SSC CGL 2017 · 2017-08-10 · Shift 3

The relationship in the first pair is cubing: \(3^3 = 27\). Applying the same rule to 4: \(4^3 = 64\). Hence the missing term is 64.

Clearly, 11 triangles are there in the given figure.

Represent the statements as set inclusions: Cups ⊂ Vegetables ⊂ Pens. Conclusion I ("Some pens are vegetable"): "All vegetables are pens" guarantees that vegetables form a non-empty subset of pens, so some pens are indeed vegetables. Follows. Conclusion II ("Some pens are cups"): chaining gives "All cups are pens", which similarly implies some pens are cups. Follows. Hence both conclusions follow.

Find each code by spotting common words across pairs of sentences. "you" is common to sentences 1 and 2, and code 4 is the shared number — so you = 4. "are" is common to sentences 1 and 3, sharing code 2 — so are = 2. "they" is common to sentences 2 and 3, sharing code 5 — so they = 5. In "they are dangerous" = "2 9 5", the remaining word "dangerous" maps to the remaining code 9.

The pattern is to subtract 1 from each digit and concatenate. Check: \(4*7*2 \rightarrow (4-1)(7-1)(2-1) = 361\) and \(5*9*1 \rightarrow (5-1)(9-1)(1-1) = 480\). Both match. Apply to \(2*1*3\): digits become \((2-1)(1-1)(3-1) = 1,0,2\), giving 102.

From dice 1 and 3 (from left to right) we conclude that 1, 6, 5 and 4 lie adjacent to 3. Therefore, 2 must lie opposite 3. From positions 2 and 3 (from left to right) we conclude that 1, 3, 2 and 5 lie adjacent to 4. Therefore, 6 must lie opposite 4. So face which will be opposite to face ‘5’ is ‘1’. Hence, the answer is ‘1’.

The letters available in BIOLOGICAL are B, I, O, L, O, G, I, C, A, L — i.e. {B, I, O, L, G, C, A}. No letter E is present. "Logic" (L, O, G, I, C), "Bail" (B, A, I, L) and "Bill" (B, I, L, L — two Ls available) can all be formed. "Globe" requires the letter E, which is absent, so it cannot be formed.

The rule is to shift each letter 2 places backward in the alphabet. Check: K−2=I, L−2=J, M−2=K, N−2=L, giving IJKL. Apply to TUVW: T−2=R, U−2=S, V−2=T, W−2=U, giving RSTU.

Given, Circle represents Musical. Square represents Electronic. Rectangle represents Toy. Shaded region in the figure is the common area between the Musical and Toy. Musical toys → 28 + 14 = 42 Hence, there are ‘42’ musical toys.

Hence, the unfolded figure will be similar to option 3.

A vertical mirror is placed, so the object at the left will appear at the right. The option 1 is the same as the question figure hence it is eliminated. Also, in option 3 and 4 cube is reversed so it is also not possible. Hence, the correct answer is option 2.

Read each code as (row, column). Verify the option against the letters of SPEAK in order. Option 1 — 10, 12, 11, 66, 58: In Matrix-I, row 1 col 0 = S, row 1 col 2 = P, row 1 col 1 = E. In Matrix-II, row 6 col 6 = A, row 5 col 8 = K. All five letters match S, P, E, A, K in sequence. The other options break down — e.g. Option 2 has 86 = R (not K); Option 3 has 56 = O (not A); Option 4 has 97 = O (not A). Hence the correct set is 10, 12, 11, 66, 58.

Substitute P=×, R=−, S=+, Q=÷ and apply BODMAS. Option 2: \(15 + 16 \div 2 \times 4 = 15 + 8 \times 4 = 15 + 32 = 47\). Matches the RHS. Quick check on the rest: Option 1 gives \(18 - 4 + 2 = 16 \ne 8\); Option 3 gives \(15 - 6 = 9 \ne 6\); Option 4 gives \(15 + 14 = 29 \ne 27\). Hence Option 2 is the true equation.

Step 1 — Set up variables: Let present ages be \(P = 5x\) and \(Q = 8x\). Step 2 — Apply the future ratio: \[\frac{5x+3}{8x+3} = \frac{8}{11}\] Step 3 — Solve: \[11(5x+3) = 8(8x+3) \implies 55x + 33 = 64x + 24 \implies 9x = 9 \implies x = 1\] Therefore Q's present age \(= 8x = \mathbf{8}\) years.

Decode the clue: "Her mother's only grandson is my son" means the lady's mother has exactly one grandson, and that grandson is Diwakar's son. Since the lady is unmarried, she has no son. So her mother's grandson must come from the lady's sibling. For that grandson to be Diwakar's son, Diwakar himself must be the lady's brother (the only male child of her mother who produced the grandson). Hence the lady is Diwakar's sister.

