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JEE Main 2022 June 24th Shift 1 Maths Question Paper with Answer Keys and Solutions

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Download JEE Main 2022 Maths 24 June Question Paper with Solutions PDF (Morning Session)

Category:

JEE Main Question Papers

Content-Type:

Text, Images, Videos and PDF

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. ABCD is a parallelogram having an area of 60 cm2.P is a point on CD. Calculate the area of APB.

Ans:

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ar(APB)=12×ar( parallelogram ABCD)

(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. PQRS is a rectangle in which PQ=12 cm and PS=8 cm. Calculate the area of PRS.

Ans:

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Since PQRS is a rectangle, therefore PQ=SR.

\[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, PT is parallel to SR. QTSR is a parallelogram and PQSR is a rectangle. If the area of QTS is 60 cm2, find:

(i) the area of gm QTSR

(ii) the area of the rectangle PQRS

(iii) the area of the triangle PQS.

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Ans:

(i) ar(QTS)=12×ar( parallelogram QTSR )

(The area of a triangle is half of a parallelogram on the same base and between the same parallel lines)

ar (parallelogram QTSR) =2×ar( QTS )

\[\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)

PQ=QT....(iii)

In PSQ and QST

QS=QS

PQ=QT (from (iii))

PQS=SQT=90

Therefore, PSQQST

The area of two congruent triangles is equal.

Hence, ar(PSQ)=ar(QTS)=60 cm2


4. In the given figure area of gmPQRS is 30 cm2. Find the height of gmPQFE if PQ=6 cm.

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Ans: Area (gm PQRS )= Area ( gm PQFE )

( IIgm on same base PQ and between same parallel lines)

Area (gmPQFE)=30 cm2

Base × Height =30

6× Height =30

 Height =306=5 cm

The height of a parallelogram PQFE is 5 cm.


5. In the given figure, PQRS is a gm. A straight line through P cuts SR at point T and QR produced at N. Prove that area of triangle QTR is equal to the area of triangle STN.

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Ans: Triangle PQT and parallelogram PQRS are on the same base PQ and are between the same parallel lines PQ and SR

Area (PQT)=12 Area ( parallelogram PQRS) (i)

PSN and parallelogram PQRS are on the same base PS and between the same parallel lines PS and QN.

Area(PSN)=12Area( parallelogram PQRS).....(ii)

Adding equations (i) and (ii), we get

Area (PQT)+Area(PSN)=Area( parallelogram PQRS)

Area (quad. PSNQ)Area(QTN)=Area( parallelogram PQRS)

Area( quad. PSNQ )Area(QTN)=Area( quad. PSNQ )Area(SRN)

Area(QTN)=Area(SRN)

Subtracting Area(RTN) from both the sides, we get

Area(QTN)Area(ΔRTN)=Area(ΔSRN)Area(ΔRTN)

Area(QTR)=Area(ΔSTN)


6. In the given figure, ST PR. Prove that: area of quadrilateral PQRS= area of PQT.

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Ans: We have,

Area(PSR)=Area(PTR)

(Both the triangles on the same base PR and between the same

parallel lines PR and ST )

Adding Area(PQR) on both sides, we get

Area(PSR)+Area(PQR)=Area(PTR)+Area(PQR)

Area( Quadrilateral PQRS)=Area(PQT)


7. In the figure, ABCD is a parallelogram, and APD is an equilateral triangle of side 80 cm, Calculate the area of the parallelogram ABCD.

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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 gmPQRS is 84 cm2; find the area of

(i) gmPQMN

(ii) PQS

(iii) PQN

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Ans: (i) Area of both rectangle and parallelogram on the same base is equal.

Here,

For rectangle PQMN, base =PQ

For parallelogram PQRS, base =PQ

So, we get, Area of rectangle PQMN= Area of parallelogram PQRS

Area of rectangle PQMN=84 cm2

(ii) ar(PQS)=12×ar( parallelogram PQRS)

ar(PQS)=12×84 cm2

\[\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, PQR is a straight line. SQ is parallel to Tp. Prove that the quadrilateral PQST is equal in area to the PSR.

