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This document contains a mathematics examination paper for intermediate first-year students, focusing on coordinate geometry and calculus, with a total of 75 marks. It is divided into three sections: very short answer questions, short answer questions, and long answer questions, covering various mathematical concepts and problem-solving techniques. The paper includes questions on limits, derivatives, equations of lines and curves, and geometric properties.
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0% found this document useful (0 votes)
61 views22 pagesPrevious Year's Quetion Papers
This document contains a mathematics examination paper for intermediate first-year students, focusing on coordinate geometry and calculus, with a total of 75 marks. It is divided into three sections: very short answer questions, short answer questions, and long answer questions, covering various mathematical concepts and problem-solving techniques. The paper includes questions on limits, derivatives, equations of lines and curves, and geometric properties.
Uploaded by
sri varshaCopyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF or read online on Scribd
/ 22
PCRS La LO ROL
INTERMEDIATE — FIRST YEAR
MARCH - 2020 |.P.E.PAPER (T.S
| PART - Ill: MATHEMATICS — I(B) (E.M.)
[Time : 3 Hours] (Coordinate Geometry and Calculus) [Max. Marks : 75
Note : This question paper consists of three sections A, Band C.
SECTION - A
I. Very Short Answer Type Questions : 10x 2=20
i) Answer all questions.
ii) Each question carries two marks.
1. Find the value of p, if the straight lines x + p = 0, y + 2 = 0 and 3x + 2y+5 =O are
concurrent.
2. Find the length of the perpendicular drawn from the point (3, 4) to the straight line
3x-4y+10=0.
3. Show that the points (1, 2, 3), (7,0, 1) and (~ 2,3, 4) are collinear.
4, Find the direction cosines of the normal to the plane x + 2y + 22-4 = 0.
one
5. Compute : Lim
x0 x,t!
6. Compute: Lim 81*!*3* —
x3= 31xl—2x
7. If f(s = x Tan” x, then find f(@).
8. Ify = ae™ + be, then prove that y" = n’y.
9. Find Ay and dy for the function y = 5x? + 6x + 6 at x = 2 when Ax = 0.001.
10. Define strictly increasing function and strictly decreasing function on an interval |.
SECTION - B
Il. Short Answer Type Questions : 3x4=20
i) Attempt any five questions.
ii) Each question carries four marks.
11. A@,3) and BC3, 4) are two given points. Find the equation of locus of P so that the
area of the triangle PAB is 8.5.
12. When the axes are rotated through an angle 45°, the transformed equation of a curve|
16xy + 17y? = 225. Find the original equation of the curve.
is 17x?
13, Find the image of the point (1, 2) in the straight line 3x + 4y- 1 = 0.
it
S-Matevial —— = -
DB 14.
15.
16.
17.
mn.
18.
19.
21.
22.
23.
24,
[Coun uae Von
Cos ax ~ Cos bx
e
if x40
Show that f=}
3 -a’) if x=0
where a and b are real constants, is continuous at
Find the derivative of Sin 2x from the First Principle.
Find the lengths of subtangent, subnormal at a point 't’ on the curve
x= a(Cos t + t Sint), y = a(Sin t -t Cos t).
The volume of a cube is increasing at a rate of 9 cm*/sec. How fast is the surface area
increasing when the length of the edge is 10 cm?
SECTION-C
Long Answer Type Questions : 5x7 = 35
i) Attempt any five questions.
li) Each question carries seven marks.
If p and q are the lengths of the perpendiculars from the origin to the straight lines
x Sec 0. + y Cosec @ = a and x Cos a.-y Sin a = a Cos 20, prove that 4p? + q? = a2.
If the equation S = ax? + 2hxy + by? + 2gx + 2fy + c = 0 represents a pair.of parallel
straight lines, then show that i) h? = ab, ii) af? = bg? and iii) the distance between the
7 g’-ac be
= 2 Oe
prt ines f= og [Fe
Show that the lines joining the origin to the points of intersection of the curve
x?—xy +y?+3x+3y-2 =O and the straight line x-y~ /2 = 0 are mutually perpendicular,
Find the angle between the lines whose direction cosines are given by the equations
31 +m +5n = 0 and 6mn ~ 2nl + Sim = 0.
2x ‘3x-x'
= Tan! + Tan!
Ify = Tan’ (25) Tan’ (33)
Find the angle between the curves 2y?- 9x = 0, 3x? + 4y = 0 (in the 4!" quadrant),
From a rectangular sheet of dimensions 30 cm x 80 em four equal squares of side x cm
are removed at the corners, and the sides are then turned up so as to form an open
rectangular box. Find the value of x, so that the volume of the box is greatest.