Hence, option 1 will complete the given question figure.

Check each number for primality: 31, 37 and 43 are all prime numbers (divisible only by 1 and themselves). However, \(49 = 7 \times 7 = 7^2\), which is composite and a perfect square. Therefore 49 is the odd one out.

The relationship in Fire : Burn is entity-to-its-characteristic-action: fire's defining action is to burn. Applying the same rule: ice's characteristic action/state is to freeze. Water does not drink (drink is what someone does with water), wood comes from trees (source relation), and rose is a kind of flower (category relation). Only Ice : Freeze matches.

The question figure is embedded in the following figure; Hence, the given question figure is embedded in option 2.

Compare letter by letter. All five words start with S; only Satire (5) has second letter 'a', so it comes first. Among the remaining 'St-' words, third letters are: Stamped 'a', Storm 'o', Strangle 'r', Strap 'r'. So Stamped (4) is next, then Storm (1). Between Strangle and Strap, the fifth letters are 'n' and 'p' respectively, so Strangle (3) precedes Strap (2). Final order: 5, 4, 1, 3, 2 — option 54132.

Ludo, Chess and Carrom are all indoor board games played on a board or table. Polo is an outdoor team sport played on horseback on a large field, making it the odd one out.

Check the gap between consecutive letters in each group using positions (A=1, B=2, ...): DGJ: 4, 7, 10 — gaps \(+3, +3\). KNQ: 11, 14, 17 — gaps \(+3, +3\). RUX: 18, 21, 24 — gaps \(+3, +3\). ILN: 9, 12, 14 — gaps \(+3, +2\), which breaks the pattern. Hence ILN is the odd one.

The rule for each row is: \(\text{result} = a \times b \times c + c\), where a, b, c are the first three entries. Row 1: \(3 \times 10 \times 6 + 6 = 180 + 6 = 186\). Row 2: \(9 \times 5 \times 3 + 3 = 135 + 3 = 138\). Row 3: \(5 \times 7 \times 1 + 1 = 35 + 1 = 36\). Row 4: \(3 \times 2 \times 5 + 5 = 30 + 5 = 35\). Therefore the missing number is 35.

Split the series into two interleaved sub-series. Odd positions: 3, 5, 8, 12, 17 — differences \(+2, +3, +4, +5\) (not needed for the answer). Even positions: 8, 27, 64, 125, ? — these are \(2^3, 3^3, 4^3, 5^3\). The missing term is the 5th cube: \(6^3 = 216\).

First letters: B, D, F — each advances by \(+2\), so the next is H. Second letters: T, R, P — each goes back by \(-2\), so the next is N. Hence the missing term is HN.

The 49th Parallel North forms a long stretch of the international boundary between the USA and Canada, running from the Lake of the Woods westward to the Strait of Georgia. It was fixed by the Treaty of 1818 and the Oregon Treaty of 1846. The North-South Vietnam boundary was the 17th Parallel; Germany-France and Brazil-Chile borders are not defined by lines of latitude.

La Niña is the cool phase of the El Niño-Southern Oscillation (ENSO). Stronger-than-normal trade winds push warm surface water toward the western Pacific, allowing cold deep water to upwell along the central and eastern equatorial Pacific. This produces below-average sea surface temperatures — in short, La Niña cools down the temperature of the water in the Pacific Ocean.

Deoxygenated (impure) blood from the body is collected by the superior and inferior vena cava and emptied into the right atrium of the heart. The aorta carries oxygenated blood away from the heart, and the pulmonary vein carries oxygenated blood from the lungs to the left atrium — neither carries impure blood to the heart.

Cryolite (\(Na_3AlF_6\), sodium aluminium fluoride) is an aluminium-bearing mineral and is used as a flux in the Hall-Heroult process to dissolve alumina during aluminium extraction. The other options are ores of different metals: Galena (\(PbS\)) is an ore of lead, Cinnabar (\(HgS\)) is an ore of mercury, and Epsom salt (\(MgSO_4 \cdot 7H_2O\)) is a magnesium compound.

Fundamental Duties are listed in Part IV A of the Indian Constitution under Article 51A. They were added by the 42nd Constitutional Amendment Act, 1976 on the recommendation of the Swaran Singh Committee. Originally 10 duties were prescribed; the 11th duty (parents/guardians providing education to children aged 6-14) was added by the 86th Amendment Act, 2002. Part II covers Citizenship, Part III covers Fundamental Rights, and Part V covers the Union Government.

