The volume of a spherical shell is given by
- A$\frac{4}{3} \pi\left(R^3-r^3\right)$
- B$\pi\left(R^3-r^3\right)$
- C$\frac{4}{3} \pi\left(R^2-r^2\right)$
- D$4 \pi\left(R^3-r^3\right)$
Question types
Model Paper 8 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
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 8 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $\left(3 x+\frac{1}{2}\right)\left(3 x-\frac{1}{2}\right)=9 x^2-p$ then the value of $p$ is
- ✓$\frac{1}{4}$
- B$-\frac{1}{4}$
- C$0$
- D$\frac{1}{2}$
Answer: A.
View full solution →In $\triangle A B C$ and $\triangle D E F$ its is given that $\angle B=\angle E$ and $\angle C=\angle F$ in order that $\triangle A B C \cong \triangle D E F$ we must have
- A$BC = EF$
- B$\angle A=\angle D$
- C$A B=D F$
- D$AC = DE$
The positive solutions of the equation $a x+b y+c=0$ always lie in the
- A3rd quadrant
- B4th quadrant
- C2nd quadrant
- D1st quadrant
Read the statements carefully.
Statement-1: The product of a rational and an irrational number is an irrational number.
Statement 2: Reciprocal of every rational number is a rational number.
Statement-1: The product of a rational and an irrational number is an irrational number.
Statement 2: Reciprocal of every rational number is a rational number.
- AStatement-1 is false but Statement-2 is true.
- BStatement-1 is true but Statement-2 is false.
- CBoth Statement-1 and Statement-2 are false.
- DBoth Statement-1 and Statement-2 are true.
Assertion (A): There are infinite number of lines which passes through $(2,14)$.
Reason (R): A linear equation in two variables has infinitely many solutions.
Reason (R): A linear equation in two variables has infinitely many solutions.
- 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.
- CA is true but R is false.
- DA is false but R is true.
Assertion $(A): $The perimeter of a right angled triangle is $60 \ cm$ and its hypotenuse is $26 \ cm .$ The other sides of the triangle are $10 \ cm$ and $24 \ cm .$ Also, area of the triangle is $120 \ cm^2$.
Reason $(R):(\text { Base })^2+(\text { Perpendicular })^2=(\text { Hypotenuse })^2$
Reason $(R):(\text { Base })^2+(\text { Perpendicular })^2=(\text { Hypotenuse })^2$
- ✓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 →Express the linear equation $y -2=0$ in the form $ax + by + c =0$ and indicate the value of $a , b$ and c in case.
View full solution →Write two solutions of the form $x=0, y=a$ and $x=b, y=0:-4 x+3 y=12$
View full solution →In figure, OA and OB are respectively perpendiculars to chords CD and EF of a circle whose centre is O . If $OA =$ OB, prove that $\overline{E C} \cong \overline{D F}$.
View full solution →If $O$ is the centre of the circle, find the value of $x$ in the given figure:
View full solution →If a line intersects two concentric circles $($circles with the same centre$)$ with centre $O$ at $A , B , C$ and $D ,$ prove that $A B=C D$.
View full solution →In fig. write the Co-ordinates of the points and if we join the points write the name of fig. formed. Also write Co-ordinate of intersection point of AC and BD .
In the given figure, the side $BC$ of $\triangle ABC$ has been produced to a point $D$ . If the bisectors of $\angle ABC$ and $\angle ACD$ meet at point $E$ then prove that $\angle B E C=\frac{1}{2} \angle B A C$.
View full solution →In given figure, it is given that $\text{AB = CF , EF = BD}$ and $\angle AFE =\angle CBD$. Prove that $\triangle AFE \cong \triangle CBD$.
View full solution →A cylinder, a cone and a sphere are of the same radius and same height. Find the ratio of their curved surface.
Find the cost of laying grass in a triangular field of sides $50 m, 65 m$ and $65 m$ at the rate of $Rs.7$ per $m ^2$.
View full solution →The lengths of 62 leaves of a plant are measured in millimetres and the data is represented in the following table:
Number of leaves
| Length (in mm) | Number of leaves |
| 118-126 | 8 |
| 127-135 | 10 |
| 136-144 | 12 |
| 144-153 | 17 |
| 154-162 | 7 |
| 163-171 | 5 |
| 172-180 | 3 |
In the given figure, $O P, O Q, O R$ and $O S$ are four rays. Prove that $\angle POQ +\angle ROQ +\angle SOR +\angle POS =360^{\circ}$.
View full solution →If is given that $\angle X Y Z=64^{\circ}$ and $X Y$ is produced to point $P.$ Draw a figure from the given information. If ray $YQ$ bisects $\angle ZYP$, find $\angle XYQ$ and reflex $\angle QYP$.
View full solution →In the adjoining figure, name:
i. Two pairs of intersecting lines and their corresponding points of intersection
ii. Three concurrent lines and their points of intersection
iii. Three rays
iv. Two line segments
i. Two pairs of intersecting lines and their corresponding points of intersection
ii. Three concurrent lines and their points of intersection
iii. Three rays
iv. Two line segments
Find the values of $a$ and $b$ if $\frac{7+3 \sqrt{5}}{3+\sqrt{5}}-\frac{7-3 \sqrt{5}}{3-\sqrt{5}}=a+b \sqrt{5}$.
