Gated take · Anannt /10

Anannt Free Diagnostic. TMUA-style (shortened).

Anannt Education / Anannt Training Institute. Original practice diagnostic. Not the official TMUA. Not affiliated with UAT or Cambridge Assessment. Time allowed: about 30 minutes. No calculator. Choose one option for each question. Stems are verbatim from Ramesh SIGN-OFF Student.md. Raw Anannt check only.

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Q1 | Application — quadratic functions

The function \(f\) is defined by \(f(x)=x^{2}-6x+11\) for all real \(x\).

If \(f(a)=f(b)\) and \(a\neq b\), what is the value of \(a+b\)?

A) \(3\) B) \(5\) C) \(6\) D) \(11\) E) Cannot be determined from the information given

Q2 | Application — sequences

A sequence is defined by \(u_{1}=1\) and \(u_{n+1}=2u_{n}+1\) for every integer \(n\ge 1\).

What is the value of \(u_{5}\)?

A) \(15\) B) \(31\) C) \(32\) D) \(63\) E) \(64\)

Q3 | Application — number properties

How many positive integers \(n\) with \(n\le 40\) have the property that \(n^{2}\) leaves a remainder of \(1\) when divided by \(5\)?

A) \(8\) B) \(12\) C) \(16\) D) \(20\) E) \(24\)

Q4 | Application — trigonometry in a triangle

In \(\triangle ABC\), \(AB=AC=5\) and \(BC=6\).

What is \(\cos\angle BAC\)?

A) \(\dfrac{1}{5}\) B) \(\dfrac{7}{25}\) C) \(\dfrac{11}{25}\) D) \(\dfrac{3}{5}\) E) \(\dfrac{4}{5}\)

Q5 | Application — exponential equations

The equation \(4^{x}-5\cdot 2^{x}+4=0\) has two real solutions \(x_{1}\) and \(x_{2}\).

What is the value of \(x_{1}+x_{2}\)?

A) \(0\) B) \(1\) C) \(2\) D) \(3\) E) \(4\)

Q6 | Application — counting

A committee of \(3\) students is to be chosen from a group of \(9\) students. The order of selection does not matter.

In how many different ways can the committee be chosen?

A) \(27\) B) \(84\) C) \(168\) D) \(504\) E) \(729\)

Q7 | Reasoning — inequalities

Suppose \(x>0\), \(y>0\), and \(xy=4\).

Which of the following must be true?

A) \(x+y=4\) B) \(x+y>4\) C) \(x+y\ge 4\) D) \(x-y\ge 2\) E) \(x^{2}+y^{2}=8\)

Q8 | Reasoning — logic / counterexamples

Consider the statement:

“If the product \(ab\) is even, then both \(a\) and \(b\) are even.”

Which of the following is correct?

A) The statement is true for all integers \(a\) and \(b\). B) The statement is false; a counterexample is \(a=2\), \(b=4\). C) The statement is false; a counterexample is \(a=2\), \(b=3\). D) The statement is false; a counterexample is \(a=1\), \(b=1\). E) The statement is true for positive integers only.

Q9 | Application — graphs of quadratics

The curve \(y=(x-4)(x+2)\) intersects the \(x\)-axis at \(P\) and \(Q\), and has vertex \(V\).

What is the \(y\)-coordinate of \(V\)?

A) \(-9\) B) \(-8\) C) \(-4\) D) \(0\) E) \(8\)

Q10 | Reasoning — functional conditions

The function \(g\) is defined for all real \(x\neq 0\) by

\[g(x)=\dfrac{x+3}{x}.\]

For how many real numbers \(k\) is it true that the equation \(g(x)=k\) has no real solution \(x\)?

A) \(0\) B) \(1\) C) \(2\) D) \(3\) E) Infinitely many

End of diagnostic.