In additive colour mixing (mixing of lights), the primary colours are Red, Green and Blue (RGB). Combining any two primaries gives a secondary colour: Red + Green = Yellow, Red + Blue = Magenta, Green + Blue = Cyan; all three together give White. Hence mixing Red and Green light produces Yellow.

The ESIC scheme launched for beneficiaries in the Delhi region is "Kahin bhi kabhi bhi" ("Anywhere, Anytime"). Under it, an insured person registered at one ESIC dispensary can avail OPD services at any other ESIC dispensary or hospital, helping migrant workers and those away from their home station. The other options are not ESIC schemes; Jeevan Arogya is an LIC health plan.

'My Story' is the autobiography of former Australian cricket captain Michael Clarke, published in 2012. Clarke led Australia to victory in the 2015 ICC Cricket World Cup. Kumar Sangakkara's memoir is "Winning: Twenty Years in Cricket"; Kevin Pietersen wrote "KP: The Autobiography"; and Brendon McCullum's autobiography is titled "Declared".

Guru Har Krishan, the eighth Sikh Guru, before his death indicated his successor with the words "Baba Bakale", pointing to the village of Bakala. The Sikhs accordingly recognised Guru Tegh Bahadur, his grand-uncle, as the ninth Guru (1665-1675). Guru Angad Dev (2nd) succeeded Guru Nanak, Guru Amar Das was the 3rd Guru, and Guru Hargobind was the 6th Guru.

JPEG stands for Joint Photographic Experts Group - the committee that created the widely used lossy compression standard for digital photographs (file extensions .jpg / .jpeg). The other options mix in terms like "Protocol", "Programming" and "Project" that are not part of the official name.

Construction permits for housing in East Jerusalem are issued by Israeli planning authorities (Jerusalem municipality and the Higher Planning Committee). The approval of over 500 new settler homes in East Jerusalem was therefore made by Israel. The USA, UAE and UK have no jurisdiction over building permits in East Jerusalem; settlement expansion there is widely viewed as illegal under international law, but is approved by Israel.

SAARC (South Asian Association for Regional Cooperation), founded in Dhaka on 8 December 1985, has eight member states: Afghanistan, Bangladesh, Bhutan, India, Maldives, Nepal, Pakistan and Sri Lanka. Nepal, Maldives and Afghanistan are all members (Afghanistan joined in 2007). China only holds observer status, not full membership, and is therefore the odd one out.

The Bhopal gas tragedy (2-3 December 1984) was caused by the leak of Methyl Isocyanate (MIC, \(CH_3NCO\)) from the Union Carbide India Ltd. pesticide plant in Bhopal. MIC is a highly toxic and volatile liquid used as an intermediate in producing carbamate pesticides such as Sevin (carbaryl). The other options (sodium, nitrogen, potassium isothiocyanates) are unrelated chemicals and were not released.

Microeconomics deals with individual economic units (a consumer, a firm, a single market), while macroeconomics deals with economy-wide aggregates. Consumer's Equilibrium - the situation where an individual consumer maximises utility from a given income and prices (condition \(\frac{MU_x}{P_x}=\frac{MU_y}{P_y}\)) - is a micro-economic variable. National Income, Aggregate Supply and Employment are all economy-wide aggregates and hence macro-economic variables.

A Gambit is a chess opening in which a player deliberately sacrifices material (usually a pawn) early on to gain rapid development, central control or attacking chances. Well-known examples include the Queen's Gambit, King's Gambit and Evans Gambit. The term is not used in basketball, boxing or golf, so the answer is Chess.

Under Article 157 of the Constitution, a person is eligible for appointment as Governor of a State only if he is a citizen of India and has completed 35 years of age. This matches the minimum age fixed for the President (Art. 58) and Vice-President (Art. 66). 25 years is the minimum for Lok Sabha/MLA, and 30 years for Rajya Sabha/MLC, so those options do not apply to the Governor's office.

Insulin is a peptide hormone secreted by the beta cells of the islets of Langerhans in the pancreas. Its role is to regulate blood glucose by promoting cellular uptake and storage of glucose as glycogen. Although chemically it is a small protein (two polypeptide chains linked by disulfide bonds), in physiology it is classified by function as a hormone. It is not an enzyme (no catalytic action) and not a vitamin (not an essential micronutrient).

An endothermic reaction is one in which the system absorbs heat from the surroundings, so the surroundings cool down and the enthalpy change \(\Delta H > 0\). Examples include melting of ice, photosynthesis and dissolving ammonium nitrate in water. Option 1 describes an exothermic reaction (heat released). Option 3 describes a thermoneutral process, which is a separate category.

