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A-Level CAIE Mathematics A21.1 QuadraticsQuestion Bank

1.1 Quadratics

Question 1

[Maximum number: 3]

Solve the equation 3x+2=2x−13 x+2=\frac{2}{x-1}.

Question 1

Question 1(a)

(a)

Express 16x2−24x+1016 x^{2}-24 x+10 in the form (4x+a)2+b(4 x+a)^{2}+b.

[ 2 ]

Question 1(b)

(b)

It is given that the equation 16x2−24x+10=k16 x^{2}-24 x+10=k, where k is a constant, has exactly one root.

Find the value of this root.

[ 2 ]

Question 1

Question 1(a)

(a)

Express x2+6x+5x^{2}+6 x+5 in the form (x+a)2+b(x+a)^{2}+b, where a and b are constants.

[ 2 ]

Question 1

[Maximum number: 5]

The function f is defined by f(x)=x2−4x+8\mathrm{f}(x)=x^{2}-4 x+8 for x∈Rx \in \mathbb{R}.

Question 1(i)

(a)

Express x2−4x+8x^{2}-4 x+8 in the form (x−a)2+b(x-a)^{2}+b.

[ 2 ]

Question 1(ii)

(b)

Hence find the set of values of x for which f(x)<9, giving your answer in exact form.

[ 3 ]

Question 1

[Maximum number: 3]

Express 2x2−12x+72 x^{2}-12 x+7 in the form a(x+b)2+ca(x+b)^{2}+c, where a, b and c are constants.

Question 1

Question 1(a)

(a)

Express x2−8x+11x^{2}-8 x+11 in the form (x+p)2+q(x+p)^{2}+q where p and q are constants.

[ 2 ]

Question 1(b)

(b)

Hence find the exact solutions of the equation x2−8x+11=1x^{2}-8 x+11=1.

[ 2 ]

Question 1

[Maximum number: 3]

Showing all necessary working, solve the equation 4x−11x12+6=04 x-11 x^{\frac{1}{2}}+6=0.

Question 1

[Maximum number: 3]

Express 3x2−12x+73 x^{2}-12 x+7 in the form a(x+b)2+ca(x+b)^{2}+c, where a, b and c are constants.

Question 1

Question 1(a)

(a)

Express 3y2−12y−153 y^{2}-12 y-15 in the form 3(y+a)2+b3(y+a)^{2}+b, where a and b are constants.

[ 2 ]

Question 1(b)

(b)

Hence find the exact solutions of the equation 3x4−12x2−15=03 x^{4}-12 x^{2}-15=0.

[ 3 ]

Question 1

Question 1(i)

(a)

Express x2+6x+2x^{2}+6 x+2 in the form (x+a)2+b(x+a)^{2}+b, where a and b are constants.

[ 2 ]

Question 1(ii)

(b)

Hence, or otherwise, find the set of values of x for which x2+6x+2>9x^{2}+6 x+2>9.

[ 2 ]
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