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246 views12 pagesMaths - 1B
Uploaded by
Anjaneyulu VeerankiCopyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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/ 12
MATHEMATICS - IB
Time : 3 Hours] [Max. Marks : 75
iq WEEE Move PAPER
Note : This question paper consists of three sections A, B and C
SECTION =A
1. Very Short Answer Type Questions : 10«2=20
i) Answer ALL the questions.
li) Each question carries TWO marks.
Q.No: 1 Straight Line:
1. Find the value of s#if the slope of the line passing through (2, 5) and (x, 3) is 2.
2, Find the equation of the straight line passing through (— 4, 5) and cutting off e
zero intercepts on the coordinate axes.
3, Find the condition for the points (a, 0), (h, k) and (0, b), where ab #0, to be collinear.
. x
4, If the product of the intercepts made by the straight line x tan a + ysec a= 1 (0 Sa< 5) on
the coordinate axes is equal to sin a, find «.
5, Find the equation of the straight line passing through the points (at,”, 2at) and (at *, 2atz).
Find the value of y, if the line joining the points (3, y) and (2, 7) is parallel to the line joining the
points (-1, 4) and (0, 6).
7. Ifthe area of the triangle formed by the straight lines, x = 0, y = 0 and 3x + 4y = a(a> 0) is’6’. Find
the value of ‘a’.
8 Transform the equation x + y + 1 = 0 into normal form.
Find the equation of the straight line passing through (-2, 4) and making non-zero intercepts
whose sum is zero.
10. Transform the equation into the form Ly +AL2 = 0 and find the point of concurrency of the
family of straight lines represented by the equations (2 + 5k) x-3(1 + 2k) y + (2-k) = 0.
(Qo: 2 Straight Line
1. Find the distance between the parallel lines 5x - 3y - 4 = 0, 10x - 6y - 9 = 0.
2. Find the equation of the straight line parallel to the line 2x + 3y + 7 = O and passing through the
point (5, 4).
3. Find the angle which the straight line y = Vx -4 makes with the y-axis.
4. If2x-3y—5 « is the perpendicular bisector of the line segment joing (3, ~ 4) and (ce, B) find c+ B.
5. ACO, 4), BCA, 9) and C(-2, -1) are the vertices of a triangle. Find the equation of the median
through 2
10.
2
ae
10.
Q.No : 4 The Plane :
prepress wn
10.
i.
Find the area of the triangle formed by the following straight lines and the coordinates axes}
XCosa+ysina~ p(p> 0),
Find the value of p, if the straight lines x « p= 0,y + 2= and 3x + 2y +5 = 0 are concurrent.
Find the value of k, if the straight lines y ~ 3kx + 4 - 0 and (2k ~ 1)x = Bk = 1) y= 6 = 0 are|
perpendicular,
Find the point of intersection of the straight lines * + 7 = land Ree 21(a4+b).
a'b b
WACO, 4), BCA, 9) and C(-2,-1) are the vertices of a triangle, Find the equation of the altitude
through B.
Find the coordinates of the vertex 'C’ of AABC If Its centroid Is the origin and the vertices A, B
are (1,1, 1) and (2, 4, 1) respectively.
(3, 2,1), (4, 1, 1) and (6, 2, 5) are three vertices and (4, 2, 2) is the centroid ofa tetrahedron,
find the fourth vertex.
Find the ratio in which the XZ-plane divides the line joining A 2, 3, 4) and B(1, 2, 3).
Find the ratio in which XY-plane divides the line joining A(2, 4,5) and B(3, 5, ~4). Also find the
point of intersection.
Find the fourth vertex of the parallelogram whose consecutive vertices are (2, 4,-1), 6 6,-1) and (4,5, 1).!
Show that the points A(I, 2, 3), B(7, 0, 1) and C2, 3, 4) are collinear.
Find the centroid of the tetrahedron whose vertices are (2, 3, - 4), (3, 3,- 2), - 1,4, 2) and (3,5, 1).