Madhuri Dixit is formally trained in Kathak, the classical dance form of North India characterised by intricate footwork (tatkar), rapid spins (chakkars) and expressive abhinaya. She received training under Pandit Birju Maharaj and has showcased Kathak in films like Devdas and Dedh Ishqiya. Bharatnatyam (Tamil Nadu), Kuchipudi (Andhra Pradesh) and Kathakali (Kerala) are not her classical specialisation.

The RBI's instruments of credit control fall under quantitative tools (Bank Rate, Open Market Operations, CRR, SLR, Repo/Reverse Repo) and qualitative tools (Rationing of Credit, Margin Requirements, Moral Suasion, Direct Action). Rationing of credit, Direct Action and Open Market Operations are all standard instruments. The phrase "Variable cost reserve ratios" is not a recognised tool — the correct term is Variable Reserve Ratio (i.e., CRR/SLR), with no element of "cost". Hence it is the odd one out.

Cinnamon is the spice obtained from the dried inner bark of trees of the genus Cinnamomum (chiefly C. verum for Ceylon cinnamon and C. cassia for Cassia). The bark is peeled, fermented and rolled into quills, with the aroma coming mainly from cinnamaldehyde. The stem wood, roots and fruits are not the commercial source of the spice.

Lord Cornwallis, Governor-General of Bengal (1786-1793), introduced the Permanent Settlement of Bengal in 1793 (also called the Zamindari System), under which zamindars were recognised as landowners and the land revenue payable to the Company was fixed permanently. It covered Bengal, Bihar and Odisha. The Ryotwari settlement of Madras is associated with Thomas Munro and Captain Alexander Read, not Cornwallis.

Matching each scientist to the discovery they are credited with: Joseph J. Thomson identified the electron in 1897 via cathode-ray experiments — so 1-c. Alexander Fleming discovered penicillin in 1928 from the mould Penicillium notatum — so 2-d. Narinder Singh Kapany, the "Father of Fibre Optics", pioneered optical fibre — so 3-a. Marie Curie, with Pierre Curie, isolated radium (and polonium) in 1898 — so 4-b. The correct combination is 1-c, 2-d, 3-a, 4-b.

Light rays from the bottom of the tank travel from water (optically denser) to air (rarer), and on crossing the interface they bend away from the normal. The eye traces these refracted rays back in straight lines, locating the bottom at a virtual image closer to the surface. The relation is \(\mu = \dfrac{\text{real depth}}{\text{apparent depth}}\); since \(\mu_{water} \approx 1.33 > 1\), the apparent depth is less than the real depth. This effect is refraction — not reflection, diffraction, or total internal reflection.

The Sanjay Chopra Award is one of the National Bravery Awards instituted by the Indian Council for Child Welfare (ICCW), given to a boy for exceptional courage. For the year 2016, it was conferred on Arjun Singh of Madhya Pradesh, who saved fellow passengers when their school bus plunged into a canal. The award is presented annually by the Prime Minister on the eve of Republic Day.

Step 1 — Ratio of one-day work: \[A:B:C = \tfrac{1}{8}:\tfrac{1}{10}:\tfrac{1}{12}\] Multiplying through by LCM(8,10,12) = 120: \[A:B:C = 15:12:10\] Step 2 — Share of B from total wages: Since all three work together for the same time, wages split in the ratio of efficiencies. Total parts \(=15+12+10=37\). \[\text{B's share} = \dfrac{12}{37}\times 7400 = 12\times 200 = 2400\] Therefore B receives Rs. 2400.

Dividing numerator and denominator of the expression by \(x\) reduces it to: \[\dfrac{x^{3}+\tfrac{1}{x^{3}}}{\left(x+\tfrac{1}{x}\right)-3}\] Step 1 — Denominator: \[\left(x+\tfrac{1}{x}\right)-3 = 17-3 = 14\] Step 2 — Numerator using \(x^{3}+\tfrac{1}{x^{3}} = k^{3}-3k\) where \(k=17\): \[x^{3}+\tfrac{1}{x^{3}} = 17^{3}-3(17) = 4913-51 = 4862\] Step 3 — Final value: \[\dfrac{4862}{14} = \dfrac{2431}{7}\] Therefore the expression equals \(\tfrac{2431}{7}\).

Step 1 — Total runs in each set: \(45\times 5=225\) runs in the first 5 matches; \(15.75\times 4=63\) runs in the next 4 matches. Step 2 — Overall average: \[\text{Average}=\frac{225+63}{9}=\frac{288}{9}=32\] Therefore the bowler's average over 9 matches is 32.

