In Figure 5, ∆ABC is right angled at B,
Question: In Figure 5, ABC is right angled at B, BC = 7 cm and AC AB = 1 cm. Find the value of cos A sin A. Solution: It is given thatis right angled atB,BC= 7 cm andAC AB= 1 cm then we have to find the value of The following diagram is given ACAB= 1 (1) Now, apply the Pythagoras theorem in, we get $A C^{2}=A B^{2}+B C^{2}$ $\Rightarrow \quad A C^{2}-A B^{2}=B C^{2}$ $\Rightarrow(A C-A B)(A C+A B)=7^{2}$ $\Rightarrow \quad A C+A B=9$..........(2) Now add the equation (1) and (2), we get $2 A C=5...
Read More →In the given figure, if AB = AC, prove that BE = CE.
Question: In the given figure, ifAB=AC, prove that BE =CE. Solution: Given, $A B=A C$ We know that the tangents from an external point are equal. $\therefore A D=A F, B D=B E$ and $C F=C E \ldots \ldots . .(i)$ Now, $A B=A C$ $\Rightarrow A D+D B=A F+F C$ $\Rightarrow A F+D B=A F+F C \quad[$ from $(\mathrm{i})]$ $\Rightarrow D B=F C$ $\Rightarrow B E=C E \quad[$ from (i) $]$ Hence proved....
Read More →In a resonance tube experiment when the tube is filled
Question: In a resonance tube experiment when the tube is filled with water up to a height of $17.0 \mathrm{~cm}$ from bottom, it resonates with a given tuning fork. When the water level is raised the next resonance with the same tuning fork occurs at a height of $24.5 \mathrm{~cm}$. If the velocity of sound in air is $330 \mathrm{~m} / \mathrm{s}$, the tuning fork frequency is :$2200 \mathrm{~Hz}$$550 \mathrm{~Hz}$$1100 \mathrm{~Hz}$$3300 \mathrm{~Hz}$Correct Option: 1 Solution: (1) Here, $l_{1...
Read More →The sum of two rational numbers is −8.
Question: The sum of two rational numbers is $-8$. If one of the numbers is $\frac{-15}{7}$, find the other. Solution: It is given that the sum of two rational numbers is $-8$, where one of the number $s$ is $\frac{-15}{7}$. Let the other rational number be $x$. $\therefore x+\left(\frac{-15}{7}\right)=-8$ $\Rightarrow x=\frac{-8}{1}-\frac{-15}{7}$ $\Rightarrow x=\frac{-56}{7}-\frac{-15}{7}$ $=\frac{-56-(-15)}{7}$ $=\frac{-56+15}{7}$ $=\frac{-41}{7}$ Therefore, the other rational number is $\fra...
Read More →Prove that
Question: Prove that $\frac{\tan \theta-\cot \theta}{\sin \theta \cos \theta}=\tan ^{2} \theta-\cot ^{2} \theta$. Solution: Here we have to prove that $\frac{\tan \theta-\cot \theta}{\sin \theta \cos \theta}=\tan ^{2} \theta-\cot ^{2} \theta$ First we takeLHSand use the identities $\tan \theta=\frac{\sin \theta}{\cos \theta}$ and $\cot \theta=\frac{\cos \theta}{\sin \theta}$ $L H S=\frac{\tan \theta-\cot \theta}{\sin \theta \cos \theta}$ $=\frac{\frac{\sin \theta}{\cos \theta}-\frac{\cos \theta}...
Read More →Prove that the tangents drawn at the ends of the diameter of a circle are parallel.
Question: Prove that the tangents drawn at the ends of the diameter of a circle are parallel. Solution: Here, $P T$ and $Q S$ are the tangents to the circle with centre $O$ and $A B$ is the diameter. Now, radius of a circle is perpendicular to the tangent at the point of contact. $\therefore \mathrm{OA} \perp \mathrm{AT}$ and $\mathrm{OB} \perp \mathrm{BS}$ (since tangents drawn from an external point are perpendicular to the radius at point of contact) $\cdot \angle O A T=\angle O B O=90^{\circ...
