Quadratics, functions and admissible solutions: check 1
Difficulty: ★★☆Info
Maths4U Original · Short practice
Let \(f(x)=x^2-6x+13\), \(x\ge3\). Find the lower endpoint of the domain of \(f^{-1}\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Quadratics, functions and admissible solutions: check 2
Difficulty: ★★☆Info
Maths4U Original · Short practice
For \(f(x)=(x-3)^2+4\), \(x\ge3\), find \(f^{-1}(13)\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Quadratics, functions and admissible solutions: check 3
Difficulty: ★★☆Info
Maths4U Original · Short practice
For \(f(x)=x^2\), \(g(x)=x-5\), what is \(f(g(x))\)?Choose one answer.
Quadratics, functions and admissible solutions: check 4
Difficulty: ★★☆Info
Maths4U Original · Short practice
For \(f(x)=(x-2)^2+3\), which \(k\) gives two distinct solutions of \(f(x)=k\) over all real x?Choose one answer.
Lines, circles and derivatives: check 1
Difficulty: ★★☆Info
Maths4U Original · Short practice
For \(y=x^2+2x\), find the tangent gradient at \(x=1\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Lines, circles and derivatives: check 2
Difficulty: ★★☆Info
Maths4U Original · Short practice
The tangent to \(y=x^2+2x\) at \((1,3)\) has equation \(y=4x+c\). Find \(c\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Lines, circles and derivatives: check 3
Difficulty: ★★☆Info
Maths4U Original · Short practice
Find the gradient of the tangent to \(x^2+y^2=25\) at \((3,4)\), using radius geometry.Enter one number or fraction. Round decimals to at least six decimal places when needed.
Lines, circles and derivatives: check 4
Difficulty: ★★☆Info
Maths4U Original · Short practice
The normal to \(y=x^2\) at \((1,1)\) meets the y-axis at \((0,b)\). Find \(b\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Trigonometric branches and sector models: check 1
Difficulty: ★★☆Info
Maths4U Original · Short practice
A sector has radius 6 cm and angle \(\displaystyle \frac{\pi }{3}\). Its arc length is \(k\pi\) cm. Find \(k\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Trigonometric branches and sector models: check 2
Difficulty: ★★☆Info
Maths4U Original · Short practice
For radius 6 cm and angle \(\displaystyle \frac{\pi }{3}\), the minor segment area is \(6\pi-k\sqrt3\) square cm. Find \(k\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Trigonometric branches and sector models: check 3
Difficulty: ★★☆Info
Maths4U Original · Short practice
A sector angle is \(120^\circ\). Which radian value belongs in \(A=\tfrac12r^2\theta\)?Choose one answer.
Trigonometric branches and sector models: check 4
Difficulty: ★★☆Info
Maths4U Original · Short practice
A chord of a radius-8 cm circle subtends \(\displaystyle \frac{\pi }{3}\). Find its length in cm.Enter one number or fraction. Round decimals to at least six decimal places when needed.
Coefficients as sequence data: check 1
Difficulty: ★★☆Info
Maths4U Original · Short practice
An AP has \(u_n=3n+2\). Find the coefficient of \(x\) in \((1+x)^{u_2}\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Coefficients as sequence data: check 2
Difficulty: ★★☆Info
Maths4U Original · Short practice
A GP begins \(2,6,18,\ldots\). Its third term is \(a\). Find the coefficient of \(x^2\) in \((1+ax)^3\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Coefficients as sequence data: check 3
Difficulty: ★★☆Info
Maths4U Original · Short practice
The coefficients of \(1,x,x^2\) in \((1+x)^4\) form a short sequence. Find their sum.Enter one number or fraction. Round decimals to at least six decimal places when needed.
Coefficients as sequence data: check 4
Difficulty: ★★☆Info
Maths4U Original · Short practice
Are the first three coefficients \(1,4,6\) of \((1+x)^4\) an AP or GP?Choose one answer.
Recovered curves, regions and accumulated quantities: check 1
Difficulty: ★★☆Info
Maths4U Original · Short practice
A model has \(Q(t)=t^3-3t^2+12\). Find its instantaneous rate at \(t=3\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Recovered curves, regions and accumulated quantities: check 2
Difficulty: ★★☆Info
Maths4U Original · Short practice
The signed rate is \(q(t)=3t^2-6t\). Find the net change from \(t=0\) to \(t=3\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Recovered curves, regions and accumulated quantities: check 3
Difficulty: ★★☆Info
Maths4U Original · Short practice
For \(Q(t)=t^3-3t^2+12\), find the minimum value on \(0\le t\le3\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Recovered curves, regions and accumulated quantities: check 4
Difficulty: ★★☆Info
Maths4U Original · Short practice
A rate is measured in litres per second and integrated with respect to time in seconds. What units does the integral have?Choose one answer.
Quadratics, functions and admissible solutions: review variant 1
Difficulty: ★★☆Info
Maths4U Original · Short practice
For \(f(x)=(x+2)^2+7\), x at least -2, enter the lower endpoint of the inverse domain.Enter one number or fraction. Round decimals to at least six decimal places when needed.
