
Download JEE Main 2022 Maths 24 June Question Paper with Solutions PDF (Morning Session)
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Exam:
JEE Main
Academic Session:
2024
Medium:
English Medium
Subject:
Maths
Year:
2022
Date and Month:
24th- June
Shift:
Shift-1 Morning
Available Material:
JEE Main Previous Year Question Papers with PDF
Access Frank Solutions for Maths Class 9 Chapter 21 Areas Theorems on Parallelograms
1.
Ans:
(The area of a triangle is half of a parallelogram on the same base and between the same parallel lines)
\[\Rightarrow \operatorname{ar}(\triangle A P B)=\dfrac{1}{2} \times 60 \mathrm{~cm}^2 \\
\Rightarrow \operatorname{ar}(\triangle A P B)=30 \mathrm{~cm}^2\]
2.
Ans:
Since
\[S R=12 \mathrm{~cm} \\
P S=8 \mathrm{~cm} \\
\Rightarrow \operatorname{ar}(\triangle P R S)=\dfrac{1}{2} \times \text { base } x \text { height } \\
\Rightarrow \operatorname{ar}(\triangle P R S)=\dfrac{1}{2} \times S R \times P S \\
\Rightarrow \operatorname{ar}(\triangle P R S)=\dfrac{1}{2} \times 12 \times 8 \\
\Rightarrow \operatorname{ar}(\triangle P R S)=48 \mathrm{~cm}^2\]
3. In the figure,
(i) the area of
(ii) the area of the rectangle PQRS
(iii) the area of the triangle PQS.
Ans:
(i)
(The area of a triangle is half of a parallelogram on the same base and between the same parallel lines)
\[\Rightarrow \operatorname{ar}(\text { parallelogram QTSR })=2 \times 60 \mathrm{~cm}^2 \\
\Rightarrow \operatorname{ar}(\text { parallelogram QTSR })=120 \mathrm{~cm}^2 \\
\text { (ii) } \operatorname{ar}(\triangle Q T S)=\dfrac{1}{2} \times \operatorname{ar}(\text { parallelogram QTSR) } \\
\operatorname{ar}(\triangle Q T S)=\operatorname{ar}(\triangle R S Q)=60 \mathrm{~cm}^2\]
Now,
\[\operatorname{ar}(\triangle R S Q)=\dfrac{1}{2} \times \operatorname{ar}(\text { rectangle } P Q R S) \\
\Rightarrow \operatorname{ar}(\text { rectangle } P Q R S)=2 \times \operatorname{ar}(\triangle R S Q) \\
\Rightarrow \operatorname{ar}(\text { rectangle } P Q R S)=2 \times 60 \mathrm{~cm}^2 \\
\Rightarrow \operatorname{ar}(\text { rectangle } P Q R S)=120 \mathrm{~cm}^2\]
(iii) Since PQRS is a rectangle,
Therefore RS = PQ ......(i)
QTSR is a parallelogram,
Therefore, RS = QT ......(ii)
From (i) and (ii)
In
Therefore,
The area of two congruent triangles is equal.
Hence,
4. In the given figure area of
Ans: Area
( IIgm on same base PQ and between same parallel lines)
5. In the given figure,
Ans: Triangle PQT and parallelogram
Adding equations (i) and (ii), we get
Subtracting
6. In the given figure, ST
Ans: We have,
(Both the triangles on the same base PR and between the same
parallel lines
Adding
7. In the figure,
Ans: \[\Rightarrow \operatorname{ar}(\triangle A P D)=\dfrac{\sqrt{3} s^2}{4} \\
\Rightarrow \operatorname{ar}(\triangle A P D)=\dfrac{\sqrt{3} \times 8^2}{4} \\
\Rightarrow \operatorname{ar}(\triangle A P D)=\dfrac{\sqrt{3} \times 64}{4} \\
\Rightarrow \operatorname{ar}(\triangle A P D)=\sqrt{3} \times 16 \\
\Rightarrow 16 \sqrt{3} \mathrm{~cm}^2 \operatorname{ar}(\triangle A P D) \\
\Rightarrow \dfrac{1}{2} \times \operatorname{ar}(\text { parallelogram } A B C D)\]
(The area of a triangle is half of a parallelogram on the same base and between the same parallel lines)
\[\Rightarrow \operatorname{ar}(\text { parallelogram } A B C D)=2 \times \operatorname{ar}(\triangle A P D) \\
\Rightarrow \operatorname{ar}(\text { parallelogram } A B C D)=2 \times 16 \sqrt{3} \mathrm{~cm}^2 \\
\Rightarrow \operatorname{ar}(\text { parallelogram } A B C D)=32 \sqrt{3} \mathrm{~cm}^2\]
8. In the figure, if the area of
(i)
(ii)
(iii)
Ans: (i) Area of both rectangle and parallelogram on the same base is equal.
