Direction cosines of a line perpendicular to both x-axis and z-axis are
- ✓0, 1, 0
- B0, 0, 1
- C1, 1, 1
- D1, 0, 1
Answer: A.
View full solution →Question types
Model Paper 1 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 1 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $f(x)=\left\{\begin{array}{cl}\frac{1}{1+e^{1 x}} & , x \neq 0 \\ 0 & , x=0\end{array}\right.$ then $f ( x )$ is
- ✓none of these
- Bdifferentiable but not continuous at $x = 0$
- Ccontinuous but not differentiable at $x = 0$
- Dcontinuous as well as differentiable at $x = 0$
Answer: A.
View full solution →If the position vectors of P and Q are $\hat{i}+3 \hat{j}-7 \hat{k}$ and $5 \hat{i}-2 \hat{j}+4 \hat{k}$ respectively, then the cosine of the angle between $\overrightarrow{P Q}$ and $y$-axis is
- A$\frac{4}{\sqrt{162}}$
- B$\frac{11}{\sqrt{162}}$
- C$\frac{5}{\sqrt{162}}$
- ✓$-\frac{5}{\sqrt{162}}$
Answer: D.
View full solution →Which of the following is the general solution of $\frac{d^2 y}{d x^2}-2 \frac{d y}{d x}+y=0 ?$
- A$y=A \cos x+B \sin x$
- B$y=(A x+B) e^{-x}$
- C$y=A e^x+B e^{-x}$
- ✓$y=(A x+B) e^x$
Answer: D.
View full solution →You are given that $A$ and $B$ are two events such that $P ( B )=\frac{3}{5}, P ( A \mid B )=\frac{1}{2}$ and $P ( A \cup B )=\frac{4}{5}$, then $P ( A )$ equals
- ✓$\frac{1}{2}$
- B$\frac{1}{5}$
- C$\frac{3}{5}$
- D$\frac{3}{10}$
Answer: A.
View full solution →Assertion $(A):$ The function $f : R ^* \rightarrow R ^*$ defined by $f(x)=\frac{1}{x}$ is one$-$one and onto, where $R*$ is the set of all non$-$zero real numbers.
Reason $(R):$ The function $g : N \rightarrow R ^*$ defined by $f(x)=\frac{1}{x}$ is one$-$one and onto.
Reason $(R):$ The function $g : N \rightarrow R ^*$ defined by $f(x)=\frac{1}{x}$ is one$-$one and onto.
- 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.
- D$A$ is false but $R$ is true.
Answer: C.
View full solution →Assertion $(A):f(x)=2 x^3-9 x^2+12 x-3$ is increasing outside the interval $(1, 2).$
Reason $(R): f ^{\prime}( x )<0$ for $x \in(1,2)$
Reason $(R): f ^{\prime}( x )<0$ for $x \in(1,2)$
- 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 $A$
- C$A$ is true but $R$ is false.
- D$A$ is false but $R$ is true.
Answer: B.
View full solution →Show that the function $f(x)=x^3-3 x^2+6 x-100$ is increasing on $R.$
View full solution →Evaluate $\int_{-1}^2(|x+1|+|x|+|x-1|) d x$
View full solution →Find the intervals in which $f(x)=\frac{4 x^2+1}{x}$ is increasing or decreasing.
View full solution →Find the intervals in which the function f given by $f(x)=x^2-4 x+6$ is
a. increasing
b. decreasing
View full solution →a. increasing
b. decreasing
A man $1.6m$ tall walks at the rate of $0.3\ m/s$ away from a street light is $4 m$ above the ground. At what rate is the tip of his shadow moving? At what rate is his shadow lengthening?
View full solution →If $x = a \left(\cos t+\log \tan \frac{t}{2}\right), y = a \sin t$, then evaluate $\frac{d^2 y}{d x^2}$ at $t=\frac{\pi}{3}$.
View full solution →Solve the Linear Programming Problem graphically: Maximize $Z = 3x + 4y$ Subject to $\begin{array}{l}2 x+2 y \leq 80 \\ 2 x+4 y \leq 120\end{array}$
View full solution →In Fig, the feasible region $($shaded$)$ for a $\text{LPP}$ is shown. Determine the maximum and minimum value of $Z =x+2y$
Solve the Linear Programming Problem graphically:
Maximize $Z = 3x + 4y$ Subject to
$2 x+2 y \leq 80$
$2 x+4 y \leq 120$
View full solution →Solve the Linear Programming Problem graphically:
Maximize $Z = 3x + 4y$ Subject to
$2 x+2 y \leq 80$
$2 x+4 y \leq 120$
In the differential equation show that it is homogeneous and solve it : $x^2 \frac{d y}{d x}= x ^2+ xy + y ^2$
View full solution →Find the general solution of the differential equation $\left(1+x^2\right) \frac{d y}{d x}-x=2 \tan ^{-1} x$
View full solution →Show that the lines $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-1}{5}$ and $\frac{x-2}{2}=\frac{y-1}{3}=\frac{z+1}{-2}$ intersect and find their point of intersection.
