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Practice · Oxbridge maths · Oxford MAT

MAT practice questions

40 questions with worked solutions. Multiple choice at the level of the Mathematics Admissions Test: algebra, calculus, sequences, trigonometry and counting.

18 easy · 21 medium · 1 hard · MAT

Practise timed: 10 questions · 50 min All Oxbridge maths topics

1 Easy MAT
What is the minimum value of x2−6x+13x^2-6x+13 for real xx?
  1. 2
  2. 3
  3. 4
  4. 5
  5. 9
Show answer
Answer: C. 4
Complete the square: (x−3)2+4(x-3)^2+4. The square is at least 0, so the minimum is 4, at x=3x=3.
2 Medium MAT
∑k=1991k(k+1)=\displaystyle\sum_{k=1}^{99}\frac{1}{k(k+1)}=
  1. 9899\displaystyle \frac{98}{99}
  2. 99100\displaystyle \frac{99}{100}
  3. 11
  4. 100101\displaystyle \frac{100}{101}
  5. 101100\displaystyle \frac{101}{100}
Show answer
Answer: B. 99100\displaystyle \frac{99}{100}
Partial fractions: 1k(k+1)=1k−1k+1\tfrac1{k(k+1)}=\tfrac1k-\tfrac1{k+1}. The sum telescopes to 1−11001-\tfrac1{100}.
3 Medium MAT
How many real roots does x3−3x+1=0x^3-3x+1=0 have?
  1. 0
  2. 1
  3. 2
  4. 3
  5. 4
Show answer
Answer: D. 3
f′(x)=3x2−3f'(x)=3x^2-3 vanishes at x=±1x=\pm1. f(−1)=3>0f(-1)=3>0 and f(1)=−1<0f(1)=-1<0, and f→∓∞f\to\mp\infty as x→∓∞x\to\mp\infty. So ff crosses zero once before −1-1, once in (−1,1)(-1,1) and once after 1.
4 Medium MAT
∫01xex dx=\displaystyle\int_0^1 xe^{x}\,dx=
  1. 11
  2. e−1e-1
  3. e2\displaystyle \frac e2
  4. 22
  5. ee
Show answer
Answer: A. 11
By parts with u=xu=x, dv=ex dxdv=e^x\,dx: [xex]01−∫01ex dx=e−(e−1)=1\big[xe^x\big]_0^1-\int_0^1e^x\,dx=e-(e-1)=1.
5 Medium MAT
The line y=mxy=mx is tangent to the circle (x−4)2+y2=4(x-4)^2+y^2=4. What is m2m^2?
  1. 14\displaystyle \frac14
  2. 13\displaystyle \frac13
  3. 12\displaystyle \frac12
  4. 23\displaystyle \frac23
  5. 34\displaystyle \frac34
Show answer
Answer: B. 13\displaystyle \frac13
Tangency means the distance from the centre (4,0)(4,0) to the line mx−y=0mx-y=0 equals the radius: ∣4m∣m2+1=2\tfrac{|4m|}{\sqrt{m^2+1}}=2. So 16m2=4m2+416m^2=4m^2+4 and m2=13m^2=\tfrac13.
6 Medium MAT
For how many integers nn with 1≤n≤1001\le n\le100 is n2+nn^2+n divisible by 6?
  1. 33
  2. 50
  3. 66
  4. 67
  5. 100
Show answer
Answer: C. 66
n(n+1)n(n+1) is always even, so we need 3∣n(n+1)3\mid n(n+1): n≡0n\equiv0 or 2(mod3)2\pmod3. There are 33 of each in the range (3 to 99 and 2 to 98), so 66.
