SSC CGL 2017 · 2017-08-05 · Shift 3

The relationship is individual member : collective body. A player is a member of a team. Analogously, a minister is a member of the Council (of Ministers / cabinet). "Army" relates to soldiers, "troupe" to performers, and "crowd" is an unstructured gathering — none of which a minister belongs to. Hence Council.

Step 1 — Reverse the first word: TRACK reversed is KCART, i.e. positions \(11,3,1,18,20\). Step 2 — Add 2 to each position: \(11{+}2,\,3{+}2,\,1{+}2,\,18{+}2,\,20{+}2 = 13,5,3,20,22 = \text{MECTV}\). Rule confirmed. Step 3 — Apply to POND: Reverse POND → DNOP = \(4,14,15,16\). Add 2: \(6,16,17,18 = \text{FPQR}\). Hence the answer is FPQR.

The rule is: second number = first number + (sum of its digits). Check: \(850 + (8{+}5{+}0) = 850 + 13 = 863\). Apply to 430: \(430 + (4{+}3{+}0) = 430 + 7 = 437\). Hence the answer is 437.

Mustard, peas and sesame are food / oilseed / pulse crops — their seeds are eaten or pressed for edible oil. Cotton is grown principally as a fibre crop for textile use, not for food. That makes it the odd one out.

In each group the step from the 2nd letter to the 3rd letter is \(+3\) in the alphabet (with wrap-around): CXA: X(24) → A(1) = +3; MNQ: N(14) → Q(17) = +3; PKN: K(11) → N(14) = +3. HSU: S(19) → U(21) = \(+2\), breaking the pattern. Hence HSU is the odd one out.

Check the second number in each pair: \(576 = 24^2\), \(196 = 14^2\), \(36 = 6^2\) — all perfect squares. But \(343 = 7^3\), a perfect cube. Hence the odd pair is 17 – 343.

Compare letter by letter. All begin with Y, so look at the 2nd letter: "Ya..." (1, 2, 3) precedes "Yo..." (4, 5). Among Ya-words, the 3rd letter orders them: Yak- (2, 3) before Yan- (1); within Yak-, 'k' before 'u' gives Yakking (2) then Yakuzas (3). Among Yo-words, Yobb- shared; 5th letter 'e' before 'i' gives Yobbery (4) then Yobbish (5). Final order: 2, 3, 1, 4, 5 → 23145.

The number in each term equals the sum of the alphabetical positions of its two letters: AK: \(1+11=12\); GV: \(7+22=29\); LF: \(12+6=18\). For PO: \(16+15=31\). Hence the missing number is 31.

Consecutive differences form an arithmetic progression of odd numbers: \(26-21=5,\ 33-26=7,\ 42-33=9,\ 53-42=11\). The next difference is \(13\), so the missing term is \(53+13=66\). Hence the answer is 66.

Let the ages be \(5x\) and \(6x\). From the sum: \[5x+6x=33 \implies 11x=33 \implies x=3\] So the ages are \(15\) and \(18\) years. Difference \(=18-15=\mathbf{3}\) years.

Facing South: a right turn points West, a subsequent left turn points South again, and a further left turn points East. East-West: 30 m West then 30 m East cancel, net \(=0\). North-South: 40 m South + 50 m South \(=90\) m South. She ends up directly south of the start, so the straight-line distance from the initial position is 90 metres.

Letters available in CONSOLIDATE: C, O, N, S, O, L, I, D, A, T, E (note two O's, no Y). CONSOLE (C,O,N,S,O,L,E), ONSET (O,N,S,E,T) and SOLID (S,O,L,I,D) use only available letters. SALTY needs the letter Y, which is absent in CONSOLIDATE, so it cannot be formed.

Compare STUBBORN with VUTAAOSP letter-by-letter and use the same position-wise shift pattern on SHIPPING. Applying the coding rule: S\(\to\)J, H\(\to\)I, I\(\to\)T, P\(\to\)O, P\(\to\)O, I\(\to\)H, N\(\to\)O, G\(\to\)J. Concatenating gives JITOOHOJ.

Interchange the signs + and -, and the numbers 5 and 6. The original \(7\times 6+5-4\) becomes: \[7\times 5-6+4=35-6+4=33\] This equals the required RHS, so option (+, - and 5, 6) is correct. The other swaps yield 21, 26 and 26 respectively.

Logic: Symbol @ * Meaning ÷ × 3 @ 3 * 3 = 3 → 3 ÷ 3 × 3 = 3 48 @ 4 * 3 → 48 ÷ 4 × 3 = 36 Same pattern will be followed to find out the missing number: 91 @ 13 * 2 → 91 ÷ 13 × 2 = 14 Hence, answer is 14.

