Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle whose sides are
Question: Construct a triangle with sides $5 \mathrm{~cm}, 6 \mathrm{~cm}$ and $7 \mathrm{~cm}$ and then another triangle whose sides are $\frac{7}{5}$ of the corresponding sides of first triangle. Solution: Steps of Construction :Step 1. Draw a line segment BC = 4 cm.Step 2. With B as centre, draw an angle of 90o.Step 3. With B as centre and radius equal to 3 cm, cut an arc at the right angle and name it A.Step 4. Join AB and AC.Thus, △ ABC is obtained . Step 5. Extend $B C$ to $D$, such that $...
Read More →Construct a ∆PQR, in which PQ = 6 cm, QR = 7 cm and PR = 8 cm.
Question: Construct a $\triangle \mathrm{PQR}$, in which $\mathrm{PQ}=6 \mathrm{~cm}, \mathrm{QR}=7 \mathrm{~cm}$ and $\mathrm{PR}=8 \mathrm{~cm}$. Then, construct another triangle whose sides are $\frac{4}{5}$ times the corresponding sides of $\triangle \mathrm{PQR}$. Solution: Steps of ConstructionStep 1. Draw a line segment QR = 7 cm.Step 2. With Q as centre and radius 6 cm, draw an arc.Step 3. With R as centre and radius 8 cm, draw an arc cutting the previous arc at P.Step 4. Join PQ and PR....
Read More →As shown in the figure, a particle of mass 10 kg is placed at a point A.
Question: As shown in the figure, a particle of mass $10 \mathrm{~kg}$ is placed at a point $\mathrm{A}$. When the particle is slightly displaced to its right, it starts moving and reaches the point $\mathrm{B}$. The speed of the particle at $\mathrm{B}$ is $\mathrm{xm} / \mathrm{s}$. (Take $\left.\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)$ The value of 'x' to the nearest integer is_____ Solution: (10) Using work energy theorem, $\mathrm{W}_{\mathrm{g}}=\Delta \mathrm{K} . \mathrm{E}$ $(1...
Read More →Draw a line segment of length 8 cm and divide it internally in the ratio 4 : 5.
Question: (i) Draw a line segment of length 8 cm and divide it internally in the ratio 4 : 5. (ii) Draw a line segment of length 7.6 cm and divide it in the ratio 5 : 8. Measure the two parts. Solution: (i)Steps of Construction:Step 1. Draw a line segment AB = 8 cmStep 2. Draw a ray AX, making an acute angleBAX.Step 3. Along AX, mark (4 + 5 =) 9 points A1, A2, A3, A4, A5, A6, A7, A8and A9such thatAA1= A1A2= A2A3=A3A4=A4A5=A5A6=A6A7= A7A8= A8A9Step 4. Join A9B. Step 5. From $A_{4}$, draw $A_{4} D...
Read More →A constant power delivering machine has towed a box, which was initially at rest,
Question: A constant power delivering machine has towed a box, which was initially at rest, along a horizontal straight line. The distance moved by the box in time ' $t$ ' is proportional to :-$t^{2 / 3}$$\mathrm{t}^{3 / 2}$$\mathrm{t}$$t^{1 / 2}$Correct Option: , 2 Solution: (2) $\mathrm{P}=\mathrm{C}$ $\mathrm{FV}=\mathrm{C}$ $\mathrm{M} \frac{\mathrm{dV}}{\mathrm{dt}} \mathrm{V}=\mathrm{C}$ $\frac{\mathrm{V}^{2}}{2} \propto \mathrm{t}$ $\mathrm{V} \propto \mathrm{t}^{1 / 2}$ $\frac{\mathrm{dx...
Read More →Draw a line segment AB of length 7 cm.
Question: Draw a line segment $A B$ of length $7 \mathrm{~cm}$. Using ruler and compasses, find a point $P$ on $A B$ such that $\frac{A P}{A B}=\frac{3}{5}$. Solution: Steps of Construction:Step 1. Draw a line segment AB = 7 cm.Step 2. Draw a ray AX, making an acute angleBAX.Step 3. Along AX, mark 5 points (greater of 3 and 5) A1, A2, A3, A4and A5such thatAA1= A1A2= A2A3=A3A4=A4A5Step 4. Join A5B.Step 5. From A3, draw A3P parallel to A5B (draw an angle equal toAA5B), meeting AB in P. Here, $\mat...