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Ans: In quadrilateral PQST,

ar(PQS)=12×ar( quadrilateral PQST)

ar( quadrilateral PQST)=2ar(PQS)(i)

In PSR,

ar(PSR)=ar(PQS)+ar(QSR)

but ar(PQS)=ar(QSR)( since QS is median as QSTP)

ar(PSR)=2ar(PQS).(ii)

From (i) and (ii)

ar( quadrilateral PQST)=ar(PSR)


10. In the given figure, if ABDCFG and AE is a straight line. Also,

ADFC. Prove that: area of gmABCD= area of gm BFGE.

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Ans: Joining AC and FE, we get

Triangle AFC and Triangle AFE are on the same base AF and both are between the same parallels AF and CE.

Area(AFC)=Area(AFE)

Area(ABF)+Area(ABC)=Area(ABF)+Area(BFE)

Area(ABC)=Area(BFE)

12Area( parallelogram ABCD)=12 Area (parallelogram BFGE)

A( parallelogram ABCD)= Area (parallelogram BFGE)


11. In the given figure, the perimeter of parallelogram PQRS is 42 cm. Find the lengths of PQ and PS.

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Ans: Area of gmPQRS=PQ×6

Also,

Area of gmPQRS=PS×8

\[\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 gmPQRS=PQ+QR+RS+PS

42=2PQ+2PS (opposite sides of a parallelogram are equal)

21=PQ+PS

4PS3+PS=21 from (i)

4PS+3PS3=21

7PS =63

PS=9 cm

Now,

PQ=4PS3=4×93=12 cm

PQ=12 cm and PS=9 cm.


12. In the given figure, PTQR and QTRS. Show that: area of PQR= area of TQS.

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Ans: By joining TR, we get

Triangle PQR and Triangle QTR are on the same base QR and between the same parallel lines QR and PT.

Ar(PQR)=Ar(QTR) (i)

QTR and TQS are on the same base QT and between the same parallel lines

QT and RS.

Ar(QTR)=Ar(TQS)(ii)

From (i) and (ii), we get

Ar(PQR)=Ar(TQS)


13. In the given figure, PQR is right-angled at P.PABQ and QRST are squares on the side PQ and hypotenuse QR. If PNTS, show that:

(a) QRBPQT

(b) Area of square PABQ= area of rectangle QTNM.

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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.

A(PQT)=12A( rect. QTNM )

A( rect. QTNM )=2×A(PQT)

A( rect. QTNM )=2×ar(BQR) From (ii) ... (iii)

Triangle BQR and square PABQ are on the same base BQ and both are between the same parallel lines BQ and AR.2×A(BQR)=A (sq. PABQ) .... (iv)

From (iii) and (iv),

A( sq. PABQ)=A( rect. QTNM )


14. In the figure, AB=BE. Prove that the area of triangle ACE is equal in area to the parallelogram ABCD.

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Ans: In parallelogram ABCD,

ar(ABC)=12×ar( parallelogram ABCD)

(The area of a triangle is half that of a parallelogram that lies on the same base

and between the same parallels)

ar( parallelogram ABCD)=2ar(ABC)(i)

In ACE,

ar(ACE)=ar(ABC)+ar(BCE)

but ar(ABC)=ar(BCE) (since BC is median)

ar(ACE)=2ar(ABC).........(ii)

From (i) and (ii)

ar( parallelogram ABCD)=ar(ACE)


15. The diagonals of a parallelogram ABCD intersect at O. A line through O meets AB in P and CD in Q. Show that

(a) Area of APQD=12 area of gmABCD

(b) Area of APQD= Area of BPQC

Ans:

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(a) A diagonal divides a parallelogram into two triangles of equal areas.

A(ADB)=12A(gmABCD) (i)

In DOQ and BOP,

DOQ=BOP (Vertically opposite)

DO=BO (Diagonals of a gm )

ODQ=OBP (Alternate angles)

DOQBOP (ASA test of congruency)

Ar(DOQ)=Ar(BOP)

Adding Ar(DOPA) on both sides, we get

Ar(DOQ)+Ar(DOPA)=Ar(BOP)+Ar(DOPA)

Area of APQD=Ar(ADB)

Area of APQD=12Ar(gmABCD)( From (i)) (ii)

(b) Ar(ABC)=12Ar(gmABCD)

In COQ and AOP,

COQ=AOP (Vertically opposite angles)

CO=AO (Diagonals of a gm bisect

OCQ=OAP (Alternate angles)

COQAOP (ASA test of congruency)

Ar(COQ)=Ar(AOP)

Adding Ar(COPB) on both sides, we get

Ar(COQ)+Ar(COPB)=Ar(AOP)+Ar(COPB)

Area of BPQC=A(ABC)

Area of BPQC=12A(gmABCD)... (iii)

Area of APQD= Area of BPQC From (ii) and (iii)


16. Prove that the median of a triangle divides it into two triangles of equal area.

Ans:

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Draw AL perpendicular to BC.