By section formula find the point which divides the line joining the points A(2, - 3, 1) and
BG, 4, —5) in the ratio 1:3.
Show that the points (1, 2, 3), (2, 3, 1) and (3, 1, 2) form an equilateral triangle.
Find the ratio in which the point (6,~17,-4) divides the line segment joining (2, 3, 4) (3,-2, 2).
Write the equation of the plane 4x - dy + 2z + § = 0 in the intercept form.
Find the intercepts of the plane on the coordinate axes 4x + 3y - 2z + 2=0.
Find the equation of the plane whose intercepts on x, y, z- axes are 1, 2, 4 respectively.
Find the directions of the normal to the plane x + 2y + 2z-4=0.
Reduce the equation x + 2y - 3z.~6 = 0 of the plane to the normal form.
Find the angle between the planes x + 2y + 22-5 = 0 and 3x + 3y + 22-8 =0,
Find the angle between the planes, 2x—y + z= 6 and x+y +2z=7.
Find the equation of the plane if the foot of the perpendicular from origin to the plane is (1, 3, -5).
Find the constant k so that the planes x - 2y + kz = 0 and 2x + Sy z= 0 are at right angles.
Find the equation to the plane parallel to the ZX - plane and passing through (0, 4, 4).
Find the equation of the plane passing through the point (1, 1, 1) and parallel to the plane|
x4 2y+32-7-0. _J Go: 5 Limits And Contin
Eee
lity :
1. Find Lt =)
x40]
tan(&— 9) J3, compute Lt sles
2. Evaluate Lt 3
xan x? a x50 XCOS X
sin
loge (1+ 5x)
x
db
4. Evaluate Lt, 5.Compute Lt
x0 OX
b=]
zx-2
6. Compute Lt,
x!
7. Evaluate Lt, “05%
ey
2
8. Show that Lt = 1. }9. Show that us, ( 5
10. Find the right and the left timits of the function (G0 = {*/? <2) atthe poiat a= 2,
x" /3 (x22)
QNo: 6 Limits And Continuity?
1. Compute Lt (yx?+x-x), 2. Compute Lt (rT - vx),
—cos bx a) tan? (=
3. Bvaluate Lt (S222 08 2), evaluate Lp Gta) tant (x= a)
aio x aa
sina bx) ~ sin (abo) 1
5. Evaluate Lt,
6. Compute Lt, 7 (a> 0), (b> 0), (b¥1).
x »
2x43 Lp Leos 2mx
7. Evaluate | Lt . 8. Evaluate Uj aPay (msn 2).
SE
in (n cos*x) ur SlL+3x
9. Compute Lr SRE. 10. Evaluate Ut 3h 5 +
Q.No : 7 Differentiation :
1. Ify = cos (log (cot x)), then find x.
2. Find the derivative of 20!8t2"® with reference to x.
3. Find the derivative of 7*’ + with reference tox.
4, Find the derivatives of sin“! x want. x.
5. Find the derivatives of sin
6. Find the derivative of sec (tan x) want. x find {' @.
7. Iy = log (sin (log x)), then find x . 8. Find the derivative of log (tan 5x) w.r.t. x.
9. Find the derivative of =~ = Wer 10, Ity =x e*sinx, find x.
11. Find the derivative of f(x) = e* @? + 1D. 12, 1fx = acos* t, y= asin't, then find w,
rae eam QNo #8 Differentiation:
1. Find the derivative of log (sin“!(e*)) wert.
Py
a
2 Ifx= tame then show that 5
3. Find the derivative of cosec™! (e*** !) want. x,
+x
4. My = {cot (x), then find x,
Find the derivative of tan“! (2) Winx,
Trax
3x
6. Find the derivative of sinh“! (2) wart. x,
oy
7, Find the derivative of y = sec™! (es) find Ge
8. Differentiation f(x) = e* with respect to g(t) = Vx
¢
9 My =x* (> 0), then = =x¥ (1 +logx).
a
10. If x3 + y3—3axy = 0, then find a
QNo : 9 Errors And Approximations :
1. Find Ay and dy ify = 5x? + 6x + 6, x = 2, Ax = 0.001,
2. Find Ay and dy ify = f(x) =x" + x,x= 10 and Ax =0.1.
3.. Find Ay and dy ify = x° + 3x +6,x= 10, Ax= 0.01.
