Question : In $\triangle P Q R$, $\angle Q=90^{\circ}$, $PQ=8$ cm and $\angle P R Q=45^{\circ}$ Find the length of $QR$.

Option 1: 6 cm

Option 2: 3 cm

Option 3: 5 cm

Option 4: 8 cm

Question : It is given that ABC $\cong$ PQR, AB = 5 cm, $\angle$B = $40^{\circ}$, and $\angle$A = $80^{\circ}$. Which of the following options is true?

Option 1: PQ = 5 cm and $\angle$R = $60^{\circ}$

Option 2: QR = 5 cm and $\angle$R = $60^{\circ}$

Option 3: QR = 5 cm and $\angle$Q = $60^{\circ}$

Option 4: PQ = 5 cm and $\angle$P = $60^{\circ}$

Question : In a right-angled triangle $\Delta PQR, PR$ is the hypotenuse of length 20 cm, $\angle PRQ = 30^{\circ}$, the area of the triangle is:

Option 1: $50\sqrt{3}\text{ cm}^{2}$

Option 2: $100\sqrt{3}\text{ cm}^{2}$

Option 3: $25\sqrt{3}\text{ cm}^{2}$

Option 4: $\frac{100}{\sqrt{3}}\text{ cm}^{2}$

Question : PQR is a triangle right-angled at Q and PQ : QR = 3 : 4. What is the value of $\sin P + \sin Q + \sin R?$

Option 1: $\frac{4}{5}$

Option 2: $\frac{3}{5}$

Option 3: $\frac{12}{5}$

Option 4: $\frac{2}{5}$

Question : In a right-angled triangle PQR, right-angled at Q, the length of the side PR is 17 units, the length of the base QR is 8 units, and the length of the side PQ is 15 units. If $\angle RPQ = \sin\alpha $, then $\sin\alpha + \cos\alpha$ is:

Option 1: $\frac{18}{17}$

Option 2: $\frac{23}{17}$

Option 3: $\frac{21}{17}$

Option 4: $\frac{15}{17}$