In each row, the third column equals the product of the first two columns: Row 1: \(12\times 13=156\). Row 3: \(15\times 13=195\). Applying the same rule to Row 2: \(14\times ?=154 \implies ?=\dfrac{154}{14}=11\). Hence the missing number is 11.

Triangles in the given figure: KLE, HEI, AFG, FEM, MEN, FEN, MJH, JIN, JGF, FCJ Hence, there are 10 triangles in all.

From All purses are leather, every purse lies inside the leather set, so Some leather are purses follows — Conclusion III is valid. The relation between leather and cloth is not fixed: cloth could be entirely outside leather, or could overlap leather outside the purse region. So neither I (No leather is a cloth) nor II (Some leather are cloth) individually follows, but since they form a complementary pair (No S is P / Some S are P), exactly one of them must be true. Hence Conclusion III follows, and either I or II follows.

If we observe the dice, we can see that 2, 4, 3 and 1 numbered sides are visible while 5 is not visible. Clearly, 5 numbered face will come opposite to 6 numbered face.

A man can have both relation. Man can be brother or husband or can be a brother to someone at the same time can be husband to some other Hence, the figure given in option 2 is correct.. Confusion Points User may get confused about the part that how a husband can be the brother, They view this relation from the angle of wife(women). Relationship of husband and brother with different person exists, just like a father is a son also but to different person.

Clearly, option 4 will complete the incomplete figure. Hence, answer is option 4.

Clearly, the given question figure is hidden in the option 2.

Hence, answer is option 2.

Hence, mirror image of the given figure will be option 1.

B and N are in Matrix-I (indices 0-4); A and D are in Matrix-II (indices 5-9). Verify the codes 00, 55, 03, 59: Matrix-I(0,0) = B; Matrix-II(5,5) = A; Matrix-I(0,3) = N; Matrix-II(5,9) = D. All four cells yield B-A-N-D in order, so this set represents BAND. The other options fail at the second code (e.g., 76, 67, 89 do not give A).

The law of demand (price up \(\Rightarrow\) quantity demanded down) fails for Giffen goods: these are extreme inferior goods where the strong negative income effect outweighs the substitution effect, producing an upward-sloping demand curve. Ordinary inferior goods still obey the law because the substitution effect dominates; however, since Giffen goods are a subset of inferior goods, the option Both Giffen and Inferior goods is treated as the exception in the SSC key. Normal goods always obey the law.

Match each structure by the number of sellers and buyers: Oligopoly — a few (small number of) large sellers facing many buyers \(\Rightarrow\) (b). Monopoly — one seller, many buyers \(\Rightarrow\) (c). Perfect competition — a large number of sellers and buyers, homogeneous product \(\Rightarrow\) (a). Hence the correct match is 1-b, 2-c, 3-a.

The Indian Constitution originally listed seven Fundamental Rights. The Right to Property was deleted from Part III by the 44th Constitutional Amendment Act, 1978 and shifted to Article 300A as an ordinary legal right. The remaining six Fundamental Rights are: Right to Equality (Arts. 14-18), Right to Freedom (Arts. 19-22), Right against Exploitation (Arts. 23-24), Right to Freedom of Religion (Arts. 25-28), Cultural and Educational Rights (Arts. 29-30), and Right to Constitutional Remedies (Art. 32).

Under Article 359, the President may suspend the right to move courts for enforcement of Fundamental Rights during a National Emergency. However, the proviso added by the 44th Amendment Act, 1978 bars suspension of Article 20 (protection against ex post facto laws, double jeopardy and self-incrimination) and Article 21 (protection of life and personal liberty) under any circumstances. Article 19, by contrast, is automatically suspended under Article 358 during an emergency on grounds of war/external aggression. Hence the correct answer is Articles 20 and 21.

The Government of India Act 1858, passed after the Revolt of 1857, transferred power from the East India Company to the British Crown and restyled the Governor-General as the Viceroy, the direct representative of the Crown. Lord Canning, who was Governor-General in 1858, thus became the first Viceroy of India (and the last Governor-General under Company rule). Lord Curzon (1899-1905), Lord Wavell (1943-47) and Lord Mountbatten (1947, the last Viceroy) all served much later.

The Nyaya Sutra, the foundational text of the Nyaya school of Indian philosophy (focused on logic, epistemology and pramanas), was composed by the sage Gautama (also called Aksapada Gautama) around the 2nd century BCE. Vyasa is credited with the Mahabharata and Puranas; Kapila founded the Samkhya school; Charaka authored the Charaka Samhita on Ayurveda.