Read More →Assume that the displacement (s) of air is proportional to the pressure difference
Question: Assume that the displacement $(s)$ of air is proportional to the pressure difference $(\Delta p)$ created by a sound wave. Displacement $(s)$ further depends on the speed of sound (v), density of air $(\rho)$ and the frequency $(f)$. If $\Delta p \sim 10 \mathrm{~Pa}$, $v \sim 300 \mathrm{~m} / \mathrm{s}, p \sim q \mathrm{~kg} / \mathrm{m}^{3}$ and $f \sim 1000 \mathrm{~Hz}$, then $s$ will be of the order of (take the multiplicative constant to be 1 )$\frac{3}{100} \mathrm{~mm}$$10 \m...
Read More →The sum of two numbers is
Question: The sum of two numbers is $\frac{-4}{3}$. If one of the numbers is $-5$, find the other. Solution: It is given that the sum of two numbers is $\frac{-4}{3}$, where one of the number $s$ is $-5$. Let the other number be $x$. $\therefore x+(-5)=\frac{-4}{3}$ $\Rightarrow x=\frac{-4}{3}-\left(\frac{-5}{1}\right)$ $\Rightarrow x=\frac{-4}{3}-\frac{-15}{3}$ $\Rightarrow x=\frac{-4-(-15)}{3}$ $\Rightarrow x=\frac{-4+15}{3}=\frac{11}{3}$...
Read More →Prove that the length of two tangents drawn from an external point to a circle are equal.
Question: Prove that the length of two tangents drawn from an external point to a circle are equal. Solution: Given two tangents AP and AQ are drawn from a point A to a circle with centre O. To prove: $A P=A Q$ Join $O P, O Q$ and $O A$. $A P$ is tangent at $P$ and $O P$ is the radius. $\therefore O P \perp A P($ since tangents drawn from an external point are perpendicular to the radius at the point of contact) Similarly, $O Q \perp A Q$ In the right $\triangle O P A$ and $\triangle O Q A$, we ...
Read More →The sum of two numbers is
Question: The sum of two numbers is $\frac{-1}{3}$. If one of the numbers is $\frac{-12}{3}$, find the other. Solution: It is given that the sum of two numbers is $\frac{-1}{3}$, where one of the number $s$ is $\frac{-12}{3}$. Let the other number be $x$. $\therefore x+\frac{-12}{3}=\frac{-1}{3}$ $\Rightarrow x=\frac{-1}{3}-\frac{-12}{3}$ $\Rightarrow x=\frac{-1-(-12)}{3}=\frac{-1+12}{3}=\frac{11}{3}$...
Read More →The sum of the numerator and the denominator
Question: The sum of the numerator and the denominator of a fraction is 8 . If 3 is added to both the numerator and the denominator, the fraction becomes $\frac{3}{4}$. Find the fraction. Solution: Here we are assuming that numerator and denominator are $x$ and $y$ respectively, then fraction will be $\frac{x}{y}$ and we have to find the value of $\frac{x}{y}$. From the given condition, x + y= 8 (1) If 3 are added in numerator and denominator then fraction will be $\frac{3}{4}$, from this we hav...
Read More →The driver of a bus approaching a big wall notices that the frequency
Question: The driver of a bus approaching a big wall notices that the frequency of his bus's horn changes from $420 \mathrm{~Hz}$ to $490 \mathrm{~Hz}$ when he hears it after it gets reflected from the wall. Find the speed of the bus if speed of the sound is $330 \mathrm{~ms}^{-1}$.$91 \mathrm{kmh}^{-1}$$81 \mathrm{kmh}^{-1}$$61 \mathrm{kmh}^{-1}$$71 \mathrm{kmh}^{-1}$Correct Option: 1 Solution: (1) From the Doppler's effect of sound, frequency appeared at wall $f_{w}=\frac{330}{330-v} \cdot f$ ...
Read More →The sum of the two numbers is
Question: The sum of the two numbers is $\frac{5}{9}$. If one of the numbers is $\frac{1}{3}$, find the other. Solution: It is given that the sum of two numbers is $\frac{5}{9}$, where one of the number $s$ is $\frac{1}{3}$. Let the other number be $x$. $\therefore \quad x+\frac{1}{3}=\frac{5}{9}$ $\Rightarrow x=\frac{5}{9}-\frac{1}{3}$ $\Rightarrow x=\frac{5}{9}-\frac{3}{9}$ $\Rightarrow x=\frac{5-3}{9}$ $\Rightarrow x=\frac{2}{9}$...