View full solution →Read the following text carefully and answer the questions that follow:
There is a Diwali celebration in the $\text{DPS}$ school Janakpuri New Delhi. Girls are asked to prepare Rangoli in a triangular shape. They made a rangoli in the shape of triangle $\text{ABC}$ . Dimensions of $\triangle \text{ABC}$ are $26 \ cm, 28 \ cm, 25 \ cm$ .
$i.$ In fig $R$ and $Q$ are mid$-$points of $\text{AB}$ and $\text{AC}$ respectively. Find the length of $\text{RQ}$.
$ii.$ Find the length of Garland which is to be placed along the side of $\triangle \text{QPR}$.
$iii.$ R, $P$ and $Q$ are the mid$-$points of $\text{AB, BC}$, and $\text{AC}$ respectively. Then find the relation between area of $\triangle P Q R$ and area of $\triangle \text{ABC}$.
OR
$\text{R, P, Q}$ are the mid$-$points of corresponding sides $\text{AB, BC, CA}$ in $\triangle \text{ABC}$, then name the figure so obtained $\text{BPQR.}$
View full solution →There is a Diwali celebration in the $\text{DPS}$ school Janakpuri New Delhi. Girls are asked to prepare Rangoli in a triangular shape. They made a rangoli in the shape of triangle $\text{ABC}$ . Dimensions of $\triangle \text{ABC}$ are $26 \ cm, 28 \ cm, 25 \ cm$ .
$i.$ In fig $R$ and $Q$ are mid$-$points of $\text{AB}$ and $\text{AC}$ respectively. Find the length of $\text{RQ}$.
$ii.$ Find the length of Garland which is to be placed along the side of $\triangle \text{QPR}$.
$iii.$ R, $P$ and $Q$ are the mid$-$points of $\text{AB, BC}$, and $\text{AC}$ respectively. Then find the relation between area of $\triangle P Q R$ and area of $\triangle \text{ABC}$.
OR
$\text{R, P, Q}$ are the mid$-$points of corresponding sides $\text{AB, BC, CA}$ in $\triangle \text{ABC}$, then name the figure so obtained $\text{BPQR.}$
Read the following text carefully and answer the questions that follow:
Ajay lives in Delhi, The city of Ajay's father in laws residence is at Jaipur is $600 \ km$ from Delhi. Ajay used to travel this $600 \ km$ partly by train and partly by car.
He used to buy cheap items from Delhi and sale at Jaipur and also buying cheap items from Jaipur and sale at Delhi. Once From Delhi to Jaipur in forward journey he covered $2 x \ km$ by train and the rest $y \ km$ by taxi.But, while returning he did not get a reservation from Jaipur in the train. So first $2 y \ km$ he had to travel by taxi and the rest $x \ km$ by Train. From Delhi to Jaipur he took 8 hrs but in returning it took $10 hr$ .
$i$. Write the above information in terms of equation.
$ii$. Find the value of $x$ and $y$ ?
$iii$. Find the speed of Taxi?
OR
Find the speed of Train?
View full solution →Ajay lives in Delhi, The city of Ajay's father in laws residence is at Jaipur is $600 \ km$ from Delhi. Ajay used to travel this $600 \ km$ partly by train and partly by car.
He used to buy cheap items from Delhi and sale at Jaipur and also buying cheap items from Jaipur and sale at Delhi. Once From Delhi to Jaipur in forward journey he covered $2 x \ km$ by train and the rest $y \ km$ by taxi.But, while returning he did not get a reservation from Jaipur in the train. So first $2 y \ km$ he had to travel by taxi and the rest $x \ km$ by Train. From Delhi to Jaipur he took 8 hrs but in returning it took $10 hr$ .
$i$. Write the above information in terms of equation.
$ii$. Find the value of $x$ and $y$ ?
$iii$. Find the speed of Taxi?
OR
Find the speed of Train?
Read the following text carefully and answer the questions that follow:
In Agra in a grinding mill, there were installed $5$ types of mills. These mills used steel balls of radius $5 \ mm, 7\ mm , 10 \ mm, 14 \ mm$ and $16 \ mm$ respectively. All the balls were in the spherical shape.
For repairing purpose mills need $10 $ balls of $7\ mm$ radius and $20 $ balls of $3.5\ mm$ radius. The workshop was having $20000\ mm^3$ steel. This $20000 \ mm^3$ steel was melted and $10$ balls of $7\ mm$ radius and $20$ balls of $3.5\ mm$ radius were made and the remaining steel was stored for future use.
$i$. What was the volume of one ball of $3.5\ mm$ radius?
$ii$. What was the surface area of one ball of $3.5\ mm$ radius?
$iii$. What was the volume of $10$ balls of radius $7\ mm$?
OR
How much steel was kept for future use?
View full solution →In Agra in a grinding mill, there were installed $5$ types of mills. These mills used steel balls of radius $5 \ mm, 7\ mm , 10 \ mm, 14 \ mm$ and $16 \ mm$ respectively. All the balls were in the spherical shape.
For repairing purpose mills need $10 $ balls of $7\ mm$ radius and $20 $ balls of $3.5\ mm$ radius. The workshop was having $20000\ mm^3$ steel. This $20000 \ mm^3$ steel was melted and $10$ balls of $7\ mm$ radius and $20$ balls of $3.5\ mm$ radius were made and the remaining steel was stored for future use.
$i$. What was the volume of one ball of $3.5\ mm$ radius?
$ii$. What was the surface area of one ball of $3.5\ mm$ radius?
$iii$. What was the volume of $10$ balls of radius $7\ mm$?
OR
How much steel was kept for future use?
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