7 Easy MAT
log⁡23⋅log⁡34⋅log⁡45⋯log⁡6364=\log_2 3\cdot\log_3 4\cdot\log_4 5\cdots\log_{63}64=
  1. 3
  2. 4
  3. 5
  4. 6
  5. 7
Show answer
Answer: D. 6
Change of base: log⁡ab=ln⁡bln⁡a\log_ab=\tfrac{\ln b}{\ln a}. The product telescopes to ln⁡64ln⁡2=log⁡264=6\tfrac{\ln 64}{\ln 2}=\log_2 64=6.
8 Easy MAT
What is the area of the region enclosed by y=x2y=x^2 and y=2xy=2x?
  1. 23\displaystyle \frac23
  2. 11
  3. 43\displaystyle \frac43
  4. 32\displaystyle \frac32
  5. 83\displaystyle \frac83
Show answer
Answer: C. 43\displaystyle \frac43
The curves meet at x=0x=0 and x=2x=2, with the line above. ∫02(2x−x2) dx=4−83=43\int_0^2(2x-x^2)\,dx=4-\tfrac83=\tfrac43.
9 Medium MAT
What is the remainder when x100x^{100} is divided by x2−1x^2-1?
  1. 0
  2. 1
  3. xx
  4. x+1x+1
  5. 100x100x
Show answer
Answer: B. 1
Write x100=(x2−1)q(x)+ax+bx^{100}=(x^2-1)q(x)+ax+b. At x=1x=1: 1=a+b1=a+b. At x=−1x=-1: 1=−a+b1=-a+b. So a=0a=0, b=1b=1.
10 Medium MAT
How many solutions does 2sin⁡2x+3cos⁡x=32\sin^2x+3\cos x=3 have with 0≤x<2π0\le x<2\pi?
  1. 1
  2. 2
  3. 3
  4. 4
  5. 6
Show answer
Answer: C. 3
Use sin⁡2x=1−cos⁡2x\sin^2x=1-\cos^2x: 2cos⁡2x−3cos⁡x+1=02\cos^2x-3\cos x+1=0, so cos⁡x=1\cos x=1 or 12\tfrac12. cos⁡x=1\cos x=1 gives x=0x=0; cos⁡x=12\cos x=\tfrac12 gives π3\tfrac\pi3 and 5π3\tfrac{5\pi}3. Three solutions.
11 Easy MAT
For how many real values of kk does x2+kx+k=0x^2+kx+k=0 have a repeated root?
  1. 0
  2. 1
  3. 2
  4. 3
  5. Infinitely many
Show answer
Answer: C. 2
Repeated root iff the discriminant k2−4k=0k^2-4k=0: k=0k=0 or k=4k=4.
12 Easy MAT
What is the coefficient of x3x^3 in the expansion of (1+2x)6(1+2x)^6?
  1. 20
  2. 80
  3. 120
  4. 160
  5. 240
Show answer
Answer: D. 160
(63)(2x)3=20⋅8x3=160x3\binom63(2x)^3=20\cdot8x^3=160x^3.
13 Medium MAT
1+23+39+427+⋯=1+\frac23+\frac39+\frac4{27}+\cdots=
  1. 32\displaystyle \frac32
  2. 22
  3. 94\displaystyle \frac94
  4. 33
  5. 278\displaystyle \frac{27}8
Show answer
Answer: C. 94\displaystyle \frac94
This is ∑n≥0(n+1)rn\sum_{n\ge0}(n+1)r^n with r=13r=\tfrac13. Differentiating ∑rn=11−r\sum r^{n}=\tfrac1{1-r} gives ∑(n+1)rn=1(1−r)2=94\sum(n+1)r^n=\tfrac1{(1-r)^2}=\tfrac94.
14 Easy MAT
In how many distinct ways can the letters of OXFORD be arranged?
  1. 60
  2. 120
  3. 180
  4. 360
  5. 720
Show answer
Answer: D. 360
Six letters with O repeated twice: 6!2!=360\tfrac{6!}{2!}=360.
15 Easy MAT
What is the largest value of sin⁡x+3cos⁡x\sin x+\sqrt3\cos x?