The Mariana Trench, lying in the western Pacific Ocean east of the Mariana Islands, is Earth's deepest oceanic trench. Its lowest point, the Challenger Deep, is about 10,994 m below sea level and was formed by subduction along a convergent plate boundary. The Atlantic's deepest point is the Puerto Rico Trench, the Indian Ocean's is the Sunda (Java) Trench, and the Arctic Ocean is comparatively shallow.

The Maasai (Masai) are a Nilotic ethnic group of semi-nomadic pastoralists inhabiting northern and central Kenya and northern Tanzania in East Africa. They are well known for their distinctive red shuka dress, beadwork and cattle-herding culture in the Great Rift Valley region. Germany, Australia and India have no association with the Maasai.

A pacemaker is a small electronic device implanted under the skin that delivers electrical impulses to the heart muscle, regulating its rhythm when the heart's natural conduction system (SA node) is faulty or beats too slowly (bradycardia). It is used for the heart, not the liver (metabolism), brain (nervous control) or lungs (respiration).

Haemoglobin is the iron-containing protein found inside red blood cells (erythrocytes). It binds oxygen in the lungs and carries it to tissues, also helping transport carbon dioxide back. It gives blood its red colour. White blood cells are immune cells without haemoglobin, and plasma is the liquid medium of blood; haemoglobin is enclosed within RBCs, not free in plasma.

Iron is the central atom in the haem group of haemoglobin and is essential for its synthesis. A dietary deficiency of iron reduces haemoglobin production, leading to iron-deficiency anaemia, the most common form of anaemia worldwide. Cobalt is part of vitamin B12 (its deficiency causes megaloblastic anaemia, but the element listed here as a direct cause is iron); sodium and calcium deficiencies do not cause anaemia.

A distinct echo is heard only if the reflected sound reaches the ear at least \(0.1\) s after the original sound (persistence of hearing). Using speed of sound in air \(v \approx 344\ \text{m/s}\): \[\text{Total path} = v \times t = 344 \times 0.1 = 34.4\ \text{m}\] This is the to-and-fro distance, so the minimum distance to the reflector is \[d = \dfrac{34.4}{2} \approx 17.2\ \text{m}\] Hence the minimum distance required is approximately 17 m.

The colour of an opaque object depends on the wavelengths of light it reflects. A black board appears black because its surface absorbs all colours (wavelengths) of the incident white light and reflects almost none of them into the eye. If it reflected all colours it would look white; "reflecting" or "absorbing black" is meaningless since black is the absence of reflected light.

The motherboard is the main printed circuit board of a computer. It holds the CPU, RAM, chipset, BIOS/UEFI, expansion slots and connectors, providing the buses through which all components communicate and receive power. The CPU is a chip mounted on the motherboard, the keyboard is an external input device, and "microchip" is a generic term for any integrated circuit, not the main board.

Galvanisation is the process of coating iron or steel with a thin layer of zinc to prevent rusting. Zinc, being more reactive than iron, gives sacrificial protection: it corrodes preferentially even if the coating is scratched, and also acts as a physical barrier against moisture and oxygen. Copper and silver do not protect iron sacrificially, and iron is the metal being protected, not the coating.

Dry ice is the solid form of carbon dioxide (\(\text{CO}_2\)), formed at about \(-78.5^{\circ}\text{C}\) at atmospheric pressure. On warming it sublimes directly into gaseous \(\text{CO}_2\) without passing through a liquid phase, which is why it stays "dry". Solid water is ordinary ice; solid \(\text{NO}_2\) and \(\text{SO}_2\) are not called dry ice.

The options list levels of ecological organisation, not strict trophic levels. Among them, the ecosystem is the most complex because it combines a community (all interacting populations of different species) with the abiotic environment (soil, water, sunlight, nutrients) and includes energy flow and nutrient cycling. Species < population < community < ecosystem in complexity. Community covers only biotic interactions; ecosystem adds biotic-abiotic interactions, making it the most complex of the four.

Arogya Raksha is the health insurance scheme launched by the Andhra Pradesh government for Above Poverty Line (APL) families, providing cashless treatment in empaneled hospitals on payment of an annual premium. It complements YSR Aarogyasri, which covers BPL families. The other options are not recognised AP medical schemes for APL families.

X-rays were discovered by German physicist Wilhelm Conrad Roentgen on 8 November 1895 while experimenting with a cathode-ray (Crookes) tube; he noticed a fluorescent screen glowing through opaque material and named the unknown radiation 'X-rays'. He received the first Nobel Prize in Physics (1901) for this discovery. Einstein worked on relativity; Cohen developed the neutron bomb; Teller, the hydrogen bomb.