Step 1 — Initial quantities: Milk \(=10\%\text{ of }80=8\) L; water \(=72\) L. Water stays constant when milk is added. Step 2 — Equation: Let x litres of milk be added. Water must be 80% of the new total: \[72=0.8\,(80+x)\] Step 3 — Solve: \[90=80+x \implies x=10\] Therefore 10 litres of milk must be added.

Step 1 — Prime factorisation: \[5000=2^{3}\times 5^{4}\] Step 2 — Make every exponent even: The exponent of 2 is 3 (odd); the exponent of 5 is 4 (even). Removing one factor of 2 leaves \(2^{2}\times 5^{4}=2500=50^{2}\), a perfect square. Therefore the least divisor is 2.

Step 1 — Equalise B in A:B and B:C: LCM of 5 and 4 is 20. \(A:B=8:20\) and \(B:C=20:15\), so \(A:B:C=8:20:15\). Step 2 — Equalise C with C:D = 2:1: LCM of 15 and 2 is 30. Scale the first chain by 2 and C:D by 15: \[A:B:C:D=16:40:30:15\] Dropping B gives \(A:C:D=\mathbf{16:30:15}\).

Step 1 — CP and SP of one pen: \(CP=\tfrac{1}{25}\) rupee, \(SP=\tfrac{1}{15}\) rupee. Step 2 — Profit %: \[\text{Profit \%}=\frac{SP-CP}{CP}\times 100=\left(\frac{SP}{CP}-1\right)\times 100=\left(\frac{25}{15}-1\right)\times 100\] \[=\frac{10}{15}\times 100=\frac{200}{3}=66\tfrac{2}{3}\%\] Therefore the profit is \(\mathbf{66\tfrac{2}{3}\%}\).

The perpendicular from the centre bisects the chord, so half-chord \(=30\) cm. With the radius as hypotenuse, Pythagoras gives: \[r^{2}=16^{2}+30^{2}=256+900=1156\] \[r=\sqrt{1156}=34\text{ cm}\] Therefore the radius is 34 cm.

Total time taken to make the car = 300 hours ⇒ Total time taken in paint and frame = 300 × (13 + 22)/100 = 35 × 3 = 105 ∴ the correct option is 3)

When the altitude is dropped from the right-angle vertex to the hypotenuse, the geometric-mean (altitude-on-hypotenuse) theorem gives: \[AD^{2}=BD\times CD=7\times 28=196\] \[AD=\sqrt{196}=14\text{ cm}\] Therefore \(AD=\mathbf{14}\) cm.

Since, 15% of total time is spent on quality check. Percentage of time taken for quality check from all other processes = 15/6 = 2.5% ⇒ Percent of time taken by seats = 6 - 2.5 = 3.5% ⇒ Percent of time taken by glass = 8 - 2.5 = 5.5% ⇒ Percent of time taken by both seats and glass = 3.5 + 5.5 = 9% Required sectorial angle = (9/100) × (360) = 32.4° ∴ the correct option is 2.

Given: cosec 6A – cot 6A – 3 cosec 2A cot 2A Let a = cosec 2A, b = cot 2A ⇒ (a – b) 3= a 3– b 3– 3ab(a – b) ⇒ (cosec 2A – cot 2A) 3= cosec 6A – cot 6A – 3 cosec 2A cot 2A(cosec 2A – cot 2A) Since, cosec 2A – cot 2A = 1, we have ⇒ 1 3= cosec 6A – cot 6A – 3 cosec 2A cot 2A (1) ∴ the correct option is 4)

Step 1 — Find the rate from scenario 1: \(SI_{1}=448-400=48\), so \[R=\frac{100\times 48}{400\times 2}=6\%\] Step 2 — Apply to scenario 2: \(SI_{2}=682-550=132\), so \[T=\frac{100\times 132}{550\times 6}=\frac{13200}{3300}=4\text{ years}\] Therefore the required time is 4 years.

Let \(y=\sqrt{\dfrac{1+x}{x}}\); then \(\sqrt{\dfrac{x}{1+x}}=\dfrac{1}{y}\), and the equation becomes: \[y-\frac{1}{y}=\frac{1}{\sqrt 6}\implies \sqrt 6\,y^{2}-y-\sqrt 6=0\] The quadratic formula gives \(y=\dfrac{1\pm 5}{2\sqrt 6}\). Since \(y\ge 0\), take \(y=\dfrac{\sqrt 6}{2}\). Squaring: \[\frac{1+x}{x}=\frac{3}{2}\implies 2+2x=3x\implies x=2\] Therefore \(x=\mathbf{2}\).