  1. 1
  2. 2\displaystyle \sqrt2
  3. 3\displaystyle \sqrt3
  4. 2
  5. 1+3\displaystyle 1+\sqrt3
Show answer
Answer: D. 2
sin⁡x+3cos⁡x=2sin⁡ ⁣(x+π3)\sin x+\sqrt3\cos x=2\sin\!\left(x+\tfrac\pi3\right), with maximum 2. The two terms cannot both be at their maximum at once, so 1+31+\sqrt3 is too big.
16 Easy MAT
What is the local maximum value of f(x)=x3−6x2+9xf(x)=x^3-6x^2+9x?
  1. 0
  2. 4
  3. 6
  4. 9
  5. 18
Show answer
Answer: B. 4
f′(x)=3(x−1)(x−3)f'(x)=3(x-1)(x-3); f′′(1)=−6<0f''(1)=-6<0, so the maximum is at x=1x=1: f(1)=4f(1)=4.
17 Easy MAT
How many real numbers xx satisfy ∣x2−4∣=3|x^2-4|=3?
  1. 0
  2. 2
  3. 3
  4. 4
  5. 6
Show answer
Answer: D. 4
x2−4=3x^2-4=3 gives x=±7x=\pm\sqrt7; x2−4=−3x^2-4=-3 gives x=±1x=\pm1. Four solutions.
18 Easy MAT
If a>0a>0 and ∫0a(2x−1) dx=6\displaystyle\int_0^a(2x-1)\,dx=6, then a=a=
  1. 2
  2. 6\displaystyle \sqrt6
  3. 3
  4. 4
  5. 6
Show answer
Answer: C. 3
a2−a=6a^2-a=6, so (a−3)(a+2)=0(a-3)(a+2)=0 and a=3a=3.
19 Easy MAT
The circles x2+y2=1x^2+y^2=1 and (x−3)2+(y−4)2=r2(x-3)^2+(y-4)^2=r^2 touch externally. What is rr?
  1. 2
  2. 3
  3. 4
  4. 5
  5. 6
Show answer
Answer: C. 4
The centres are 5 apart. External tangency means 1+r=51+r=5, so r=4r=4.
20 Easy MAT
How many of the numbers 2102^{10}, 373^7, 545^4, 10310^3 are greater than 1000?
  1. 0
  2. 1
  3. 2
  4. 3
  5. 4
Show answer
Answer: C. 2
210=10242^{10}=1024 and 37=21873^7=2187 exceed 1000; 54=6255^4=625 does not, and 103=100010^3=1000 is not greater than 1000.
21 Medium MAT
What is the smallest positive integer nn for which n!n! is divisible by 1000?
  1. 10
  2. 12
  3. 15
  4. 20
  5. 25
Show answer
Answer: C. 15
1000=23⋅531000=2^3\cdot5^3. Factors of 2 are plentiful; the fives come from 5, 10, 15. So n=15n=15 (14!14! has only two).
22 Medium MAT
A sequence has a1=1a_1=1 and an+1=an1+ana_{n+1}=\dfrac{a_n}{1+a_n}. What is a100a_{100}?
  1. 1299\displaystyle \frac1{2^{99}}
  2. 1101\displaystyle \frac1{101}
  3. 1100\displaystyle \frac1{100}
  4. 199\displaystyle \frac1{99}
  5. 100
Show answer
Answer: C. 1100\displaystyle \frac1{100}
Take reciprocals: 1an+1=1an+1\tfrac1{a_{n+1}}=\tfrac1{a_n}+1. So 1an=n\tfrac1{a_n}=n and a100=1100a_{100}=\tfrac1{100}.
23 Easy MAT
For which value of kk is x2+y2−4x+6y=kx^2+y^2-4x+6y=k a circle of radius 5?
