The marks scored by 40 students of class VIII in mathematics are given below:
Question: The marks scored by 40 students of class VIII in mathematics are given below: 81, 55, 68, 79, 85, 43, 29, 68, 54, 73, 47, 35, 72, 64, 95, 44, 50, 77, 64, 35, 79, 52, 45, 54, 70, 83, 62, 64, 72, 92, 84, 76, 63, 43, 54,38, 73, 68, 52, 54. Prepare a frequency distribution with class size of 10 marks. Solution: The frequency table of the marks scored by 40 students of class VIII in mathematics is given below:...
Read More →For what value of k is the following function continuous at x = 2?
Question: For what value ofkis the following function continuous atx = 2? $f(x)=\left\{\begin{array}{rc}2 x+1 ; \text { if } x2 \\ k ; x=2 \\ 3 x-1 ; x2\end{array}\right.$ Solution: Given: $f(x)=\left\{\begin{array}{c}2 x+1, \text { if } \mathrm{x}2 \\ k, x=2 \\ 3 x-1, x2\end{array}\right.$ We have $(\mathrm{LHL}$ at $x=2)=\lim _{x \rightarrow 2^{-}} f(x)=\lim _{h \rightarrow 0} f(2-h)=\lim _{h \rightarrow 0}(2(2-h)+1)=5$ (RHL at $x=2$ ) $=\lim _{x \rightarrow 2^{+}} f(x)=\lim _{h \rightarrow 0}...
Read More →Construct a frequency table with class-intervals 0-5
Question: Construct a frequency table with class-intervals 0-5 (5 not included) of the following marks obtained by a group of 30 students in an examination. 0, 5, 7, 10, 12, 15, 20, 22, 25, 27, 8, 11, 17, 3, 6, 9, 17, 19, 21, 29, 31, 35, 37, 40, 42,45, 49, 4, 50, 16. Solution: The frequency table with class intervals $0-5,5-10,10-15, \ldots, 50-55$ is given below:...
Read More →Write down all subsets of each of the following sets:
Question: Write down all subsets of each of the following sets: (i) $A=\{A\}$ (ii) $B=\{a, b\}$ (iii) $\mathrm{C}=\{-2,3\}$ (iv) $D=\{-1,0,1\}$ (v) $E=\phi$ (vi) $F=\{2,\{3\}\}$ (vii) $G=\{3,4,\{5,6\}\}$ Solution: (i) The subsets of $\{\mathrm{A}\}$ are $\phi$ and $\{\mathrm{A}\}$ (ii) The subsets of $\{a, b\}$ are $\phi,\{a\},\{b\}$, and $\{a, b\}$. (iii) The subsets of $\{-2,3\}$ are $\phi,\{-2\},\{3\}$, and $\{-2,3\}$. (iv) The subsets of $\{-1,0,1\}$ are $\phi,\{-1\},\{0\},\{1\},\{-1,0\},\{0...
Read More →Construct a frequency table for the following weights (in gm) of 35 mangoes using the equal class intervals,
Question: Construct a frequency table for the following weights (in gm) of 35 mangoes using the equal class intervals, one of them is 40-50 (45 not included): 30, 40, 45, 32, 43, 50, 55, 62, 70, 70, 61, 62, 53, 52, 50, 42, 35, 37, 53, 55, 65, 70, 73, 74, 45, 46, 58, 59, 60, 62, 74, 34, 35, 70, 68. (i) What is the class mark of the class interval 40-45? (ii) What is the range of the above weights? (iii) How many classes are there? Solution: The frequency table for the given weights (in gm) of 35 ...
Read More →If in two
Question: If in two $\triangle \mathrm{PQR}, \frac{A B}{Q R}=\frac{B C}{P R}=\frac{C A}{P Q}$,then (a)Δ PQR~Δ CAB (b) Δ PQR ~ Δ ABC (c)Δ CBA ~ Δ PQR (d) Δ BCA ~ Δ PQR Solution: (a) Given, in two $\triangle \mathrm{ABC}$ and $\triangle \mathrm{PQR}, \frac{A B}{Q R}=\frac{B C}{P R}=\frac{C A}{P Q}$ which shows that sides of one triangle are proportional to the side of the other triangle, then their corresponding angles are also equal, so by SSS similarity, triangles are similar. i.e., Δ CAB Δ PQR...
Read More →For what value of λ is the function
Question: For what value of is the function $f(x)=\left\{\begin{array}{cl}\lambda\left(x^{2}-2 x\right), \text { if } x \leq 0 \\ 4 x+1, \text { if } x0\end{array}\right.$ continuous at $x=0 ?$ What about continuity at $x=\pm 1 ?$ Solution: The given function $f$ is $f(x)= \begin{cases}\lambda\left(x^{2}-2 x\right), \text { if } x \leq 0 \\ 4 x+1, \text { if } x0\end{cases}$ If $f$ is continuous at $x=0$, then $\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)=f(0)$ $\Rightarrow ...
