if
Question: If $A=\left[\begin{array}{rrr}1 0 -2 \\ 3 -1 0 \\ -2 1 1\end{array}\right], B=\left[\begin{array}{rrr}0 5 -4 \\ -2 1 3 \\ -1 0 2\end{array}\right]$ and $C=\left[\begin{array}{rrr}1 5 2 \\ -1 1 0 \\ 0 -1 1\end{array}\right]$ , verify that $A(B-C)=A B-A C$. Solution: Given : $A(B-C)=A B-A C$ $\Rightarrow\left[\begin{array}{ccc}1 0 -2 \\ 3 -1 0 \\ -2 1 1\end{array}\right]\left(\left[\begin{array}{ccc}0 5 -4 \\ -2 1 3 \\ -1 0 2\end{array}\right]-\left[\begin{array}{ccc}1 5 2 \\ -1 1 0 \\ 0...
Read More →In the given figure, ABCD is a rectangle inscribed in a quadrant of a circle of radius 10 cm.
Question: In the given figure, $A B C D$ is a rectangle inscribed in a quadrant of a circle of radius $10 \mathrm{~cm}$. If $A D=2 \sqrt{5} \mathrm{~cm}$, then area of the rectangle is (a) $32 \mathrm{~cm}^{2}$ (b) $40 \mathrm{~cm}^{2}$ (c) $44 \mathrm{~cm}^{2}$ (d) $48 \mathrm{~cm}^{2}$ Solution: (b) $40 \mathrm{~cm}^{2}$ Radius of the circle,AC= 10 cmDiagonal of the rectangle,AC= 10 cm Now, $A B=\sqrt{A C^{2}-B C^{2}}=\sqrt{10^{2}-(2 \sqrt{5})^{2}}=\sqrt{80}=4 \sqrt{5} \mathrm{~cm}$ $\therefor...
Read More →How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make
Question: How many terms of the G.P. $3,3 / 2,3 / 4, \ldots$ be taken together to make $\frac{3069}{512} ?$ Solution: Here,a= 3 Common ratio, $r=\frac{1}{2}$ $S_{n}=\frac{3069}{512}$ $\therefore S_{n}=3\left\{\frac{1-\left(\frac{1}{2}\right)^{n}}{1-\frac{1}{2}}\right\}$ $\Rightarrow \frac{3069}{512}=3\left\{\frac{1-\frac{1}{2^{n}}}{\frac{1}{2}}\right\}$ $\Rightarrow \frac{3069}{512}=6\left\{1-\frac{1}{2^{n}}\right\}$ $\Rightarrow \frac{3069}{3072}=1-\frac{1}{2^{n}}$ $\Rightarrow \frac{1}{2^{n}}=...
Read More →For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e. A (B + C) = AB + AC:
Question: For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e.A(B+C) =AB+AC: (i) $A=\left[\begin{array}{rr}1 -1 \\ 0 2\end{array}\right], B=\left[\begin{array}{rr}-1 0 \\ 2 1\end{array}\right]$ and $C=\left[\begin{array}{rr}0 1 \\ 1 -1\end{array}\right]$ (ii) $A=\left[\begin{array}{rr}2 -1 \\ 1 1 \\ -1 2\end{array}\right], B=\left[\begin{array}{ll}0 1 \\ 1 1\end{array}\right]$ and $C=\left[\begin{array}{rr}1 -1 \\ 0 1\end{array}\right]$. Solutio...
Read More →In the given figure, a || gm ABCD and a rectangle ABEF are of equal area. Then,
Question: In the given figure, a || gmABCDand a rectangleABEFare of equal area. Then, (a) perimeter ofABCD= perimeter ofABEF(b) perimeter ofABCD perimeter ofABEF(c) perimeter ofABCD perimeter ofABEF (d) perimeter of $A B C D=\frac{1}{2}$ (perimeter of $A B E F$ ) Solution: (c) perimeter ofABCD perimeter ofABEFParallelogramABCDand rectangleABEFlie on the same baseAB, i.e., one side is common in both the figures.In ||gmABCD, we have:ADis the hypotenuse of right angled triangleADF.So,ADAF Perimeter...
Read More →In the given figure, points A, B, C and D are the centres of four circles that each have
Question: In the given figure, pointsA,B,CandDare the centres of four circles that each have a radius of length one unit. If a point is selected at random from the interior of squareABCD.What is the probability that the point will be chosen from the shaded region.\ Solution: Given:A, B, C, and D are the centers of four circles that each have a radius of length one unit. If a point is selected at random from the interior of square ABCD To find:Probability that the point will be chosen from the sh...
