ST. LAWRENCE COLLEGE Chabahil, Kathmandu Department of Mathematics Model Set:
GroupA (Short Questions) 1. a. b. c. 2. a. b. c. 3. a. b. c. 4. a. b. c. 5. a. b. c. 6. a. b. c.
If A = [-2,3) and B = (1,4], find AUB and B-A. If -a < x < a; prove that |x| < a. Show that (r1 - r)(r2+r3) = a2. Prove that cosec-1x + sec-1x = π/2. The length of perpendicular drawn from the point (a,3) on the line 3x+4y+5=0 is 4, find the value of a. Find the equations of the lines whose joint equation is x2+2xysecθ+y2 = 0. 1 −2 3 2 3 if A = and B = are given matrices. Explain why (AB) is defined but (BA) is −1 2 1 3 1 not? Calculate AB. Solve by Cramer's rule: -x+y=-9 x-3y=5 Determine the half-plane given by the inequality 2x-y < 2. Express the complex number (-2,-2√3) in the polar form. Find the value of k so that the equation 3x2+7x+6-k=0 has one root equal to zero. 5x + 1 Find the partial fractions of 2 . x ( x + 2) Evaluate: lim x →∞
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)
x − x −3 .
Find the angle between the lines whose direction cosines are proportional to 1,2,4 and -2,1,5 dy 1 if y = .sin x . Find dx x 4 x − 2 for x ≥ 1 Let f:R→R be defined as: f ( x ) = , find f(2), f(1), f(0). for x < 1 2x Find the area bounded by the x-axis, the following curve and the ordinates: y=eax, x=b, x=c. 3 Integrate: ∫ sec x. dx .
Group B (Long Questions) 7. a. b. 8. a. b.
9. a.
b.
A College awarded 38 medals in football, 15 in Basketball and 20 in Cricket. If these three medals went to a total 58 men and only three men got medals in all three games, how many received medals in exactly two of the three games. Let Q be the set of rational numbers. Is the function f:Q→Q defined by f(x)=4x-7; xєQ one - one and onto? Find the formula for f-1. Prove that: tan(2tan-1x) = 2tan(tan-1x+tan-1x3). OR, Solve: 2sin3x-2sinx+5cos2x=0. Show that cosA+cosB+cosC-1 = r/R. OR, If three sides are proportional to 2:√6:√3+1, find all the angles.
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2 2 Show that the product of the perpendicular drawn from the points ± a − b , 0 upon a line
x y cos θ + sin θ = 1 is b2. a b Show that the two straight lines x2(tan2θ+cos2θ)-2xytanθ+y2sin2θ=0 makes with the x-axis angles such that the difference of their tangent is 2.
10.a. b.
11.a. b. 12.a. b.
13.a.
b. 14.a.
b.
a+x b c a b c b+ y c = xyz 1 + + + . Prove that: a x y z a b c+ z Solve by using row equivalent matrix method: x-2y+2z = 0 x-2y+3z = -1 2x-2y+z = -3 OR, Solve by inverse matrix method: -2x+4y = 3 3x-7y = 1. If a, b, c are rational and a+b+C=0, show that the roots of (b+c-a)x2+(c+a-b)x+(a+b-c)=0 are rational. 1 3 Find the forth roots of − + i . 2 2 x4 + 2 x4 + 3 . Resolve into partial fractions: ( x + 3)( x + 2) x sin y − y sin x . Define limit of a function at a point. Evaluate: lim x→ y x− y x2 − x − 6 ;x ≠ 3 2 OR, A function is defined by: f ( x) = x − 2 x − 3 5 ; x=3 3 Is the function f(x) continuous at x=3? If not why? State how can you make it continuous. Find from definition the derivative of log(ax+b). OR, A man who has 144m of fencing material wishes to enclose a rectangular garden, find the maximum area he can enclose. 2 Integrate: ∫ x .sin x.dx . OR, Find the area of the region enclosed by the curves y2 = 4ax and x2 = 4ay. Show that the lines whose direction cosines are given by l+m+n=0 and 2mn+3nl-5lm=0 are perpendicular to each other. OR, What is the projection of a line on the given line? Find the projection of the line joining the points (x1,y1,z1) and (x2,y2,z2) on the line whose direction cosines are l,m,n. Maximize and minimize the function f = 34x+6y subjected to the constraints, x+y ≥ 1 x+y ≤ 6 and 1 ≤ x ≤ 3.
