2025 A-level H2 Mathematics (9758) Specimen Paper 2 Suggested Solutions

(a)(i)

Thus, we can observe that has a minimum point at .
Therefore, .

(a)(ii)

(b)(i)

Largest value of for g to be a one-to-one function in order for the inverse to exist.

(b)(ii)

Let


Since , then
.

(a)

If is a quadratic polynomial, then



.
Observe here that will be a linear function. Clearly, the given sequence is not linear. Thus, cannot be a quadratic polynomial in .

(b)

Using GC,

(c)

(a)

Using GC, or

(b)(i)

Using Ratio Theorem,


Since and are non-parallel, we can compare coefficients,


Using GC,

(b)(ii)

Thus is parallel to with common point , i.e., are collinear and .

(a)(i)

where

(a)(ii)

When

(b)

(c)

Comparing coefficients, .
Required term .

(a)

Number of ways ways

(b)

Number of ways ways

(c)

Number of ways ways

(d)

Number of ways ways

(a)(i)

(a)(ii)

As increases, is increasing at an increasing rate. Thus a linear model should not be used.

(b)

Using GC,
For the model
For the model
Thus the model is the better model as the r value is close to 1 which suggest a stronger positive linear correlation between and .
Using GC, .

(c)(i)(A)

kg

(c)(i)(B)

kg

(c)(ii)

The estimate for (A) is more reliable as the value of used is within the given data range while the value of used in (b) is out of the given data range.

(a)

Let be the number of yellow sweets in a randomly chosen packet, out of 8 sweets.

Required probability

(b)

Let be the number of packets out of 70 packets that contain no more than one yellow sweet.

Required probability

(c)

Since the modal value is 3, then is the greatest.

(a)

Required probability

(b)

(c)

Using GC,

(a)

Let be the radius of a randomly chosen bolt.

Required probability
Thus the required percentage is .

(b)

Let be the interior radius of a randomly chosen washer.

Let
Let be the required circumference.

Using GC, .

(c)

Required probability

(d)

Let be the exterior radius of a randomly chosen washer.





Using GC, .

(a)

The production manager should use a 2 tailed test as he is only interested to test whether there is a difference in the average time taken.

(b)

The production manager is able to carry out a hypothesis test as the sample average time taken to manufacture the control panels follows an approximate normal distribution by Central Limit Theorem with a sample size larger than 30.

(c)(i)

unbiased estimate of population mean
unbiased estimate of population variance

(c)(ii)

Let be the population mean time taken to manufacture a control panel using the alternative process.
Test
against at level of significance.
Under , by central limit theorem, approximately.
Using GC,

Since , we do not reject and conclude that there is insufficient evidence at level of significance that the alternative process will give a different average manufacture time of a control panel.

(d)

Let be the population mean time taken to manufacture a control panel using the alternative process.
To test against at 10% level of significance.
Under , by central limit theorem, approximately.
To not reject if ,

Thus, if the population variance is such that , the finance manager will conclude that the average time taken for the manufacture of a control panel under the alternative process is shorter at 10% level of significance.