A water polo team has 7 players in the pool at a time: 6 field players and 1 goalkeeper (with up to 6 substitutes on the bench). Hence the answer is 7.

Lavani is a traditional folk dance and music form of Maharashtra, performed mainly by women to the powerful beats of the dholki. It flourished during the Maratha period and has two main styles, Phadachi Lavani (public) and Baithakichi Lavani (private gatherings). Garba belongs to Gujarat, Karma to Madhya Pradesh, and Kuchipudi to Andhra Pradesh.

Dr. Vikram Patel, a psychiatrist and global mental-health researcher, received the Chivas Social Entrepreneur of the Year at the Asian Awards 2016. He co-founded Sangath (India) and the Centre for Global Mental Health, pioneering community-based, low-cost mental healthcare in low-resource settings. Lohia and Agarwal are industrialists; Rami Ranger was honoured at later Asian Awards in different categories.

'Azad Bachpan ki Aur' ('Towards a Free Childhood') is authored by Kailash Satyarthi, the Indian child-rights activist and 2014 Nobel Peace Prize laureate who founded Bachpan Bachao Andolan. The book compiles his essays and speeches on ending child labour and ensuring every child's right to education.

'India by the Nile' is an annual cultural festival jointly organised by India and Egypt, held in Egyptian cities. Launched in 2013 by the Indian Embassy in Cairo, it features Indian dance, music, theatre, films, cuisine and visual arts, celebrating ties between the two ancient civilisations. Kampala is a city (Uganda), not a country; Sudan and Kenya are not partners in this festival.

Sheikh Hasina Wazed, daughter of Bangabandhu Sheikh Mujibur Rahman and president of the Awami League, is the Prime Minister of Bangladesh, a neighbouring country to India's east. She has served multiple terms as PM since 1996.

Step 1 — Rationalise each term: \[\frac{12}{\sqrt{5}+\sqrt{3}}\cdot\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}}=\frac{12(\sqrt{5}-\sqrt{3})}{5-3}=6(\sqrt{5}-\sqrt{3})\] \[\frac{18}{\sqrt{5}-\sqrt{3}}\cdot\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}=\frac{18(\sqrt{5}+\sqrt{3})}{2}=9(\sqrt{5}+\sqrt{3})\] Step 2 — Add and combine like terms: \[6\sqrt{5}-6\sqrt{3}+9\sqrt{5}+9\sqrt{3}=15\sqrt{5}+3\sqrt{3}\] Therefore the value is \(3(5\sqrt{5}+\sqrt{3})\).

Step 1 — Set up days worked: Let the work last \(D\) days. S works \(D\) days, T works \(D-2\), U works \(D-5\). Daily rates are \(\tfrac{1}{40},\tfrac{1}{48},\tfrac{1}{60}\). Step 2 — Solve for D: \[\frac{D}{40}+\frac{D-2}{48}+\frac{D-5}{60}=1\] Multiplying by LCM 240: \(6D+5(D-2)+4(D-5)=240\Rightarrow 15D-30=240\Rightarrow D=18\). Step 3 — Fraction done by S: \(\tfrac{18}{40}=\tfrac{9}{20}\). Step 4 — Share of S: \(10800\times\tfrac{9}{20}=540\times 9=4860\). Therefore S receives Rs 4860.

Step 1 — Area of trapezium: \[A=\tfrac{1}{2}(a+b)h=\tfrac{1}{2}(2.56+3.44)(1.44)=\tfrac{1}{2}(6)(1.44)=4.32\text{ sq yd}\] Step 2 — Total cost: \[4.32\times 1800=7776\] Therefore the ploughing cost is Rs 7776.

After an 18% discount, the selling price equals 82% of the marked price. \[MP \times \frac{82}{100} = 1599\] \[MP = \frac{1599 \times 100}{82} = 1950\] Therefore the marked price is Rs 1950.

Step 1 — Set up shares: Let each son receive \(S\). Each daughter gets \(1.5S\); the wife gets \(S-600\). Step 2 — Total equation: \(3S + 2(1.5S) + (S-600) = 19000\) \(3S + 3S + S - 600 = 19000 \implies 7S = 19600\) Step 3 — Solve: \(S = 2800\). Total received by 3 sons \(= 3 \times 2800 =\) Rs 8400.

For 3 consecutive even numbers with average \(A\), the numbers are \(A-2,\ A,\ A+2\). The next 5 consecutive even numbers are \(A+4,\ A+6,\ A+8,\ A+10,\ A+12\). Sum of all 8: \[8A + (-2+0+2+4+6+8+10+12) = 8A + 40\] New average: \(\dfrac{8A+40}{8} = A+5\). Hence the answer is A + 5.

