DTM University Entrance Test (Uzbekistan) · Mathematics

Calculus, series and word problems

28 practice questions, each with the answer and an explanation.

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All 28 questions

A box contains 7 red and 6 blue balls. Two balls are taken at random without replacement. Find the probability that they are of different colours.
A straight line passes through $(- 2, 5)$ and $(1, 8)$ . Find the value of $y$ when $x = 9$ .
Evaluate $x \to 7 lim \frac{x ^{2} - 49}{x - 7}$ .
If $α$ and $β$ are the roots of $x^{2} + 11 x + 30 = 0$ , find $α^{2} + β^{2}$ .
Find the coordinates of the centre of the circle $x^{2} + y^{2} + 2 x - 4 y - 20 = 0$ .
Evaluate $lo g_{2} 32 + lo g_{2} 8 - lo g_{2} 4$ .
If $y = (x + 2) (x^{2} + 3)$ , find $\frac{d y}{d x}$ when $x = 1$ .
Evaluate $\int_{1}^{4} (4 x^{2} - 3 x) d x$ .
If $α$ and $β$ are the roots of $x^{2} + 3 x - 10 = 0$ , find $α^{3} + β^{3}$ .
Find the radius of the circle $x^{2} + y^{2} - 8 x - 10 y - 8 = 0$ .
Solve $lo g_{2} (x + 2) = 3$ .
If $α$ and $β$ are the roots of $x^{2} + 3 x - 4 = 0$ , find $α^{3} + β^{3}$ .
Find the radius of the circle $x^{2} + y^{2} - 4 x + 10 y + 13 = 0$ .
Evaluate $lo g_{2} 16 + lo g_{2} 8 - lo g_{2} 4$ .
If $f (x) = 2 x^{3} + 4 x^{2} + 5 x + 1$ , find $f^{'} (- 2)$ .
If $α$ and $β$ are the roots of $x^{2} - 7 x + 10 = 0$ , find $\frac{1}{α} + \frac{1}{β}$ .
Find the radius of the circle $x^{2} + y^{2} + 8 x + 2 y + 1 = 0$ .
Evaluate $lo g_{5} 3125 + lo g_{5} 25 - lo g_{5} 5$ .
If $f (x) = 2 x^{3} + 5 x^{2} - 4 x + 4$ , find $f^{'} (1)$ .
Evaluate $\int_{1}^{4} (4 x^{2} + 2 x + 4) d x$ .
Find the magnitude of $2 i + 6 j + 9 k$ .
Find the radius of the circle $x^{2} + y^{2} + 6 x - 6 y - 18 = 0$ .
Solve $3^{x} \cdot 3^{1} = 243$ .
If $y = ln (3 x + 5)$ , find $\frac{d y}{d x}$ at $x = 0$ .
Evaluate $lo g_{5} 125 + lo g_{5} 625 - lo g_{5} 25$ .
If $y = (5 x + 1)^{2}$ , find $\frac{d y}{d x}$ when $x = 1$ .
Find the coordinates of the centre of the circle $x^{2} + y^{2} - 4 x + 6 y + 4 = 0$ .
Evaluate $lo g_{3} 243 + lo g_{3} 9 - lo g_{3} 3$ .

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