4, Find Ay and dy ify =e" +x, x =5 and Ax = 0.02.
5.
6.
7.
Find the approximatioin of Y65-
Find the approximation of /82 «
Ifthe increase in the side of a square is 2%, then find the approximate percentage of increase|
inits area.
8 Ifthe increase in the side of a square is 4%, then find the approximate percentage of increase
in the area of the square.
9. Ifthe radius of a sphere is increased from 7em to 7.02 cm then find the approximate increase in
the volume of the sphere.
10, The side of a square is increased from 3 cm to 3.01 cm. Find the approximate increase in the|
area of the square.
No: 10 Rolle’s Theorem Ang Lagrange’s Mean Value Theorem
24 4in[-3,3].
2. Find the value of ‘c’ Rolle's theorem for the function x*- 1 on [2, 3].
3. Verify Rolle’s theorem for the following function x*-1 on [-1, 1}.
4, It is given that Rolle’s theroem holds for the function f(x) = x* + bx? + ax on [1, 3] with)
1
c=2+ 7g. Find the values of a and b.
1. Verify Rolle’s theorem for the function y = (x) Q.No? 11 Locus:
1
Seenrne
Q.No : 12 Transformation of Axes
1
Let f(x) = (x= 1) (X= 2) (X=3). Prove that there is more than one ‘ec’ in (1, 3) such that f(c) = 0.
;. Find a point on the graph of the curve y = x" where the tangent {s parallel to the chord Joining)
On the curve y = x7, find a point at which the tangent is parallel to the chord joining (0, 0) and (1, 1).
. Show that there is no real number k, for which the equation x? — 3x + k = 0 has two distinct
)._ Verify the Rolle’s theorem for the function (x? ~ 1) (x2) on [- 1, 2}. Find a point in the interval
|. Find ‘c’ so that ["(c) =
|. Short Answer Type Questions : 4x5=20
. The ends of the hypotenuse of a right angled triangle are (0, 6) and (6, 0). Find the equation of
. ACI, 2), B2,- 3) and CC 2, 3) are three points. A point P moves such that PA? + PB? = 2PC2,
. AG, 3) and B(3,- 2) are two fixed points. Find the equation of the locus of P, so that the area of
. Find the equation of the locus of P, if the ratio of the distances from P to A(5, - 4) and B(7, 6) is 2:3.
1. Find the equation of the locus of P, if A = (2, 3), B = (2, - 3) and PA + PB = 8.
). Find the equation of the locus of P, If A = (4, 0), B = (- 4, 0) and| PA-PB |= 4.
. When the origin Is shifted to the point (2, 3), the transformed equation of a curve Is x” + Sxy—
;. When the axes are rotated through an angle a find the transformed equation of x2 + 2/3 xy
(1, 1) and (3, 27),
roots in (0, 1]
where the derivative vanishes.
13
fhe ue) im the cases {(x) = x9-3x-1;a= 4 b-=.
N-B
i) Answer ANY FIVE questions.
ii) Each question carries FOUR marks.
Find the equation of locus of P, if the line segment Joining (2, 3) and (-1, 5) subtends a right
angle at P.
locus of its third vertex.
Show that the equation to the locus of P is 7x- 7y + 4=0.
triangle PAB Is 9 sq.units.
A@,3) and BC-3, 4) are two given points. Find the equation of locus of P. So that the area of the
triangle PAB Is 8.5 sq. units,
Find the equation of the locus of a point, the sum of whose distances from (0, 2) and (0, — 2) is 6.
Find the equation of the locus of a point, the difference of whose distances from (~5, 0) and
6,0) is 8.
When the origin is shifted to (- 1, 2) by the translation of axes, find the transformed equation
of 2x? +y?—4x + 4y=0.