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 →Given $A=\left[\begin{array}{ccc}2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5\end{array}\right], B=\left[\begin{array}{ccc}1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2\end{array}\right]$, find $B A$ and use this to solve the system of equations $y + 2 z=7, x-y=3,2 x+3 y+4 z=17$
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 →$i$. reflexive
$ii$. symmetric
$iii$. transitive
Give reason in each case.
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 →Read the following text carefully and answer the questions that follow:
Mrs. Maya is the owner of a high-rise residential society having $50$ apartments. When he set rent at $₹ 10000 $ month, all apartments are rented. If he increases rent by $₹ 250$ / month, one fewer apartment is rented. The maintenance cost for each occupied unit is $₹ 500$ / month.
$i.$ If $P$ is the rent price per apartment and $N$ is the number of rented apartments, then find the profit. $(1)$
$ii.$ If $x$ represents the number of apartments which are not rented, then express profit as a function of $x. (1)$
$iii.$ Find the number of apartments which are not rented so that profit is maximum. $(2)$
OR
Verify that profit is maximum at critical value of $x$ by second derivative test. $(2)$
View full solution →Mrs. Maya is the owner of a high-rise residential society having $50$ apartments. When he set rent at $₹ 10000 $ month, all apartments are rented. If he increases rent by $₹ 250$ / month, one fewer apartment is rented. The maintenance cost for each occupied unit is $₹ 500$ / month.
$i.$ If $P$ is the rent price per apartment and $N$ is the number of rented apartments, then find the profit. $(1)$
$ii.$ If $x$ represents the number of apartments which are not rented, then express profit as a function of $x. (1)$
$iii.$ Find the number of apartments which are not rented so that profit is maximum. $(2)$
OR
Verify that profit is maximum at critical value of $x$ by second derivative test. $(2)$
Read the following text carefully and answer the questions that follow:
Team $P, Q, R$ went for playing a tug of war game. Teams $P, Q, R$ have attached a rope to a metal ring and is trying to pull the ring into their own areas $($team areas when in the given figure below$)$. Team $P$ pulls with force$F _1=4 \hat{i}+0 \hat{j} KN$
Team $Q$ pull with force $F _2=-2 \hat{i}+4 \hat{j} KN$
Team $R$ pulls with force $F _3=-3 \hat{i}-3 \hat{j} KN$
$i$. What is the magnitude of the teams combined force? $(1)$
$ii$. Find the magnitude of Team $B. (1)$
$iii$. Which team will win the game? $(2)$
OR
Find the probability that she gets grade $A$ in at least one subject. $(2)$
View full solution →Team $P, Q, R$ went for playing a tug of war game. Teams $P, Q, R$ have attached a rope to a metal ring and is trying to pull the ring into their own areas $($team areas when in the given figure below$)$. Team $P$ pulls with force$F _1=4 \hat{i}+0 \hat{j} KN$
Team $Q$ pull with force $F _2=-2 \hat{i}+4 \hat{j} KN$
Team $R$ pulls with force $F _3=-3 \hat{i}-3 \hat{j} KN$
$i$. What is the magnitude of the teams combined force? $(1)$
$ii$. Find the magnitude of Team $B. (1)$
$iii$. Which team will win the game? $(2)$
OR
Find the probability that she gets grade $A$ in at least one subject. $(2)$
Read the following text carefully and answer the questions that follow:
Shama is studying in class $XII$. She wants do graduate in chemical engineering. Her main subjects are mathematics, physics, and chemistry. In the examination, her probabilities of getting grade $A$ in these subjects are $0.2, 0.3,$ and $0.5$ respectively.
$1$. Find the probability that she gets grade $A$ in all subjects.$ (1)$
$2.$ Find the probability that she gets grade $A$ in no subjects. $(1)$
$3$. Find the probability that she gets grade $A$ in two subjects. $(2)$
OR
Find the probability that she gets grade $A$ in at least one subject. $(2)$
View full solution →Shama is studying in class $XII$. She wants do graduate in chemical engineering. Her main subjects are mathematics, physics, and chemistry. In the examination, her probabilities of getting grade $A$ in these subjects are $0.2, 0.3,$ and $0.5$ respectively.
$1$. Find the probability that she gets grade $A$ in all subjects.$ (1)$
$2.$ Find the probability that she gets grade $A$ in no subjects. $(1)$
$3$. Find the probability that she gets grade $A$ in two subjects. $(2)$
OR
Find the probability that she gets grade $A$ in at least one subject. $(2)$
Generate a Model Paper 1 paper free
Pick question groups from the list above, set marks and difficulty, and export a branded PDF with step-by-step answer keys. First 3 chapters free — no signup.