Read More →The following is the distribution of weights (in kg) of 52 persons:
Question: The following is the distribution of weights (in kg) of 52 persons: (i) What is the lower limit of class 50-60? (ii) Find the class marks of the classes 40-50, 50-60. (iii) What is the class size? Solution: (i) The lower limit of the class $50-60$ is 50 . (ii) Class mark for the class $40-50$ : $\frac{40+50}{2}$ $=\frac{90}{2}$ $=45$ Again, class mark for the class $50-60$ : $\frac{50+60}{2}$ $=\frac{110}{2}$ $=55$ (iii) Here the class size is $40-30$, i.e. 10 ....
Read More →If ΔABC ~ ΔEDF and ΔABC is not similar
Question: If ΔABC ~ ΔEDF and ΔABC is not similar to ΔDEF, then which of the following is not true? (a) BC EF = AC FD (b) AB EF = AC DE (c) BC DE = AB EF (d) BC DE = AB FD Solution: (c) Given, $\triangle A B C \sim \triangle E D F$ $\therefore$ $\frac{A B}{E D}=\frac{B C}{D F}=\frac{A C}{E F}$ Taking first two terms, we get $\frac{A B}{E D}=\frac{B C}{D F}$ $\Rightarrow$ $A B \cdot D F=E D \cdot B C$ Or $B C \cdot D E=A B \cdot D F$ So, option (d) is true. Taking last two terms, we get $\frac{B C...
Read More →If A = {1} and B = {{1}, 2} then show that A ⊄ B.
Question: If A = {1} and B = {{1}, 2} then show that A B. Hint 1 ϵ A but 1 B. Solution: There is only one element in set A. Now, 1 is not an element of B. Therefore, A B....
Read More →Examine whether the following statements are true of false:
Question: Examine whether the following statements are true of false: (i) {a, b} {b, c, a} (ii) {a}ϵ {a, b, c} (iii) ϕ {a, b, c} (iv) {a, e} {x : x is a vowel in the English alphabet} (v) {x : x ϵ W, x + 5 = 5} =ϕ (vi) a ϵ {{a}, b} (vii) {a} {{a}, b} (viii) {b, c} {a, {b, c}} (ix) {a, a, b, b} = {a, b} (x) {a, b, a, b, a, b, .} is an infinite set. (xi) If A = set of all circles of unit radius in a plane and B = set of all circles in the same plane then AB. Solution: (i) False Explanation: Since ...
Read More →If the lengths of the diagonals of
Question: If the lengths of the diagonals of rhombus are 16 cm and 12 cm. Then, the length of the sides of the rhombus is (a) 9 cm (b) 10 cm (c) 8 cm (d) 20 cm Solution: (b)We know that, the diagonals of a rhombus are perpendicular bisector of each other. Given, AC = 16 cm and BD = 12 cm [let] AO = 8cm, SO = 6cm and AOB = 90 In right angled AOB,...
Read More →The marks scored by 20 students in a test are given below:
Question: The marks scored by 20 students in a test are given below: 54, 42, 68, 56, 62, 71, 78, 51, 72, 53, 44, 58, 47, 64, 41, 57, 89, 53, 84, 57. Complete the following frequency table: Solution: The frequency table can be completed as follows: The class interval with the greatest frequency (8) is 50-60...
Read More →In figure, if ∠BAC =90° and AD⊥BC. Then,
Question: In figure, if BAC =90 and ADBC. Then, (a) BD.CD = BC (b) AB.AC = BC (c) BD.CD=AD (d) AB.AC =AD Solution: (c)InΔADBandΔADC,...
Read More →The marks obtained by 40 students of class VIII in an examination are given below:
Question: The marks obtained by 40 students of class VIII in an examination are given below: 16, 17, 18, 3, 7, 23, 18, 13, 10, 21, 7, 1, 13, 21, 13, 15, 19, 24, 16, 3, 23, 5, 12, 18, 8, 12, 6, 8, 16, 5, 3, 5, 0, 7, 9, 12, 20, 10, 2, 23. Divide the data into five groups, namely 0-5, 5-10, 10-15, 15-20 and 20-25 and prepare a grouped frequency table. Solution: The frequency table for the marks of 40 students of class VIII in an examination is given below:...