Read More →Find the sum of the following series:
Question: Find the sum of the following series: (i) 5 + 55 + 555 + ... tonterms; (ii) 7 + 77 + 777 + ... tonterms; (iii) 9 + 99 + 999 + ... tonterms; (iv) 0.5 + 0.55 + 0.555 + ... tonterms. (v) 0.6 + 0.66 + 0.666 + .... tonterms Solution: (i) We have, 5 + 55 + 555+ ...nterms Taking 5 as common: $S_{n}=5[1+11+111+\ldots n$ terms $]$ $=\frac{5}{9}(9+99+999+\ldots n$ terms $)$ $=\frac{5}{9}\left\{(10-1)+\left(10^{2}-1\right)+\left(10^{3}-1\right)+\ldots+\left(10^{n}-1\right)\right\}$ $=\frac{5}{9}\...
Read More →ABCD is a quadrilateral whose diagonal AC divides it into two parts, equal in area, then ABCD is
Question: ABCDis a quadrilateral whose diagonal AC divides it into two parts, equal in area, thenABCDis(a) a rectangle(b) a || gm(c) a rhombus(d) all of these Solution: (d) all of theseIn all the mentioned quadrilaterals, a diagonal divides them into two triangles of equal areas....
Read More →For the following matrices verify the associativity of matrix multiplication i.e. (AB) C = A (BC):
Question: For the following matrices verify the associativity of matrix multiplication i.e. (AB)C=A(BC): (i) $A=\left[\begin{array}{rrr}1 2 0 \\ -1 0 1\end{array}\right], B=\left[\begin{array}{rr}1 0 \\ -1 2 \\ 0 3\end{array}\right]$ and $C=\left[\begin{array}{r}1 \\ -1\end{array}\right]$ (ii) $A=\left[\begin{array}{lll}4 2 3 \\ 1 1 2 \\ 3 0 1\end{array}\right], B=\left[\begin{array}{rrr}1 -1 1 \\ 0 1 2 \\ 2 -1 1\end{array}\right]$ and $C=\left[\begin{array}{rrr}1 2 -1 \\ 3 0 1 \\ 0 0 1\end{arra...
Read More →In the given figure ABCD is a trapezium in which AB || DC such that AB = a cm and DC = b cm.
Question: In the given figureABCDis a trapezium in whichAB||DCsuch thatAB=acm andDC=bcm. IfEandFare the midpoints ofADandBCrespectively. Then, ar(ABFE) : ar(EFCD) = ? (a)a:b(b) (a+ 3b) : (3a+b)(c) (3a+b) : (a+ 3b)(d) (2a+b) : (3a+b) Solution: (c) (3a+b) : (a+3b) Clearly, EF $=\frac{1}{2}(a+b)$ [Mid point theorem] Letdbe the distance betweenABandEF.Thendis the distance betweenDCandEF. Now, we have : $\operatorname{ar}(\operatorname{trap} A B E F)=\frac{1}{2}\left(a+\frac{a+b}{2}\right) d=\frac{(3...
Read More →Let
Question: Let $A=\left[\begin{array}{rrr}-1 1 -1 \\ 3 -3 3 \\ 5 5 5\end{array}\right]$ and $B=\left[\begin{array}{rrr}0 4 3 \\ 1 -3 -3 \\ -1 4 4\end{array}\right]$, compute $A^{2}-B^{2}$. Solution: Given : $A=\left[\begin{array}{ccc}-1 1 -1 \\ 3 -3 3 \\ 5 5 5\end{array}\right]$ Now, $A^{2}=A A$ $\Rightarrow A^{2}=\left[\begin{array}{ccc}-1 1 -1 \\ 3 -3 3 \\ 5 5 5\end{array}\right]\left[\begin{array}{ccc}-1 1 -1 \\ 3 -3 3 \\ 5 5 5\end{array}\right]$ $\Rightarrow A^{2}=\left[\begin{array}{ccc}1+3-...
Read More →Evaluate the following:
Question: Evaluate the following: (i) $\sum_{n=1}^{11}\left(2+3^{n}\right)$ (ii) $\sum_{k=1}^{n}\left(2^{k}+3^{k-1}\right)$ (iii) $\sum_{n=2}^{10} 4^{n}$ Solution: (i) $S_{11}=\sum_{n=1}^{11}\left(2+3^{n}\right)$ $\Rightarrow S_{11}=\sum_{n=1}^{11} 2+\sum_{n=1}^{11} 3^{n}$ $\Rightarrow S_{11}=2 \times 11+\left(3+3^{2}+3^{3}+\ldots+3^{11}\right)$ $=22+3\left(\frac{3^{11}-1}{3-1}\right)$ $=22+\left(\frac{177147-1}{2}\right)$ $=22+265719$ $=265741$ (ii) $S_{n}=\sum_{k=1}^{n}\left(2^{k}+3^{k-1}\righ...