Tan’
3
—tx= 4x", then show that © _
1-6x? +x" dx 14x
oR REE EN
INTERMEDIATE — FIRST YEAR
MAY - 2019 |.P.E.PAPER (T.S)
PART - Ill : MATHEMATICS - I(B) (E.M.)
Time : 3 Hours] [Max. Marks : 75
Note : This question paper consists of three sections A, B and C.
SECTION - A
L. Very Short Answer Type Questions : 10x2=20
i) Answer all questions.
ii) Each question carries two marks.
1. Find the equation of the straight line which makes an angle 60° with the positive
X-axis measured counter-clockwise and passing through the point (1, 2).
2. Find the value of p, if the straight lines 3x + 7y- 1 = 0 and 7x~ py +3 = 0 are mutually
perpendicular.
3. Find the distance between the mid point of the line segment AB and the point
(,-1, 2), where A = (6, 3,-4) and B = © 2,-1,2).
4, Find the angle between the planes x + 2y + 22-5 = 0 and 3x + 3y + 22-8=0.
5. Find the left hand limit and right hand limit of the function,
x-2|
foo) = ESF atx =2.
6 Compute: Lim 2*5**2.
: * xove Ox? 5x41
7. Find the derivative of the function, f(x) = 5 Sin x + e* Log x.
8. Itx= eS", find 2.
dx
9. Find dy and Dy of y = f(x) = x? + x at x = 10, when Dx = 0.1,
10. Verify Rolle's theorem for the function y = £() = x" ~ 1 on [= 1, 1].
Il. Short Answer Type Questions : 5x4=20
i) Attempt any five questions.
ii) Each question carries four marks.
11. AG, 3) and BG, 2) are two fixed points. Find the equation of the locus of a P, so that
the area of triangle PAB is 9.
S-Matetiat —— z 12.
13.
14.
1
23.
24,
When the axes are rotated through an angle i find the transformed equation of!
3x2 + 10xy + 3y? = 9,
Transform the equation 24% = 1 into the normal form when a > 0 and b > 0. If the
Perpendicular distance of the straight line from the origin is p, deduce that
11,1
Check the continuity of the function f given by
f(x) =
15. Wy = Tan? (2) (ixI<1), then find &Y.,
1-x dx
16.
mn
A stone is dropped into a quiet lake and ripples move in circles at the speed of 5 cm/
Tl.
18.
19.
20.
21.
3n = O and 77 + 5m? —3n? = 0.
22,
Find the angle between the curves, y? = 4x ; x? + y= 5,
Awindow is in the shape of a rectangle surmounted by a semicircle. If the perimeter
of the window is 20 ft., find the maximum area.
Pa a
» O a
log.
6. Evaluate : lim bao)
| 7. Wf) =1+x4x? +... +x! then find (1).
8. Find the angle which the straight line y = V3 x- 4 makes with the Y-axis.
9. Verify Rolle's theorem for the function y = f(x) = x? + 4 in [-3, 3].
10. Find Ay and dy for the function y = cos x at x = 60° with Ax =i,
(cos 6i = 0.4848, i = 0.0174 radians).
SECTION - B
I. Short Answer Type Questions : 5x4=20
i) Attempt ANY FIVE questions.
if) Each question carries FOUR marks.
| 50-4) if O if x>2
12. AQ, 2), B(, - 3) and C(- 2, 3) are three points. If a point P moves such that
MARCH - 2019 |.P.E.PAPER
ES
INTERMEDIATE — FIRST YEAR
18)
PART - II: MATHEMATICS — (B) (E.M.)
“Time : 3 Hours] ~ [Max. Marks
Note : This question paper consists of three sections A, B and C.
L
| Compute : lim 2—!
. Find the value of p, if the straight lines 3x + py - 1 = 0, 7x- 3y + 3 = 0 are mutually
. If £@¥) = log (tan e*), then find f(x).
|. Find the ratio in which the xz-plane divides the line joining AC 2, 3, 4) and B(,, 2, 3).
. Reduce the equation of the plane x + 2y - 3z— 6 = 0 to the normal form.
(SECTION-A
Very Short Answer Type Questions : 10 x 2=20
i) Attempt ALL questions.
ii) Each question carries TWO marks.
x0 bY -1
perpendicular,
PA2 + PB? = 2PC?, then show that the equation to the locus of P is 7x - Ty + 4 = 0. 13.