Read More →If α and β are zeroes of the quadratic polynomial
Question: If and are zeroes of the quadratic polynomialx2 6x+a; find the value of 'a' if 3 + 2 = 20. Solution: Given that:andare the zeroes of the quadratic polynomialx2 6x+aand 3+ 2= 20, then we have to find the value ofa. We have the following procedure. $\alpha+\beta=-\frac{-6}{1}$ $=6$...............(1) $\alpha \beta=\frac{a}{1}$ $=a$.........(2) $3 \alpha+2 \beta=20$ $\Rightarrow 2(\alpha+\beta)+\alpha=20$ $\Rightarrow \quad 2 \times 6+\alpha=20$ $\Rightarrow \quad \alpha=8$ Now, we are put...
Read More →Fill in the blanks.
Question: Fill in the blanks.(i) A line intersecting a circle at two distinct points is called a ....... .(ii) A circle can have ....... parallel tangents at the most.(iii) The common point of a tangent to a circle and the circle is called the ....... .(iv) A circle can have ...... tangents. Solution: (i) A line intersecting a circle at two distinct points is called asecant.(ii) A circle can havetwoparallel tangents at the most.(iii) The common point of a tangent to a circle and the circle is ca...
Read More →Evaluate each of the following:
Question: Evaluate each of the following: (i) $\frac{2}{3}-\frac{3}{5}$ (ii) $-\frac{4}{7}-\frac{2}{-3}$ (iii) $\frac{4}{7}-\frac{-5}{-7}$ (iv) $-2-\frac{5}{9}$ (v) $\frac{-3}{-8}-\frac{-2}{7}$ (vi) $\frac{-4}{13}-\frac{-5}{26}$ (vii) $\frac{-5}{14}-\frac{-2}{7}$ (viii) $\frac{13}{15}-\frac{12}{25}$ (ix) $\frac{-6}{13}-\frac{-7}{13}$ (x) $\frac{7}{24}-\frac{19}{36}$ (xi) $\frac{5}{63}-\frac{-8}{21}$ Solution: (i) $\frac{2}{3}-\frac{3}{5}=\frac{10}{15}-\frac{9}{15}=\frac{10-9}{15}=\frac{1}{15}$ (...
Read More →Two tangents BC and BD are drawn to a circle with centre O,
Question: Two tangentsBCandBDare drawn to a circle with centreO,such that CBD= 120. Prove thatOB= 2BC. Solution: Here, $O B$ is the bisector of $\angle C B D$. ( $\mathrm{T}$ wo tangents are equally inclined to the line segment joining the centre to that point) $\therefore \angle C B O=\angle D B O=\frac{1}{2} \angle C B D=60^{\circ}$ $\therefore$ From $\Delta B O D, \angle B O D=30^{\circ}$ Now, from right - angled $\Delta B O D$, $\frac{B D}{O B}=\sin 30^{\circ}$ $\Rightarrow \frac{B D}{O B}=\...
Read More →Use Euclid's division algorithm
Question: Use Euclid's division algorithm to find the HCF of 10224 and 9648. Solution: Here we have to find the HCF of the numbers 10224 and 9648 by using Euclids division algorithm. We know that If we divideabybandris the remainder andqis the quotient, Euclids Lemma says that $A=b q+r$, where $0 \leq rb$ And HCF of (a, b) = HCF of (b, r) Here $a=10224$ and $b=9648$ Therefore, we have the following procedure, $10224=9648 \times 1+576$ Now, we apply the division algorithm on 9648 and 576. $9648=5...
Read More →In the given figure, PA and PB are two tangents from an external point P to a circle with centre O.
Question: In the given figure, PA and PB are two tangents from an external point P to a circle with centre O. If PBA = 65∘, find OABand APB Solution: We know that tangents drawn from the external point are congruent. PA = PBNow, In isoceles triangle APBAPB + PBA + PAB = 180∘ [Angle sum property of a triangle]⇒ APB + 65∘+ 65∘= 180∘ [∵PBA = PAB = 65∘]⇒ APB = 50∘We know that the radius and tangent are perperpendular at their point of contactOBP = OAP = 90∘Now, In quadrilateral AOBPAOB + OBP + APB +...