Here,
For rectangle
For parallelogram
So, we get, Area of rectangle
Area of rectangle
(ii)
\[\Rightarrow \operatorname{ar}(\triangle P Q S)=42 \mathrm{~cm}^2 \\
\text { (iii) } \operatorname{ar}(\triangle P Q N)=\dfrac{1}{2} \times \operatorname{ar}(\text { rectangle } P Q M N) \\
\Rightarrow \operatorname{ar}(\triangle P Q N)=\dfrac{1}{2} \times 84 \mathrm{~cm}^2 \\
\Rightarrow \operatorname{ar}(\triangle P Q N)=42 \mathrm{~cm}^2\]
9. In the figure,
Ans: In quadrilateral PQST,
In
but
From (i) and (ii)
10. In the given figure, if
Ans: Joining
Triangle
11. In the given figure, the perimeter of parallelogram PQRS is
Ans: Area of
Also,
Area of
\[\therefore P Q \times 6=P S \times 8 \\
\Rightarrow P Q=\dfrac{8 P S}{6} \\
\Rightarrow P Q=\dfrac{4 P S}{3} \ldots(i)\]
The perimeter of
Now,
12. In the given figure,
Ans: By joining TR, we get
Triangle
QT and RS.
From (i) and (ii), we get
13. In the given figure,
(a)
(b) Area of square
Ans: \[\angle B Q R=\angle B Q P+\angle P Q R \\
\Rightarrow \angle B Q R=90^{\circ}+\angle P Q R \\
\angle P Q T=\angle T Q R+\angle P Q R \\
\Rightarrow \angle P Q T=90^{\circ}+\angle P Q R \\
\Rightarrow \angle B Q R=\angle P Q T \ldots .(i) \\
\text { (a) In } \triangle Q R B \text { and } \triangle P Q T, \\
B Q=P Q \ldots .(\text { sides of a square } P A B Q \text { ) } \\
Q R=Q T \ldots \text { (sides of a square QRST) } \\
\angle B Q R=\angle P Q T \text { From (i) } \\
\therefore \triangle Q R B \cong \triangle P Q T \text { (by SAS congruence criterion) } \\
\Rightarrow A(\triangle B Q R)=\text { A( } \triangle P Q T) \ldots \text { (ii) }\]
(b) Triangle PQT and rectangle QTNM are on the same base QT and both are between the same parallel lines QT and PN.
Triangle
From (iii) and (iv),
14. In the figure,
Ans: In parallelogram
(The area of a triangle is half that of a parallelogram that lies on the same base
and between the same parallels)
In
but
From (i) and (ii)
15. The diagonals of a parallelogram
(a) Area of
(b) Area of
Ans:
(a) A diagonal divides a parallelogram into two triangles of equal areas.
In
Adding
Area of
(b)
In
Adding
16. Prove that the median of a triangle divides it into two triangles of equal area.
Ans:
Draw AL perpendicular to
Since
\[\Rightarrow B D=D C \\
\Rightarrow B D \times A L=D C \times A L(\text { multiplying by } A L) \\
\Rightarrow \dfrac{1}{2}(B D \times A L)=\dfrac{1}{2}(D C \times A L) \\
\Rightarrow \operatorname{ar}(\triangle A B D)=\operatorname{ar}(\triangle A D C)\]
17.
