Opposite & Adjacent Sides in a Right Angled Triangle

Q.

Let $f\left(x\right)=\int \frac{\sqrt{x}}{{\left(1+x\right)}^2}dx,x\ge 0$. Then $f\left(3\right)-f\left(1\right)$ is equal to?

1. $-\frac{\mathrm{\pi }}{6}+\frac{1}{2}+\frac{\sqrt{3}}{4}$

2. $\frac{\mathrm{\pi }}{6}+\frac{1}{2}-\frac{\sqrt{3}}{4}$

3. $-\frac{\mathrm{\pi }}{12}+\frac{1}{2}+\frac{\sqrt{3}}{4}$

4. $\frac{\mathrm{\pi }}{12}+\frac{1}{2}-\frac{\sqrt{3}}{4}$

Q.

The sides of a right angled triangle are 5 cm and 12 cm, find the hypotenuse.

Q.

If $3\cot \left(A\right)=4$, check whether$\frac{(1-{\tan }^{2}A)}{(1+{\tan }^{2}A)}={\cos }^{2}A–{\sin }^{2}A$ or not.

Q.

If $\angle Aand\angle B$ are acute angles such that sin A = sin B then prove that $\angle A=\angle B$.

Q.

If$\tan y=\frac{2t}{(1-t^2)}$ and $\sin x=\frac{2t}{(1+t^2)}$, then $\frac{dy}{dx}$ =

1. $\frac{2}{\left(1-t^2\right)}$

2. $\frac{1}{\left(1+t^2\right)}$

3. $1$

4. $2$

Q.

In the figure given below, if DE || BC, then x equals:

1. 4 cm

2. 2 cm

3. 6.7 cm

4. 3 cm

Q. In a right triangle ABC right-angled at B, if P and Q are points on the sides AB and AC respectively, then

(a) AQ² + CP² = 2(AC² + PQ²)
(b) 2(AQ² + CP²) = AC² + PQ²
(c) AQ² + CP² = AC² + PQ²
(d) $\mathrm{AQ}+\mathrm{CP}=\frac{1}{2}\left(\mathrm{AC}+\mathrm{PQ}\right)$

Q.

A man walks 8 km due North, then 5 km to East and 4 km to North. How far is he from the starting point?

Q.

In the given figure, T and B are right angles. If the lengths of AT, BC and AS (in centimeters) are 15, 16 and 17 respectively, then the length of TC (in centimeters) is:

1. 16

2. 18

3. 19

4. 12

Q.

The general value of $\theta$ is the equation $\cos \theta =\frac{1}{\sqrt{2}}$, $\tan \theta =-1$ is

1. $2n\mathrm{\pi }+\frac{\mathrm{\pi }}{6},\mathrm{n}\in \mathrm{I}$

2. $2n\mathrm{\pi }+\frac{7\mathrm{\pi }}{4},\mathrm{n}\in \mathrm{I}$

3. $2n\mathrm{\pi }+{\left(-1\right)}^n\frac{\mathrm{\pi }}{3},\mathrm{n}\in \mathrm{I}$

4. $2n\mathrm{\pi }+{\left(-1\right)}^n\frac{\mathrm{\pi }}{4},\mathrm{n}\in \mathrm{I}$

Q. If E is a point on side CA of an equilateral triangle ABC such that BE ⊥ CA, then AB² + BC² + CA² =

(a) 2 BE²
(b) 3 BE²
(c) 4 BE²
(d) 6 BE²

Q.

If cos $cos\theta =\frac{7}{25}$, find the values of all T-ratios of $\theta$

Q.

In figure below, ΔABC is right-angled at B and tan A = ⁴/₃. If AC = 15 cm, then the length of AB is :

1. 4 cm

2. 3 cm

3. 12 cm

4. 9 cm

Q.

Prove the following: If $\tan A=\frac{3}{4}$, then $\sin A·\cos A=\frac{12}{25}$

Q. In triangle ABC and triangle XYZ if angle A and angle X are acute angles such that cosA=cosX then show that angle A=angle X

Q.

$\tan \left({\cos }^{-1}x\right)$is equal to?

1. $\frac{\sqrt{1-x^2}}{x}$

2. $\frac{x}{1+x^2}$

3. $\frac{\sqrt{1+x^2}}{x}$

4. $\sqrt{1+x^2}$

Q. A 1.6 m tall girl stands at a distance of 3.2 m from a lamp-post and casts a shadow of 4.8 m on the ground. Find the height of the lamp-post by using (i) trignometric ratios (ii) property of similar triangles.

Q.

For the given statements select the correct option.

Assertion : The value of $sin\theta =\frac{4}{3}$ in not possible.

Reason : Hypotenuse is the largest side in any right angled triangle.

1. Assertion is false but reason is true

2. Assertion is true but reason is false.

3. Both assertion and reason are true but reason is not the correct explanation of assertion.

4. Both assertion and reason are true and reason is the correct explanation of assertion.

Q. In ∆ABC, point M is the midpoint of side BC.
If, AB² + AC² = 290 cm², AM = 8 cm, find BC.

Q. If triangle ABC is right angled at B and
$sinA=\frac{3}{5}$, then find the value of cos A.

1. $\frac{3}{5}$
2. $\frac{5}{4}$
3. $\frac{4}{5}$
4. $\frac{5}{3}$

Q.

The solution of $\tan \left(2\theta \right)\tan \left(\theta \right)=1$ is

1. $\frac{\mathrm{\pi }}{3}$

2. $(6n\pm 1)\frac{\mathrm{\pi }}{6}$

3. $(4n\pm 1)\frac{\mathrm{\pi }}{6}$

4. $(2n\pm 1)\frac{\mathrm{\pi }}{6}$

Q. In right-angled triangle ABC is which ∠C = 90°, if D is the mid-point of BC, prove that AB² = 4 AD² − 3AC².

Q. ABC is right Δ, right angled at C and D is the mid point of BC. Prove that AC²=4AD²-3AB².

Q.

In a right angled triangle, with respect to an acute angle $\theta$, the opposite side is twice the adjacent side. Find the value of cosec$\theta$.

1. √5/2

2. √3/2

3. √2/3

4. 5/2

Q.

In the given right triangle ABC, right angled at B, the hypotenuse, base and perpendicular are _____________ respectively.

1. BC, AC and AB

2. BC, AB and AC

3. AB, BC and AC

4. AC, BC and AB

Q. In a right angled triangle ABC, right angled at C, if $\tan A=\cot B$, then what is value of A + B?

Q.

ABC is a triangle with AB=13cm, BC=15cm and CA=14cm, AD and BE are altitude from A and B to BC and AC respectively .H is the point of intersection of AD and BE then the ratio 5HD/HB is

Q.

ABCD is a rectangle such that $\angle CAB={60}^{\circ }$ and AC = $a$ units. The area of rectangle ABCD is $\frac{\sqrt{3}}{2}{a}^2$

1. True

2. False

Q. Two poles of height 6 m and 11 m stand vertically upright on a plane ground. If the distance between their foot is 12 m, the distance between their tops is

(a) 12 m
(b) 14 m
(c) 13 m
(d) 11 m

Q.

In the given figure, D is a point on hypotenuse AC of ΔABC, DM ⊥ BC and DN ⊥ AB, Prove that:

(i) DM² = DN.MC

(ii) DN² = DM.AN