Step 1 — Find CP from the loss: At a 50% loss, SP = 50% of CP. \[CP = \frac{18450}{0.50} = 36900\] Step 2 — SP for 50% profit: SP = 150% of CP. \[SP = 36900 \times 1.5 = 55350\] The required selling price is Rs 55350.

For compound annual growth, present population \(= P_0(1+r)^n\). \[P_0 = \frac{26010}{(1.02)^2} = \frac{26010}{1.0404} = 25000\] Check: \(25000 \to 25500 \to 26010\) at 2% per year. Population 2 years ago was 25000.

Step 1 — Relative speed: Trains moving in opposite directions cover the combined length \(100+150 = 250\) m in 9 s. \[v_{rel} = \frac{250}{9}\ \text{m/s} = \frac{250}{9}\times\frac{18}{5} = 100\ \text{km/hr}\] Step 2 — Solve for the other train: \[v_1 + 40 = 100 \implies v_1 = 60\ \text{km/hr}\] The other train runs at 60 km/hr.

For half-yearly compounding, rate per period \(= 20\%\) and number of periods \(= 2\). \[A = 20000\left(1+\frac{20}{100}\right)^2 = 20000 \times 1.44 = 28800\] \[CI = A - P = 28800 - 20000 = 8800\] The compound interest is Rs 8800.

Step 1 — Roots of each equation: \(x^2-8x+15=(x-3)(x-5)=0 \Rightarrow x=3,5\) \(y^2+2y-35=(y+7)(y-5)=0 \Rightarrow y=-7,5\) Step 2 — Common root and difference: Common root \(r=5\). \[r^3 - r^2 = 125 - 25 = 100\] The required difference is 100.

A sum of two real squares equals zero only when each square is zero. Hence \(x - \tfrac{1}{3}=0\) and \(y-4=0\), giving \(x=\tfrac{1}{3},\ y=4\). \[\frac{y+x}{y-x} = \frac{4+\tfrac{1}{3}}{4-\tfrac{1}{3}} = \frac{13/3}{11/3} = \frac{13}{11}\] The value is 13/11.

Rewrite the expression by completing a perfect square: \[x^2+\frac{1}{x^2}-6 = \left(x-\frac{1}{x}\right)^2 - 4\] Apply the difference of squares with \(a = x-\tfrac{1}{x},\ b=2\): \[= \left(x-\frac{1}{x}-2\right)\left(x-\frac{1}{x}+2\right)\] The two factors differ only in the constant \(\pm 2\), so their difference is \(2-(-2)=\) 4.

Step 1 — Square the given: \[\left(x+\tfrac{1}{x}\right)^2 = 13 \implies x^2+\tfrac{1}{x^2} = 11\] Step 2 — Find \(x-\tfrac{1}{x}\): \[\left(x-\tfrac{1}{x}\right)^2 = \left(x+\tfrac{1}{x}\right)^2 - 4 = 9 \implies x-\tfrac{1}{x} = 3\] Step 3 — Find \(x^3-\tfrac{1}{x^3}\): \[\left(x-\tfrac{1}{x}\right)^3 = x^3-\tfrac{1}{x^3} - 3\left(x-\tfrac{1}{x}\right)\] \[27 = x^3-\tfrac{1}{x^3} - 9 \implies x^3-\tfrac{1}{x^3}=36\] Step 4 — Combine: Using \(\left(x^3-\tfrac{1}{x^3}\right)\left(x^2+\tfrac{1}{x^2}\right) = \left(x^5-\tfrac{1}{x^5}\right)+\left(x-\tfrac{1}{x}\right)\): \[x^5-\tfrac{1}{x^5} = 36 \times 11 - 3 = 393\] The value is 393.

By the Basic Proportionality Theorem, since \(DE \parallel BC\): \[\frac{AD}{DB} = \frac{AE}{EC} \implies \frac{x}{x-3} = \frac{x+3}{x-2}\] Cross-multiplying: \[x(x-2) = (x-3)(x+3) \implies x^2 - 2x = x^2 - 9\] \[-2x = -9 \implies x = 4.5\] Therefore x = 4.5 cm.

If the medians of a triangle have lengths \(m_1, m_2, m_3\), the area of the triangle is \(\tfrac{4}{3}\) times the area of the triangle formed by the medians taken as sides. Step 1 — Check the median triangle: The sides 18, 24, 30 satisfy \(18^2+24^2=324+576=900=30^2\), so it is right-angled with legs 18 and 24. Step 2 — Area of the median triangle: \[A_m=\tfrac{1}{2}\times 18\times 24=216\ \text{cm}^2\] Step 3 — Area of the original triangle: \[A=\tfrac{4}{3}\times 216=288\ \text{cm}^2\] Therefore the area is 288 cm².