Ans:
Therefore, Area
Now PD is the median of
Therefore, Area
Subtracting equation (2) from equation (1), we get
18. In the given figure
Ans: \[\text { In } \triangle A B C \\
A F=F B \text { and } E F \| B C \ldots \text { (given) } \\
\therefore A E=E C \ldots \text { (Converse of Midpoint theorem) ...(i) } \\
\text { In } \triangle A E F \text { and } \triangle C E D, \\
\angle F E A=\angle D E C \ldots \text { (Vertically opposite angles) } \\
C E=A E \text {...From (i) } \\
\angle F A E=\angle D C E \ldots \text { (Alternate angles) } \\
\therefore \triangle F A E \cong \triangle C E D \ldots(\text { ASA test of congruency) } \\
\Rightarrow A r(\triangle A E F)=A r(\triangle C E D) \ldots . \text { (ii) } \\
A r(\triangle A B C)=A r(\triangle A E F)+A r(E F B C) \\
=A r(\triangle C E D)+A r(E F B C) \ldots . \text { (From (ii)) } \\
\therefore A(\triangle A B C)=A\left(\|^{g m} D C B F\right) \\
\Rightarrow A\left(\|^{g m} D C B F\right)=30 \text { sq. units. }\]
19. Prove that the diagonals of a parallelogram divide it into four triangles of equal area.
Ans:
The diagonals of the parallelogram bisect each other.
So,
Therefore, it will divide two triangles into equal areas.
In
From (i), (ii), and (iii)
Hence, the diagonals of a parallelogram divide it into four triangles of equal areas.
20. The diagonals
Ans:
In
In
Adding (i) and (ii)
Therefore,
21. Prove that the area of a rhombus is equal to half the rectangle contained by its diagonals.
Ans:
The diagonals of a rhombus intersect each other at right angles,
Therefore,
Now,
Therefore, the area of a rhombus is equal to half the rectangle contained by its diagonals.
22. PQRS is a parallelogram and
Ans:
Let us draw a line
In parallelogram
From equations (1) and (2), we obtain
Therefore, quadrilateral PQLK is a parallelogram.
It is clear that
Similarly, for
Area
Adding equations (3) and (4), we obtain
Area
Area
Let us draw a line segment
In parallelogram PQRS,
MN || PS (By construction) ... (6)
PQRS is a parallelogram.
From equations (6) and (7), we obtain
MN || PS and PN || SN
Therefore, quadrilateral PNMS is a parallelogram.
It can be clear that
Similarly, for
Area
Adding equations (8) and (9), we obtain
By comparing (5) and (10), we obtain
23. A quadrilateral
Ans:
Join
Thus
Therefore, their corresponding altitudes
Now, in
Therefore,
24. In the given figure,
(a) If the area of
(b) If the area of
Ans: (a) Triangles on the same base and between the same parallels are equal in area.
Now,
Given,
(b)
Given,
25. In the given figure,
quadrilateral
Ans: Let
Now,
Similarly,
Quad rilaterals ROSD, APOS and POQB are parallelograms.
parallel lines.
Similarly,
Adding equations (i), (ii), (iii) and (iv), we get
26.
Ans:
Join PS. Suppose PS and QR intersect at O. Draw PM and SN both perpendicular to
Thus
Therefore, their corresponding altitudes
Now, in
Therefore,
27. If the medians of a
Ans:
Property: The median of a triangle divides it into two triangles of equal areas.
In
In
Subtracting (ii) from (i),
Similarly,
From (iii) and (iv),
But
Therefore,
Hence,
28. In
Ans:
Therefore,
Also,
From (i) and (ii)
Therefore,
Hence,
triangles of equal areas)
Similarly, QCRP and QRAP are
From (v) and (vi)
Now,
29. In the given figure,
Ans: \[S M \| P N \\
\Rightarrow S M \| T N \\
\text { Also, SR | MN } \\
\Rightarrow S T \| M N\]
Hence, SMNT is a parallelogram.
\[S M \| P N \\
\Rightarrow S M \| P O \\
\text { Also, PS | QM } \\
\Rightarrow P S \| O M\]
Hence, SMOP is a parallelogram.
Now, parallelograms SMNT and SMOP are on the same base SM and both are between the same parallel lines SM and PN.
Similarly, we can prove that quadrilaterals
Now, parallelograms PQRS and SMOP are on the same base PS, and both are between the same parallel lines
From (i) and (ii), we have
31. In
Ans:
Area
Since
Therefore, area
Substituting in (i)
Area
32. In the figure,
Ans: The diagonals of a parallelogram divide it into four triangles of equal area Therefore, the area of
In
Therefore, area
In Parallelogram BPCD, BC is the diagonal
Therefore, area
From (iii) and (ii)
area
adding (i) and (iv)
Area of
33. The medians
Ans:
Join two points MN.