Since AD is the median of Triangle ABC. Therefore, D is the mid-point line of BC.

\[\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. AD is a median of a ABC.P is any point on AD. Show that the area of ABP is equal to the area of ACP.

Ans:

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AD is the median of ABC. So, it will divide Triangle ABC into two triangles of equal areas.

Therefore, Area (ABD)=area(ACD) (1)

Now PD is the median of PBC.

Therefore, Area (PBD)= area (PCD)...(2)

Subtracting equation (2) from equation (1), we get

Area(ABD)area(PBD)=Area(ACD)Area(PCD)

Area(ABP)=area(ACP)


18. In the given figure AF=BF and DCBF is a parallelogram. If the area of ABC is 30 square units, find the area of the parallelogram DCBF.

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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:

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The diagonals of the parallelogram bisect each other.

So, O is the mid-point of lines AC and BD.

BO is the median in ABC.

Therefore, it will divide two triangles into equal areas.

ar(AOB)=ar(BOC)..(i)

In BCD,CO is the median.

From (i), (ii), and (iii)

ar(AOB)=ar(BOC)=ar(COD)=ar(AOD)

Hence, the diagonals of a parallelogram divide it into four triangles of equal areas.


20. The diagonals AC and BC of a quadrilateral ABCD intersect at O. Prove that if BO=OD, then areas of ABC and ADC are equal.

Ans:

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In ABD,

BO=OD

O is the mid-point of BD

AO is a median.

ar(AOB)=ar(AOD)(i)

In CBD,O is the mid-point of BD

CO is a median.

ar(COB)=ar(COD)...(ii)

Adding (i) and (ii)

ar(AOB)+ar(COB)=ar(AOD)+ar(COD)

Therefore, ar(ABC)=ar(ADC)


21. Prove that the area of a rhombus is equal to half the rectangle contained by its diagonals.

Ans:

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The diagonals of a rhombus intersect each other at right angles,

Therefore, OBAC and ODAC

Now, ar( rhombus ABCD)=ar(ABC)+ar(ADC)

=12(AC×BO)+12(AC×DO)

=12{AC×(BO+DO)}

=12(AC×BD)

Therefore, the area of a rhombus is equal to half the rectangle contained by its diagonals.


22. PQRS is a parallelogram and O is any point in its interior. Prove that: area( POQ)+area(ROS)area(QOR)+area(SOP)=12area(gmPPRS)

Ans:

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Let us draw a line KL which passes through point O and parallel to line PQ.

In parallelogram PQRS,

PQKL (By construction) ... (1)

PQRS is a parallelogram.

PSQR (Opposite sides of a parallelogram)

PKQL...(2)

From equations (1) and (2), we obtain

PQKL and PKQL

Therefore, quadrilateral PQLK is a parallelogram.

It is clear that POQ and parallelogram PQLK are lying on the same base PQ and between the same parallel lines PK and QL.

Area (POQ)=12 Area (parallelogram PQLK) ... (3)

Similarly, for ROS and parallelogram KLRS,

Area (ROS)=12 Area (parallelogram KLRS) ... (4)

Adding equations (3) and (4), we obtain

Area (POQ)+Area(ROS)=12Area( parallelogram PQLK)+

12 Area (parallelogram KLRS)

Area (POQ)+ Area (ROS)=12 Area (PQRS) ......(5)

Let us draw a line segment MN which is passing through point OP and is parallel to line segment PS.

In parallelogram PQRS,

MN || PS (By construction) ... (6)

PQRS is a parallelogram.

PQRS (Opposite sides of a parallelogram)

PN SN (7)

From equations (6) and (7), we obtain

MN || PS and PN || SN

Therefore, quadrilateral PNMS is a parallelogram.