Read More →Express each of the following as a rational number of the form
Question: Express each of the following as a rational number of the form $\frac{p}{q}$ : (i) $\frac{-8}{3}+\frac{-1}{4}+\frac{-11}{6}+\frac{3}{8}-3$ (ii) $\frac{6}{7}+1+\frac{-7}{9}+\frac{19}{21}+\frac{-12}{7}$ (iii) $\frac{15}{2}+\frac{9}{8}+\frac{-11}{3}+6+\frac{-7}{6}$ (iv) $\frac{-7}{4}+0+\frac{-9}{5}+\frac{19}{10}+\frac{11}{14}$ (v) $\frac{-7}{4}+\frac{5}{3}+\frac{-1}{2}+\frac{-5}{6}+2$ Solution: (i) $\frac{-8}{3}+\frac{-1}{4}+\frac{-11}{6}+\frac{3}{8}-3$ $=\frac{-64}{24}+\frac{-6}{24}+\fra...
Read More →A boy is rolling a 0.5kg ball on the frictionless floor
Question: A boy is rolling a $0.5 \mathrm{~kg}$ ball on the frictionless floor with the speed of $20 \mathrm{~ms}^{-1}$. The ball gets deflected by an obstacle on the way. After deflection it moves with $5 \%$ of its initial kinetic energy. What is the speed of the ball now?$19.0 \mathrm{~ms}^{-1}$$4.47 \mathrm{~ms}^{-1}$$14.41 \mathrm{~ms}^{-1}$$1.00 \mathrm{~ms}^{-1}$Correct Option: , 2 Solution: (2) Given, $\mathrm{m}=0.5 \mathrm{~kg}$ and $\mathrm{u}=20 \mathrm{~m} / \mathrm{s}$ Initial kine...
Read More →PQ is a chord of length 16 cm of a circle of radius 10 cm.
Question: PQ is a chord of length 16 cm of a circle of radius 10 cm. The tangent at P and Q intersect at a point T as shown in the figure. Find the length of TP Solution: Let TR =yand TP =xWe know that the perpendicular drawn from the centre to the chord bisects it. PR = RQNow, PR + RQ = 16⇒ PR + PR = 16⇒ PR = 8Now, in right triangle PORBy Using Pyhthagoras theorem, we havePO2= OR2+ PR2⇒ 102= OR2+ (8)2⇒ OR2= 36⇒ OR = 6Now, in right triangle TPRBy Using Pyhthagoras theorem, we haveTP2= TR2+ PR2⇒x...
Read More →Prove that the angles between the two tangents drawn form an external point to a circle is supplementary to the angle subtended
Question: Prove that the angles between the two tangents drawn form an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact at the centre. Solution: Given,PAandPBare the tangents drawn from a pointPto a circle with centreO. Also, the line segmentsOAandOBare drawn. To prove : $\angle A P B+\angle A O B=180^{\circ}$ We know that the tangent to a circle is perpendicular to the radius through the point of contact. $\therefore P A \perp...
Read More →In Fig. 1, the graph of a polynomial p (x) is shown.
Question: In Fig. 1, the graph of a polynomial p (x) is shown. The number of zeroes of p (x) is(a) 4(b) 1(c) 2(d) 3 Solution: In the following graph we observe that it intersectsx-axis atx =1. So it has only one zero. Hence the correct option is (b)...
Read More →Solve the following
Question: Evaluate: $\frac{\sec \theta \operatorname{cosec}\left(90^{\circ}-\theta\right)-\tan \theta \cot \left(90^{\circ}-\theta\right)+\sin ^{2} 55^{\circ}+\sin ^{2} 35^{\circ}}{\tan 10^{\circ} \tan 20^{*} \tan 60^{\circ} \tan 70^{\circ} \tan 80^{\circ}}$ Solution: Here we to evaluate the value of the expression given as follow $\frac{\sec \theta \operatorname{cosec}\left(90^{\circ}-\theta\right)-\tan \theta \cot \left(90^{\circ}-\theta\right)+\sin ^{2} 55^{\circ}+\sin ^{2} 35^{\circ}}{\tan 1...
Read More →A wire of length L and mass
Question: A wire of length $\mathrm{L}$ and mass per unit length $6.0 \times 10^{-3}$ $\mathrm{kgm}^{-1}$ is put under tension of $540 \mathrm{~N}$. Two consecutive frequencies that it resonates at are: $420 \mathrm{~Hz}$ and $490 \mathrm{~Hz}$. Then $\mathrm{L}$ in meters is:$2.1 \mathrm{~m}$$1.1 \mathrm{~m}$$8.1 \mathrm{~m}$$5.1 \mathrm{~m}$Correct Option: 1 Solution: (1) Fundamental frequency, $f=70 \mathrm{~Hz}$. The fundamental frequency of wire vibrating under tension $T$ is given by $f=\f...
