The order and degree of a differential equation $\frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^4+x^{\frac{1}{5}}=0$; respectively, are
- A2 and 4
- ✓2 and 1
- C2 and 3
- D3 and 3
Answer: B.
View full solution →Question types
CBSE Model Paper 1 question types
44 questions across 6 question groups — pick any mix to generate a Applied Maths paper with step-by-step answer keys.
44
Questions
6
Question groups
5
Question types
01
MCQ
18 Q→02Assertion (A) & Reason (B) MCQ
2 Q→032 Marks Questions
7 Q→043 Marks Question
8 Q→055 Marks Questions
6 Q→06Case study (4 Marks)
3 Q→Sample Questions
CBSE 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.
(where QT is the transpose of the matrix Q), P-Q is: - A
- B
- C
- D
For a $3 \times 3$ matrix if $\operatorname{adj} A=2 A^{-1}$, find $\left|3 A A^T\right|$
- ✓108
- B12
- C54
- D8
Answer: A.
View full solution →Shown below is a curve.
$L_1$ is the tangent to any point $(x, y)$ on the curve.
$L_2$ is the line that connects the point $(x, y)$ to the origin.
The slope of $L_1$ is one third of the slope of $L_2$.
Then the differential equation, using the given conditions is:
$L_1$ is the tangent to any point $(x, y)$ on the curve.
$L_2$ is the line that connects the point $(x, y)$ to the origin.
The slope of $L_1$ is one third of the slope of $L_2$.
Then the differential equation, using the given conditions is:
- ✓$\frac{d y}{d x}=\frac{y}{3 x}$
- B$\frac{d y}{d x}=\frac{y}{x}$
- C$\frac{d y}{d x}=\frac{x}{3 y}$
- D$\frac{d y}{d x}=\frac{3 y}{x}$
Answer: A.
View full solution →The present value of a sequence of payments of ₹ 800 made at the end of every 6 month and continuing forever. If money is worth 4%per annum compounded semi-annually, then the present value of the sequence is:
- A₹ 20000
- ✓₹ 40000
- C₹ 60000
- D₹ 80000
Answer: B.
View full solution →Assertion(A): $A=\left[a_{i j}\right]=\left\{\begin{array}{r}m ; i=j \\ 0 ; i \neq j\end{array}\right.$
where $m$ is a scalar, is an identity matrix if $m=1$
Reason (R): Every identity matrix is not a scalar matrix
where $m$ is a scalar, is an identity matrix if $m=1$
Reason (R): Every identity matrix is not a scalar matrix
- 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): The effective rate of interest equivalent to a nominal rate of 6% when compounded continuously is equal to $e^{0.06}-1=6.18 \%$.
Reason (R): The relation between effective rate $\left(r_{e f f}\right)$ of interest and nominal rate $(r)$ of interest: $r _{\text {eff }}= e ^r - 1$; where ' $e$ ' - Euler's number (approximate value is $2 . 7 1 8 2 8$ ), when compounded continuously.
Reason (R): The relation between effective rate $\left(r_{e f f}\right)$ of interest and nominal rate $(r)$ of interest: $r _{\text {eff }}= e ^r - 1$; where ' $e$ ' - Euler's number (approximate value is $2 . 7 1 8 2 8$ ), when compounded continuously.
- ✓Both $(A)$ 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).
- C(A) is true but (R) is false.
- D(A) is false but (R) is true.
Answer: A.
View full solution →If $A=\left[\begin{array}{ll}\alpha & 0 \\ 1 & 1\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 0 \\ 5 & 1\end{array}\right]$, then find the value of $\alpha$ (if exists) for which $A^2=B$.
View full solution →The lifetime of an item produced by a machine has a normal distribution with mean 12 months and standard deviation of 2 months. Find the probability of an item produced by this machine will last
(i) less than $7$ months
(ii) between $7$ and $1 4$ months.
(Given $P\left(Z<\frac{5}{2}\right)=0.9938$ and $\left.P(Z<1)=0.8413\right)$
View full solution →(i) less than $7$ months
(ii) between $7$ and $1 4$ months.
(Given $P\left(Z<\frac{5}{2}\right)=0.9938$ and $\left.P(Z<1)=0.8413\right)$
The incidence of occupational disease in an industry is such that the workers have a 20% chance of suffering from it. What is the probability that out of six workers 4 or more will catch the disease?
A boat takes thrice as long to go upstream to a point as to return downstream to the starting point. If the speed of the stream is $5 km / h$, find the speed of the boat in still water.
View full solution →In a 200 𝑚 race, A can give a start of 18 𝑚 to B and a start of 31 𝑚 to C. In a race of 350 𝑚, how much start can B give to C?