2y? + 17x~7y~11 = 0. Find the original equation of the curve.
-y*= 202,
7 —7 4. When the axes are rotated through an angle a, find the transformed equation of, x cos «+ y sin a= P|
5. When the axes are rotated through an angle ©, find the transformed equation of 3x? + Oxy +
ay?
6. When the axes are rotated through an angle 45°, the transformed equation of a curve Is
1ix?—16xy + 17y? = 225. Find the original equation of the curve.
7. Prove that the angle of rotation of the axes to eliminate xy term from the equation|
2 aby? Ois 2 tant (2) whe x itae
ax? + 2hxy + by’ 30s Sita (3 where a#b and 4 ifa=b.
8 When the origin is shifted to the point @, ~4), the transformed equation of a curve is x? + y? = 4,
Find the original equation of the curve.
9. When the origin is shifted to (1, 2) by the translation of axes, find the transformed equation|
of 2x? + y?— dx + dy = 0.
10, When the origin is shifted to (1, 2) by the translation of axes {ind the transformed equation of
xtey?s x-dy+ 1-0,
Q.No: 13 Straight Line te
1, Transform the equation 2 + = = 1 into the normal form when a> 0 and b > 0, If the perpendicu-
lar distance of the straight line from the origin is p, deduce that
2. Find the value of k, ifthe lines 2x—3y +k +0, 3x—4y-13 = 0 and 8x- 1ly-33= 0 are concurrent.
3. Show that the lines 2x + y-3=0, 3x + 2y~2= 0 and 2x~3y-23 = 0 are concurrent and find the
point of concurrent.
4, I the straight lines ax + by + = 0, bx + cy +a =O and cx + ay + b = 0 are concurrent, then prove
that a° + b? + c3 = abe.
5. Find the value of p, if thé lines 3x + 4y = 5, 2x + 3y = 4, px + 4y = 6 are concurrent.
Find the equation of, ifthe angle between the straight lines 4x-y + 7 = 0 and lot -5y-9=0is 45°.
7. x-3y-5 = 0 is the perpendicular bisector of the line segment joining the points A, B. If
A=(-1,-3) find the coordinates of B.
4
8 Astraight line passing through A (1,—2) makes an angle tan! (3) with the positive direction
of the X - axis in the anti-clockwise sense. Find the points on the straight line whose distance
from Ais 5.
9. Find the points on the line 3x—4y—1 = 0 which are at a distance of 5 units from the point (3, 2).
10. Astraight line through Q(/,2) makes an angle £ with the positive direction of the X- axis.
Ifthe straight line intersects the line /3x —4y +8 = 0 at P, find the distance PQ.
11. Astraight line with slope 1 passes through Q(-3, 5) meets the line x + y- 6 = 0 at P, Find the
distance PQ. @.No: 14 Limits And Continuity:
Yi+x-
x
1. Compute lim.
2. Compute Jim Sanasinx |
xaa
3, Compute lim S2S2%=008 bx | .
x30 ne
MO2
cos ax cos bx”
iixe0
6, Show thatd(a) « x
where a and b are real constants, is continuous at'0’.
fr=at) tx=0
sinx ifxs0
. 2
7. Find real constants a, b so that the function f given by {(¢) = a abe is continuous
-3 itx>3
onR,
4-x, If xs0
. . xe 5, if O
between the lines is given by cos 0 =
Prove that the product of the perpendicular from (a, ) to the pair of lines ax? + 2hxy + by? = 0
fac? + 2haf + bB"|
1S
Va=py san?
3. Prove that the area of trlangle formed by the lines ax? + 2hxy + by ~ 0 and &x + my +n = Os
ath aD seyamils
lam? — 2him + bi*|
4. Show that the equation to the pair of bisectors of the angle between the pair of lines
ax? + 2hxy + by? = Ois h[x?-y?] = (a-b) xy.
