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Class X Session 2023-24 Question 28-optional

 MATH SAMPLE QUESTION PAPER Class X Session 2023-24 MATHEMATICS STANDARD (Code No.041) Question 28 Solve:2x+3y=2;4xβˆ’9y=βˆ’1,x,y>0 Explanation:   Let's consider : a=1xandb=1y=βˆ’1  Now equation becomes  2a+3b = 2 -----------------(1) 4a - 9b = -1 -------------------------(2) now solve both equations (1) and (2) First, multiply equation (1) by 2 to align the coefficients of a and subtract from each other  2x(2a+3b) - (4a -9b ) = 4 -(-1) 2Γ—(2a+3b)βˆ’(4aβˆ’9b)=4βˆ’(βˆ’1) or4a+6bβˆ’4a+9b=4+1 or15b=5 orb=515 orb=13 Now, substitute b in equation (1) β‡’2a+3Γ—13=2  or2a+1=2  or2a=1 ∴a=12 Now we have a and b so we can find the values of x and y  \[\ a = \fr...

Class X Session 2023-24 Question 28

MATH SAMPLE QUESTION PAPER Class X Session 2023-24 MATHEMATICS STANDARD (Code No.041) Question 28 The sum of a two digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, find the number. How many such numbers are there? Explanation:   \begin{flalign} & Let's\; consider\; the \; unit\; digit\; be\; a\; and \; ten\; digit\; be \; b\; &\ & so, Original \; number \; is\; a+10b ---(1) \; where\; a\; \neq b &\ & By \;reversing\; the\; digit\,\; we\; get\; 10b+a ----(2) &\ & According \;to\; the\; question\;,\; we \; get & \ & (a+10b) + (10b+a) = 66 &\ &\Rightarrow 11a +11b = 66 & \ & or \; a+b = 6 --(3) & \ & since\; given \; that\; digit \;of \;the\; number \;differ \;2 &\ & \therefore a-b = 2 ---(4) &\ & by \; adding \; equations \;2\; and \;3\; we \;get &\ & (a-b) +(a+b) = 2+6 &\ & 2a = 8 &\ & \therefore a = 4 ...

Class X Session 2023-24 Question 27

 MATH SAMPLE QUESTION PAPER Class X Session 2023-24 MATHEMATICS STANDARD (Code No.041) Question 27 if,Ξ±Ξ²areZeroesofquadraticpolynomial5x2+5x+10findthevalueof(1)Ξ±2+Ξ²2,(2)Ξ±βˆ’1+Ξ²βˆ’1  Explanation:   \begin{flalign} & since\; \alpha\; and \; \beta\; are \;the \;root\; of\; 5x^2 + 5x + 10 as \;we \;know\; that\; if \; quadratic\; equation;&\ & ax^2 + bx +c =0 \;then \; SUM \;of \; zeroes\; \alpha + \beta\ = -\frac{b}{a}; and \;product\; of \;Zeroes\; =\; \frac{c}{a}&\ & since\; \alpha\; and \; \beta\; are \;the \;root\; of\;5x^2 + 5x + 10 \;similarly \;\ from\; the \;given\; equation\; \alpha + \beta = & \ & \frac{-5}{5} and \; \alpha \beta \;= \frac{1}{5} , \; now \; lets \;find\; \alpha^2 + \beta^2 &\ &\Rightarrow (\alpha + \beta)^2 = \a...

Class X Session 2023-24 Question 26

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 MATH SAMPLE QUESTION PAPER Class X Session 2023-24 MATHEMATICS STANDARD (Code No.041) Question 26 Find the area of the unshaded region shown in the given figure.  Solution: modified view of shared image The total horizontal or vertical extent of the region is 8 cm and extent includes the side length of the square (π‘Ž)  and the diameters of the semicircles on either side of the square. Given that each semicircle has a radius π‘Ÿ, the side length of the square is π‘Ž=8βˆ’2π‘Ÿ. Given the figure, the radius π‘Ÿ of each semicircle is 2cm Side Length of the Square π‘Ž = 8-2x2  = 4 cm Area of the Square:  π‘Ž^2 = 4^2 = 16 Area of One Semicircle = 12Ο€r2=2Ο€cm2wherer=2 Combined Area of Four Semicircles: 4Γ—2Ο€cm2 Area of the unshaded region=Area of the square + 4 x Area of  one semicircle = 16cm2+4Γ—2Ο€cm2 (16+8Ο€)cm2 

Class X Session 2023-24 Question 25

 MATH SAMPLE QUESTION PAPER Class X Session 2023-24 MATHEMATICS STANDARD (Code No.041) Question 25 Find the value of x if 2cosec2(30)+xsin2(60)βˆ’34tan2(30)=10  Explanation:    Now simplify the equation below  2cosec2(30)+xsin2(60)βˆ’34tan2(30)=10 β‡’2Γ—(2)2+x(32)2βˆ’34(13)2=10 we know that value of cosec(30) , sin(60)and tan(30) putting their values β‡’2Γ—4+x34βˆ’(34Γ—13)=10 OR8+3x4βˆ’14=10 OR3x4=14+(10βˆ’8) OR3x4=14+2β‡’94 OR3x4=94 β‡’3x=9 ∴x=3

Class X Session 2023-24 Question 24

 MATH SAMPLE QUESTION PAPER Class X Session 2023-24 MATHEMATICS STANDARD (Code No.041)  Question 24 iftan⁑(A+B)=3andtan⁑(Aβˆ’B)=13 , 0Β° < A + B < 90Β°; A > B, find A and B. Explanation:   Given that : tan⁑(A+B)=3andtan⁑(Aβˆ’B)=13 First, recognize the angles for which tan⁑(ΞΈ)=3andtan(ΞΈ)=13 we know that tan⁑(60∘)=3andtan(30∘)=13  so,A+B=60∘andAβˆ’B=30∘ now add the two equation , we get   A+B=60∘,βˆ’βˆ’βˆ’(1)   Aβˆ’B=30∘,βˆ’βˆ’βˆ’(2)  so,(A+B)+(Aβˆ’B)=60∘+30∘  or,2A=90∘  or,A=45∘ Now Put A value in equation (1) and we get   from(1),B=60βˆ˜βˆ’45∘ ∴B=15∘ \[Now \; A\;= 45^{\circ}\; and \; B\;= 15^{\circ}\...

Class X Session 2023-24 Question 23

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 MATH SAMPLE QUESTION PAPER Class X Session 2023-24 MATHEMATICS STANDARD (Code No.041) SECTION B - Question 23 From an external point P, two tangents, PA and PB are drawn to a circle with centre O. At a point E on the circle, a tangent is drawn to intersect PA and PB at C and D, respectively. If PA = 10 cm, find the perimeter of βˆ†PCD . . Explanation: Given an external point  𝑃 from which two tangents π‘ƒπ΄   and 𝑃𝐡 are drawn to a circle with center 𝑂, we know the following: 𝑃𝐴 = 10 cm PB=10 cm (since the tangents drawn from an external point to a circle are equal)  PerimeterofΞ”PCD=PC+CD+PD β‡’PC+CE+ED+PD β‡’PC+CA+DB+PD β‡’PA+PB β‡’PA+PB ORPA+PA OR2PA =2(10) PerimeterofΞ”PCDis20cm,whereCE=CA,DE=DB,PA=PB Note: Tangents from the internal point to a circle are equal