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MATHEMATICS 9709

Cambridge International AS and A Level

 Name of student Date Adm. number Year/grade Stream Subject Mechanics 2 (M2) Variant(s) P41, P43 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 8 9 10 11 12 Total
Marks 3 5 5 6 6 5 6 8 9 8 9 12 82
Score

Get Mathematics 9709 Topical Questions (2010-2021) $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/51/O/N/10/1, Topic: - A horizontal circular disc rotates with constant angular speed$9 \mathrm{rad} \mathrm{s}^{-1}$about its centre$O$. A particle of mass$0.05 \mathrm{~kg}$is placed on the disc at a distance$0.4 \mathrm{~m}$from$O$. The particle moves with the disc and no sliding takes place. Calculate the magnitude of the resultant force exerted on the particle by the disc.$[3]$Question 2 Code: 9709/53/O/N/11/2, Topic: - An object is made from two identical uniform rods$A B$and$B C$each of length$0.6 \mathrm{~m}$and weight$7 \mathrm{~N}$. The rods are rigidly joined to each other at$B$and angle$A B C=90^{\circ}$.$\text{(i)}$Calculate the distance of the centre of mass of the object from$B$.$[1]$The object is freely suspended at$A$and a force of magnitude$F \mathrm{~N}$is applied to the rod$B C$at$C$. The object is in equilibrium with$A B$inclined at$45^{\circ}$to the horizontal.$\text{(ii)}~~\text{(a)}$![9709-53-O-N-11-2a.PNG] Calculate$F$given that the force acts horizontally as shown in Fig. 1.$[2]\text{(b)}$![9709-53-O-N-11-2b.PNG] Calculate$F$given instead that the force acts perpendicular to the rod as shown in Fig. 2.$[2]$Question 3 Code: 9709/53/O/N/15/2, Topic: - One end of a light inextensible string of length$0.5 \mathrm{~m}$is attached to a fixed point$A$. A particle$P$of mass$0.2 \mathrm{~kg}$is attached to the other end of the string.$P$moves with constant speed in a horizontal circle with centre$O$which is$0.4 \mathrm{~m}$vertically below$A$.$\text{(i)}$Show that the tension in the string is$2.5 \mathrm{~N}$.$[2]\text{(ii)}$Find the speed of$P$.$[3]$Question 4 Code: 9709/53/O/N/18/2, Topic: - Question 5 Code: 9709/53/O/N/13/3, Topic: - A particle$P$of mass$0.5 \mathrm{~kg}$moves in a horizontal circle on the smooth inner surface of a hollow cone which is fixed with its axis vertical and its vertex downwards.$P$moves with angular speed$5 \mathrm{rad} \mathrm{s}^{-1}$in a circle of radius$0.4 \mathrm{~m}$(see diagram). Show that the semi-vertical angle of the cone is$45^{\circ}$and calculate the magnitude of the force exerted on$P$by the surface of the cone.$[6]$Question 6 Code: 9709/51/O/N/15/3, Topic: - A particle$P$of mass$0.3 \mathrm{~kg}$moves in a straight line on a smooth horizontal surface.$P$passes through a fixed point$O$of the line with velocity$8 \mathrm{~m} \mathrm{~s}^{-1}.$A force of magnitude$2 x \mathrm{~N}$acts on$P$in the direction$P O$, where$x \mathrm{~m}$is the displacement of$P$from$O$.$\text{(i)}$Show that$v \displaystyle\frac{\mathrm{d} v}{\mathrm{~d} x}=k x$and state the value of the constant$k$.$[2]\text{(ii)}$Find the value of$x$at the instant when$P$comes to instantaneous rest.$[3]$Question 7 Code: 9709/51/O/N/19/3, Topic: - A smooth horizontal surface has two fixed points$O$and$A$which are$0.8 \mathrm{~m}$apart. A particle$P$of mass$0.25 \mathrm{~kg}$is projected with velocity$3 \mathrm{~m} \mathrm{~s}^{-1}$horizontally from$A$in the direction away from$O$. The velocity of$P$is$v \mathrm{~m} \mathrm{~s}^{-1}$when the displacement of$P$from$O$is$x \mathrm{~m}$. A force of magnitude$k v^{2} x^{-2} \mathrm{~N}$opposes the motion of$P$.