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8.2 Quiz 1

Please submit solutions to the following via moodle by 23:59 on Wednesday 23rd November. Each question is worth [2] marks. See Week06Logic.pdf for background relevant to Q1.4-1.5.

Q1.1.Substitution 1. What is the value of the integral

I=∫01x1+x2⁢e1+x2⁢d⁢x⁢?I=\int_{0}^{1}\frac{x}{\sqrt{1+x^{2}}}e^{\sqrt{1+x^{2}}}\,dx\;\;\mbox{?}

(A) 2-1\sqrt{2}-1; \;\;\;\; (B) 2⁢e2\sqrt{2}e^{\sqrt{2}}; \;\;\;\; (C) 2⁢e2-e\sqrt{2}e^{\sqrt{2}}-e; \;\;\;\; (D) e2-ee^{\sqrt{2}}-e; \;\;\;\; (E) e22\frac{e^{\sqrt{2}}}{\sqrt{2}}.

Q1.2.Substitution 2. The integral

J=∫-π2π2d⁢x2⁢cos⁡x+sin⁡x+3J=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{dx}{2\cos x+\sin x+3}

has which of the following values?

(A) π\pi;\;\;\;\; (B) π4\frac{\pi}{4};\;\;\;\; (C) log⁡8\log 8; \;\;\;\; (D) log⁡2\log 2;\;\;\;\; (E) 00.

Q1.3.Substitution 3. Evaluate the integral

K=∫012d⁢x(1-x2)32.K=\int_{0}^{\frac{1}{\sqrt{2}}}\frac{dx}{(1-x^{2})^{\frac{3}{2}}}.

(A) 11; \;\;\;\; (B) 2\sqrt{2}; \;\;\;\; (C) 2-1\sqrt{2}-1;\;\;\;\; (D) 1-21-\sqrt{2};\;\;\;\; (E) 2⁢(1-2)2(1-\sqrt{2}).

Q1.4.Command matching. Which of the following commands evaluates the logical statement

tan-1⁡x≤log⁡(3y)⁢and⁢tan-1⁡x≤sinh⁡(z)\tan^{-1}x\leq\log\left(\frac{3}{y}\right)\;\;\mbox{and}\;\;\tan^{-1}x\leq% \sinh(z)

for scalars xx, yy, zz assigned as objects x, y and z respectively?

  1. (A)

    atan(x) <= max( log(3/y), sinh(z) )

  2. (B)

    atan(x) <= min( log(3/y), sinh(z) )

  3. (C)

    atan(x) >= max( log(3/y), sinh(z) )

  4. (D)

    atan(x) >= min( log(3/y), sinh(z) )

  5. (E)

    cot(x) <= max( log(3/y), sinh(z) )

Q1.5.Positive elements in a sequence. Let

(a1,a2,…,a100)(a_{1},a_{2},\ldots,a_{100})

be a sequence with kk-th element

ak=sin⁡(3⁢k⁢π44).a_{k}=\sin\Bigl(\frac{3k\pi}{44}\Bigr).

Find the number of elements of the sequence for which ak>0.1a_{k}>0.1.