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### MATHEMATICS 9709

#### Cambridge International AS and A Level

 Name of student Date Adm. number Year/grade Stream Subject Mechanics 1 (M1) Variant(s) P41, P42, P43 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 8 Total
Marks 4 7 8 7 11 10 11 11 69
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 8 questions Question 1 Code: 9709/43/M/J/15/1, Topic: - A block is pulled along a horizontal floor by a horizontal rope. The tension in the rope is$500 \mathrm{~N}$and the block moves at a constant speed of$2.75 \mathrm{~m} \mathrm{~s}^{-1}$. Find the work done by the tension in$40 \mathrm{~s}$and find the power applied by the tension.$$Question 2 Code: 9709/42/M/J/12/3, Topic: - A particle$P$moves in a straight line, starting from the point$O$with velocity$2 \mathrm{~m} \mathrm{~s}^{-1}$. The acceleration of$P$at time$t \mathrm{~s}$after leaving$O$is$2 t^{\frac{2}{3}} \mathrm{~m} \mathrm{~s}^{-2}$.$\text{(i)}$Show that$t^{\frac{5}{3}}=\frac{5}{6}$when the velocity of$P$is$3 \mathrm{~m} \mathrm{~s}^{-1}$.$\text{(ii)}$Find the distance of$P$from$O$when the velocity of$P$is$3 \mathrm{~ms}^{-1}$.$$Question 3 Code: 9709/43/O/N/11/5, Topic: - A particle$P$moves in a straight line. It starts from rest at$A$and comes to rest instantaneously at$B$. The velocity of$P$at time$t$seconds after leaving$A$is$v \mathrm{~m} \mathrm{~s}^{-1}$, where$v=6 t^{2}-k t^{3}$and$k$is a constant.$\text{(i)}$Find an expression for the displacement of$P$from$A$in terms of$t$and$k$.$\text{(ii)}$Find an expression for$t$in terms of$k$when$P$is at$B$.$$Given that the distance$A B$is$108 \mathrm{~m}$, find$\text{(iii)}$the value of$k$,$\text{(iv)}$the maximum value of$v$when the particle is moving from$A$towards$B$.$$Question 4 Code: 9709/43/M/J/15/5, Topic: - ![9709-43-M-J-15-5a.PNG] Four coplanar forces of magnitudes$4 \mathrm{~N}, 8 \mathrm{~N}, 12 \mathrm{~N}$and$16 \mathrm{~N}$act at a point. The directions in which the forces act are shown in Fig. 1.$\text{(i)}$Find the magnitude and direction of the resultant of the four forces.$$ ![9709-43-M-J-15-5b.PNG] The forces of magnitudes$4 \mathrm{~N}$and$16 \mathrm{~N}$exchange their directions and the forces of magnitudes$8 \mathrm{~N}$and$12 \mathrm{~N}$also exchange their directions (see Fig. 2).$\text{(ii)}$State the magnitude and direction of the resultant of the four forces in Fig.$2.$Question 5 Code: 9709/42/O/N/19/6, Topic: - A block of mass$3 \mathrm{~kg}$is initially at rest on a rough horizontal plane. A force of magnitude$6 \mathrm{~N}$is applied to the block at an angle of$\theta$above the horizontal, where$\cos \theta=\frac{24}{25}$. The force is applied for a period of$5 \mathrm{~s}$, during which time the block moves a distance of$4.5 \mathrm{~m}$.$\text{(i)}$Find the magnitude of the frictional force on the block.$\text{(ii)}$Show that the coefficient of friction between the block and the plane is$0.165$, correct to 3 significant figures.$\text{(iii)}$When the block has moved a distance of$4.5 \mathrm{~m}$, the force of magnitude$6 \mathrm{~N}$is removed and the block then decelerates to rest. Find the total time for which the block is in motion.$$Question 6 Code: 9709/41/O/N/11/7, Topic: - A particle$P$starts from a point$O$and moves along a straight line.$P$'s velocity$t$s after leaving$O$is$v \mathrm{~m} \mathrm{~s}^{-1}$, where $$v=0.16 t^{\frac{3}{2}}-0.016 t^{2}$$$P$comes to rest instantaneously at the point$A$.$\text{(i)}$Verify that the value of$t$when$P$is at$A$is 100.$\text{(ii)}$Find the maximum speed of$P$in the interval$0< t <100$.$\text{(iii)}$Find the distance$O A$.$\text{(iv)}$Find the value of$t$when$P$passes through$O$on returning from$A$.$$Question 7 Code: 9709/42/O/N/11/7, Topic: - A tractor travels in a straight line from a point$A$to a point$B$. The velocity of the tractor is$v \mathrm{~m} \mathrm{~s}^{-1}$at time$t \mathrm{~s}$after leaving$A$.$\text{(i)}$ The diagram shows an approximate velocity-time graph for the motion of the tractor. The graph consists of two straight line segments. Use the graph to find an approximation for$\text{(a)}$the distance$A B$,$\text{(b)}$the acceleration of the tractor for$0< t <400$and for$400< t <800$.$\text{(ii)}$The actual velocity of the tractor is given by$v=0.04 t-0.00005 t^{2}$for$0 \leqslant t \leqslant 800$.$\text{(a)}$Find the values of$t$for which the actual acceleration of the tractor is given correctly by the approximate velocity-time graph in part$\text{(i)}$.$$For the interval$0 \leqslant t \leqslant 400$, the approximate velocity of the tractor in part$\text{(i)}$is denoted by$v_{1} \mathrm{~m} \mathrm{~s}^{-1}\text{(b)}$Express$v_{1}$in terms of$t$and hence show that$v_{1}-v=0.00005(t-200)^{2}-1$.$\text{(c)}$Deduce that$-1 \leqslant v_{1}-v \leqslant 1$.$$Question 8 Code: 9709/41/M/J/13/7, Topic: - A car driver makes a journey in a straight line from$A$to$B$, starting from rest. The speed of the car increases to a maximum, then decreases until the car is at rest at$B$. The distance travelled by the car$t$seconds after leaving$A$is$0.0000117\left(400 t^{3}-3 t^{4}\right)$metres.$\text{(i)}$Find the distance$A B$.$\text{(ii)}$Find the maximum speed of the car.$\text{(iii)}$Find the acceleration of the car$\text{(a)}$as it starts from$A$,$\text{(b)}$as it arrives at$B$.$\text{(iv)}$Sketch the velocity-time graph for the journey.$\$

Worked solutions: P1, P3 & P6 (S1)

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