14,
15.
16.
17.
I.
20.
ah
22.
23.
24.
18.
19.
TED
x
A straight line through Q(J/3 . 2) makes an angle of ¢ with the positive direction of
the X-axis. If the straight line intersects the line 3 x - 4y + 8 = 0 at P, then find the
distance of PQ.
When the axes are rotated through an angle o, find the transformed equation of
Xcos 0+ y sin @=p.
Show that the tangent at any point @ on the curve
x=csec 0, y = ¢ tan @is y sin @ = x-c cos @
Find the derivative of cos? x from the first principle.
A container is in the shape of an inverted cone has height 8 m and radius 6 m at the
top. If it is filled with water at the rate of 2 m°/minute, how fast is the height of water’
changing when the level is 4m.
Long Answer Type Questions : 5x7=35
i) Attempt ANY FIVE questions.
ii) Each question carries SEVEN marks.
Find the orthocentre of the triangle whose vertices are (5, - 2), (- 1, 2) and (1, 4).
Show that the area of the triangle formed by the lines ax’ + 2hxy + by* = 0 and
Find the angle between the lines whose direction cosines satisfy the equations :
(+m+n-=0,? +m?
dy _y(1-logxlogy
If x!ogy = log x, then show that : — =| —~——>>= |"
oe dx x (logx)
If the tangent at any point on the curve x? +y?/3 = a”/3 intersects the co-ordinate axes’
in A and B, then show that the length AB is a constant.
Find the values of k, if the lines joining the, origin to the points of intersection of the|
curve 2x? — 2xy + 3y?+ 2x-y +1 = 0and the line x + 2y = k are mutually perpendicular.
Find the maximum area of the rectangle that can be formed with fixed perimeter 20.
eR ROK OK L
MAY - 2018 I.P.E.PAPER (T.S)
. Find the distance between the parallel lines 5x — 3y - 4 = 0, 10x- 6y-9 = 0.
. Find the coordinates of the vertex 'C’ of ABC, if its centroid is the origin and the
. Find the intercepts of the plane 4x + 3y - 2z + 2 = 0 on the coordinate axes.
a : Lt|———
compte: (=)
. IEf@) = x. e%, Sin x, then find f(@).
. Ify = e°>""*, then prove that s
ix
|. If the increase in the side of a square is 4%, then find the approximate percentage of
|. Define Lagrange's mean value theorem.
|. Short Answer Type Questions : 5x4 =20
. Find the equation of the locus of a point, the difference of whose distances from
INTERMEDIATE — FIRST YEAR
MAY
Ne
PART - Ill: MATHEMATICS — 1(B) (E.M.)
L— - |
Time : 3 Hours] [Max. Marks : 75]
Note : This question paper consists of three sections A, B and C.
|
|
|
Very Short Answer Type Questions : 10 x2=20 |
i) Answer all questions.
ii) Each question carries two marks.
Find the equation of the straight line passing through (- 4, 5) and cutting off equal
and nonzero intercepts on the coordinate axes.
vertices A, B are (1, 1, 1) and (- 2, 4, 1) respectively.
Evaluate u (+4)
increase in the area of the square.
i) Attempt any five questions.
ii) Each question carries four marks.
5, 0) and (6, 0) is 8.
S-Materiat an woh. 13.
15.
22,
23,
24,
. Iff, given by f(x)
- Find the circumcentre of the triangle whose vertices are (1, 3), (0,~2) and (3, 1). |
19.
20.
~ Ifa ray makes angles «, B, y and 8 with the four di
n ° ansformed equation of a curye|
12. When the axes are rotated through an angle 45°, the transfo1 a |
is 17x? - 16xy + 17y? = 225. Find the original equation of the curve. |
, re hen |
Astraight line through P(3, 4) makes an angle of 60° with the positive, eh of the
X-axis, Find the coordinates of the points on the line which are 5 units away from P,
K’x-k, ifx21
2 tick
ofk. |
Find the derivative of the function Cos(ax) from the first principle. * |
, is a continuous function on R, then find the value
}. Find the equations of tangent and normal to the curve xy = 10 at (2, 5).
17.
The volume of a cube is increasing at the rate 8 cm3/sec. How fast is the surface areal
increasing, when the length of an edge is 12 cm ?
SECTION - C
Long Answer Type Questions : 5x7=35
i) Attempt any five questions.
ii) Each question carries seven marks.