It can be clear that POS and parallelogram PNMS are lying on the same base PS and between the same parallel lines PS and MN.

Area (SOP)=12 Area (PNMS) ... (8)

Similarly, for QOR and parallelogram MNQR,

Area (QOR)=12 Area (MNQR)(9)

Adding equations (8) and (9), we obtain

Area(SOP)+ Area (QOR)=12 Area (PNMS)+12 Area (MNQR)

Area(SOP)+Area(QOR)=12 Area (PQRS)

By comparing (5) and (10), we obtain

Area(POQ)+Area(ROS)=Area(SOP)+Area(QOR)= 12Area( || gmPQRS)


23. A quadrilateral ABCD is such that diagonals BD divide its area into two equal parts. Prove that BD bisects AC.

Ans:

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Join AC. Suppose AC and BD intersect at O. Draw lines AL and CM both perpendicular to BD.

ar(ABD)=ar(BDC)

Thus ABD and ABC are on the same base AB and have equal areas.

Therefore, their corresponding altitudes AL=CM are equal.

Now, in ALO and CMO,

1=2 (vertically opposite angles)

ALO=CMO (right angles)

AL=CM

Therefore, ALOCMO (AAS axiom)

AO=OC

BD bisects AC


24. In the given figure, BCDE.

(a) If the area of ADC is 20sq. units, find the area of AEB.

(b) If the area of BFD is 8 square units, find the area of CEF

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Ans: (a) Triangles on the same base and between the same parallels are equal in area. Ar(DBC)=Ar(ECB).... (i)

Now,

Ar(ABC)=Ar(ADC)+Ar(DBC)=Ar(AEB)+Ar(ECB)

Ar(ADC)+Ar(DBC)=Ar(AEB)+Ar(ECB)

Ar(ADC)=Ar(AEB).... (ii) from (i)

Given, Ar(ADC)=20 sq. units

Ar(AEB)=20 sq. units

(b)

Ar(ADC)=Ar(AEB) From (ii)

Ar(ADC)Ar(ADFE)=Ar(AEB)Ar(ADFE)

Ar(CEF)=Ar(BFD)

Given, Ar(BFD)=8 sq. units

Ar(CEF)=8sq. units


25. In the given figure, ABSQDC and ADPRBC. If the area of

quadrilateral ABCD is 24 square units, find the area of quadrilateral PQRS.

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Ans: Let SQ and PR intersect at point O.

Now,

DCSQ and RPBC

RCOQ and ROQC

Quadrilateral ROQC is a parallelogram.

Similarly,

Quad rilaterals ROSD, APOS and POQB are parallelograms.

ROQ and parallelogram ROQC are on the same base and between the same

parallel lines.

A(ROQ)=12×A(gmROQC)( i)

Similarly,

A(POQ)=12×A(gmPOQB) (ii)

A(POS)=12×A(gmAPOS)( iii)

A(SOR)=12×A(gmROSD) (iv)

Adding equations (i), (ii), (iii) and (iv), we get

A(ROQ)+A(POQ)+A(POS)+A(SOR)

=12[A(gmROQC)+A(gmPOQB)+A(gmAPOS)+A(gmROSD)

=12×24

=12sq. units


26. PQR and SQR are on the same base QR with P and S on opposite sides of line QR, such that area of PQR is equal to the area of SQR. Show that QR bisects PS.

Ans:

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Join PS. Suppose PS and QR intersect at O. Draw PM and SN both perpendicular to QR.

ar(PQR)=ar(SQR)

Thus PQR and SQR are on the same base QR and have equal areas.

Therefore, their corresponding altitudes PM=SN are equal.

Now, in PMO and SNO,

1=2 (vertically opposite angles)

PMO=SNO (right angles)

PM=SN

Therefore, PMOSNO (AAS axiom)

PO=OS

QR bisects PS


27. If the medians of a ABBC intersect at G, show that

ar(AGB)=ar(AGC)=ar(BGC)=13ar(ABC)

Ans:

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Property: The median of a triangle divides it into two triangles of equal areas.