Read More →For a transverse wave travelling along a straight line,
Question: For a transverse wave travelling along a straight line, the distance between two peaks (crests) is $5 \mathrm{~m}$, while the distance between one crest and one trough is $1.5 \mathrm{~m}$. The possible wavelengths (in $\mathrm{m}$ ) of the waves are :$1,3,5, \ldots \ldots$$\frac{1}{1}, \frac{1}{3}, \frac{1}{5}, \ldots \ldots$$1,2,3, \ldots \ldots$$\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \ldots \ldots$Correct Option: , 2 Solution: (2) Given : Distance between one crest and one trough $=...
Read More →Prove that
Question: Prove that $\sqrt{7}$ is an irrational number. Solution: We have to prove that $\sqrt{7}$ is an irrational number We will prove this by contradiction: Let $\sqrt{7}$ be an irrational number such that $\sqrt{7}=\frac{x}{y}$ where $x$ and $y$ are co prime So' $\sqrt{7}=\frac{x}{y}$ $\Rightarrow 7=\frac{x^{2}}{y^{2}}$ $\Rightarrow \quad 7 y^{2}=x^{2}$ $\Rightarrow \quad 7$ divides $x^{2}$ $\Rightarrow \quad 7$ divieds $x$.......(1) This means that: $x=7 k$ where $\mathrm{k}$ is any positi...
Read More →In the given figure, PA and PB are the tangents to a circle with centre O
Question: In the given figure,PAandPBare the tangents to a circle with centreO.Show that the pointsA,O,B,P are concyclic. Solution: Here, $\mathrm{OA}=\mathrm{OB}$ And $\mathrm{OA} \perp \mathrm{AP}, \mathrm{OA} \perp \mathrm{BP}$ (Since tangents drawn from an external point are perpendicular to the radius at the point of contact) $\therefore \angle \mathrm{OAP}=90^{\circ}, \angle \mathrm{OBP}=90^{\circ}$ $\therefore \angle \mathrm{OAP}+\angle \mathrm{OBP}=90^{\circ}+90^{\circ}=180^{\circ}$ $\th...
Read More →Subtract the first rational number from the second in each of the following:
Question: Subtract the first rational number from the second in each of the following: (i) $\frac{3}{8}, \frac{5}{8}$ (ii) $\frac{-7}{9}, \frac{4}{9}$ (iii) $\frac{-2}{11}, \frac{-9}{11}$ (iv) $\frac{11}{13}, \frac{-4}{13}$ (v) $\frac{1}{4}, \frac{-3}{8}$ (vi) $\frac{-2}{3}, \frac{5}{6}$ (vii) $\frac{-6}{7}, \frac{-13}{14}$ (viii) $\frac{-8}{33}, \frac{-7}{22}$ Solution: (i) $\frac{5}{8}-\frac{3}{8}=\frac{5-3}{8}=\frac{2}{8}=\frac{1}{4}$ (ii) $\frac{4}{9}-\frac{-7}{9}=\frac{4-(-7)}{9}=\frac{4+7}...
Read More →Find the mode of the following data:
Question: Find the mode of the following data: Solution: We have to find the mode of the following distribution, The class (40-60) has the maximum frequency; therefore this is the modal class. Lower limit of the modal class $x_{k}=40$ Width of the class intervalh= 20 Frequency of the modal classfk= 18 Frequency of the class preceding the modal classfk1= 6 Frequency of the class succeeding the modal classfk+1= 10 Now, we have the following formula to find the value of mode. Mode $=x_{k}+h\left(\f...
Read More →A uniform thin rope of length 12m and mass 6kg hangs vertically
Question: A uniform thin rope of length $12 \mathrm{~m}$ and mass $6 \mathrm{~kg}$ hangs vertically from a rigid support and a block of mass $2 \mathrm{~kg}$ is attached to its free end. A transverse short wave-train of wavelength $6 \mathrm{~cm}$ is produced at the lower end of the rope. What is the wavelength of the wavetrain (in $\mathrm{cm}$ ) when it reaches the top of the rope?36129Correct Option: , 3 Solution: (3) Using, $V=f \lambda$ $\frac{V_{1}}{\lambda_{1}}=\frac{V_{2}}{\lambda_{2}} \...
Read More →