Prove that (23+3) +23 lies between -(23+15) and (23+15). - Vidyadip

Question

Prove that $(2\sqrt{3}+3)\sin\text{x}+2\sqrt{3}\cos\text{x}$ lies between $-(2\sqrt{3}+\sqrt{15})$ and $(2\sqrt{3}+\sqrt{15}).$

✓

Answer

Let $\text{f(x)}=(2\sqrt{3}+3)\sin\text{x}+2\sqrt{3}\cos\text{x}$
We know that,
$-\sqrt{(2\sqrt{3}+3)^2+(2\sqrt{3})^2}\le\text{f(x)}\le\sqrt{(2\sqrt{3}+3)^2+(2\sqrt{3})^2}$
$\Rightarrow-\sqrt{12+9+12\sqrt{3}+12}\le\text{f(x)}\le\sqrt{12+9+12\sqrt{3}+12}$
$\Rightarrow-\sqrt{33+12\sqrt{3}}\le\text{f(x)}\le\sqrt{33+12\sqrt{3}}$
Disclaimer: Instead of $-(2\sqrt{3}+\sqrt{15})$ and $(2\sqrt{3}+\sqrt{15}),$ it should be $-\sqrt{33+12\sqrt{3}}$ and $\sqrt{33+12\sqrt{3}}.$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Trigonometric Ratios Of Multiple And Sub Multiple Angles→Trigonometric Ratios Of Multiple And Sub Multiple Angles — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Prove that $\cos \frac{2 \pi}{15} \cdot \cos \frac{4 \pi}{15} \cdot \cos \frac{8 \pi}{15} \cdot \cos \frac{16 \pi}{15}=\frac{1}{16}$

Evaluate the following limit:
$\lim\limits_{\text{h}\rightarrow0}\frac{\text{(a}+\text{h})^2\sin(\text{a}+\text{h})-\text{a}^2\sin\text{a}}{\text{h}}$

The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.

Find the term independent of $x$ in the expansion of the following expressions:
$\Big(\frac{1}{2}\text{x}^{\frac{1}{3}}+\text{x}^{-\frac{1}{5}}\Big)^{8}$

Prove that:
$\frac{\sin\text{A}\sin2\text{A}+\sin3\text{A}\sin6\text{A}}{\sin\text{A}\cos2\text{A}+\sin3\text{A}\cos6\text{A}}=\tan5\text{A}$

Prove that:
$\tan20^\circ\tan30^\circ\tan40^\circ\tan80^\circ=1$

If the coefficients of $(2r + 4)$th and $(r - 2)$th teram in the expansion of $(1 + x)^{18}$ are equal, find r.

Prove the following by the principle of mathematical induction:
$1 + 2 + 2^2 + ... + 2^n = 2^{n+1} - 1$ for all $\text{n}\in\text{N}.$

If $^\text{n+5}\text{P}_\text{n+1}=-\frac{11(\text{n-1})}{2}\ ^\text{n+3}\text{P}_\text{n}$ find n.

Prove the following by using the principle of mathematical induction for all n ∈ N:$\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{\text{n(n+1)(n+2)}}=\frac{\text{n(n+3)}}{4(\text{n+1})(\text{n}+2)}.$