write the interval for the principal value of function and draw its graph: $\cos ^{-1} x$.
View full solution →Find the matrix $X$ for which $:\left[\begin{array}{ll}3 & 2 \\ 7 & 5\end{array}\right] X\left[\begin{array}{ll}-1 & 1 \\ -2 & 1\end{array}\right]=\left[\begin{array}{ll}2 & -1 \\ 0 & 4\end{array}\right]$
View full solution →Show that the function given by $f(x)=\sin x$ is neither increasing nor decreasing in $(0, \pi)$
View full solution →Show that $f(x)=\cos \left(2 x+\frac{\pi}{4}\right)$ is an increasing function on $\left(\frac{3 \pi}{8}, \frac{7 \pi}{8}\right)$
View full solution →A particle moves along the curve $6 y=x^3+2$. Find the points on the curve at which $y-$ coordinates is changing $2$ times as fast as $x -$ coordinates.
View full solution →Find all points of discontinuity of $f$ where $f$ is defined as follows$, f(x)$
View full solution →Evaluate the integral: $\sec ^{-1} \sqrt{x} d x$
View full solution →Let $\vec{a}=i+4 j+2 k, b=3 i-2 j+7 k$ and $c=2 i-j+4 k$. Find a vector $d$ which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c} . \vec{d}=15$.
View full solution →The two adjacent sides of a parallelogram are $2 i-4 j-5 k$ and $2 i+2 j+3 k$. Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram.
View full solution →Solve differential equation: $\frac{d y}{d x}-y \tan x=e^x$
View full solution →Show that the semi$-$vertical angle of a cone of maximum volume and given slant height is $\tan ^{-1} \sqrt{2}$ or $\cos ^{-1} \frac{1}{\sqrt{3}}$.
View full solution →Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.
Three shopkeepers $\text{A, B}$ and $C$ go to a store to buy stationery. A purchases $12$ dozen notebooks, $5$ dozen pens and $6$ dozen pencils. $B$ purchases $10$ dozen notebooks, $6$ dozen pens and $7$ dozen pencils. $C$ purchases $11$ dozen notebooks, $13$ dozen pens and $8$ dozen pencils. A notebook costs $40$ paise, a pen costs ₹ $1.25$ and a pencil costs $35$ paise. Use matrix multiplication to calculate each individual's bill.
View full solution →Find the area of the region $\left\{(x, y): 0 \leqslant y \leqslant\left(x^2+1\right), 0 \leqslant y \leqslant(x+1), 0 \leqslant x \leqslant 2\right\}$
View full solution →Let $A =\{1,2,3\}$ and $R =\left\{( a , b ): a , b \in A\right.$ and $\left|a^2-b^2\right| \leq 5$. Write R as set of ordered pairs. Mention whether $R$ is i. reflexive
$ii.$ symmetric
$iii.$ transitive
Give reason in each case.
View full solution →$ii.$ symmetric
$iii.$ transitive
Give reason in each case.
Read the following text carefully and answer the questions that follow:
If $a_1, b_1, c_1$ and $a_2, b_2, c_2$ are direction ratios of two lines say $L_1$ and $L_2$ respectively.
Then $L_1 \| L_2$ iff $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$ and $L _1 \perp L_2$ if $a _1 a _2+ b _1 b_2+ c _1 c _2=0$.
$i.$ Find the coordinates of the foot of the perpendicular drawn from the point $A (1,2,1)$ to the line joining $B (1$, $4,6)$ and $C(5,4,4) \cdot(1)$
$ii.$ Find the direction ratios of the line which is perpendicular to the lines with direction ratios proportional to $(1, -2,-2)$ and $(0,2,1) \cdot(1)$
$iii.$ What is the relation between lines $\frac{x-2}{3}=\frac{y+1}{-2}=\frac{z-2}{0}$ and $\frac{x-1}{1}=\frac{y+\frac{3}{2}}{3}=\frac{z+5}{2} \cdot(2)$
$OR$
If $l _1, m_1, n _1$ and $l _2, m_2, n _2$ are direction cosines of $L _1$ and $L _2$ respectively, then what is the condition for $L _1$ parallel to $L _2. (2)$
View full solution →If $a_1, b_1, c_1$ and $a_2, b_2, c_2$ are direction ratios of two lines say $L_1$ and $L_2$ respectively.
Then $L_1 \| L_2$ iff $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$ and $L _1 \perp L_2$ if $a _1 a _2+ b _1 b_2+ c _1 c _2=0$.
$i.$ Find the coordinates of the foot of the perpendicular drawn from the point $A (1,2,1)$ to the line joining $B (1$, $4,6)$ and $C(5,4,4) \cdot(1)$
$ii.$ Find the direction ratios of the line which is perpendicular to the lines with direction ratios proportional to $(1, -2,-2)$ and $(0,2,1) \cdot(1)$
$iii.$ What is the relation between lines $\frac{x-2}{3}=\frac{y+1}{-2}=\frac{z-2}{0}$ and $\frac{x-1}{1}=\frac{y+\frac{3}{2}}{3}=\frac{z+5}{2} \cdot(2)$
$OR$
If $l _1, m_1, n _1$ and $l _2, m_2, n _2$ are direction cosines of $L _1$ and $L _2$ respectively, then what is the condition for $L _1$ parallel to $L _2. (2)$
Read the following text carefully and answer the questions that follow:
A building contractor undertakes a job to construct 4 flats on a plot along with parking area. Due to strike the probability of many construction workers not being present for the job is 0.65 . The probability that many are not present and still the work gets completed on time is 0.35 . The probability that work will be completed on time when all workers are present is 0.80 .
Let: $E _1$ : represent the event when many workers were not present for the job;
$E _2$ : represent the event when all workers were present; and
E: represent completing the construction work on time.
i. What is the probability that all the workers are present for the job? (1)
ii. What is the probability that construction will be completed on time? (1)
iii. What is the probability that many workers are not present given that the construction work is completed on time? (2)
OR
What is the probability that all workers were present given that the construction job was completed on time?(2)
View full solution →A building contractor undertakes a job to construct 4 flats on a plot along with parking area. Due to strike the probability of many construction workers not being present for the job is 0.65 . The probability that many are not present and still the work gets completed on time is 0.35 . The probability that work will be completed on time when all workers are present is 0.80 .
Let: $E _1$ : represent the event when many workers were not present for the job;
$E _2$ : represent the event when all workers were present; and
E: represent completing the construction work on time.
i. What is the probability that all the workers are present for the job? (1)
ii. What is the probability that construction will be completed on time? (1)
iii. What is the probability that many workers are not present given that the construction work is completed on time? (2)
OR
What is the probability that all workers were present given that the construction job was completed on time?(2)
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