Two dice are rolled simultaneously.
Question: Two dice are rolled simultaneously. The probability that they show different faces is (a) $\frac{2}{3}$ (b) $\frac{1}{6}$ (c) $\frac{1}{3}$ (d) $\frac{5}{6}$ Solution: GIVEN: A pair of dice is thrown TO FIND: Probability of getting different faces 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), (5...
Read More →Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
Question: Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it. Solution: Here, first term,a= 1 Common ratio =r $\therefore a_{n}=\left[a_{n+1}+a_{n+2}+a_{n+3}+\ldots \ldots \infty\right] \forall n \in N$ $\Rightarrow a r^{n-1}=a r^{n}+a r^{n-1}+\ldots \ldots \infty$ $\Rightarrow r^{n-1}=\frac{r^{n}}{1-r} \quad$ [Putting $a=1$ ] $\Rightarrow r^{n-1}(1-r)=r^{n}$ $\Rightarrow 1-r=r$ $\Rightarrow 2 r=1$ $\Rightarrow r=\frac{1}{2}$ Thus, the infinte G...
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Question: If $A=\left[\begin{array}{rr}3 1 \\ -1 2\end{array}\right]$, show that $A^{2}-5 A+7 l=0$ use this to find $A^{4}$. Solution: Given : $A=\left[\begin{array}{cc}3 1 \\ -1 2\end{array}\right]$ Now, $A^{2}=A A$ $\Rightarrow A^{2}=\left[\begin{array}{cc}3 1 \\ -1 2\end{array}\right]\left[\begin{array}{cc}3 1 \\ -1 2\end{array}\right]$ $\Rightarrow A^{2}=\left[\begin{array}{cc}9-1 3+2 \\ -3-2 -1+4\end{array}\right]$ $\Rightarrow A^{2}=\left[\begin{array}{cc}8 5 \\ -5 3\end{array}\right]$ $A^...
Read More →Assertion: In the given figure, ABCD is a || gm in which DE ⊥ AB and BF ⊥ AD.
Question: Assertion:In the given figure,ABCDis a || gm in whichDEABandBFAD. IfAB= 16 cm,DE= 8 cm andBF= 10 cm, thenADis 12 cm. Reason:Area of a || gm = base height.(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.(c) Assertion is true and Reason is false.(d) Assertion is false and Reason is true. Solution: (d) Assertion is false and Reason is true. Clearly, reason (R...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following:A box contains 90 discs, numbered from 1 to 90. If one disc is drawn at random from the box, the probability that it bears a prime number less than 23, is (a) $\frac{7}{90}$ (b) $\frac{10}{90}$ (C) $\frac{4}{45}$ (d) $\frac{9}{89}$ [CBSE 2013] Solution: There are 90 discs numbered from 1 to 90. Total number of outcomes = 90The prime numbers less than 23 are 2, 3, 5, 7, 11, 13, 17 and 19.So, the favourable number of outcomes are 8. $...
Read More →One side of an equilateral triangle is 18 cm.
Question: One side of an equilateral triangle is 18 cm. The mid-points of its sides are joined to form another triangle whose mid-points, in turn, are joined to form still another triangle. The process is continued indefinitely. Find the sum of the (i) perimeters of all the triangles. (ii) areas of all triangles. Solution: According to the midpoint theorem, the sides of each triangle formed by joining the midpoints of an equilateral triangle are half of the sides of the equilateral triangle. In ...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following:The probability of getting an even number, when a die is thrown once is (a) $\frac{1}{2}$ (b) $\frac{1}{3}$ (c) $\frac{1}{6}$ (d) $\frac{5}{6}$ [CBSE 2013] Solution: In a single throw of a die, the possible outcomes are 1, 2, 3, 4, 5 and 6. Total number of outcomes = 6The favourable outcomes are 2, 4 and 6.So, the number of favourable outcomes are 3. $\therefore \mathrm{P}$ (getting an even number) $=\frac{\text { Favourable number ...
Read More →Assertion: The area of a trapezium whose parallel sides measure 25 cm and 15 cm respectively and the distance between them is 6 cm, is 120 cm2.
Question: Assertion:The area of a trapezium whose parallel sides measure 25 cm and 15 cm respectively and the distance between them is 6 cm, is 120 cm2. Reason: The area of an equilateral triangle of side $8 \mathrm{~cm}$ is $16 \sqrt{3} \mathrm{~cm}^{2}$. (a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.(c) Assertion is true and Reason is false.(d) Assertion is fals...