Read More →The students of a school decided to beautify
Question: The students of a school decided to beautify the school on the annual day by fixing colourful flags on the straight passage of the school. They have 27 flags to be fixed at intervals of every 2 m. The flags are stored at the position of the middle most flag. Ruchi was given the responsibility of placing the flags. Ruchi kept her books where the flags were stored. She could carry only one flag at a time. How much distance she did cover in completing this job and returning back to collec...
Read More →State in each case whether A ⊂ B or A ⊄ B.
Question: State in each case whether A B or A B. A = {x : x is an even natural number less than 8}, B = {x : x is a natural number which divides 32} Solution: A B Explanation: we have, A = {2,4,6} and B = {1,2,4,8,16,32}. Thus , A B , since 6 B...
Read More →State in each case whether A ⊂ B or A ⊄ B.
Question: State in each case whether A B or A B. A = {x : x is a triangle in a plane}, B = {x : x is a rectangle in the same plane} Solution: A B We have, A = set of triangles and B = set of rectangles Now, we can see elements of A does not belong to B since the set of rectangles does not include a set of triangles....
Read More →Jaspal Singh repays his total loan of ₹ 118000
Question: Jaspal Singh repays his total loan of ₹ 118000 by paying every month starting with the first installment of ₹ 1000. If he increases the installment by ₹ 100 every month, what amount will be paid by him in the 30th installment? What amount of loan does he still have to pay after the 30th installment? Solution: Given that, Jaspal singh takes total loan = ₹ 118000 He repays his total loan by paying every month. His first installment = ₹ 1000 Second installment = 1000 + 100 = ₹ 1100 Third ...
Read More →State in each case whether A ⊂ B or A ⊄ B.
Question: State in each case whether A B or A B. A = {x : x is a square in a plane}, B = {x : x is a rectangle in the same plane} Solution: A B Explanation: Set of squares is a subset of set of rectangles since all squares are rectangles....
Read More →Solve the equation
Question: Solve the equation 4 + (-1) + 2 + + x = 437. Solution: Given equation is, 4 -1 + 2 + + x = 437 (i) Here, 4-1 + 2 + + x forms an AP with first term =-4, common difference =3, an= l = x $\because$ nth term of an AP, $a_{n}=l=a+(n-1) d$ $\Rightarrow \quad x=-4+(n-1) 3$ $\Rightarrow \quad \frac{x+4}{3}=n-1 \Rightarrow n=\frac{x+7}{3}$ ....(ii) $\therefore \quad$ Sum of an AP, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$ $S_{n}=\frac{x+7}{2 \times 3}\left[2(-4)+\left(\frac{x+4}{3}\right) \cdot 3\right]...
Read More →In the middle of a rectangular field measuring 30m × 20m, a well of 7 m diameter and 10 m depth is dug.
Question: In the middle of a rectangular field measuring 30m 20m, a well of 7 m diameter and 10 m depth is dug. The earth so removed is evenly spread over the remaining part of the field. Find the height through which the level of the field is raised. Solution: Letr m be the radius andhm be the depth of the well that is dug. Volumeof the well $=\pi r^{2} h=$$\frac{22}{7}$ (3.5 m)2 (10 m) = 385m3 Volume of the well = Volume of the rectangular field Volume of the rectangular field =385m3= 30 m 20 ...
Read More →State in each case whether A ⊂ B or A ⊄ B.
Question: State in each case whether A B or A B. A = {x : x is an isosceles triangle in the plane}, B = {x : x is an equilateral triangle in the same plane} Solution: A B Explanation: since all isosceles triangles are not equilateral triangles. Therefore set of isosceles triangle is not contained in the set of equilateral triangle....
Read More →State in each case whether A ⊂ B or A ⊄ B.
Question: State in each case whether A B or A B. A = {x : x is a real number,}, B = {x : x is a complex number} Solution: A B Explanation: we have, A = set of real numbers and B = set of complex numbers, a combination of the real and imaginary number in the form of a+ib, where a and b are real, and i is imaginary. Since, any real number can be expressed as complex number, A B....
Read More →What length of a solid cylinder 2 cm in diameter must be taken to recast into a hollow cylinder of length 16 cm,
Question: What length of a solid cylinder 2 cm in diameter must be taken to recast into a hollow cylinder of length 16 cm, external diameter 20 cm and thickness 2.5 mm? Solution: Here,r= Internal radius R= External radius = 10 cm h = Length of the cylinder t = Thickness = 0.25 cm Volume of the hollow cylinder = h(R2-r2) = (16) (102- (10-0.25)2) = 79 cm3 Volume of the solid cylinder = Volume of the hollow cylinder We know that the radius of the solid cylinder is 1 cm. (12)h= 79 h= 79 cm Hence, ...
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