Since the line segment joining the mid-points of two sides of a triangle is parallel to the third side.
Therefore,
Clearly,
Therefore,
We know that a median of a triangle divides it into two triangles of equal areas.
Therefore, area
(i))
34.
a) In the given figure,
If
b) In the given figure,
If
Area of
Ans: (a)
so it will divide
Similarly, ED is the median of
Subtracting equation (ii) from (i), we have
(b)
So, it will divide
Similarly, ED is the median of
Subtracting equation (ii) from (i), we have
Since
Now,
From (i), (ii), (iii) and (iv), we get
Now,
35. In a parallelogram
Ans:
In a parallelogram PQRS,
So, it bisects the parallelogram.
Again,
36. In a parallelogram
Ans:
Construction: First, Join QR.
Let the diagonals
Since diagonals of a parallelogram bisect each other and
Now, medians PR and QS of a triangle divide it into two triangles of equal area.
In
Similarly, OT is the median of
Subtracting equation (ii) from (i), we have
37.
Prove that: area
Ans:
Since the diagonals of a parallelogram bisect each other at the point of intersection.
Therefore,
In
Therefore, area
In
Therefore, area
Adding (i) and (ii)
area
JEE Main 2022 24 June morning shift Maths Question Paper Analysis
Mentioned below is the summarised analysis of the mathematics question paper.
The mathematics section was deemed moderately challenging.
The mathematics section was comparatively more difficult than physics and chemistry.
The questions had lengthy calculations.
Calculus and algebra were the important topics that had the most weightage in the question paper.
Apart from it vector and induction was given importance.
Binomial Theorem, Permutations and Combinations, Coordinate Geometry were some of the other important topics covered in the examination.
Regular practice of JEE Main 2022 mock test and JEE Main previous year question paper were the most important aspects of preparation which helped students to perform well in the examination.
Maths is the most scoring part of JEE Main exams so solve and practice the JEE Main 2022 Maths question paper 24 June Morning shift to get an added advantage in the exam. By going through the exam, you will be able to understand the types of questions asked and the level of difficulty of the paper. So download our JEE Main 24 June morning 2022 Maths question paper with solutions to be completely exam ready! The solved JEE Main 2022 Maths Question Paper for the other days of exam are also provided on Vedantu.
For more information on JEE Main 2022 Maths, visit our website and check below-given the JEE Main 2022 Maths question paper FAQs.
FAQs on JEE Main 2022 June 24th Shift 1 Maths Question Paper with Answer Keys and Solutions
1. In how many sessions will the JEE Main 2022 exam be held?
The National Testing Agency (NTA) has scheduled to conduct the JEE Main 2022 exams in two sessions this year.
Session 1: 23, 24, 25, 26, 27, 28, and 29 June 2022
Session 2: 21, 22, 23, 24, 25, 26, 27, 28, 29 and 30 July 2022
2. How should I prepare for JEE Main 2022 Maths exam?
Maths is the most scoring section in the JEE Main exam and you can ace it by following the right preparation strategy and study resources, the plethora of which you can find on Vedantu. The main technique is to analyse your weaknesses and strengths and prepare accordingly. You can do so with our JEE Main Maths solutions 2022 24 June shift 1 and solving JEE Main Previous Year Question Papers.
3. What are some of the most important topics covered in JEE Main 2022 24 June morning shift Maths question paper?
Some of the most important topics that were covered in the JEE main mathematics question paper of 2022 included the following:
Calculus
Coordinate Geometry
Algebra
Vector
3D Geometry
Calculus
Vectors
4. What was the overall difficulty level of the JEE main 2022 maths question paper for the 24 June morning shift?
According to the question paper analysis the paper can be considered moderately difficult. The mathematics section was deemed to be the most challenging as compared to physics and chemistry. The paper constituted lengthy calculative questions, the weightage was primarily given to calculus, algebra and vector. The students who regularly practised the mock paper and questions would have performed well in the examination.

