Read More →If a triangle and a parallelogram are on the same base and between the same parallels
Question: If a triangle and a parallelogram are on the same base and between the same parallels, then the ratio of the area of the triangle to the area of the parallelogram is (a) 1 : 2(b) 1 : 3(c) 1 : 4(d) 3 : 4 Figure Solution: (a) 1:2 If a triangle and a parallelogram are on the same base and between the same parallels, then the area of the triangles is half the area of the parallelogram. i.e., area of triangle $=\frac{1}{2} \times$ area of parallelogram $\therefore$ Required ratio = area of ...
Read More →A target shown in the given figure consists of three concentric circle of radii 3, 7
Question: A target shown in the given figure consists of three concentric circle of radii 3, 7 and 9 cm respectively. A dart is thrown and lands on the target. What is the probability that the dart will land on the shaded region? Solution: Given:A target is shown in figure consists of three concentric circles of radius 3, 7, and 9 cm. A dart is thrown and lands on the target To find:Probability that the dart will land in shaded region? Total area of circle with radius 9 cm Area of circle with ra...
Read More →Find the sum of the following geometric series:
Question: Find the sum of the following geometric series: (i) 0.15 + 0.015 + 0.0015 + ... to 8 terms; (ii) $\sqrt{2}+\frac{1}{\sqrt{2}}+\frac{1}{2 \sqrt{2}}+\ldots$ to 8 terms; (iii) $\frac{2}{9}-\frac{1}{3}+\frac{1}{2}-\frac{3}{4}+\ldots$ to 5 terms; (iv) (x+y) + (x2+xy+y2) + (x3+x2y+xy2+y3) + ... tonterms; (v) $\frac{3}{5}+\frac{4}{5^{2}}+\frac{3}{5^{3}}+\frac{4}{5^{4}}+\ldots$ to $2 n$ terms; (vi) $\frac{a}{1+i}+\frac{a}{(1+i)^{2}}+\frac{a}{(1+i)^{3}}+\ldots+\frac{a}{(1+i)^{n}}$. (vii) 1, a,a...
Read More →In ∆ABC, it is given that D is the midpoint of BC; E is the midpoint of BD and O is the midpoint of AE.
Question: In ∆ABC, it is given thatDis the midpoint ofBC;Eis the midpoint ofBDandOis the midpoint ofAE. Then, ar(∆BOE) = ? (a) $\frac{1}{3} \operatorname{ar}(\Delta A B C)$ (b) $\frac{1}{4} \operatorname{ar}(\Delta A B C)$ (c) $\frac{1}{6} \operatorname{ar}(\Delta A B C)$ (d) $\frac{1}{8} \operatorname{ar}(\Delta A B C)$ Solution: (d) $\frac{1}{8} \operatorname{ar}(\Delta A B C)$ Given:Dis the midpoint ofBC,Eis the midpoint ofBDandOis the mid point ofAE.SinceDis the midpoint ofBC,ADis the median...
Read More →if
Question: If $A=\left[\begin{array}{rrr}2 -3 -5 \\ -1 4 5 \\ 1 -3 -4\end{array}\right]$ and $B=\left[\begin{array}{rrr}2 -2 -4 \\ -1 3 4 \\ 1 -2 -3\end{array}\right]$, show that $A B=A$ and $B A=B$. Solution: Given : $A B=\left[\begin{array}{ccc}2 -3 -5 \\ -1 4 5 \\ 1 -3 -4\end{array}\right]\left[\begin{array}{ccc}2 -2 -4 \\ -1 3 4 \\ 1 -2 -3\end{array}\right]$ $\Rightarrow A B=\left[\begin{array}{ccc}4+3-5 -4-9+10 -8-12+15 \\ -2-4+5 2+12-10 4+16-15 \\ 2+3-4 -2-9+8 -4-12+12\end{array}\right]$ $\...
Read More →In the accompanying diagram a fair spinner is placed at the centre O of the circle Diameter
Question: In the accompanying diagram a fair spinner is placed at the centreOof the circle DiameterAOBand radiusOCdivide the circle into three regions labelledX,YandZ. It BOC = 45. What is the probability that the spinner will land in the regionX? (in the given figure). Solution: Given:A fair spinner is placed at the centre O of the circle. Diameter AOB and radius OC divide the circle into three regions labeled X, Y and Z and angle To find:Probability that the spinner will land in X region? Tota...