In ABC,AD is the median

ar(ABD)=ar(ACD)(i)

In GBC,GD is the median

ar(ΔGBD)=ar(ΔGCD)...(ii)

Subtracting (ii) from (i),

ar(ABD)ar(GBD)=ar(ACD)ar(GCD)

ar(AGB)=ar(AGC) (iii)

Similarly, ar(AGB)=ar(BGC) (iv)

From (iii) and (iv),

ar(AGB)=ar(BGC)=ar(AGC)(v)

But ar(AGB)+ar(BGC)+ar(AGC)=ar(ABC)

Therefore, 3ar(AGB)=ar(ABC)

ar(AGB)=13ar(ABC)

Hence, ar(AGB)=ar(AGC)=ar(BGC)=13ar(ABC).


28. In ABC, the mid-points of AB,BC and AC are P,Q and R respectively. Prove that BQRP is a parallelogram and that its area is half of ABC.

Ans:

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P and R are mid-points of AB and AC respectively given in the question.

Therefore, PRBC and PR=12BC........(i)

Also, Q is the mid-point of BC,

QC=12BC...(ii) 

From (i) and (ii)

PRBC and PR=QC

PRQC and PR=QC

Q and R are also mid-points of BC and AC respectively as given.

Therefore, QRBP and QR=BP

Hence, BQRP is a parallelogram.

PQ is a diagonal of gmBQRP

ar(PQR)=ar(BQP)(v) (as diagonal of a gm divides it into two

triangles of equal areas)

Similarly, QCRP and QRAP are gm and

ar(PQR)=ar(QCR)=ar(APR)(vi)

From (v) and (vi)

ar(PQR)=ar(BQP)=ar(QCR)=ar(APR)

Now, ar(ABC)=ar(PQR)+ar(BQP)+ar(QCR)+ar(APR)

ar(ABC)=ar(PQR)+ar(PQR)+ar(PQR)+ar(PQR)

ar(ABC)=4ar(PQR)

ar(PQR)=14ar(ABC)...(vii )

ar(gmBQRP)=ar(PQR)+ar(BQP)

ar(lgmBRP)=ar(PQR)+ar(PQR)( from (v))

ar(lgBQRP)=2ar(PQR)

ar(lgBQRP)=2×14ar(ABC) (from (vii))

ar(gmBQRP)=12ar(ABC)


29. In the given figure, PQSRMN,PSQM and SMPN. Prove that: ar. (SMNT)= ar. (PQRS).

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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.

Area ( parallelogram SMNT) = Area (parallelogram SMOP) ....(i)

Similarly, we can prove that quadrilaterals PQRS is a parallelograms.

Now, parallelograms PQRS and SMOP are on the same base PS, and both are between the same parallel lines PS and QM.

A( parallelogram PQRS)=A( parallelogram SMOP)....(ii)

From (i) and (ii), we have

A( parallelogram SMNT )=A( parallelogram PQRS)


31. In PQR,PS is a median. T is the mid-point of SR and M is the mid-point of PT. Prove that: PMR=18PQR.

Ans:

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Area (PQR)=area(PQS)+area(PSR)(i)

Since PS is the median of PQR and median divides a triangle into two triangles of equal area.

Therefore, area (PQS)=area(PSR)...(ii)

Substituting in (i)

Area (PQR)=area(PSR)+ area (PSR) Area (PQR)


32. In the figure, ABCD is a parallelogram and CP is parallel to DB. Prove that: Area of OBPC=34 area of ABCD

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Ans: The diagonals of a parallelogram divide it into four triangles of equal area Therefore, the area of AOD= area BOC= area ABO= area CDO. area BOC=14 area (lgmABCD)........(i)

In gmABCD,BD is the diagonal

Therefore, area (ABD)=area(BCD)

area (BCD)=12 area (lgmABCD).. (ii)

In Parallelogram BPCD, BC is the diagonal

Therefore, area (BCD)=area(BPC).. (iii)

From (iii) and (ii)

area (BPC)=12 area (gmABCD) (iv)

adding (i) and (iv)

area(BPC)+ area BOC=12 area (gmABCD)+14 area (gmABCD)

Area of OBPC=34 area of ABCD


33. The medians QM and RN of PQR intersect at O. Prove that: area of ROQ= area of quadrilateral PMON.

Ans:

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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, MNQR

Clearly, QMN and RNM are on the same base MN and between the same parallel lines.