Read More →The following table shows the ages of 100 persons of a locality.
Question: The following table shows the ages of 100 persons of a locality. Represent the above as the less than type frequency distribution and draw an ogive for the same. Solution: The given frequency distribution is We have to change the above distribution as less than type frequency distribution and also we have to draw its ogive. We have the following procedure to change the given distribution in to less than type distribution. = To draw its ogive, take the number of persons on y-axis and ag...
Read More →Simplify each of the following and write as a rational number of the form
Question: Simplify each of the following and write as a rational number of the form $\frac{p}{q}$ : (i) $\frac{3}{4}+\frac{5}{6}+\frac{-7}{8}$ (ii) $\frac{2}{3}+\frac{-5}{6}+\frac{-7}{9}$ (iii) $\frac{-11}{2}+\frac{7}{6}+\frac{-5}{8}$ (iv) $\frac{-4}{5}+\frac{-7}{10}+\frac{-8}{15}$ (v) $\frac{-9}{10}+\frac{22}{15}+\frac{13}{-20}$ (vi) $\frac{5}{3}+\frac{3}{-2}+\frac{-7}{3}+3$ Solution: (i) $\frac{3}{4}+\frac{5}{6}+\frac{-7}{8}$ $=\frac{18}{24}+\frac{20}{24}+\frac{-21}{24}$ $=\frac{18+20+(-21)}{2...
Read More →Check graphically whether the pair of linear equation
Question: Check graphically whether the pair of linear equation 4xy 8 = 0 and 2x 3y+ 6 = 0 is consistent. Also, find the vertices of the triangle formed by these lines with thex-axis. Solution: Here we have to draw the graph between two equations given by $4 x-y-8=0$.........(1) $2 x-3 y+6=0$............(2) Also we have to find the vertices of the triangle form by thex-axis and these lines. The first equation can written as follow $y=4 x-8$ ............(3) Now we are going to find the value ofya...
Read More →There harmonic waves having equal frequency v and same
Question: There harmonic waves having equal frequency $v$ and same intensity $\mathrm{I}_{0}$, have phase angles $0, \frac{\pi}{4}$ and $-\frac{\pi}{4}$ respectively. When they are superimposed the intensity of the resultant wave is close to:$5.8 \mathrm{I}_{0}$$0.2 \mathrm{I}_{0}$$3 \mathrm{I}_{0}$$\mathrm{I}_{0}$Correct Option: 1 Solution: (1)...
Read More →A transverse wave travels on a taut steel wire
Question: A transverse wave travels on a taut steel wire with a velocity of $v$ when tension in it is $2.06 \times 10^{4} \mathrm{~N}$. When the tension is changed to $T$, the velocity changed to $v / 2$. The value of $T$ is close to:$2.50 \times 10^{4} \mathrm{~N}$$5.15 \times 10^{3} \mathrm{~N}$$30.5 \times 10^{4} \mathrm{~N}$$10.2 \times 10^{2} \mathrm{~N}$Correct Option: , 2 Solution: (2) The velocity of a transverse wave in a stretched wire is given by $v=\sqrt{\frac{T}{\mu}}$ Where, $T=$ T...
Read More →If 2 cos θ − sin θ = x and cos θ − 3 sin θ = y.
Question: If 2 cos sin =xand cos 3 sin =y. Prove that 2x2+y2 2xy= 5. Solution: Given that: $2 \cos \theta-\sin \theta=x$ $\cos \theta-3 \sin \theta=y$ Then we have to prove that First we take theLHSand put the value ofxandy, we get $2 x^{2}+y^{2}-2 x y=2(2 \cos \theta-\sin \theta)^{2}+(\cos \theta-3 \sin \theta)^{2}-2(2 \cos \theta-\sin \theta)(\cos \theta-3 \sin \theta)$ $=2\left(4 \cos ^{2} \theta+\sin ^{2} \theta-4 \sin \theta \cos \theta\right)+\left(\cos ^{2} \theta+9 \sin ^{2} \theta-6 \si...