  1. −12
  2. 5
  3. 12
  4. 13
  5. 25
Show answer
Answer: C. 12
Complete the squares: (x−2)2+(y+3)2=k+13(x-2)^2+(y+3)^2=k+13. Radius 5 needs k+13=25k+13=25, so k=12k=12.
24 Medium MAT
What is the maximum value of xx2+1\dfrac{x}{x^2+1} over real xx?
  1. 14\displaystyle \frac14
  2. 13\displaystyle \frac13
  3. 12\displaystyle \frac12
  4. 1
  5. There is no maximum
Show answer
Answer: C. 12\displaystyle \frac12
For x>0x>0, AM–GM gives x2+1≥2xx^2+1\ge2x, so xx2+1≤12\tfrac{x}{x^2+1}\le\tfrac12, with equality at x=1x=1. Negative xx give negative values.
25 Easy MAT
How many integers nn with 1≤n≤1001\le n\le100 have n2n^2 ending in the digit 1?
  1. 10
  2. 20
  3. 25
  4. 40
  5. 50
Show answer
Answer: B. 20
A square ends in 1 exactly when the number ends in 1 or 9: ten of each in the range.
26 Medium MAT
What is the coefficient of x2x^2 in (1+x+x2)3(1+x+x^2)^3?
  1. 3
  2. 6
  3. 9
  4. 12
  5. 27
Show answer
Answer: B. 6
Take x2x^2 from one bracket and 1 from the others (3 ways), or xx from two brackets (3 ways): 3+3=63+3=6.
27 Medium MAT
What is the smallest value of 3sin⁡x+4cos⁡x+53\sin x+4\cos x+5?
  1. 0
  2. 1
  3. 2
  4. 5
  5. 10
Show answer
Answer: A. 0
3sin⁡x+4cos⁡x=5sin⁡(x+α)3\sin x+4\cos x=5\sin(x+\alpha) with tan⁡α=43\tan\alpha=\tfrac43, so it ranges over [−5,5][-5,5] and the sum has minimum 0.
28 Medium MAT
How many pairs of real numbers (x,y)(x,y) satisfy both x2+y2=1x^2+y^2=1 and x+y=1x+y=1?
  1. 0
  2. 1
  3. 2
  4. 3
  5. Infinitely many
Show answer
Answer: C. 2
Substitute y=1−xy=1-x: 2x2−2x=02x^2-2x=0, so x=0x=0 or x=1x=1, giving (0,1)(0,1) and (1,0)(1,0) — the line is a chord of the circle.
29 Easy MAT
∫1eln⁡xx dx=\displaystyle\int_1^{e}\frac{\ln x}{x}\,dx=
  1. 12\displaystyle \frac12
  2. 1
  3. e−1e-1
  4. e2\displaystyle \frac e2
  5. e2−12\displaystyle \frac{e^2-1}{2}
Show answer
Answer: A. 12\displaystyle \frac12
With u=ln⁡xu=\ln x the integral is ∫01u du=12\int_0^1u\,du=\tfrac12.
30 Medium MAT
If f(x)=x1−xf(x)=\dfrac{x}{1-x}, what is f(f(x))f(f(x))?
  1. xx
  2. x1−2x\displaystyle \frac{x}{1-2x}
  3. x1+x\displaystyle \frac{x}{1+x}
  4. −x-x
  5. 1−xx\displaystyle \frac{1-x}{x}
Show answer
Answer: B. x1−2x\displaystyle \frac{x}{1-2x}
f(f(x))=x1−x1−x1−x=x(1−x)−x=x1−2xf(f(x))=\dfrac{\frac{x}{1-x}}{1-\frac{x}{1-x}}=\dfrac{x}{(1-x)-x}=\dfrac{x}{1-2x}.
31 Medium MAT
How many real solutions does 2x=x+22^{x}=x+2 have?