Therefore, area(QMN)=area(RNM)

Area(QMN)area(ONM)=area(RNM)area(ONM)

area(ΔQON)=area(ROM)(i)

We know that a median of a triangle divides it into two triangles of equal areas.

Therefore, area (QMR)=area(PQM)

area(ROQ)+area(ROM)=area( quad. PMON)+area(QON)

area (ROQ)+area(ROM)= area (quad. PMON )+ area (ROM) (from

(i))

area(ROQ)=area( quad. PMON)


34.

a) In the given figure, ABC is a triangle, and AD is the median.

If E is any point on the median AD. Show that: Area of ABE= Area of ACE.

b) In the given figure, ABC is a triangle, and AD is the median.

If E is the midpoint of the median AD, prove that: Area of ABC=4×

Area of ABE

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Ans: (a) AD is the median of ABC

so it will divide ABC into two triangles of equal areas.

Area(ABD)=Area(ACD).. (i)

Similarly, ED is the median of EBC.

Area(DEBD)=Area(DECD) (ii)

Subtracting equation (ii) from (i), we have

Area(ABD)Area(EBD)=Area(ACD)Area(ECD)

Area(ABE)=Area(ACE)

(b) AD is the median of ABC.

So, it will divide ABC into two triangles of equal areas.

Area(ABD)=Area(ACD).. (i)

Similarly, ED is the median of EBC.

Area(EBD)=Area(ECD) (ii)

Subtracting equation (ii) from (i), we have

Area(ABD)Area(EBD)=Area(ACD)Area(ECD)

Area(ABE)=Area(ACE) (iii)

Since E is the mid-point of median AD,

AE=ED

Now,

ABE and BED have equal bases and a common vertex B.

Area(ABE)=Area(BED).....(iv)

From (i), (ii), (iii) and (iv), we get

Area(ABE)=A(BED)=Area(ACE)=Area(EDC)(v)

Now,

Area(ABC)=Area(ABE)+A(BED)+Area(ACE)+Area(EDC)

=4×Area(ABE) From (v)


35. In a parallelogram PQRS,M and N are the midpoints of the sides PQ and PS respectively. If the area of PMN is 20 square units, find the area of the parallelogram PQRS.

Ans:

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In a parallelogram PQRS, SQ is the diagonal.

So, it bisects the parallelogram.

Area(DPSQ)=12× Area (parallelogram PQRS )

SM is the median of PSQ.

Area(PSM)=12×Area(PSQ)=12×12× Area (parallelogram PQRS) =14× Area (parallelogram PQRS )

Again, MN is the median of PSM.

Area(PMN)=12×Area(PSM)=12×14× Area (parallelogram PQRS) =18× Area (parallelogram PQRS)

20=18× Area(parallelogram PQRS)

Area (parallelogram PQRS) =160 square units


36. In a parallelogram PQRS,T is any point on the diagonal PR. If the area of PTQ is 18 square units find the area of PTS.

Ans:

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Construction: First, Join QR.

Let the diagonals PR and QS intersect each other at point O.

Since diagonals of a parallelogram bisect each other and O is the mid-point of both lines PR and QS.

Now, medians PR and QS of a triangle divide it into two triangles of equal area.

In PSQ,OP is the median.

Area(POS)=Area(POQ) (i)

Similarly, OT is the median of TSQ.

Area(TOS)=Area(TOQ) (ii)

Subtracting equation (ii) from (i), we have

Area(POS)Area(TOS)=Area(POQ)Area(TOQ)

Area(PTQ)=Area(PTS)

Area(PTS)=18 square units


37. ABCD is a quadrilateral in which diagonals AC and BD intersect at a point O.

Prove that: area AOD+ area BOC+ area ABO+ area CDO.

Ans:

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Since the diagonals of a parallelogram bisect each other at the point of intersection.

Therefore, OB=OD and OA=OC

In ABC,OB is the median and the median divides the triangle into two triangles of equal areas

Therefore, area (BOC)=area(ABO)(i)

In ADC,OD is the median and the median divides the triangle into two triangles of equal areas

Therefore, area (AOD)=area(CDO) (ii)

Adding (i) and (ii)

area (AOD)+ area (BOC)=area(ABO)+ area (CDO)


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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.

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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.