If the probability of success in a single trial is $0 . 0 1$, how many minimum number of Bernoulli trials must be performed in order that the probability of at least one success is $\frac{ 1 }{ 2 }$ or more? (Use $\log _{10} 2=0.3010$ and $\log _{10} 99=1.9956$ )
View full solution →An octagonal prism is a three-dimensional polyhedron bounded by two octagonal bases and eight rectangular side faces. It has 24edges and 16vertices.
The prism is rolled along the rectangular faces and number on the bottom face (touching the ground) is noted. Let Xdenotes the number obtained on the bottom face and the following table gives the probability distribution of X.
On the above context, answer the following questions.
(i) Find the value of $p$.
(ii) Find the mean, $E ( x )$.
View full solution →The prism is rolled along the rectangular faces and number on the bottom face (touching the ground) is noted. Let Xdenotes the number obtained on the bottom face and the following table gives the probability distribution of X.
| X: | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| P(X): | p | 2p | 2p | p | 2p | $p^2$ | $2 p^2$ | $7 p^2+p$ |
(i) Find the value of $p$.
(ii) Find the mean, $E ( x )$.
A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 25. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 14. If the profit on a necklace is ₹ 100 and that on a bracelet is ₹ 300, formulate an L.P.P. for finding how many of each should be produced daily to maximize the profit? It is being given that at least one of each must be produced.
Mr Rohit invested ₹ 5000 in a fund at the beginning of year 2021 and by the end of year 2021 his investment was worth ₹ 9000. Next year market crashed and he lost ₹ 3000 and ending up with ₹ 6000 at the end of year 2022. Next year i.e. 2023 he gained ₹ 4500 and ending up with ₹ 10500 at the end of the year. Find CAGR (Compounded Annual Growth Rate) of his investment.
$\left(\operatorname{Use}(2.1)^{1 / 3}=1.2805\right)$
View full solution →$\left(\operatorname{Use}(2.1)^{1 / 3}=1.2805\right)$
A traffic engineer records the number of bicycle riders that use a particular cycle track. He records that an average of 3.2 bicycle riders use the cycle track every hour. Given that the number of bicycles that use the cycle track follow a Poisson distribution, what is the probability that 2 or less bicycle riders will use the cycle track within an hour? Also find the mean expectation and variance for the random variable.
(Given $e^{-3.2}=0.041$ )
View full solution →(Given $e^{-3.2}=0.041$ )
In 4 years, a mobile costing ₹ 36,000 will have a salvage value of ₹ 7200.
The following graph shows the depreciation of a mobile’s value over 4 years.
A new mobile at that time (i.e., after 4 years) is expected to cost for ₹ 55,200. In order to provide funds for the difference between the replacement cost and the salvage cost, a sinking fund is set up into which equal payments are placed at the end of each year. If the fund earns interest at the rate 7% compounded annually, how much should each payment be? Also find the amount of Annual Depreciation of the mobile’s value over 4 years and find the rate of depreciation (under straight line method).
Use (1.07)4=1.3107.
The following graph shows the depreciation of a mobile’s value over 4 years.
A new mobile at that time (i.e., after 4 years) is expected to cost for ₹ 55,200. In order to provide funds for the difference between the replacement cost and the salvage cost, a sinking fund is set up into which equal payments are placed at the end of each year. If the fund earns interest at the rate 7% compounded annually, how much should each payment be? Also find the amount of Annual Depreciation of the mobile’s value over 4 years and find the rate of depreciation (under straight line method).
Use (1.07)4=1.3107.
Supply and demand curves of a tyre manufacturer company is given below:
The above graph showing the demand and supply curves of a tyre manufacturer company which are linear. ‘ABC’ tyre manufacturer sold 25 units every month when the price of a tyre was ₹ 20000 per units and ‘ABC’ tyre manufacturer sold 125 units every month when the price dropped to ₹ 15000 per unit. When the price was ₹ 25000 per unit, 180 tyres were available per month for sale and when the price was only ₹ 15000 per unit, 80 tyres remained. Find the demand function. Also find the consumer surplus if the supply function is given to be 𝑺(𝒙) = 𝟏𝟎𝟎 𝒙 + 𝟕𝟎𝟎𝟎
The above graph showing the demand and supply curves of a tyre manufacturer company which are linear. ‘ABC’ tyre manufacturer sold 25 units every month when the price of a tyre was ₹ 20000 per units and ‘ABC’ tyre manufacturer sold 125 units every month when the price dropped to ₹ 15000 per unit. When the price was ₹ 25000 per unit, 180 tyres were available per month for sale and when the price was only ₹ 15000 per unit, 80 tyres remained. Find the demand function. Also find the consumer surplus if the supply function is given to be 𝑺(𝒙) = 𝟏𝟎𝟎 𝒙 + 𝟕𝟎𝟎𝟎
A toy rocket is fired, from a platform, vertically into the air, its height above the ground after $t$ seconds is given by $s(t)= a t ^2+ b t + c$, where $a , b , c \in R ; a \neq 0$ and $s(t)$ is measured in
metres. After $1 0$ second, the rocket is $1 6 ~ m$ above the ground; after $2 0$ seconds, $2 2 ~ m$; after 30 seconds, $2 5$ m.
