#ssc 10th
- \(1 + \tan^2\theta = \) Answer: C. sec²θ Dividing sin²θ + cos²θ = 1 by cos²θ gives tan²θ + 1 = sec²θ . 1
- \(\sin 30° + \cos 60° = \) Answer: A. 1 1/2 + 1/2 = 1 . 1
- If \(\sin\theta = \frac{3}{5}\) and \(\theta\) is acute, then \(\cos\theta = \) Answer: B. 4/5 cosθ = √(1 - 9/25) = √(16/25) = 4/5 . 1
- \((1 - \sin^2\theta)\sec^2\theta = \) Answer: D. 1 (1 - sin²θ)sec²θ = cos²θ × 1/cos²θ = 1 . 1
- If \(\sec\theta = \frac{25}{7}\) and \(\theta\) is acute, then \(\tan\theta = \) Answer: A. 24/7 tan²θ = sec²θ - 1 = 625 - 49/49 = 576/49 , so tanθ = 24/7 . 1
- When an observer looks at an object above the horizontal line through the eye, the angle between the line of vision and the horizontal line is called the Answer: C. angle of elevation Looking up from the horizontal gives an angle of elevation; looking down gives an angle of depression. 1
- From a point on the ground 30 m from the foot of a tower, the angle of elevation of the top of the tower is 60°. The height of the tower is Answer: B. 30√(3) m tan 60° = h/30 ⇒ h = 30√(3) m. 1
- A vertical pole 6 m high casts a shadow \(6\sqrt{3}\) m long. The angle of elevation of the sun is Answer: D. 30° tanθ = 6/6√(3) = 1/√(3) ⇒ θ = 30° . 1
- If \(\sec\theta + \tan\theta = 3\), then \(\sec\theta - \tan\theta = \) Answer: B. 1/3 (secθ + tanθ)(secθ - tanθ) = sec²θ - tan²θ = 1 , so secθ - tanθ = 1/3 . 1
- The valency of carbon is Answer: A. 4 Carbon has 4 valence electrons (2, 4) and forms 4 covalent bonds. 1
- The functional group present in alcohols is Answer: B. –OH Alcohols have the hydroxyl group –OH. 1
- The general formula of alkanes is Answer: C. C_nH_(2n+2) Alkanes are saturated hydrocarbons, C_nH_(2n+2) (e.g. CH₄, C₂H₆ ). 1
- The IUPAC name of \(\text{CH}_3\text{CH}_2\text{OH}\) is Answer: A. ethanol It has two carbon atoms (eth-), single bonds (-an-) and an –OH group (-ol): ethanol. 1
- Two successive members of a homologous series differ by Answer: C. CH₂ Successive homologues differ by a CH₂ unit (molecular mass 14 u). 1
- The number of covalent bonds in a molecule of ethane \(\text{C}_2\text{H}_6\) is Answer: A. 7 Ethane has one C–C bond and six C–H bonds: 1 + 6 = 7 . 1
- The height of an equilateral triangle of side 6 cm is Answer: C. 3√(3) cm Height = √(3)/2 × 6 = 3√(3) cm. 1
- In \(\triangle PQR\), \(\angle PQR = 90°\) and seg \(QS \perp\) hypotenuse PR. If \(PS = 4\) and \(SR = 9\), then \(QS = \) Answer: A. 6 By the theorem of geometric mean, QS² = PS × SR = 4 × 9 = 36 , so QS = 6 . 1
- In \(\triangle ABC\), AD is a median, \(BC = 10\) and \(AD = 6\). Then \(AB^2 + AC^2 = \) Answer: C. 122 By Apollonius' theorem, AB² + AC² = 2AD² + 2BD² = 2(36) + 2(25) = 122 (since BD = 5 ). 1
- In isosceles \(\triangle ABC\), \(AB = AC = 13\) cm and \(BC = 10\) cm. The length of the perpendicular from A to BC is Answer: A. 12 cm The perpendicular from A bisects BC, so BD = 5 . Height = √(13² - 5²) = √(144) = 12 cm. 1
- Two triangles have equal heights and their bases are 9 cm and 6 cm. The ratio of their areas is Answer: B. 3 : 2 For triangles with equal heights, the ratio of the areas equals the ratio of the bases: 9 : 6 = 3 : 2 . 1
- Which of the following is NOT a test of similarity of triangles? Answer: A. SSA AAA (or AA), SAS and SSS are tests of similarity. SSA is not a test. 1
- The areas of two similar triangles are 25 cm² and 36 cm². The ratio of their corresponding sides is Answer: C. 5 : 6 Ratio of sides = √(25) : √(36) = 5 : 6 . 1
- \(\triangle ABC \sim \triangle PQR\) and \(AB : PQ = 2 : 3\). Then \(A(\triangle ABC) : A(\triangle PQR) = \) Answer: C. 4 : 9 The ratio of the areas of similar triangles equals the ratio of the squares of corresponding sides: 2² : 3² = 4 : 9 . 1
- In \(\triangle ABC\), seg \(DE \parallel\) side \(BC\), with D on AB and E on AC. If \(AD = 4\), \(DB = 6\) and \(AE = 6\), then \(EC = \) Answer: C. 9 By the basic proportionality theorem, AD/DB = AE/EC ⇒ 4/6 = 6/EC ⇒ EC = 9 . 1