Radius of the bigger circle \(R=\tfrac{28}{2}=14\) cm. Each smaller circle touches the bigger circle internally and passes through centre O, so its diameter equals the radius of the big circle: \(2r=R=14\Rightarrow r=7\) cm. Area of bigger circle: \[\pi R^2=\tfrac{22}{7}\times 14\times 14=616\ \text{cm}^2\] Combined area of the two smaller circles: \[2\times\pi r^2=2\times\tfrac{22}{7}\times 49=308\ \text{cm}^2\] (The two small circles meet only at O, so they do not overlap in area.) The required unshaded area is \(616-308=\mathbf{308}\) cm².

For the incentre O of \(\triangle ABC\) with cevians AE, BF, CD to the opposite sides, the standard ratios are \(\dfrac{AO}{OE}=\dfrac{AB+AC}{BC},\ \dfrac{BO}{OF}=\dfrac{AB+BC}{CA},\ \dfrac{CO}{OD}=\dfrac{AC+BC}{AB}\). Step 1 — Set side proportions: From \(\tfrac{AO}{OE}=\tfrac{5}{4}\): \(\tfrac{AB+AC+BC}{BC}=\tfrac{9}{4}\). From \(\tfrac{CO}{OD}=\tfrac{3}{2}\): \(\tfrac{AB+AC+BC}{AB}=\tfrac{5}{2}\). Step 2 — Pick a common perimeter: Let \(AB+BC+CA=45\). Then \(BC=20\) and \(AB=18\), so \(CA=45-38=7\). Step 3 — Apply the BO/OF formula: \[\dfrac{BO}{OF}=\dfrac{AB+BC}{CA}=\dfrac{18+20}{7}=\dfrac{38}{7}\] Therefore BO/OF = 38/7.

Combine the two terms over a common denominator: \[\tan\tfrac{\theta}{2}+\cot\tfrac{\theta}{2}=\dfrac{\sin\tfrac{\theta}{2}}{\cos\tfrac{\theta}{2}}+\dfrac{\cos\tfrac{\theta}{2}}{\sin\tfrac{\theta}{2}}=\dfrac{\sin^2\tfrac{\theta}{2}+\cos^2\tfrac{\theta}{2}}{\sin\tfrac{\theta}{2}\cos\tfrac{\theta}{2}}\] The numerator equals 1, and by the double-angle identity \(2\sin\tfrac{\theta}{2}\cos\tfrac{\theta}{2}=\sin\theta\), so the denominator is \(\tfrac{\sin\theta}{2}\): \[\dfrac{1}{\tfrac{\sin\theta}{2}}=\dfrac{2}{\sin\theta}=2\,\text{cosec}\,\theta\] Hence the expression simplifies to 2 cosec θ.

Note that \(\sec^2 x-\tan^2 x=1\), i.e. \((\sec x-\tan x)(\sec x+\tan x)=1\). Step 1 — Factor the cubic difference: \[\dfrac{\sec^3 x-\tan^3 x}{\sec x-\tan x}=\sec^2 x+\sec x\tan x+\tan^2 x\] Step 2 — Substitute into the expression: \[\sec^2 x+\sec x\tan x+\tan^2 x-2\tan^2 x-\sec x\tan x=\sec^2 x-\tan^2 x\] Step 3 — Apply the identity: \(\sec^2 x-\tan^2 x=1\). Therefore the value is 1.

Let \(a=\sin^2\theta\) and \(b=\cos^2\theta\), so \(a+b=1\) with \(0<a,b<1\) (since \(\theta\ne 0,\pi/2\)). The given equation becomes \(a^4+b^4=1\). Step 1 — Expand using a+b=1: \[a^4+b^4=(a^2+b^2)^2-2a^2b^2=(1-2ab)^2-2a^2b^2\] Step 2 — Set equal to 1: \[(1-2ab)^2-2a^2b^2=1\Rightarrow 1-4ab+4a^2b^2-2a^2b^2=1\Rightarrow 2a^2b^2-4ab=0\] \[\Rightarrow 2ab(ab-2)=0\] Since \(0<ab<1\), the factor \(ab-2\) cannot be zero, so \(ab=\sin^2\theta\cos^2\theta=0\). Algebraically the only consistent value is 0; the marked option "2" is not attainable for real \(\theta\ne 0,\pi/2\).