Read More →Examine each of the following statements and comment:
Question: Examine each of the following statements and comment:(i) If two coins are tossed at the same time, there are 3 possible outcomestwo heads, two tails, or one of each. Therefore, for each outcome, the probability of occurrence is 1/3.(ii) If a die in thrown once, there are two possible outcomes an odd number or an even number. Therefore, the probability of obtaining an odd number is 1/2 and the probability of obtaining an even number is 1/2. Solution: (i) Incorrect When two coins are tos...
Read More →if
Question: If $A=\left[\begin{array}{rrr}0 c -b \\ -c 0 a \\ b -a 0\end{array}\right]$ and $B=$ $\left[\begin{array}{lll}a^{2} a b a c \\ a b b^{2} b c \\ a c b c c^{2}\end{array}\right]$, show that $A B=B A=O_{3 \times 3}$ Solution: Here, $A B=\left[\begin{array}{ccc}0 c -b \\ -c 0 a \\ b -a 0\end{array}\right]\left[\begin{array}{lll}a^{2} a b a c \\ a b b^{2} b c \\ a c b c c^{2}\end{array}\right]$ $\Rightarrow A B=\left[\begin{array}{ccc}0+a b c-a b c 0+b^{2} c-b^{2} c 0+b c^{2}-b c^{2} \\ -a^...
Read More →Fill in the blanks:
Question: Fill in the blanks:(i) Probability of a sure event is ...............(ii) Probability of an impossible event is ................(iii) The probability of an event (other than sure and impossible event) lies between .................(iv) Every elementary event associated to a random experiment has .............. probability.(v) Probability of an event A + Probability of event 'not A' = ................(vi) Sum of the probabilities of each outcome in an experiment is ................ Solu...
Read More →Find the sum of the following geometric progressions:
Question: Find the sum of the following geometric progressions: (i) 2, 6, 18, ... to 7 terms; (ii) 1, 3, 9, 27, ... to 8 terms; (iii) 1, 1/2, 1/4, 1/8, ... to 9 terms; (iv) $\left(a^{2}-b^{2}\right),(a-b),\left(\frac{a-b}{a+b}\right), \ldots$ to $n$ terms; (v) 4, 2, 1, 1/2 ... to 10 terms. Solution: (i) Here,a= 2 andr= 3. $\therefore S_{7}=a\left(\frac{r^{7}-1}{r-1}\right)$ $=2\left(\frac{3^{7}-1}{3-1}\right)$ $=2187-1$ $=2186$ (ii) Here,a= 1 andr= 3. $\therefore S_{8}=a\left(\frac{r^{8}-1}{r-1}...
Read More →Two dice are thrown simultaneously. What is the probability that:
Question: Two dice are thrown simultaneously. What is the probability that:(i) 5 will not come up on either of them?(ii) 5 will come up on at least one?(iii) 5 will come up at both dice? Solution: GIVEN: Two dice are thrown TO FIND: Probability of the following: Let us first write the all possible events that can occur (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2...
Read More →The king, queen and jack of clubs are removed form a deck of 52 playing cards and
Question: The king, queen and jack of clubs are removed form a deck of 52 playing cards and the remaining cards are shuffled. A card is drawn from the remaining cards. Find the probability of getting a card of (i) heart (ii) queen (iii) clubs. Solution: Given: King, Queen and Jack of Clubs are removed from a deck of 52 playing cards and the remaining cards are shuffled and a card is drawn at random from the remaining cards TO FIND: Probability of getting a card (i) Heart (ii) Queen (iii) Clubs A...
Read More →The vertex A of ∆ABC is joined to a point D on BC.
Question: The vertexAof ∆ABCis joined to a pointDonBC. IfEis the midpoint ofAD, then ar(∆BEC) = ? (a) $\frac{1}{2} \operatorname{ar}(\Delta A B C)$ (b) $\frac{1}{3} \operatorname{ar}(\Delta A B C)$ (c) $\frac{1}{4} \operatorname{ar}(\Delta A B C)$ (d) $\frac{1}{6} \operatorname{ar}(\Delta A B C)$ Solution: (a) $\frac{1}{2} \operatorname{ar}(\Delta A B C)$ Since E is the midpoint of AD, BE is a median of ∆ABD.We know that a median of a triangle divides it into two triangles of equal areas. i.e., ...
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