Multiply numerator and denominator inside the radical by \((\sec A-1)\) and use \(\sec^{2}A-1=\tan^{2}A\): \[\sqrt{\frac{(\sec A-1)^{2}}{\sec^{2}A-1}}=\frac{\sec A-1}{\tan A}\] Converting to sine and cosine: \[\frac{\sec A-1}{\tan A}=\frac{1-\cos A}{\sin A}=\csc A-\cot A\] Therefore the simplified value is cosec A − cot A.

Given: time taken in seats = 192 hours ⇒ 6% of total hours = 192 ⇒ Total hours = (192 × 100)/6 = 3200 hours So, time taken in glass = 3200 × (8/100) = 256 hours ∴ the correct option is 1)

Step 1 — Rewrite division and factorise: \(1-x^{4}=(1-x)(1+x)(1+x^{2})\), so the expression becomes \[\frac{1+x}{(1-x)(1+x)(1+x^{2})}\times\frac{1+x^{2}}{x^{2}}\times x(1-x)\] Step 2 — Cancel common factors: \((1+x)\), \((1+x^{2})\), \((1-x)\) and one \(x\) all cancel, leaving \[\frac{1}{x}\] Therefore the value is \(\mathbf{1/x}\).

Take \(CP=100\). Marked price = 120 (20% above CP). After a 5% discount: \[SP=120\times\frac{95}{100}=114\] Profit = \(114-100=14\), so \[\text{Profit \%}=\frac{14}{100}\times 100=14\%\] Therefore the profit percentage is 14%.

Circumradius = (AB × BC × AC)/ 4 × (1/2) × BC × AH ⇒ Circumradius = (a × 10 × 6)/4 × (1/2) × a × 4 ⇒ Circumradius = 15/ 2 ⇒ Circumradius = 7.5 cm ∴ Answer is 7.5 cm ALTERNATE METHOD : Join BD ⇒ ∠ADB = ∠ΔACB, (Angles made by the same chord AB) Also, ∠DBA = ∠CEA = 90° So, ΔABD = ΔAEC are similar triangles ⇒ AD/AB = AC/AE ⇒ AD/10 = 6/4 ⇒ AD = 15 cm Since, AD = 2 × circumradius ⇒ Circumradius = 15/2 = 7.5 cm ∴ the correct option is 2)

Distances covered per hour form an AP: first term \(a=21\), common difference \(d=3\). Set the sum to 252 km. Step 1 — AP sum equation: \[S_n=\tfrac{n}{2}\bigl[2a+(n-1)d\bigr]=252\] Step 2 — Substitute and simplify: \[\tfrac{n}{2}\bigl[42+3(n-1)\bigr]=252 \implies n(3n+39)=504\] Step 3 — Solve the quadratic: \[n^2+13n-168=0 \implies (n-8)(n+21)=0\] Rejecting the negative root, n = 8 hours.

Step 1 — Compute tan 2A using the double-angle formula: \[\tan 2A=\frac{2\tan A}{1-\tan^2 A}=\frac{2\cdot\tfrac{1}{2}}{1-\tfrac{1}{4}}=\frac{1}{\tfrac{3}{4}}=\frac{4}{3}\] Step 2 — Apply tan(X+Y) with X = 2A, Y = B: \[\tan(2A+B)=\frac{\tan 2A+\tan B}{1-\tan 2A\tan B}=\frac{\tfrac{4}{3}+\tfrac{1}{3}}{1-\tfrac{4}{3}\cdot\tfrac{1}{3}}=\frac{\tfrac{5}{3}}{\tfrac{5}{9}}\] Therefore \(\tan(2A+B)=\dfrac{5}{3}\cdot\dfrac{9}{5}=\mathbf{3}\).

Percent of total hours taken by engine and tyres = 40 + 11 = 51% ⇒ 51% of total hours = 127.5 hours ⇒ Total hours = (127.5 × 100)/51 ⇒ Total hours = 250 hours Difference in time taken by frame and glass = 250(22 – 8)/100 = 250 × 14/100 = 35 ∴ the correct option is 4)

Let \(y=x-\dfrac{1}{x}\). Squaring gives the standard identity: \[x^2+\frac{1}{x^2}=y^2+2\] Substitute into the equation: \[2(y^2+2)-2y-8=0 \implies 2y^2-2y-4=0 \implies y^2-y-2=0\] Factorise: \[(y-2)(y+1)=0 \implies y=2 \text{ or } y=-1\] Hence \(x-\dfrac{1}{x}=\mathbf{-1 \text{ or } 2}\).

