1. (j) Find the value of x, iffthe slope of the line passing through (2, 5) (x. 3) is
v2.
v3.
v4
(ii) Find the equation ofthe straight line passing through (~4 , 5) and cutting of equal
intercepts on co-ordinate axes.
(ili) Find the equation of the straight Hnopassing through the points (at), 2at,), (at), 2at,)
(iv) Transform the equation J/3x + y =
8) Slope - intercept form bh) Intercept form )Normal form.
(¥) Transform the equation x +y +1 = 0 into normal form.
(vi) Find the area of the triangle formed by the line x cos a +y sin
(Find the value of pif the lines 3x + dy = 5, 2x +3y=4, px-+4y~6 are concurrent.
Find the point of concurrence of (2+ 5k) x—3 (1 +2k)y+(2—k) =0
(iii) Find the value of , if the angle between the lines kx + y +9=0,3x-y+4=0is ¥/
(iv) Find the distance between the parallel lines 5x ~3y ~ 4 =0 and 10x — 6y -9= 0.
(~) Find the value of pif the straight lines 3x + 7y — 1 = 0, 7x— py +3 = 0 are mutually
perpendicular.
(vi) Find the equation of the straight line perpendicular to the line Sx—3y+1=0 and
passing through (4, 3)
(O’ Find x, ifthe distance between (5, ~1, 7) and (x, 5, 1) is 9 units.
(ii) Show that the points (1, 2, 3), (2, 3, 1), (3, 1, 2) form an equilateral triangle.
For what values of tif the points (2,1, 3)(3,~5,t) 1, 11, 9) are collinear ?
(iv) In AABC, the centroid is the origin and the vertices of A, B are (1, 1, 1) and (~2, 4, 1)
respectively then find C.
(WIE, 2/1) (4,1, 1) and (6, 2,5) are three vertices and (4, 2,2) is the centroid of a
tetrahedron then find the fourth vertex.
(vi) Find the fourth vertex of the parallelogram whose consecutive vertices are
2.4,-1),6,6-1) and (4,5, 1)
(® Reduce the equation x + 2y ~3z—6=0 of the plane to normal form.
(Gi) Find the d.c's of the normal to the plane x-+2y +22~4
= pwith co-ordinate axes.
(ii) Write the equation of the plane 4x ~ 4y +2z+ 5 =0 into intercept form.
(iv) Find the angle between the planes x + 2y-+2z~5=0 and 3x +3y +22~8=0.
(v) Find the angle between the planes 2x—y+2=6 and x+y +2z=7
(vi) Find the equation of the plane passing thr ough (1, 1, 1) and parallel to the plane
x+2y+32-7=0. i 00
Evaluate the following :
xal-1 7
won tf
i) a) ae by Xsina-asinx
xm ksa
i 2
DT ae
| : _ (x?-a2)
| Gu) a), Sern) sifobs) sina)
tan(
Gita)
| e'=sinx-1 é
| oO» aS » ee
7 c0sax~cosbx 1~cos mx
t (vida) ea ) Moga 840
}
Evaluate the following :
1x3 3x44 V6
t b) im “~—
seater 13x3 5x? -7 » tim ea
i it Vx+I-ve tt fx2ax-:
i) ay Re et x » x +x-K
S44 ls
‘iiyay It
Gia) tan by My apexel
v eS 8 _ us
(iv) a) +x b) t (+xy (l- x)
0 Ox 0 x
2,
XY ifxsl
| = » is conti R.
(v) Is, defined by f(x) f if x> 1° 5 continuous on
OR itxeo z
(vi) If fis defined by f(x) =| x taco “imousat''?
1 ifx=
17. (Da) If f(x) =e. log (Bx + 4), then find £'(x) :
b) If f (x) = xe* sinx, then find f(x).
(i) TEA) = 1 XE + se + XI then find £°(1)
(ili) If f(x) = 72°43e then find f(x)
(iv) If f(x) = x2 2* loge, find ity
(¥) If y= log [sin (logx )] , find
dy.
(vi) If y = log (secx + tanx), find Gy Y8. @ Wty sec (Jtamr) find
(i) Find the derivative of [acted
(ii) Find the derivative of sin~! (3x ~ 4x) wart 'x’
(iv) Find the derivative of so( 5)
(©) If x=a cost, y=a sin find
(vi) If y = ae™ + be, then prove that y” =
79. (i) If y=x?+3x+6 , then find Ay and dy when x= 10, Ax =0.01
(ii) Find Ay and dy of y=f(x)=x°+x at x=10 when ax = 0.1
(iii) Find Ay and dy for y=5x?+ 6x +6 ,x=2, dx=0001
(iv) Find the approximate value of 82
(©) If the increase in the side of a square is 4% find the percentage of increase in the area of the
square.