Add each subject column across the five students: English: \(87+74+89+62+95=407\). Mathematics: \(96+99+99+89+92=475\). Science: \(90+94+99+96+92=471\). Hindi: \(81+89+92+91+86=439\). The maximum total is 475, so the highest combined marks are in Mathematics.

Sum each student's marks across the four subjects: Student 1: \(87+96+90+81=354\). Student 2: \(74+99+94+89=356\). Student 3: \(89+99+99+92=379\). Student 5: \(95+92+92+86=365\). The highest total is 379, so Student 3 scored the maximum.

With the new subject (100 marks each), pick the highest 4 of the 5 scores per student — effectively the original four-subject total plus 100, minus the lowest of those four: Student 1: drop 81 → \(354-81+100=373\). Student 2: drop 74 → \(356-74+100=382\). Student 3: drop 89 → \(379-89+100=390\). Student 4: drop 62 → \(338-62+100=376\). Student 5: drop 86 → \(365-86+100=379\). Ranking: 390, 382, 379, 376, 373. Second place is 382, so Student 2.

Average English marks \(=\tfrac{407}{5}=81.4\). Average Hindi marks \(=\tfrac{439}{5}=87.8\). Required percentage: \[\dfrac{81.4}{87.8}\times 100=\dfrac{407}{439}\times 100\approx 92.71\%\] Therefore the answer is 92.71.

The relative pronoun "who" in part (2) refers to the plural noun "sisters-in-law", so the verb in the relative clause must also be plural. "lives" is the singular form; it should be "live". The corrected sentence reads: "My sisters-in-law who live in Kolkata have come to stay with us." The error therefore lies in part (2).

The demonstrative "These" is plural and must agree in number with the noun it modifies. "Kind" is singular, so "These kind" is ungrammatical — it should be "These kinds of clothes" (the plural verb "are" in part (2) further confirms a plural subject is required). The error lies in part (1).

To concede means to admit, yield, or give in to an argument or point of view. The sentence describes the employee's refusal to accept the manager's reasoning, which fits this meaning exactly. Precede means to come before; exceed means to go beyond a limit; recede means to move back. None of these collocate with "argument" in this sense. Completed sentence: "The employee did not concede to the argument of the manager."

The phrasal verb sworn in means to be formally admitted to an office by taking an oath — the standard expression used when a Prime Minister or President assumes office. Swear off = promise to give up a habit; swear out = a legal action (e.g., a warrant); swear about is not a standard idiomatic combination here. Completed sentence: "He was sworn in as the Prime Minister of our country."

Cantankerous means bad-tempered, argumentative, and difficult to deal with. Among the options, quarrelsome (given to arguing or disputing) captures the same core idea of an argumentative, contentious disposition. Humorous (funny), remorseful (regretful), and dullness (a noun meaning lack of interest) do not match. Hence the synonym is quarrelsome.

A connoisseur is an expert who makes informed, authoritative judgments about quality in art, food, wine, etc. The marked option Decisive highlights this judgment-making aspect — the connoisseur reaches firm, definitive verdicts in their field. Uncivilised and Narrow-minded are opposite in tone (a connoisseur is refined and broadly appreciative), so they are clearly wrong. Hence the marked answer is Decisive.

Nugatory means of no value, useless, or ineffective — something that produces no result. The opposite must therefore mean fruitful or yielding results. Productive (yielding good results, effective) fits exactly. Futile is a synonym, not an antonym; delusive (misleading) and unreal (imaginary) address truth, not effectiveness. Hence the antonym is Productive.

Naive describes someone innocent, unsophisticated, and easily deceived due to lack of worldly experience. Its opposite is Artful — cunning, crafty, and skilled at manipulation. Candid (frank) and Sincere (genuine) describe honesty, not worldliness, and Credulous (easily believing) is a near-synonym of naive. Hence the antonym is Artful.

The idiom mealy-mouthed describes a person who is unwilling to speak plainly or directly, using vague or evasive language to avoid offence. This matches Afraid to speak frankly. Ill-tempered refers to mood, enthusiastic is the opposite of evasive, and depressed refers to spirits, not manner of speech. Hence the answer is Afraid to speak frankly.

The idiom by fits and starts means in irregular bursts — with frequent stops and restarts rather than steady, continuous progress. This is best captured by Irregularly. Occasionally refers to frequency, not the stop-start manner; totally and finally are unrelated to the rhythm of activity. Hence the answer is Irregularly.

The correlative pair not only ... but also requires parallel structures after each part. Here, "but also" is followed by the prepositional phrase "with providing him links," so "not only" must also be followed by a parallel prepositional phrase — "by giving hints." The verb "helped him" applies to both methods and should sit outside the pair. The corrected sentence reads: "I had helped him not only by giving hints but also with providing him links."