Read More →Find the rational numbers having the following decimal expansions:
Question: Find the rational numbers having the following decimal expansions: (i) $0 . \overline{3}$ (ii) $0 . \overline{231}$ (iii) $3.5 \overline{2}$ (iv) $0.6 \overline{8}$ Solution: (i) $0 . \overline{3}$ Let $S=0 . \overline{3}$ $\Rightarrow \mathrm{S}=0.3+0.03+0.003+0.0003+0.00003+\ldots \infty$ $\Rightarrow \mathrm{S}=0.3\left(1+10^{-1}+10^{-2}+10^{-3}+10^{-4}+\ldots \infty\right)$ $\mathrm{S}$ is a geometric series with the first term, $a$, being 1 and the common ratio, $r$, being $10^{-1...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following:A die is thrown once. The probability of getting a prime number is (a) $\frac{2}{3}$ (b) $\frac{1}{3}$ (c) $\frac{1}{2}$ (d) $\frac{1}{6}$ [CBSE 2013] Solution: In a single throw of a die, the possible outcomes are 1, 2, 3, 4, 5 and 6. Total number of outcomes = 6The favourable outcomes are 2, 3 and 5.So, the number of favourable outcomes are 3. $\therefore \mathrm{P}($ getting a prime number $)=\frac{\text { Favourable number of ou...
Read More →Assertion: The diagonals of a || gm divide it into four triangles of equal area.
Question: Assertion:The diagonals of a || gm divide it into four triangles of equal area.Reason:A diagonal of a || gm divides it into two triangles of equal area.(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.(c) Assertion is true and Reason is false.(d) Assertion is false and Reason is true. Solution: (a) Both Assertion and Reason are true and Reason is a correct ...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following:Two dice are thrown together. The probability of getting the same number on both dice is (a) $\frac{1}{2}$ (b) $\frac{1}{3}$ (c) $\frac{1}{6}$ (d) $\frac{1}{12}$ [CBSE 2012] Solution: When two dice are thrown together, all possible outcomes are(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), ...
Read More →Assertion: If ABCD is a rhombus whose one angle is 60°,
Question: Assertion: If $A B C D$ is a rhombus whose one angle is $60^{\circ}$, then the ratio of the lengths of its diagonals is $\sqrt{3}: 1$. Reason:Median of triangle divides it into two triangles of equal area.(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.(c) Assertion is true and Reason is false.(d) Assertion is false and Reason is true. Solution: (b) Both A...
Read More →if
Question: If $A=\left[\begin{array}{rr}1 -3 \\ -4 3\end{array}\right]$, find $A^{2}-3 A-7 /$ Solution: Given : $A=\left[\begin{array}{cc}1 -3 \\ -4 3\end{array}\right]$ Now, $A^{2}=A A$ $\Rightarrow A^{2}=\left[\begin{array}{cc}1 -3 \\ -4 3\end{array}\right]\left[\begin{array}{cc}1 -3 \\ -4 3\end{array}\right]$ $\Rightarrow A^{2}=\left[\begin{array}{cc}1+12 -3-6 \\ -4-12 12+9\end{array}\right]$ $\Rightarrow A^{2}=\left[\begin{array}{cc}13 -9 \\ -16 21\end{array}\right]$ $A^{2}-5 A-14 I$ $\Righta...
Read More →Two numbers 'a' and 'b' are selected successively without replacement
Question: Two numbers 'a' and 'b' are selected successively without replacement in that order from the integers 1 to 10 . The probability that $\frac{a}{b}$ is an integer, is (a) $\frac{17}{45}$ (b) $\frac{1}{5}$ (c) $\frac{17}{90}$ (d) $\frac{8}{45}$ Solution: We have a set of natural numbers from 1 to 10 whereandare two variables which can take values from 1 to 10. So, total number of possible combination of $a$ and $b$ so that $\left(\frac{a}{b}\right)$ is a fraction without replacement are: ...
Read More →Show that the matrix
Question: Show that the matrix $A=\left[\begin{array}{rr}5 3 \\ 12 7\end{array}\right]$ is root of the equation $A^{2}-12 A-1=0$ Solution: Given : $A=\left[\begin{array}{cc}5 3 \\ 12 7\end{array}\right]$ Now, $A^{2}=A A$ $\Rightarrow A^{2}=\left[\begin{array}{cc}5 3 \\ 12 7\end{array}\right]\left[\begin{array}{cc}5 3 \\ 12 7\end{array}\right]$ $\Rightarrow A^{2}=\left[\begin{array}{ll}25+36 15+21 \\ 60+84 36+49\end{array}\right]$ $\Rightarrow A^{2}=\left[\begin{array}{cc}61 36 \\ 144 85\end{arra...
Read More →Find the rational number whose decimal expansion is
Question: Find the rational number whose decimal expansion is $0.4 \overline{23}$. Solution: Let the rational number $\mathrm{S}$ be $0.4 \overline{23}$. $\because \mathrm{S}=0.4 \overline{23}=0.4+0.023+0.00023+0.0000023+\ldots \infty$ $\Rightarrow \mathrm{S}=0.4+0.023\left[1+10^{-2}+10^{-4}+\ldots \infty\right]$ Clearly, $\mathrm{S}$ is a geometric series with the first term, $a$, being 1 and the common ratio, $r$, being $10^{-2}$. $\therefore \mathrm{S}=0.4+0.023\left[\frac{1}{1-10^{-2}}\right...
Read More →A number is selected from first 50 natural numbers.