  1. 0
  2. 1
  3. 2
  4. 3
  5. Infinitely many
Show answer
Answer: C. 2
Let f(x)=2x−x−2f(x)=2^x-x-2. f(−2)=0.25>0f(-2)=0.25>0, f(0)=−1<0f(0)=-1<0 and f(2)=0f(2)=0, and ff is convex, so the curve meets the line exactly twice: once between −2-2 and 0, and again at x=2x=2.
32 Easy MAT
What is the sum of the roots of x3−6x2+11x−6=0x^3-6x^2+11x-6=0?
  1. −6
  2. 1
  3. 3
  4. 6
  5. 11
Show answer
Answer: D. 6
For x3+ax2+bx+cx^3+ax^2+bx+c the roots sum to −a-a, here 6. (They are 1, 2 and 3.)
33 Medium MAT
What is the area between y=1xy=\dfrac1x and the xx-axis from x=1x=1 to x=8x=8?
  1. ln⁡2\ln 2
  2. 2ln⁡22\ln 2
  3. 3ln⁡23\ln 2
  4. ln⁡7\ln 7
  5. 7
Show answer
Answer: C. 3ln⁡23\ln 2
∫18dxx=ln⁡8=3ln⁡2\int_1^8\tfrac{dx}{x}=\ln 8=3\ln 2.
34 Medium MAT
How many ways are there to choose three numbers from 1 to 10 so that no two are consecutive?
  1. 48
  2. 56
  3. 64
  4. 84
  5. 120
Show answer
Answer: B. 56
Subtract 0, 1 and 2 from the three chosen numbers in increasing order: this matches the choices with a bijection to any 3 numbers from 1 to 8, so (83)=56\binom83=56.
35 Easy MAT
If log⁡a2=0.3\log_a 2=0.3, what is log⁡a8\log_a 8?
  1. 0.3
  2. 0.6
  3. 0.9
  4. 2.7
  5. 3
Show answer
Answer: C. 0.9
log⁡a8=log⁡a23=3log⁡a2\log_a 8=\log_a 2^3=3\log_a 2.
36 Medium MAT
For which xx does 2log⁡3x=log⁡3(2x+3)2\log_3 x=\log_3(2x+3)?
  1. −1
  2. 1
  3. 3
  4. Both −1 and 3
  5. No solution
Show answer
Answer: C. 3
x2=2x+3x^2=2x+3 gives x=3x=3 or x=−1x=-1, but log⁡3x\log_3 x needs x>0x>0, so only x=3x=3 survives.
37 Hard MAT
For how many integers nn with 1≤n≤101\le n\le10 is n2<2nn^2<2^n?
  1. 6
  2. 7
  3. 8
  4. 9
  5. 10
Show answer
Answer: B. 7
It holds at n=1n=1, fails at n=2,3,4n=2,3,4 (where n2≥2nn^2\ge2^n), and holds from n=5n=5 onwards: 11 and 5,…,105,\dots,10, seven values.
38 Medium MAT
For how many values of aa does the tangent to y=x2y=x^2 at x=ax=a pass through (0,−1)(0,-1)?
  1. 0
  2. 1
  3. 2
  4. 3
  5. Infinitely many
Show answer
Answer: C. 2
The tangent is y=2ax−a2y=2ax-a^2. Through (0,−1)(0,-1): −a2=−1-a^2=-1, so a=±1a=\pm1.
39 Easy MAT
What is the minimum value of x+9xx+\dfrac9x for x>0x>0?
  1. 4
  2. 6
  3. 9
  4. 12
  5. There is no minimum
Show answer
Answer: B. 6
By AM–GM, x+9x≥29=6x+\tfrac9x\ge2\sqrt9=6, with equality at x=3x=3.
40 Medium MAT
How many digits does 21002^{100} have?
  1. 30
  2. 31
  3. 32
  4. 33
  5. 100
Show answer
Answer: B. 31
log⁡102100=100log⁡102≈30.103\log_{10}2^{100}=100\log_{10}2\approx30.103, so the number lies between 103010^{30} and 103110^{31}: 31 digits.

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