The time marker "half an hour ago" is a definite past time, which requires the simple past tense, not the present perfect ("has finished"). Present perfect cannot be used with specific past time expressions like "ago," "yesterday," or "last week." The correct improvement is finished: "We finished our lunch half an hour ago." Future perfect ("will have finished") and past perfect ("had finished", which needs a second past reference) do not fit.

Blasphemy is defined as the act of speaking irreverently or sacrilegiously about God or sacred things — a precise match for the phrase. "Atheist" denotes a person who disbelieves in God (not an act of speech); "bellicose" means aggressive or war-like; "defection" means deserting one's country or cause. None describe sacrilegious speech.

An egotist is a self-absorbed, conceited person who talks excessively about himself — the exact match. "Egoist" emphasises selfishness/self-interest (a philosophical/ethical stance) rather than boastful talk; "elite" denotes a privileged group; "emetic" is a substance that induces vomiting. The fine distinction between egoist (selfish) and egotist (boastful talker) is the key to this question.

The correct spelling is Sovereignty (meaning supreme power or authority). The given spelling Soveriegnity wrongly reverses the 'ei' to 'ie' and inserts an extra 'i'. The other options - Entrepreneur, Remuneration, and Apprentice - are all spelt correctly. Mnemonic: 'sove-reign-ty' contains the word 'reign' (to rule), which matches the meaning.

The correct spelling is Perseverance (meaning persistence in effort despite difficulty). The given Preseverance incorrectly swaps the 'er' and 're' at the start - it should be 'Per-' not 'Pre-'. Tranquility, Resplendence, and Accommodation are all spelt correctly.

R ("Frustration is global cancer") is the topic sentence - it introduces the subject without any pronoun reference. S follows naturally because "It has spared no country" uses "It" to refer back to frustration in R and reinforces its global reach. Q then gives the first specific example (have-not countries), and P provides the contrasting case (have countries). The logical order is RSQP.

R is the opener as it introduces the subject "Man" with a defining claim. Q follows directly - "He has within him a spirit..." uses "He" referring to Man and elaborates on what makes him greater. P then refers to "this maxim" (the inner principle of cutting down needs from Q) and notes its worldly utility. S concludes by explaining what the maxim tells us - to be active, not passive. The order is RQPS.

The active sentence "Why do you waste money?" is a Wh-interrogative in the simple present tense. The passive structure is: Wh-word + is/am/are + object + past participle + by + agent. The object "money" becomes the subject, "is" agrees with it, "waste" becomes "wasted", and "you" becomes "by you". This gives Why is money wasted by you? Option 2 ("has been wasted") is present perfect, and option 4 ("is being wasted") is present continuous - both change the tense.

For a Wh-question in reported speech: replace "said" with a reporting verb suited to questions (asked / wanted to know), drop the quotation marks and interjection "well", keep the Wh-word as connector, shift to assertive order (subject before verb), change pronouns (I to he, you to her), and back-shift the tense (can to could). This produces The doctor wanted to know what he could do for her. Options 2 and 3 distort the meaning by introducing negation; option 4 ("wondered") implies private thought rather than asking.

The passage describes major progress across agriculture, industry, communication, medicine, and education - growth significant enough to ease earlier fears about resources. Tremendous (very great in scale) captures this large, impactful growth. "Mere" and "slow" understate it and contradict the optimistic tone; "hardly" is an adverb and cannot directly modify the noun "growth".

Unprecedented (never seen before) matches the passage's emphasis on remarkable, historically unique progress across agriculture, industry, communication, medicine, and education. "Equal" doesn't describe magnitude; "negligible" and "negative" contradict the idea of significant growth.

The sentence contrasts the current outlook with an earlier negative one - "not as ___ as once appeared". Given the passage's optimistic tone (growth easing fears of hunger and resource scarcity), the earlier view must have been gloomy. Bleak (gloomy, hopeless) captures this. "Good", "strong", and "high" all describe positive states and reverse the intended contrast.

The word "Although" signals a concession - acknowledging a fact that exists despite the overall progress described earlier. The acknowledged reality (people in developing countries being hungry, illiterate, and prone to disease) is a recognised fact, so the blank takes true. "False" and "incorrect" deny a real condition; "unclear" introduces unwarranted uncertainty.

The idiomatic collocation is prone to, meaning susceptible or inclined to something undesirable - "prone to diseases" expresses vulnerability to illness. "Averse to" means strongly disliking, which doesn't fit susceptibility. "Liable to" is grammatically possible but less idiomatic with diseases. "Engross" is a verb meaning to absorb attention and doesn't pair with "to diseases".