The arithmetic mean of 1, 2, 3, 4, ..., n is:
- A$\frac{n-1}{2}$
- B$\frac{n(n+1)}{2}$
- C$\frac{n}{2}$
- ✓$\frac{n+1}{2}$
Answer: D.
View full solution →Question types
MODEL PAPER 3 (STANDARD) question types
44 questions across 6 question groups — pick any mix to generate a Maths paper with step-by-step answer keys.
44
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
8 Q→055 Marks Questions
6 Q→06Case study (4 Marks)
3 Q→Sample Questions
MODEL PAPER 3 (STANDARD) questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
An unbiased die is thrown once. The probability of getting a composite number is
- A$\frac{2}{5}$
- ✓$\frac{1}{3}$
- C$\frac{2}{3}$
- D$\frac{1}{2}$
Answer: B.
View full solution →If $P(E) = 0.05,$ what will be the probability of 'not $E\ '$?
- A$0.55$
- B$0.59$
- ✓$0.95$
- D$0.095$
Answer: C.
View full solution →The length of an arc of a sector of angle $\theta^{\circ}$ of a circle with radius R is
- A$\frac{\pi R^2 \theta}{180}$
- B$\frac{\pi R^2 \theta}{360}$
- ✓$\frac{2 \pi R \theta}{360}$
- D$\frac{2 \pi R \theta}{180}$
Answer: C.
View full solution →The length of the minute hand of a clock is $21 \ cm$. The area swept by the minute hand in $10$ minutes is
- A$252 \ cm^2$
- B$126 \ cm^2$
- ✓$231 \ cm^2$
- D$210 \ cm^2$
Answer: C.
View full solution →Assertion (A): $\sqrt{3}, 2 \sqrt{3}, 3 \sqrt{3}, 4 \sqrt{3}$ this series forms an A.P.
Reason (R): Since common difference is same and equal to $\sqrt{3}$ therefore given series is an AP.
Reason (R): Since common difference is same and equal to $\sqrt{3}$ therefore given series is an AP.
- ✓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.
- CA is true but R is false.
- DA is false but R is true.
Answer: A.
View full solution →Assrtion (A): A toy is in the form of a cone mounted on a hemisphere with the same radius. The radius of the conical portion is 4 cm and its height is 3 cm . the surface area of the toy is $163.28 cm^2$. [Take $\pi=3.14$ ]
Reason (R): Volume of hemisphere is $\frac{2}{3} \pi r^2$
Reason (R): Volume of hemisphere is $\frac{2}{3} \pi r^2$
- 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 →A chord of a circle of radius $10 \ cm$ subtends a right angle at the centre. Find the area of the corresponding:
$i$. minor segment
$ii.$ major sector.
View full solution →$i$. minor segment
$ii.$ major sector.
Prove the trigonometric identity: $\tan ^2 A \sec ^2 B-\sec ^2 A \tan ^2 B=\tan ^2 A-\tan ^2 B$
View full solution →Find the difference of the areas of a sector of angle $120^{\circ}$ and its corresponding major sector of a circle of radius $21 \ cm$ .
View full solution →If $A=B=60^{\circ}$, verify that $\sin (A-B)=\sin A \cos B-\cos A \sin B$
View full solution →A circle is inscribed in a $\triangle A B C$, touching $BC , CA$ and $AB$ at $P , Q$ and $R$ respectively, as shown in the given figure. If $AB =10 \ cm, AQ =7 \ cm$ and $CQ =5 \ cm$ then find the length of $BC$ .
View full solution →Solve algebraically the following pair of linear equations for $x$ and $y\ 31 x+29 y=33 , 29 x+31 y=27$
View full solution →$PQ$ is a chord of length $4.8 \ cm$ of a circle of radius $3 \ cm$. The tangents at $P$ and $Q$ intersect at a point $T$ as shown in the figure. Find the length of $TP$.
View full solution →Compute the median for the following cumulative frequency distribution:
| Less than 20 | Less than 30 | Less than 40 | Less than 50 | Less than 60 | Less than 70 | Less than 80 | Less than 90 | Less than 100 |
| 0 | 4 | 16 | 30 | 46 | 66 | 82 | 92 | 100 |
Prove that: $\frac{1}{\operatorname{cosec} A-\cot A}-\frac{1}{\sin A}=\frac{1}{\sin A}-\frac{1}{\operatorname{cosec} A+\cot A}$.
View full solution →In a given figure $, AB$ is a chord of length $8 \ cm$ of a circle of radius $5 \ cm$. The tangents to the circle at $A$ and $B$ intersect at $P$. Find the length of $AP$.
View full solution →From a cubical piece of wood of side $21 \ cm ,$ a hemisphere is carved out in such a way that the diameter of the hemisphere is equal to the side of the cubical piece. Find the surface area and volume of the remaining piece.
View full solution →If $x =-4$ is a root of the equation $x^2+2 x+4 p=0$, find the values of k for which the equation $x ^2+ px (1+3 k )+$ $7(3+2 k)=0$ has equal roots.
