If the line $\frac{x-2}{2 k}=\frac{y-3}{3}=\frac{z+2}{-1}$ and $\frac{x-2}{8}=\frac{y-3}{6}=\frac{z+2}{-2}$ are parallel, value of k is
- A-2
- ✓2
- C$\frac{1}{2}$
- D4
Answer: B.
View full solution →Question types
Model Paper 5 question types
45 questions across 6 question groups — pick any mix to generate a Mathematics 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 5 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $y=\left(1+\frac{1}{x}\right)^x$, then $\frac{d y}{d x}=$
- A$\left(1+\frac{1}{x}\right)^x \log \left(1+\frac{1}{x}\right)$
- B$\left(x+\frac{1}{x}\right)^x\left\{\log \left(1+\frac{1}{x}\right)+\frac{1}{x+1}\right\}$
- ✓$\left(1+\frac{1}{x}\right)^x\left\{\log \left(1+\frac{1}{x}\right)-\frac{1}{x+1}\right\}$
- D$\left(x+\frac{1}{x}\right)^x\left\{\log (x+1)-\frac{x}{x+1}\right\}$
Answer: C.
View full solution →If $\vec{a}, \vec{b}$ and $(\vec{a}+\vec{b})$ are all unit vectors and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then the value of $\theta$ is:
- ✓$\frac{2 \pi}{3}$
- B$\frac{\pi}{3}$
- C$\frac{5 \pi}{6}$
- D$\frac{\pi}{6}$
Answer: A.
View full solution →Find the particular solution for $2 x y+y^2-2 x^2 \frac{d y}{d x}=0 ; y=2$ when $x =1$
- ✓$y=\frac{2 x}{1-\log |x|}(x \neq 0, x \neq e)$
- B$y=\frac{3 x}{1-\log |x|}(x \neq 0, x \neq e)$
- C$y=\frac{3 x}{1-\log |x|}(x \neq 0, x \neq e)$
- D$y=\frac{5 x}{1+\log |x|}(x \neq 0, x \neq e)$
Answer: A.
View full solution →If for two events $A$ and $B, P(A-B)=\frac{1}{5}$ and $P(A)=\frac{3}{5}$, then $P\left(\frac{B}{A}\right)$ is equal to
- A$\frac{1}{2}$
- B$\frac{2}{5}$
- ✓$\frac{2}{3}$
- D$\frac{3}{5}$
Answer: C.
View full solution →If $R$ is the relation in the set $A=\{1,2,3,4,5\}$ given by $R=\{\{a, b):|a-b|$ is even $\}$,
Assertion (A): R is an equivalence relation.
Reason (R): All elements of $\{1,3,5\}$ are related to all elements of $\{2,4\}$.
Assertion (A): R is an equivalence relation.
Reason (R): All elements of $\{1,3,5\}$ are related to all elements of $\{2,4\}$.
- ABoth A and R are true and R is the correct explanation of A.
- BBoth A and R are true but R is not the correct explanation of A.
- ✓A is true but R is false.
- DA is false but R is true.
Answer: C.
View full solution →Assertion (A): If $3 \leq x \leq 10$ and $5 \leq y \leq 15$, then minimum value of $\left(\frac{x}{y}\right)$ is 2 .
Reason (R): If $3 \leq x \leq 10$ and $5 \leq y \leq 15$, then minimum value of $\left(\frac{x}{y}\right)$ is $\frac{1}{5}$.
Reason (R): If $3 \leq x \leq 10$ and $5 \leq y \leq 15$, then minimum value of $\left(\frac{x}{y}\right)$ is $\frac{1}{5}$.
- ABoth A and R are true and R is the correct explanation of A.
- ✓Both A and R are true but R is not the correct explanation of A.
- CA is true but R is false.
- DA is false but R is true.
Answer: B.
View full solution →Show that the function $f(x)=\frac{x}{3}+\frac{3}{x}$ decreases in the intervals $(-3,0) \cup(0,3)$.
View full solution →Evaluate: $\int \frac{(3 x+5)}{\left(x^3-x^2-x+1\right)} d x$
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 →Find the maximum and minimum values of $2 x^3-24 x+107$ on the interval $[-3,3]$.
View full solution →The volume of a cube is increasing at the rate of $7 \ cm^3 / \sec$. How fast is its surface area increasing at the instant when the length of an edge of the cube is $12 \ cm?$
View full solution →If $x=a \sin 2 t(1+\cos 2 t)$ and $y=b \cos 2 t(1-\cos 2 t)$, then find $\frac{d y}{d x}$ at $t=\frac{\pi}{4}$.
