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OL., , 03., , 0S., , 06., , , , , , , , , , Time:-1:30PM to 2:25PM, , 1. All questions are compulsory., , Class — XT, , ‘| St. Xavier’s College, Ranchi, jes Intermediate Section, , MATHEMATICS CONTEST 2021, ORGANIZED BY, ARYABHATTA CLUB, , F.M:- 100, , General Instructions:, , 2. The question paper consists of 50 questions divided into two sections A & B. Section A comprises of 40 questions, of 2 marks each and Section B comprises of 10 questions of 2 marks each., 3. Use of Google and calculators are nat permitted., , , , Sisal ra!, , Evaluate: cot{ —2cot" 3}, , (a)-V3 (b)V3_ (e)V2 (a)-V2, , 2. Find the valuc of k so that following function is, , r=2., 16x+20, , , ae, («-2), , k 2 x=2, (a) 5 (b) 6 {c)7 (a8, , 3 4] 1 ¥ 7 0, If 2 + “|= ~ then find (x- y)., 5S x] [O01 10 5,, , continuous, , , , f(x)=, , (a) 10 (bh) 12 (c)8 (d) 14, iA 2 hi petric matri, 5 “| ox43 x42 is a symunetric matrix,, then x —, (ap4 (b) (c)-4 (d)-3, The function f(x) = cot''y + x increase in the interval, (a) (1, ©) (b) (- 1, 2), (c) (0, ©) (d) (-, ©), Find the co-factors of elements ys, ay, ae,, 1 sin@ 1, respectively, of the matrix} cing 1 sind, -1 -sin® 1, , (a) 0, 2,—2sin6, (c) 2, 0, — 2sin@, , (b) 2, 0, 2sin6, (d) -2 sin6, 2,0, , 07,, , Og., , Let L denote the set of all straight lines in a plane,, Let a relation R be defined by aRB => af, ape L., Then, R is, , (a) Reflexive only, (c) Transitive only, , 7 xy 4 ! 8 w, z+6 x+y “to 6], , then find the value of (x + y + z)., , (b) Symmetric only, (d) None of these, , {ayo (b) 1 (c)2 (d)3, . Tangents to the curve y =x) + 3x ata—- land x= 1, are, , (a) parallel, , (b) intersecting obliquely but not at an angle of 45°, (c) intersecting at right angles, , (d) intersecting at an angle of 45°, , \, “G +4uan"(, 2), , @2 2 OF @=, , is equal to, , , , . Let gi) =" = 4e ~ S, then, , (a) g(x) is one-one on K, , (b) g(x) is not one-one on R, (c) g(x) is objective on RK, (d) None of these, , . If y= log, (3 ) ten 4 y equals, e dx, , 1 1, , 5 02 @, , (a) 2, Page 01

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13., , 14. If, , 15., , 16., , 18., , 20., , 21., , 22., , -IfA=|0 -2, , 5S x, y 0, (a)x=0,y=5, (c)x=y, , wa-| Janta a en, (b)x+yv=5, Ws=Oy=9, , = and iat >> then a is, , tee ae, (ajar (bj (J (a), , (5 6 -3, , If 4-4 3 2|, then find the co-factor of the, , [4-7 3, element a2, of 2" row., (a) 3 (b) -3 (c)2 (d)-2, ‘The equation of the normal to the curve y" = 8x at the, origin is, (a)2y x=0, ()x-2y=0, , (b)x y=0, Wdy=0, , 2 00, 0 |, then the value of adj A\ is, 0 0 -2, , (a) 64 (b) 16 (c) 0 (d)-8, , xv-y?, ;, , =2,then® =, <a, , , , If bso, x+y, , @-i (»), , 99y, ()-2* (a), , 99x, , Oly, 99y, 10Lx, , , , , , , , , , . Ify=e++e+1,theny, , (a) bas a local minimum, , (b) has local maximum, , (c) neither have a local minimum nor local maximum, (d) None of these, , , , The function f(x) = = increases in the interval, (a) (0, ©) {b) (0, &), () (e, ®) (dy (0), , The domain of the function, cos(loga(.x7 + Sx + 8 )) is, , (a) (2, 3] (b) [- 2, 2], ©) B, 1 (d) [-3, -2], If log y = mtan"' x , then, , (a) (1 +22) y2 + (2c +m) =0, (b) (L +7) yp + (2e-m) i = 0, (©) 1 +7) yr - (Qe 4m) 1 =0, (d) (1 +x’) yo (2e-m) yi. = 0, , 23., , 24., , 25., , 26,, , 27., , 29., , 30., , 31., , , , 12 x, Findx, if}; 1 1 | is singular, 2 1-1!, fa) (b)2 {c)3 (d)4, i yea" then the value of dy,, dx, 2 y logy, (*)1 yoga) () Tim ylogs), * logy (a) yilogy, , (2) x(1- ylog xlog y) x(1+ ylog xlog y), If the area of a triangle ABC, with vertices, , AQ, 3), B(0, 0) and C(k, 0) is 3 sq. units,, , then the value of k is, , (a)2 (b) 3 (c)4 (dy 5, , The slope of normal to the curve y = 1° +2x+6, , which is parallel to line x + l4y + 4= 0, is, , 1 1 1, @)-; ()- ©-4 (@-5, Using the principal values, the value of, , 7 ‘= i { *, sin’ {sin ==> tan! 4 tan =} =, , 6 6, (a)xw6 = (b)2w/3— (ce) w/3—— (dd) S's, , . Find the minor of 6 and co-factor of 4 respectively, 1 2, in the determinant, A- 4 5, 78 9, (a)6,6 (b)6,-6 = (c)-6,-6 — (d) -6,6, , asin x+2cos x is, , If the function f (x) = —, sin x+cos x, , strictly, increasing for all values of x, then, , faja<l (bya>1, , {c)a<2 (d)a>2, , The function f: RR given by fix) =°, xe R, is, (a) one -one but not onto, , (b) onto but not one one, , (c) both one — one and onto, , (d) neither one — one not onto, , The value(s) of x for which the function, I-x al, , F(x) =) 0-x)2Q-x),1sx52, 3-x +x>2, , fails to be continuous is (are), , {a)1 (b)2, , {c)3 (d) all real numbers, , Page 02

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32., , 33., , 34., , 36., , 37., , 38., , 39., , wa" Cana =| 3 tens, @)2 ©); Ol ©;, , It is given that atc = 1, the function x* ~ 627 + ax +9, attains its maximum value on the interval [0,2].Find, the value of a., , (a) 100 (b) 120, , (c) 140 (dp 160, , 0 2 -3, If A=|-2 O 1), then Aisa, 301 0, , (a) symmetric matrix, , (b) skew-symmetric matrix, (c) diagonal matrix, , (d) unit matrix, , . sin (cots) =, , 1 -I, On OTF, 3 1 1, Oe On, , Let R be relation on the set R of all numbers defined, by aRb iff ja — bj) = 1. Then R ts, , (a) reflexive only, , (b) symmetric only, , (c) both reflexive and symmetric, , (d) None of these, , cosa sina 0), , A=\sina cosa 0), then find adj A., 0 o oO, , cosa sina 0, , {a)| -sina cosa 0, , cosa -sina 0], (b)| sine cose 0, , 0 ool 0 o 1], , cosa sina 0 -cosa -sing =U, {c)| sina cosa 0} (d)} sina cosu 0, , 0 o ol 0 o ot, Slope of normal to the curve y =x" 1, , x, at (-1, O) is, 1 1, , (a) (b)-4 (e)5 (a4, , ah , ja BI, If A= and Av = then, ba Bo a, (a)a=a' +h, B=ab, (b)a=a +b’, B=2ab, (java+h pea-b, (d) a= 2ab, B=a’ + b*, , 40., , 41., , 42., , 43., , 45., , 46., , 47,, , 48., , 49,, , 50,, , The slope of the tangent to the curve ye = 9e9x*, at(-1, 3)is, , @-2 WS /s (ay, , , , S, Who first invented geometry?, (a) Euclid (b) Pythagoras, (c) Blaise Pascal (d) Euler, Who is known as Indian Euclid?, (a) Pingala (b) Bhaskara, (c) Aryabhata (d) Brahmagupta, , Who is the father of probability in maths?, (a) Grace Murray Hopper, , (b) Eular, , (c) Isaac Newton, , (d) Girolamo Cardano, , . Who is the true father of calculus?, , (a) Goulried Leibniz, , (b) Isaac Newton, , (c) Guillame de L'Hospital, (d) Pythagoras, , Who invented algebra?, , (a) Stephen Hawking., , (b) Muhammad ibn Musa al Khwarizmi, (c) Euclid, , (d) Muhammad ibn Musa al Zabar, , What did Rene Descartes invent ?, , Ans., , (a) Polar coordinate system., , (b) Cartesian coordinate system., , (c) Vector, , (d) Matrix, , 4) gin? cor” f= =) equals:, , dx| xf ("™, , @)-1 5 ©; w@, 2 2, , The equation of tangent to the curve y = be™* at the, point where it crosses y-axis, is :, (ajax+by=l (b)ax-by=1, x y x, @)—-+=1 (d) —+—=1, ab ab, , The function f(x) = cot! x + x increases in the, interval :, (a) (1,2), (c)(-9, ©), , (b) G1, @), (d) (0, @), , +bx*?, i ot and y = Oat» = S, then the ratio a: b is, x, , equal to :, (a) ¥5:1, , (c)3:5, , (b) S$: 2, (dy 1:2, Page 03