Volume \(V=\pi r^2 h\) must stay constant. New radius \(r'=1.25r=\tfrac{5}{4}r\). Equate old and new volumes: \[\pi r^2 h=\pi\left(\tfrac{5}{4}r\right)^2 h' \implies h'=\frac{16}{25}h\] Percentage reduction: \[\frac{h-h'}{h}\times 100=\left(1-\tfrac{16}{25}\right)\times 100=\tfrac{9}{25}\times 100=\mathbf{36\%}\]

Let radius of the smaller circle be r Here, ∠RPQ = ∠SPO, ∠PSO = ∠PRQ If we join SO and QR then, So, ΔPSO and ΔPRQ are similar triangles ⇒ PO/PS = PQ/PR ⇒ 2r/PS = 4r/34 ⇒ PS = 17 cm ∴ the correct option is 2)

The sentence lists three classic pillars of national development. "Socially" and "politically" are given, so the missing pillar is "economically" — corruption hinders growth, deters investment, and misallocates public resources. "Emotionally" and "manually" are not standard categories of national impact; "scientifically" is narrower than the social/political pairing demands.

The fixed collocation "met with" means to experience or encounter something — typically a result such as success, resistance, or failure ("met with failure"). "Caught with" implies discovery of contraband, "gone with" implies accompaniment, and "got with" is not idiomatic. Hence: "Keith has met with a failure in English examination thrice."

"Lugubrious" means looking or sounding mournful, gloomy, and dismal. Its antonym must express the opposite emotional outlook — cheerful and hopeful. Among the options, "Optimistic" (hopeful and confident about the future) is the direct opposite. "Clumsy" relates to physical coordination, "Lucid" to clarity of expression, and "Sinister" to evil intent — none oppose sadness.

A person who is "easily duped or fooled" lacks skepticism and readily believes claims without questioning them — the precise definition of "Gullible". "Bigot" means an intolerant person, "Ridicule" is the act of mocking, and "Venerable" means deserving of respect — none describe susceptibility to deception.

The idiom "do something with one's eyes closed" means to perform a task effortlessly because of repeated practice. This fits perfectly with "He has painted that picture so often…". "Arms full" and "eyes opened" do not convey effortless skill; "mind blank" is not the standard idiom here. Hence No improvement is required.

Q opens the paragraph by stating the theme — Time is the great equaliser of mankind. S follows naturally: the pronoun "It" refers back to Time, illustrating the equalising claim ("taken away the best and the worst"). R then develops a new dimension — Time offers opportunity but demands regard. P concludes with the finality ("no second chance") that justifies why Time must be regarded. The coherent order is QSRP.

The series moves outward in widening scope: society → community → ?. The next natural larger unit is the country, and the passage later refers to corruption's effect on "the nation," confirming the national-level reference. "World" and "universe" overshoot the nation-focused context, while "company" is a narrower organisational unit that breaks the widening pattern.

The blank needs a noun describing corruption as something harmful that "spreads in the mind" of people. Poison is the standard metaphor for a destructive influence that infects thought and conduct. "Havoc" names destruction caused, not the substance that spreads; "pollutant" suits the environment, not the mind; "grassroot" denotes a level, not a harmful agent.

The error is in part (1). With the uncountable noun knowledge, bare little means "hardly any" and carries a negative sense — it would contradict the rest of the sentence, which says the knowledge did prove helpful. The correct quantifier is "The little", meaning "the small amount that she possessed" — a specific, limited but positive quantity. The corrected sentence: The little knowledge of playing volleyball that she possessed proved to be helpful at the time of an inaugural match.

The error is in part (2). The phrase "I thought I was done with" is incomplete and illogical — "done with" requires an object (done with it/the homework), and the conjunction so does not yield a sensible cause-effect with the first clause about not completing the homework. A correct version would replace part (2) with something like "so I thought I would be in trouble when the". Parts (1) and (3) are grammatically sound.

Ensconce (verb) means to settle or hide someone/something securely in a safe, comfortable, or concealed place — e.g. ensconced in an armchair. Its antonym is Unveil, which means to remove a covering and reveal what was hidden — the direct opposite of concealing or settling away. Establish means to set up firmly (not an opposite); Impudence (rudeness) and Request (an act of asking) are unrelated.

The verb allow takes the pattern allow + object + to-infinitive ("allow you to have"), so "allow you her notes" is ungrammatical. The verb let, however, takes let + object + bare infinitive. Hence "let you have" her notes is correct. "Let you to have" (option 3) wrongly inserts to after let; "allow that you have" (option 1) is awkward and changes the meaning.

The phrase "Atonement for one's sins" means feeling sincere regret for wrongdoing and seeking to make amends. The single word that captures this is Repentance — sincere remorse for past wrongdoing. Elite (a superior group), Ignoramus (an ignorant person), and Incendiary (one who stirs up fires or conflict) are unrelated to sin or moral amends.