Show that the product of the perpendicular distances from a point (a, B) to the pair
of straight lines ax® + 2hxy + by? = 0 is | 207 + 2ho + bp? |
(a—b)? + 4h?
Find the angle between the lines joining the origin to the Points of intersection of the|
curve x* + 2xy +y? + 2x + 2y—5 = Oand the line 3x-y 41-0,
lagonals of a cube, then fi ? a
+ Cos? B + Cos? y+ Cos28, See
se
Ity = Tan (
for 0 < Ixi3 |
is continuous on R.
15. Find the derivative of x sin x from the first principle.
|
16. Show that at any point (x, y) on the curve y = b e%, the length of the sub-tangent is a
2
constant and the length of the subnormal is 2.
a
3t? + St — 1 where:S is
17. A particle is moving along a line according to S = f(t) = 4t3-
ion|
!
measured in metres and t is measured in seconds. Find the velocity and accelerati
at time t. At what time the acceleration is zero?
|
SECTION - C |
Ill. Long Answer Type Questions : 5x7=35
i) Answer ANY FIVE questions.
ii) Each question carries SEVEN marks.
18, Find the circumcenter of the triangle whose vertices are (1, 3), 3, 5) and GD.)
19. If the equation ax? + 2hxy + by? = 0 represents a pair of straight lines, then show that
the angle @ between the lines is given by
la+b]
cos @= —————
y(a—by? +4h?
20. Show that the lines joining the origin to the points of intersection of the curve |
xtaxy +y?+3x+3y-2-0 and the straight lines x-y - /2 = 0 are mutually perpen
dicular. |
21, Find the angle between two diagonals of a cube.
ite d
ity = tan] for0< 1x! <1,find 2.
22, Ity = tar [es ot oe
n
x) =2(az0,be0)atthe
a
23, Show that the equation of the tangent to the curve (2) + ‘
ee ieee
point (a, b) is 5+,
m x 80 cm four equal squares of sie
24. From a rectangular sheet of dimensions 30
rm al
x cm are removed at the corners, and the sides are then turned up so as to fo!
open rectangular box. Find the value of x, so that the volume of the box is the greatest
— - S-Matesidl T.
a.
Time : 3 Hours] [Max. Marks : 75
Note : This question paper consists of three Sections A, B and C. :
. Very Short Answer Type Questions.
. Transform the equation 3x + 4y + 12
. Ifthe straight lines x + p = 0, y + 2= 0 and 3x + 2y + 5 = 0 are concurrent, then find the
. Find the equation of the plane if the foot of the perpendicular from origin to the plane|
. Compute : Lim (Ix]+x).
. Evaluate Lim HSE, Lot el ee |
a
ag oY.
. Ify = Cosec"!(e**!), find =
&
. Show that, y = x + Tan x satisfies Cos?x ped + 2x = By.
). Find the approximate value of {17 «
. If0——— |
find : cos’a. + cos?B + cos?y + cos*6.
Ifx¥ + y* = a, then show that : s -|
ix
yx’ | ty" logy |
log tay! |” |
Find the lengths of subtangent and subnormal at a point ¢ on the curve
x=a(cost +t sin t) ;y = a(sin t-t cos t)
. The profit function p(x) of a company, selling x items per day is given by
P(x) = (150 - x) x - 1600.
Find the number of items that the company should sell to get maximum profit. Ais
find the maximum profit. Wed
a
INTERMEDIATE — FIRST YEAR
ARCH - 2016 I.P.E.PAPER (T.S
PART - IIL: MATHEMATICS — I(B) (E.M.)
Time : 3 Hours}
ae)
in a [Max. Marks : 75]
Note : This question paper consists of three Sections A, B and C.
| (SECTION
I. Very Short Answer Type Questions.
i) Attempt all questions.
10 x 2= 20
ii) Each question carries two marks.
1. Transform the equation /3x + y=4 into
i) Slope intercept form ii) Intercept form
Find the value of p if the straight lines 3x + 7y- 1 = 0 and 7x py + 3 = 0 are mutually|
perpendicular.
3. Show that the points (1, 2, 3), (7, 0, 1), © 2, 3, 4) are collinear.
Reduce the equation x + 2y ~ 3z ~6 = 0 of the plane to the normal form.