5. Ifs= ax? + 2hyy + by? + 2gx + 2ly + ¢ = 0 represents a pair of parallel lines, then h? = ab and
P g? —ac P-bg
bg? - af®. Also find the distance between the two parallel lines is 2, aaae (or) 2, be me 7
6. Prove that the equation 3x? + 7xy + 2y? + 5x + Sy + 2 = O represents a pair of straight lines. Find
the point of intersection. Also find the angle between them.
7. Ifax? + 2hxy + by? + 2gx + 2fy + ¢ = 0 represents a pair of lines then prove that
1) abe + 2fgh - af bg? - ch? = 0 and ii) h? > ab, g?> ac and (2 > be.
8. Show that the product of the perpendicular distances from the origin to the pair of straight,
Icl
lines represented by ax? + 2hxy + by? + 2gx + 2fy+e=0is Joona
y+ ah
(Q.No : 20 Pair of Straight Lines
1. Show that the lines joining the origin to the points of intersection of the curve x” xy + y* + 3x
+3y—2~ 0 and the straight line x—y— /2 = 0 are mutually perpendicular,
2. Let us find the lines joining the origin to the points of intersection of the curve 7x? 4xy + 8y?
+ 2x—4y—8 = Owith the straight line 3x -y = 2 and also the angle between them.
3, Find the angle between the lines joining the origin to the points of intersection of the curve
x24 Oxy +y?+ 2x + 2y—5 = 0 and the line 3x-y +10.
4, Find the value of k, if the lines joining the origin to the point of intersection of the curve|
2x? — Oxy + 3y2 + 2x—y-1-Oand the line x + 2y = k are mutually perpendicular.
5. Find the condition for the lines joining the origin to the points of intersection of x? + y”
the line fx + my = 1 to coincide.
6. Find the condition for the chord &x + my = 1 of the circle x? + y® = a? (Whose Centre is the]
origin) to subtend a right angle at the ori
and|
7. Write down the equation of the pair of straight lines joining the origin to the points of intersec-|
tion of the line 6x—y + 8 = 0 with the pair of straight lines. 3x” + 4xy—4y?- 11x + 2y + 6 = 0. Show
that the lines so obtained make equal angles with the coordinate axes, 8. Show that two pairs of lines 6x? — 5xy - 6y = 0 and 6x? - 5xy - Gy? + x + 5y—1 = 0 from a square.
9. Show that two pairs of lines 3x? + 8xy-3y’ = 0 and 3x? + 8xy -3y" + 2x—dy + 1 = 0 froma square.
10. Findk, ifthe equation 2x? + kxy-6y? + 3x + y +1 = O represents a pair of lines, Find the point of
intersection of the lines and the angle between the lines for this value of k.
‘QNo : 21 Direction Cosines And Direction Ratios :
1. Ifa ray makes angles a, B, 7, 5 with the four diagonals of a cube, find cos? « + cos* B + cos? y+
cos? 8,
2. Find the angle between two diagonals of a cube.
3. The vertices of a triangle are ACI, 4, 2), B(- 2, 1, 2), C(2, 3,-4). Find ZA, 2B, ZC.
4. Find the direction cosines of two lines which are connected by the relations [+ m + n= 0 and
mn -2ni-2Im = 0.
5. Find the direction cosines of two lines which are connected by the relations (5m + 3n =0 and
72 + 5m? -3n? =
6. Show that the lines whose direction cosines are given by / +m +n = 0, 2mn + 3n/—-5im = 0 are
perpendicular to each other.
7. Find the angle between the lines whose direction cosines satisfy the equations / + m +n = 0,
Pem?-n?=0.
8. Find the angle between the lines whose direction cosines are given by the equations 3! +m +5n = 0
and 6mn -2nl + 5im = 0.
22 Differenti:
1. y= xyaP one + at og (x + fae ex?) then show that = are.
2 fy =x'"* 4 (sin x)°**, then find x.
d
3. Ify=(sinx)!8* + x*I7¥ then find En
4. Ifx¥ + y*=aP then show that 3
5. IffGO = sin! =F and a) ‘tan™
a -y?
6 oie? + agp tenner 2 7.
then show that f(x) = g(@)x, (B