Read More →A transverse wave travels on a taut steel wire
Question: A transverse wave travels on a taut steel wire with a velocity of $v$ when tension in it is $2.06 \times 10^{4} \mathrm{~N}$. When the tension is changed to $T$, the velocity changed to $v / 2$. The value of $T$ is close to:$2.50 \times 10^{4} \mathrm{~N}$$5.15 \times 10^{3} \mathrm{~N}$$30.5 \times 10^{4} \mathrm{~N}$$10.2 \times 10^{2} \mathrm{~N}$Correct Option: , 2 Solution: (2) The velocity of a transverse wave in a stretched wire is given by $v=\sqrt{\frac{T}{\mu}}$ Where, $T=$ T...
Read More →Prove that the opposite sides of a quadrilateral circumscribing a circle
Question: Prove that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. Solution: Given, a quadrilateralABCDcircumscribes a circle with centre O. To prove: $\angle A O B+\angle C O D=180^{\circ}$ and $\angle A O D+\angle B O C=180^{\circ}$ Join $O P, O Q, O R$ and $O S$. We know that the tangents drawn from an external point of a circle subtend equal angles at the centre. $\therefore \angle 1=\angle 7, \angle 2=\angle 3, \angle...
Read More →Without using trigonometric tables, evaluate the following:
Question: Without using trigonometric tables, evaluate the following: $\frac{\sec 37^{\circ}}{\operatorname{cosec} 53^{\circ}}+2 \cot 15^{\circ} \cot 25^{\circ} \cot 45^{\circ} \cot 75^{\circ} \cot 65^{\circ}-3\left(\sin ^{2} 18^{\circ}+\sin ^{2} 72^{\circ}\right)$ Solution: We have to evaluate the following expression $\frac{\sec 37^{\circ}}{\operatorname{cosec} 53^{\circ}}+2 \cot 15^{\circ} \cot 25^{\circ} \cot 45^{\circ} \cot 75^{\circ} \cot 65^{\circ}-3\left(\sin ^{2} 18^{\circ}+\sin ^{2} 72...
Read More →Fill in the blanks:
Question: Fill in the blanks: (i) $\frac{-4}{13}-\frac{-3}{26}=. .$ (ii) $\frac{-9}{14}+\ldots=-1$ (iii) $\frac{-7}{9}+\ldots=3$ (iv) $\ldots+\frac{15}{23}=4$ Solution: (i) $\frac{-4}{13}-\frac{-3}{26}=\frac{-8}{26}-\frac{-3}{26}=\frac{-8-(-3)}{26}=\frac{-8+3}{26}=\frac{-5}{26}$ (ii) $\frac{-9}{14}+x=-1$ $\Rightarrow x=-1-\frac{-9}{14}$ $\Rightarrow x=\frac{-14}{14}-\frac{-9}{14}$ $\Rightarrow x=\frac{-14-(-9)}{14}$ $\Rightarrow x=\frac{-14+9}{14}$ $\Rightarrow x=\frac{-5}{14}$ (iii) $\frac{-7}{...
Read More →A one metre long (both ends open) organ pipe is kept in a gas that has double the density of air at STP.
Question: A one metre long (both ends open) organ pipe is kept in a gas that has double the density of air at STP. Assuming the speed of sound in air at STP is $300 \mathrm{~m} / \mathrm{s}$, the frequency difference between the fundamental and second harmonic of this pipe is____ $\mathrm{Hz}$. Solution: (106) Given : $V_{\text {air }}=300 \mathrm{~m} / \mathrm{s}, \rho_{\text {gas }}=2 \rho$ air $U \sin g, V=\sqrt{\frac{B}{\rho}}$ $\frac{V_{\text {gas }}}{V_{\text {air }}}=\frac{\sqrt{\frac{B}{...
Read More →A quadrilateral ABCD is drawn to circumscribe a circle.
Question: A quadrilateralABCD is drawn to circumscribe a circle. Prove that sums of opposite sides are equal. Solution: We know that the tangents drawn from an external point to a circle are equal. $\therefore A P=A S \ldots \ldots(i) \quad[$ tangents from $A]$ $B P=B Q \ldots \ldots \ldots(i i) \quad[$ tangents from $B]$ $C R=C Q \ldots \ldots \ldots \ldots$ (iii) [tangents from $C]$ $D R=D S \ldots \ldots \ldots . .(i v) \quad[$ tangents from $D]$ $\therefore A B+C D=(A P+B P)+(C R+D R)$ $=(A ...
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