(i) Write down a system of three linear equations in terms of $a , b$ and $c$.
(ii) Hence find the values of $a , b$ and $c$, using matrix method.
View full solution →metres. After $1 0$ second, the rocket is $1 6 ~ m$ above the ground; after $2 0$ seconds, $2 2 ~ m$; after 30 seconds, $2 5$ m.
(i) Write down a system of three linear equations in terms of $a , b$ and $c$.
(ii) Hence find the values of $a , b$ and $c$, using matrix method.
An owl was sitting at $( 0 , k ) ; k > 0$. Then it starts flying along the path whose equation is given by $y = a x ^ { 2 } + b x + c$, where $a \in R -\{0\}, b , c \in R$. It passes through the points $( 1 , 2),(2,1)$ and (4,5). Using Cramer's Rule, find the values of $a, b, c$ and hence $k$
View full solution →The quarterly profits of a small-scale industry (₹ in thousands) are as follows.
Calculate 4-quarterly moving averages.
| Year | Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 |
| 2020 | 39 | 47 | 20 | 56 |
| 2021 | 68 | 59 | 66 | 72 |
| 2022 | 88 | 60 | 60 | 67 |
A company has two factories located at P and Q and has three depots situated at A, B and C. The weekly requirement of the depots at A, B and C is respectively 5, 5 and 4 units, while the production capacity of the factories P and. Q are respectively 8 and 6 units. The cost (in ₹) of transportation per unit is given below.
Based on the above information, answer the following questions:
(i) Formulate the objective function and the constraints of the above Linear programming problem.
(ii) How many units should be transported from each factory to each depot in order that the transportation cost is minimum?
| Cost(in ₹) | |||
| TO/From | A | B | C |
| P | 160 | 100 | 150 |
| Q | 100 | 120 | 100 |
(i) Formulate the objective function and the constraints of the above Linear programming problem.
(ii) How many units should be transported from each factory to each depot in order that the transportation cost is minimum?
EQUATED MONTHLY INSTALMENTS (EMI): -
Each instalment can be considered as consisting of two parts:
(i) Interest on the outstanding loan
(ii) Repayment of part of the loan.
Methods of calculation of EMI or Instalment: -
EMI or Installment can be calculated by two methods:
1.Flat Rate Method
2. Reducing-balance method or Amortization of Loan
Rajesh purchased a house from a company for ₹2500000and made a down payment of ₹500000 He repays the balance in 25 years by monthly instalments at the rate of 9% per annum compounded monthly. $\left(\operatorname{Given}(1.0075)^{-300}=0.1062\right)$
Based on the above information, answer the following questions:
(i) Find the number of payments and find the rate of interest per month.
(ii) (a) What are the monthly payments of instalments using reducing balance method?
OR
(ii) (b) What are the monthly payments of instalments using flat rate method?
(iii) What is the total interest payment made in the process applied to calculate EMI in the above part 37 ( ii ) ?
A student Shivam is running on a playground along the curve given by y = x2+7. Another student Manita standing at point ( 3, 7 ) on playground wants to hit Shivam by paper ball when Shivam is nearest to Manita.
(i) Let at any instant while running along the curve y = x2+7 Shivam’s position be ( x, y ) Find the expression for the distance ( D ) between Shivam and Manita in terms of ' x '.
(ii) Find the critical point(s) of the distance function.
(iii) (a) What is the distance between Shivam and Manita when they are at least distance from each other.
OR
(iii) (b) Find the position of Shivam, when he is closest to Manita.
(i) Let at any instant while running along the curve y = x2+7 Shivam’s position be ( x, y ) Find the expression for the distance ( D ) between Shivam and Manita in terms of ' x '.
(ii) Find the critical point(s) of the distance function.
(iii) (a) What is the distance between Shivam and Manita when they are at least distance from each other.
OR
(iii) (b) Find the position of Shivam, when he is closest to Manita.
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