Question: A number is selected from first 50 natural numbers. What is the probability that it is a multiple of 3 or 5? (a) $\frac{13}{25}$ (b) $\frac{21}{50}$ (c) $\frac{12}{25}$ (d) $\frac{23}{50}$ Solution: GIVEN: A number is selected from 50 natural numbers TO FIND: Probability that the number selected is a multiple of 3 or 5 Total number is 50 Total numbers which are multiple of 3 or 5 up to 50 natural numbers are 3,6,5,9,10,12,15,18,20,21,24,25,27,30,33,35,36,39,40,42,45,48,50 Total number ...
Read More →Express the recurring decimal 0.125125125 ...
Question: Express the recurring decimal 0.125125125 ... as a rational number. Solution: Let the rational number $S$ be $0 . \overline{125}$. $\because \mathrm{S}=0 . \overline{125}=0.125+0.000125+0.000000125+0.000000000125+\ldots \infty$ $\Rightarrow \mathrm{S}=0.125\left[1+10^{-3}+10^{-6}+10^{-9}+\ldots \infty\right]$ Clearly, $\mathrm{S}$ is a geometric series with the first term, $a$, being 1 and the common ratio, $r$, being $10^{-3}$. $\therefore S=\frac{1}{(1-r)}$ $\Rightarrow \mathrm{S}=0....
Read More →Assertion: In a trapezium ABCD we have AB || DC and the diagonals AC and BD intersect at O. Then, ar(∆AOD) = ar(∆BOC)
Question: Assertion:In a trapeziumABCDwe haveAB||DCand the diagonalsACandBDintersect atO. Then, ar(∆AOD) = ar(∆BOC) Reason:Triangles on the same base and between the same parallels are equal in areas.(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.(c) Assertion is true and Reason is false.(d) Assertion is false and Reason is true. Solution: (a) Both Assertion and Re...
Read More →Aarushi sold 100 lottery tickets in which 5 tickets carry prizes.
Question: Aarushi sold 100 lottery tickets in which 5 tickets carry prizes. If Priya purchased a ticket, what is the probability of Priya winning a prize? (a) $\frac{19}{20}$ (b) $\frac{1}{25}$ (c) $\frac{1}{20}$ (d) $\frac{17}{20}$ Solution: GIVEN: 100 lottery tickets were sold in which 5 tickets carry prize TO FIND: Probability of Priya winning a prize Total number of tickets is100 Total number of prize carrying tickets is 5 We know that PROBABILITY = Hence probability of Priya winning a prize...
Read More →Show that the matrix
Question: Show that the matrix $A=\left[\begin{array}{ll}2 3 \\ 1 2\end{array}\right]$ satisfies the equation $A^{3}-4 A^{2}+A=0$ Solution: We have, $A=\left[\begin{array}{ll}2 3 \\ 1 2\end{array}\right]$ $\therefore A^{2}=A A$ $\Rightarrow A^{2}=\left[\begin{array}{ll}2 3 \\ 1 2\end{array}\right]\left[\begin{array}{ll}2 3 \\ 1 2\end{array}\right]$ $\Rightarrow A^{2}=\left[\begin{array}{ll}4+3 6+6 \\ 2+2 3+4\end{array}\right]$ $\Rightarrow A^{2}=\left[\begin{array}{ll}7 12 \\ 4 7\end{array}\righ...
Read More →Find the sum of the terms of an infinite decreasing G.P. in which all the terms are positive,
Question: Find the sum of the terms of an infinite decreasing G.P. in which all the terms are positive, the first term is 4, and the difference between the third and fifth term is equal to 32/81. Solution: Let $r$ be the common ratio of the given G.P. $\therefore a=4$ Sum of the geometric ifinite series: $S_{\infty}=4+4 r+4 r^{2}+\ldots \infty$ Now, $S_{\infty}=\frac{4}{1-r} \quad \ldots \ldots$ (i) The difference between the third and fifth term is $\frac{32}{81}$. $a_{3}-a_{5}=\frac{32}{81}$ $...
Read More →What is the probability that a non-leap year has 53 Sundays?
Question: What is the probability that a non-leap year has 53 Sundays? (a) $\frac{6}{7}$ (b) $\frac{1}{7}$ (c) $\frac{5}{7}$ (d) None of these Solution: GIVEN: A non leap year TO FIND: Probability that a non leap year has 53 Sundays. Total number of days in a non leap year is 365days Hence number of weeks in a non leap year is In a non leap year we have 52 complete weeks and 1 day which can be any day of the week i.e. Sunday, Monday, Tuesday, Wednesday, Thursday, Friday and Saturday To make 53 S...
Read More →Look at the statements given below:
Question: Look at the statements given below:I. A parallelogram and a rectangle on the same base and between the same parallels are equal in area.II. In a || gmABCD, it is given thatAB= 10 cm. The altitudesDEonABandBFonADbeing 6 cm and 8 cm respectively, thenAD= 7.5 cm. III. Area of a $\| \mathrm{gm}=\frac{1}{2} \times$ base $\times$ altitude. Which is true?(a) I only(b) II only(c) I and II(d) II and III Solution: (c) I and IIStatement I is true, because if a parallelogram and a rectangle lie on...
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