Sentence Q opens by defining what laws of nature really are (statements of fact, not commands). S follows as a critique of the terminology — calling them "laws" is unfortunate given Q's definition. P then proposes a better alternative term, "uniformities of nature". R closes by explaining the benefit — "This" refers to P's suggestion, which removes the fallacy that a law implies a lawgiver. The coherent order is QSPR.

Identifying the Incorrectly Spelt Word This question requires identifying the word with a spelling mistake from the provided list. We need to check the spelling of each option carefully to find the one that is incorrect. Spelling Examination of Provided Words Let's review the spelling of each word given in the options: Usable: This is a correctly spelt word. It means that something is able to be used or is suitable for use. Defense: This spelling is correct, particularly in American English. It refers to protection from attack, harm, or danger. (Note: 'Defence' is the standard British English spelling). Inventor: This word is correctly spelt. An inventor is a person who invents something, typically a product or process. Annaul: This word contains a spelling error. The correct spelling is Annual. The word 'Annual' refers to something happening once every year. The misspelling 'Annaul' incorrectly places the vowels. Determining the Misspelt Word Comparing the spellings, we find that 'Annaul' is the incorrectly spelt word among the choices. The correct spelling is 'Annual'. Therefore, 'Annaul' is the answer.

With verbs of perception (see, hear, watch) followed by an object + present participle, the passive keeps the participle: Active "X saw Y conducting Z" → Passive "Y was seen conducting Z". Applying this: "He saw him conducting the seminar..." becomes "He was seen conducting the seminar on personality development." Option 1 wrongly inserts "by me" and uses "to conduct" (the bare-infinitive pattern); options 3 and 4 keep "He" as the seer instead of converting the object "him" into the new subject.

Nefarious means wicked, villainous, or criminal — typically describing evil schemes or actions. Its closest synonym is Iniquitous, which means grossly unfair, sinful, or morally wrong; both convey deep moral wickedness. Purposeful (determined) and Suspicious (causing distrust) are unrelated; Virtuous (morally good) is the antonym, not a synonym.

A fiasco is a complete, often humiliating failure. So "ended in a fiasco" means the event or attempt concluded as a complete failure. "A successful event" is the opposite; "changed one completely" and "twisted around" describe transformation or distortion, neither of which captures the sense of utter failure.

When a Wh-question is reported, "said to" changes to asked, the quotation marks and question mark are dropped, the Wh-word itself serves as the connector (no that), the clause becomes assertive (subject before verb), and the present tense is shifts to past was. Hence: "Rashmi asked me where the telephone exchange was." Option 1 wrongly adds "that"; option 3 keeps interrogative word order ("was the telephone exchange"); option 4 changes the reporting verb unnecessarily.

The idiom "sow wild oats" refers to indulging in reckless, irresponsible, or foolish pleasure-seeking activities, typically during one's youth, before settling down. Among the options, "To waste time by doing foolish things" best captures this sense of frivolous, unproductive behaviour. "Make someone a fool", "make space", and "take revenge" have no connection to the idiom.

The sentence describes the misuse of public resources for personal gain — an act of corruption. The blank needs a word that captures the unjust, improper nature of that gain. Unfair fits perfectly: an advantage obtained by abusing public resources is not earned legitimately and violates principles of justice. Fruitful (productive) gives a positive sense that contradicts the context; wishful (full of wishing) describes a state, not an advantage; favourite (preferred) describes preference, not ethics.

The phrasal verb called for means "demanded" or "required". The sentence describes a situation where the manager was required to give an explanation of his conduct. The completed sentence reads: "The manager was called for an explanation of his conduct." Called off means cancelled, called up means phoned or summoned for military service, and called to does not form an idiomatic match with "an explanation". Hence the correct choice is called for.

Eloquent means speaking or writing in a clear, expressive, and persuasive manner. Fluent means able to express oneself easily and articulately — the closest synonym among the options. Ignorant (lacking knowledge), rude (impolite), and significant (important) all describe unrelated qualities. Hence the synonym of Eloquent is Fluent.

The standard collocation in discussions of corruption is "misuse of power and position" — i.e. the abuse of both authority and the official role one holds. This is the natural, idiomatic pairing. Growth and symbol do not fit the context of misuse, and status is rarely paired with "power" in this collocation. Hence position correctly fills the blank.

When the suffix -able is added to a word ending in a soft -ce or -ge, the silent e is retained to preserve the soft sound of c/g (e.g., notice → noticeable, manage → manageable, change → changeable). Applying this rule, trace → traceable, not tracable. The other options — Changeable, Inedible, Valuable — are all correctly spelt. Hence the incorrectly spelt word is Tracable.