View full solution →Solve the following $\text{LPP}$ by graphical method:
Minimize $Z = 20x + 10y$
Subject to $x+2 y \leq 40 , 3 x+y \geq 30 , 4 x+3 y \geq 60$ and $ x, y \geq 0$
View full solution →Minimize $Z = 20x + 10y$
Subject to $x+2 y \leq 40 , 3 x+y \geq 30 , 4 x+3 y \geq 60$ and $ x, y \geq 0$
Show that the solution set of the linear constraints is empty $x-2 y \geq 0,2 x-y \leq-2, x \geq 0$ and $y \geq 0$
View full solution →Solve : $\frac{d y}{d x}+\frac{y}{x}=\cos x +\frac{\sin x}{x}$
View full solution →Find the particular solution of the differential equation $\left(1+ y ^2\right)+\left( x -e^{\tan ^{-1} y}\right) \frac{d y}{d x}=0$ given that y = 0 when x =1
View full solution →Find the image of the point $(0, 2, 3)$ in the line $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$
View full solution →Find the length shortest distance between the lines $\frac{x-3}{3}=\frac{y-8}{-1}=z-3$ and $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}$
View full solution →Two schools $P$ and $Q$ want to award their selected students on the values of Tolerance, Kindness, and Leadership. The school $P$ wants to award $Rs. x$ each, $Rs. y$ each and $Rs. z$ each for the three respective values to $3, 2$ and $1$ students respectively with total award money of $Rs. 2200$.
School $Q$ wants to spend $Rs. 3100$ to award its $4, 1$ and $3$ students on the respective values $($by giving the sameaward money to the three values as school $P)$. If the total amount of award for one prize on each value is $Rs.1200,$ using matrices, find the award money for each value
View full solution →School $Q$ wants to spend $Rs. 3100$ to award its $4, 1$ and $3$ students on the respective values $($by giving the sameaward money to the three values as school $P)$. If the total amount of award for one prize on each value is $Rs.1200,$ using matrices, find the award money for each value
Show that the function $f: R \rightarrow\{x \in R:-1 < x<1\}$ defined by $f(x)=\frac{x}{1+|x|}, x \in R$ is one$-$one and onto function.
View full solution →Show that the relation $R$ in the set $A=\{1,2,3,4,5\}$ given by $R=\{(a, b):|a-b|$ is divisible by 2$\}$ is an equivalence relation. Write all the equivalence classes of $R$.
View full solution →Read the following text carefully and answer the questions that follow:
Naina is creative she wants to prepare a sweet box for Diwali at home. She took a square piece of cardboard of side $18 \ cm$ which is to be made into an open box, by cutting a square from each corner and folding up the flaps to form the box. She wants to cover the top of the box with some decorative paper. Naina is interested in maximizing the volume of the box.
$i$. Find the volume of the open box formed by folding up the cutting each corner with $x \ cm. (1)$
$ii$. Naina is interested in maximizing the volume of the box. So, what should be the side of the square to be cut off so that the volume of the box is maximum? $(1)$
$iii$. Verify that volume of the box is maximum at $x = 3 \ cm$ by second derivative test? $(2)$
OR
Find the maximum volume of the box. $(2)$
View full solution →Naina is creative she wants to prepare a sweet box for Diwali at home. She took a square piece of cardboard of side $18 \ cm$ which is to be made into an open box, by cutting a square from each corner and folding up the flaps to form the box. She wants to cover the top of the box with some decorative paper. Naina is interested in maximizing the volume of the box.
$i$. Find the volume of the open box formed by folding up the cutting each corner with $x \ cm. (1)$
$ii$. Naina is interested in maximizing the volume of the box. So, what should be the side of the square to be cut off so that the volume of the box is maximum? $(1)$
$iii$. Verify that volume of the box is maximum at $x = 3 \ cm$ by second derivative test? $(2)$
OR
Find the maximum volume of the box. $(2)$
Read the following text carefully and answer the questions that follow:
Once Ramesh was going to his native place at a village near Agra. From Delhi and Agra he went by flight, In the way, there was a river. Ramesh reached the river by taxi. Then Ramesh used a boat for crossing the river. The boat heads directly across the river $40 m$ wide at $4 m/s$. The current was flowing downstream at $3 m/s.$
$i.$ What is the resultant velocity of the boat? $(1)$
$ii$. How much time does it take the boat to cross the river? $(1)$
$iii$. How far downstream is the boat when it reaches the other side? $(2)$
OR
If speeds of boat and current were $1.5 m/s$ and $2.0 m/s$ then what will be resultant velocity? $(2)$
View full solution →Once Ramesh was going to his native place at a village near Agra. From Delhi and Agra he went by flight, In the way, there was a river. Ramesh reached the river by taxi. Then Ramesh used a boat for crossing the river. The boat heads directly across the river $40 m$ wide at $4 m/s$. The current was flowing downstream at $3 m/s.$
$i.$ What is the resultant velocity of the boat? $(1)$
$ii$. How much time does it take the boat to cross the river? $(1)$
$iii$. How far downstream is the boat when it reaches the other side? $(2)$
OR
If speeds of boat and current were $1.5 m/s$ and $2.0 m/s$ then what will be resultant velocity? $(2)$
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