$\text{(i)}$Show that$\displaystyle v \displaystyle\frac{\mathrm{d} v}{\mathrm{~d} x}=-4 k v^{2} x^{-2}$.$[1]\text{(ii)}$Express$v$in terms of$k$and$x$.$[5]$Question 8 Code: 9709/53/O/N/14/4, Topic: - A particle$P$is projected with speed$50 \mathrm{~m} \mathrm{~s}^{-1}$at an angle of$30^{\circ}$above the horizontal from a point$O$on a horizontal plane.$\text{(i)}$Calculate the speed of$P$when it has been in motion for$4 \mathrm{~s}$, and calculate another time at which$P$has this speed.$[5]\text{(ii)}$Find the distance$O P$when$P$has been in motion for$4 \mathrm{~s}$.$[2]$Question 9 Code: 9709/53/O/N/10/5, Topic: - A light elastic string has natural length$2 \mathrm{~m}$and modulus of elasticity$\lambda \mathrm{N}$. The ends of the string are attached to fixed points$A$and$B$which are at the same horizontal level and$2.4 \mathrm{~m}$apart. A particle$P$of mass$0.6 \mathrm{~kg}$is attached to the mid-point of the string and hangs in equilibrium at a point$0.5 \mathrm{~m}$below$A B$(see diagram).$\text{(i)}$Show that$\lambda=26$.$[4]P$is projected vertically downwards from the equilibrium position, and comes to instantaneous rest at a point$0.9 \mathrm{~m}$below$A B$.$\text{(ii)}$Calculate the speed of projection of$P$.$[5]$Question 10 Code: 9709/53/O/N/14/5, Topic: - Two light elastic strings each have one end attached to a fixed horizontal beam. One string has natural length$0.6 \mathrm{~m}$and modulus of elasticity$12 \mathrm{~N}$; the other string has natural length$0.7 \mathrm{~m}$and modulus of elasticity$21 \mathrm{~N}$. The other ends of the strings are attached to a small block$B$of weight$W \mathrm{~N}$. The block hangs in equilibrium$d \mathrm{~m}$below the beam, with both strings vertical (see diagram).$\text{(i)}$Given that the tensions in the strings are equal, find$d$and$W$.$[4]$The small block is now raised vertically to the point$0.7 \mathrm{~m}$below the beam, and then released from rest.$\text{(ii)}$Find the greatest speed of the block in its subsequent motion.$[4]$Question 11 Code: 9709/51/O/N/19/6, Topic: -$A$and$B$are two fixed points on a vertical axis with$A 0.6 \mathrm{~m}$above$B$. A particle$P$of mass$0.3 \mathrm{~kg}$is attached to$A$by a light inextensible string of length$0.5 \mathrm{~m}$. The particle$P$is attached to$B$by a light elastic string with modulus of elasticity$46 \mathrm{~N}$. The particle$P$moves with constant angular speed$8 \mathrm{rad} \mathrm{s}^{-1}$in a horizontal circle with centre at the mid-point of$A B$.$\text{(i)}$Find the speed of$P$.$[2]\text{(ii)}$Calculate the tension in the string$B P$and hence find the natural length of this string.$[7]$Question 12 Code: 9709/51/O/N/12/7, Topic: - A light elastic string has natural length$3 \mathrm{~m}$and modulus of elasticity$45 \mathrm{~N}$. A particle$P$of weight$6 \mathrm{~N}$is attached to the mid-point of the string. The ends of the string are attached to fixed points$A$and$B$which lie in the same vertical line with$A$above$B$and$A B=4 \mathrm{~m}$. The particle$P$is released from rest at the point$1.5 \mathrm{~m}$vertically below$A$.$\text{(i)}$Calculate the distance$P$moves after its release before first coming to instantaneous rest at a point vertically above$B$. (You may assume that at this point the part of the string joining$P$to$B$is slack.)$[4]\text{(ii)}$Show that the greatest speed of$P$occurs when it is$2.1 \mathrm{~m}$below$A$, and calculate this greatest speed.$[5]\text{(iii)}$Calculate the greatest magnitude of the acceleration of$P$.$[3]\$

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