(vi) If the increase in the side of a square is 2% then find the approximate percentage of increase in
its area
/10.(i) Verify Rolle’s theorem for the function x2- 1 on [- 1, 1]
(i) Verify Rolle’s theorem for the function f(x) = x? - 5x +6 in the interval [-3, 8]
(ili) Verify the condition of the Lagrange’s mean value theorem for the function
f(x) = x2= 1 on (2, 3]
(iv) Find ‘c° for the function f(x) = x? on (2, 4] by Legrange’s theorem.
(v) Verify the condition of the Legrange's theorem for the function x°- 1 on [2, 3]
(vi) Show that there is no real number k for which the equation x’— 3x +k = 0 has two
SECTION &
/11.(i) Find the equation of locus of a point P such that the distance of P from the origin is twice the
distinct roots in (0, 1)
distance of P from A(1, 2)
(ii) A(1, 2) B 2, -3) C C2, 3) are three points. A point P moves such that PA? + PB? = 2PC?
Find the Locus of P.
(iii) Find the locus of P such that the line segment joining (2, 3) & (- 1,5) subtends a rightangle
at P.
(iv) The ends of the hypotenuse of a right angled triangle are (0 , 6) and (6 , 0). Find the
equation of the locus of its third vertex.
(9) Find the equation of Locus of P, ifA(5,0) , B(~5, 0)and [PA~PB|= 8
(vi) Find the equation of locus of P, if A (4, 0), B (- 4, 0) and | PA-PB|=4.
(rT __ ee
712.() When the origin is shifted to (~1, a by the translation of axes, find the transformed
equation of x? + y* +x — 4y+1=0.
Ww) bine the origin is shifted to the point (2,3), the transformed equation of a curve is
x + 3xy—2y?+ 17x—Ty—11 = 0. Find the original equation of the curve.
(lil) When the axes are rotated through an angle 1/4 then find the transformed equation of
3x? + 10xy +3y?=9,
(iv) When the axes are rotated through a angle = = , Find the transformed equation of
x? + NBxy—y? =2a
w) = the axes are rotated through an angle 45° the transformed equation of a curve is
17x? — 16xy + 17? = 225 . Find the original equation.
(vi) Show that the axes are to rotate through an angle 5a'(z) So as to remove the xy
a- |
0, if a#b and through the angle 7 fa=b. |
term from the equation ax?+ 2hxy + by?
/13.4i) Transform the equation = + 7 = 1 into normal form. If the perpendicular distance of the
a
1
straight line from origin is ‘p! , deduce that ao z + =
P
il) Find the value of k, if the lines 2x—3y + k= 0, 3x—4y-13 = 0, &x—Lly-33=0
are concurrent.
(ii) Find the equation of the line perpendicular to the line 3x + 4y +6=0 and making an intercept—4 |
on the X axis.
(iv) Find the equation of the straight line passing through the point of intersection of the lines
x-+y+1=0 and 2x—y+5~=0 and containing the point (5,~2)
(v) Find the value ofk ifthe angle between the straight lines 4x—y +7 = 0, kx — Sy-9= is 45°
(vi) Find the image of (1, ~ 2) w.rt the straight line 2x -3y + 5 =0. |
AP 4), ifx<2
2 at the point 2.
/14,(i) Check the continuity of f given by f0) =
2-8x> ,ifx>2
(i) Check the continuity of the following function given by
2
x9 Sand x#3
vif O
3
(ui) Find the ral constants a so thatthe function f given by f) =
is continuous on R. 7
[cosax—COsPS fx #0
z where a, b are real , is conti 0.
(iv) Show that f() = y -) \ifx=0 real , is continuous at ‘0', a 1K
XIE T i a continuous function on R, then find’ k’.
() Iff, given by f(x) btiaer
-1 ds
Givens=8t+e => S=843 > SS =6r
‘ at ae
i) Velocity V = (3) =843(2)' =8412 = 20 unit/sec
per
*.
ii) Accleration a = (3 = 6(2) =12 unit/sec*
hea
——— 0.
3. Find the value of k, if the equation 2x? + ky. = Gy? + 3x + y + 1= 0 represents a pair of
straight lines. Find the point of intersection of the lines and the angle between the straight
lines for this value of k,
Sol ~ 2A—6K(1) + 2(1/2\3/2\(K/2) — 2 (1/2)? + 6 (3/2)? — 1 (2) = 0
ant B +2 By wp WABH3K=2454-K _
=) -3k-4=0 = k(k—4) +1 (k-4)=0 = (k-4) (K+ 1)=0 > k=4,-1
Let|k=4 = h=2|
< Point of Intersection =
If @ is the acute angle between the lines then
wo TEAS Bo 4 4 te
(ab)? +4n? Toro? +a" Veiri6 Yao 5
Let| k=-1 = h=-1/2|
+ Point of Intersection = [ee] - (- [Tey it]
ab-h?' ab—h
If @ is the acute angle between the lines then 6 = cos (Sry it)