View full solution →Find the mean and the median of the following data:
View full solution →| Marks | Number of Students |
| $0 - 10$ | $3$ |
| $10 - 20$ | $3$ |
| $20 - 30$ | $16$ |
| $30 - 40$ | $12$ |
| $40 - 50$ | $13$ |
| $50 - 60$ | $20$ |
| $60 - 70$ | $6$ |
| $70 - 80$ | $5$ |
A tent is in the form of a right circular cylinder surmounted by a cone. The diameter of the base of the cylinder or the cone is $24 \ m$ . The height of the cylinder is $11\ m$ . If the vertex of the cone is $16\ m$ above the ground, find the area of the canvas required for making the tent. $($Use $\pi=\frac{22}{7} )$
View full solution →In a trapezium $\text{ABCD}, A B \| D C$ and $D C=2 A B$. $E F \| A B$, where $E$ and $F$ lie on $B C$ and $A D$ respectively such that $\frac{B E}{E C}=\frac{4}{3}$. Diagonal $DB$ intersects $EF$ at $G$ . Prove that, $7 EF =11 AB$.
View full solution →Read the text carefully and answer the questions:
An observer on the top of a $40m$ tall light house $($including height of the observer$)$ observes a ship at an angle of depression $30^{\circ}$ coming towards the base of the light house along straight line joining the ship and the base of the light house.
The angle of depression of ship changes to $45^{\circ}$ after 6 seconds.
$(a)$ Find the distance of ship from the base of the light house after $6$ seconds from the initial position when angle of depression is $45^{\circ}$.
$(b)$ Find the distance between two positions of ship after $6$ seconds?
OR
Find the distance of ship from the base of the light house when angle of depression is $30^{\circ}$.
$(c)$ Find the speed of the ship?
View full solution →An observer on the top of a $40m$ tall light house $($including height of the observer$)$ observes a ship at an angle of depression $30^{\circ}$ coming towards the base of the light house along straight line joining the ship and the base of the light house.
The angle of depression of ship changes to $45^{\circ}$ after 6 seconds.
$(a)$ Find the distance of ship from the base of the light house after $6$ seconds from the initial position when angle of depression is $45^{\circ}$.
$(b)$ Find the distance between two positions of ship after $6$ seconds?
OR
Find the distance of ship from the base of the light house when angle of depression is $30^{\circ}$.
$(c)$ Find the speed of the ship?
Read the text carefully and answer the questions:
Karan went to the Lab near to his home for $\text{COVID} \ 19$ test along with his family members.
The seats in the waiting area were as per the norms of distancing during this pandemic $($as shown in the figure$)$.
His family member took their seats surrounded by red circular area.
$(a)$ What is the distance between Neena and Karan?
$(b)$ What are the coordinates of seat of Akash?
OR
Find distance between Binu and Karan.
$(c)$ What will be the coordinates of a point exactly between Akash and Binu where a person can be?
View full solution →Karan went to the Lab near to his home for $\text{COVID} \ 19$ test along with his family members.
The seats in the waiting area were as per the norms of distancing during this pandemic $($as shown in the figure$)$.
His family member took their seats surrounded by red circular area.
$(a)$ What is the distance between Neena and Karan?
$(b)$ What are the coordinates of seat of Akash?
OR
Find distance between Binu and Karan.
$(c)$ What will be the coordinates of a point exactly between Akash and Binu where a person can be?
Read the text carefully and answer the questions:
Kamla and her husband were working in a factory in Seelampur, New Delhi. During the pandemic, they were asked to leave the job.As they have very limited resources to survive in a metro city, they decided to go back to their hometown in Himachal Pradesh.After a few months of struggle, they thought to grow roses in their fields and sell them to local vendors as roses have been always in demand.Their business started growing up and they hired many workers to manage their garden and do packaging of the flowers.
In their garden bed, there are $23$ rose plants in the first row $, 21$ are in the $2^{\text {nd }}, 19$ in $3^{\text {rd }}$ row and so on. There are $5$ plants in the last row.
$(a)$ How many rows are there of rose plants?
$(b)$ Also, find the total number of rose plants in the garden.
OR
If total number of plants are $80$ in the garden, then find number of rows?
$(c)$ How many plants are there in 6th row.
View full solution →Kamla and her husband were working in a factory in Seelampur, New Delhi. During the pandemic, they were asked to leave the job.As they have very limited resources to survive in a metro city, they decided to go back to their hometown in Himachal Pradesh.After a few months of struggle, they thought to grow roses in their fields and sell them to local vendors as roses have been always in demand.Their business started growing up and they hired many workers to manage their garden and do packaging of the flowers.
In their garden bed, there are $23$ rose plants in the first row $, 21$ are in the $2^{\text {nd }}, 19$ in $3^{\text {rd }}$ row and so on. There are $5$ plants in the last row.
$(a)$ How many rows are there of rose plants?
$(b)$ Also, find the total number of rose plants in the garden.
OR
If total number of plants are $80$ in the garden, then find number of rows?
$(c)$ How many plants are there in 6th row.
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