ST. LAWRENCE COLLEGE Chabahil, Kathmandu Department of Mathematics Model Set:
GroupA (Short Questions) 1. a. b. c. 2. a. b. c. 3. a. b. c. 4. a. b. c.
Define complement of a set. If A∩B = ø, prove that A ⊂ B . Let f: A → R be given b f(x) = 2|x+2|+4 where A = {-2,0,1.2}, find the range of f. Show that the triangle is right-angled if r1 = r+r2+r3. Find the value of cos(sin-11/4 +cos-11/2) Find the equation of the sides of an equilateral triangle whose vertex is (-1,2) and base is y=0. Find the angle between the line-pair 2x2+7xy+3y2=0. 3 2 3 5 Define symmetric matrix. If A = find the matrix X such that A-3X = . 1 5 −8 2 Solve by using row-equivalent matrix method: 5x-4y=-3, and 7x+2y=6. Graph the half plane given by: y-x ≥ 1. 1 1 If α = −1 + i 3 and β = −1 − i 3 , show that α 4 + α 2 β 2 + β 4 = 0 . 2 2 Find the value of k so that the equation 3x2-(5+k)x+8=0 has roots numerically equal but opposite in sign. 3x + 1 Resolve into partial fraction: 2 . ( x + 2)
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(
6
5. a.
Calculate the limit of lim
x →64 3
)
x −2 . x −4
b.
Show that the direction cosines of the line equally inclined to the axes are ±
c.
Find
6. a. b. c.
1 1 1 ,± ,± . 3 3 3
dy , if y=4u2-3u+5 and u=2x2-3. dx
x2 − 4 at x ≠ 2 . x−2 Find the area bounded by the x-axis, following curve and ordinates: y = logx, x=e, x=1. 1 x+ 1 Evaluate: ∫ 1 − 2 .e x dx . x Test the continuity of the function f ( x) =
Group B (Long Questions) 7. a. b.
State and prove De Morgan's Law. Let f: R → R and g: R → R be defined by f(x) = x2+1 and g(x) = 2x+3, find (fg)(x) and (f-1og)(x). −1 x − 1 −1 x + 1 −1 8. a. Solve the equation: tan + tan = tan 1 . x−2 x+2 OR, Solve: 3 cos x + sin x = 2 . b. State and prove sine law of trigonometry. OR, If a=2, b= 6 , A=45o, solve the triangle. 9. a. Show that the equation of a line passing through (acos3θ, asin3 θ) and perpendicular to the line xsecθ+ycosecθ = a is xcosθ-ysinθ=acos2θ. b. Prove that the general second degree equation, ax2+2hxy+by2+2gh+2fy+c = 0 will represents two g 2 − ac parallel lines if h2=ab and bg2=af2. Also show that the distance between these lines is 2 . a ( a + b)
10.a.
a+b+c −c −b −c a+b+c −a = 2(a + b)(b + c )(c + a ) . Prove that −b −a a+b+c
b. Solve by Cramer's rule or matrix inversion method: x − 2y − z =1 2x − 3y − z = 4 2z − y + x = 9 11.a. Solve the equation z2= - 24i+7. b. If one root of the equation ax2+bx+c=0 be four times the other show that 4b2=25ca. 2x +1 12.a. Resolve into partial fractions: . ( x − 1)( x 2 + 1) b. Define continuity of a function at a point. A function f(x) is defined by sin 2 ax ,x ≠ 0 f ( x) = x 2 1, x=0 Is the function continuous at x = 0? If not why? Redefine the function in such a way that it becomes continuous at x=0. cosec θ − cot θ OR, Evaluate: lim . θ →0 θ 13.a. Find from first principle the derivative of cos23x. OR, Determine where the graph is concave upwards and downwards for the function, f(x)=x48x3+18x2-24. b.
Evaluate:
∫
π
0
x cos 2 x.dx .
x2 y2 OR, Find the area of the ellipse: + = 1. 9 4 14.a. Prove that the lines whose direction cosines are given by the relation al+bm+cn=0 and f g h fmn+ghl+hlm=0 are perpendicular if + + = 0 . a b c OR, Find the ratio in which the line joining the points (-3,4,-8) and (5,-6,4) is divided by yx-plane. Find also the coordinates of the point. b. Find the extreme values of the function, F = 2x+7y subjected to: x + 2 y ≤ 7, x − y ≤ 4, x ≥ 0, y ≥ 0.