Quadratics, functions and admissible solutions: review variant 2
Difficulty: ★★☆Info
Maths4U Original · Short practice
For \(f(x)=(x+2)^2+7\), x at least -2, find \(f^{-1}(16)\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Quadratics, functions and admissible solutions: review variant 3
Difficulty: ★★☆Info
Maths4U Original · Short practice
For \(f(x)=x^2\), \(g(x)=x+3\), find \(f(g(-1))\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Quadratics, functions and admissible solutions: review variant 4
Difficulty: ★★☆Info
Maths4U Original · Short practice
How many distinct real solutions does \((x+1)^2+5=14\) have?Enter one number or fraction. Round decimals to at least six decimal places when needed.
Lines, circles and derivatives: review variant 1
Difficulty: ★★☆Info
Maths4U Original · Short practice
For \(y=2x^2+4x\), find the tangent gradient at \(\displaystyle x=1\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Lines, circles and derivatives: review variant 2
Difficulty: ★★☆Info
Maths4U Original · Short practice
A tangent of gradient 8 passes through (1,6). It has equation \(\displaystyle y=8x+c\). Find c.Enter one number or fraction. Round decimals to at least six decimal places when needed.
Lines, circles and derivatives: review variant 3
Difficulty: ★★☆Info
Maths4U Original · Short practice
Find the tangent gradient to \(x^2+y^2=25\) at (4,3), using radius geometry.Enter one number or fraction. Round decimals to at least six decimal places when needed.
Lines, circles and derivatives: review variant 4
Difficulty: ★★☆Info
Maths4U Original · Short practice
The normal to \(y=x^2\) at (2,4) meets the y-axis at (0,b). Find b.Enter one number or fraction. Round decimals to at least six decimal places when needed.
Trigonometric branches and sector models: review variant 1
Difficulty: ★★☆Info
Maths4U Original · Short practice
A radius-8 cm sector has angle \(\displaystyle \frac{\pi }{2}\) radians. Its arc length is \(k\pi\) cm. Find \(k\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Trigonometric branches and sector models: review variant 2
Difficulty: ★★☆Info
Maths4U Original · Short practice
A radius-4 cm sector has angle \(\displaystyle \frac{\pi }{2}\). Its minor segment area is \((4\pi-k)\) square cm. Find k.Enter one number or fraction. Round decimals to at least six decimal places when needed.
Trigonometric branches and sector models: review variant 3
Difficulty: ★★☆Info
Maths4U Original · Short practice
A radius-6 cm sector has angle \(60^\circ\). Its area is \(k\pi\) square cm. Find k.Enter one number or fraction. Round decimals to at least six decimal places when needed.
Trigonometric branches and sector models: review variant 4
Difficulty: ★★☆Info
Maths4U Original · Short practice
A radius-10 cm circle has a chord subtending \(\displaystyle \frac{\pi }{3}\). Find the chord length in cm.Enter one number or fraction. Round decimals to at least six decimal places when needed.
Coefficients as sequence data: review variant 1
Difficulty: ★★☆Info
Maths4U Original · Short practice
An AP has \(u_n=4n+3\). Find the coefficient of x in \((1+x)^{u_2}\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Coefficients as sequence data: review variant 2
Difficulty: ★★☆Info
Maths4U Original · Short practice
A GP is \(3,6,12,\ldots\). Let a be its third term. Find the x-squared coefficient in \((1+ax)^4\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Coefficients as sequence data: review variant 3
Difficulty: ★★☆Info
Maths4U Original · Short practice
Find the sum of the first three coefficients, in ascending powers, of \((1+x)^5\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Coefficients as sequence data: review variant 4
Difficulty: ★★☆Info
Maths4U Original · Short practice
The first coefficients of \((1+x)^5\) are 1,5,10. Find the difference between the third and second coefficients.Enter one number or fraction. Round decimals to at least six decimal places when needed.
Recovered curves, regions and accumulated quantities: review variant 1
Difficulty: ★★☆Info
Maths4U Original · Short practice
A model has \(Q(t)=t^3-4t^2+20\). Find its instantaneous rate at \(\displaystyle t=4\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Recovered curves, regions and accumulated quantities: review variant 2
Difficulty: ★★☆Info
Maths4U Original · Short practice
A model has signed rate \(3t^2-6t\). Find its net change from \(\displaystyle t=0\) to \(\displaystyle t=4\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Recovered curves, regions and accumulated quantities: review variant 3
Difficulty: ★★☆Info
Maths4U Original · Short practice
Find the minimum of \(Q(t)=t^3-6t^2+50\) on \(0\le t\le5\).Enter one number or fraction. Round decimals to at least six decimal places when needed.
Recovered curves, regions and accumulated quantities: review variant 4
Difficulty: ★★☆Info
Maths4U Original · Short practice
A flow is constant at 4 litres/second for 7 seconds. Find the accumulated volume in litres.Enter one number or fraction. Round decimals to at least six decimal places when needed.