5. Compute Lim
ime
6. Compute Di ee aaa
7. Find the derivative of Sin"! (3x - 4x) with respect to ‘x.
8. If 2x2-3xy + y?+x+2y-8-= 0, then find s.
9. Find dy and Ay of y = f(x) = x? + x at x = 10 when Ax = 0.1.
10. Find the length of subtangent at a point on the curve y ="b Sin (2).
a
5x4=20
Il. Short Answer Type Questions.
i) Attempt any five questions.
ii) Each question carries four marks.
| 11. Find the equation of locus of a point, the sum of whose distances from (0, 2) and
,-2)is 6.
12. When the origin is shifted to the point (2, 3) the transformed equation of a curve is
x2 + xy - 2y? + 17x- Ty - 11 = 0. Find the original equation of curve.
| 13. Find the equation of the straight line parallel to the line 3x + 4y = 7 and passing through
the point of intersection of the lines x - 2y-3 = 0, x + 3y-6 = 0.
S—Materia —— 14.
15.
16.
V7.
i.
18.
19.
20.
21.
22.
23.
24.
(eee)
Oe
4-x if x<0
x-5 if O be
Find the lines joining the origin to the points of intersection of the curve
‘Tx? — dxy + 8y? + 2x - dy - 8 = 0 with the straight line 3x - y = 2 and also the angle
between them.
Find the direction cosines of the two lines which are connected by the relations
[-5m + 3n = 0, 72 + 5m? —3n? = 0.
dy (22 +y* 233
If x¥ + y*= a? then prove that 9. -~| \y Logxexy™?
If the curved surface of right circular cylinder inscribed in a sphere of radius 'r' is
maximum, show that the height of the cylinder is J2r.
If ax? + by? = 1, a,x” + b,y’ = 1, then show that the condition for orthogonality of abov®
_1 iii
curves is 75
a [Time
L
10,
i.
12.
Note : This question paper consists of three Sections A, B and C.
2. Find the value of Pif the straight lines x+ P = 0,y+2=
ARLES MAY
INTERMEDIATE — FIRST YEAR |
MAY — 2015 1|.P.E.PAPER (T.S)
PART - Ill : MATHEMATICS — 1(B) (E.M.)
3 Hours]
=
[Max. Marks : 75
SECTION 10x 2= 20
Very Short Answer Type Questions.
1) Attempt all questions.
ii) Each question carries two marks.
Find the equation of the straight line passing through the point (-2, 4) and making
non-zero intercepts on the axis of coordinates whose sum is zero.
| 3x + 2y + 5 = O are concurrent.
Find the ratio in which the XZ-plane divides the line joining A(-2, 3, 4) and B (1, 2, 3).
Find the direction cosines of the normal to the plane x + 2y + 22-4-0
compte ta( 2).
7
x
Evaluate tal
Find the derivative of tan“ (log x),
Find the derivative of sin"! (3)
Find dy and Ay of y = f(x) = x? + x at x = 10 when Ax - 0.1.
Verify Rolle's theorem for the function {(x) = x? + 4 in [-3, 3].
‘SECTIOI
B 5x4=20
Short Answer Type Questions.
i) Attempt any five questions.
fi) Each question carries four marks.
If the distance from the Point P to the points (2, 3) and (2, ~ 3) are in the ratio 2: 3,
then find the equation of the Locus of P.
When the axes are rotated through an angle 45°, the transformed equation of a curve
is 17x? - 16xy + 17y? = 225. Find the original equation of the curve.
S~Material $= _ ae 14,
15.
16,
17.
tl.
18.
19.
20.
21.
22,
23.
24,
NEUES
Find the equation of the straight line passing through the points (- 1, 2) and (5 ~ 5
and also find the area of the triangle formed by it with the axes of coordinates,
Check the continuity of the following function at 2 :
3a? 4) if O2
Find the derivative of the function sin 2x from the first principle.
Find the equation of tangent and normal to the curve y =x? + 4x’ at (1, 3).
The volume of a cube is increasing at a rate of 9(centimeters)° per second. How fast
is the surface area increasing when the length of the edge is 10 centimeters ?
SECTION - C 5x7=35
Long Answer Type Questions.
i) Attempt any five questions,
ii) Each question carries seven marks,
Find the circumcenter of the triangle whose vertices are c
2,3), 2-1) and (4, 0).
Show that the area of the triangle formed by the lines
vb? ab
ax? + 2] + by? =O and k+my+n-0is ee
sd am? ~ 2hin +b
Show that the lines joining the origin to the Points of intersection of the curve
oxy + y% 3x4 3y—2 = Oand the line x~y— = 0 are mutual
Find the direction cosines of two
+m +n=0and mn -2nl-2im = 0,
ly perpendicular.
lines which are connected by the relations: