13-07-2025

9610WPE801566250003 TM

PHYSICS

1) Evaluate the following integral

(1) ex + 7 + C
(2) e–x + 7 + C
(3) ex + 5 + 2x + C

(4)

2) Find the value of

(1) 0
(2) 1
(3) –1

(4)

3) The correct graph representing the negative slope is :

(1)

(2)

(3)

(4)

4) The correct curve of the parabola y = – 16x2 is :

(1)

(2)

(3)

(4)

5) A circle with centre of origin and radius is 10 represented by the equation :

(1) x2 + y2 = 10
(2) (x + y)2 = 10
(3) x2 + y2 = 100
(4) x2 + y2 + 2xy = 100

6) Find the length of semi-minor axis from the given equation of ellipse

(1) 24
(2) 13
(3) 26
(4) 12

7) If the radius of the circle is and centre is at origin then what is the value of P from the
given equation x2 + y2 – P = 0

(1)
(2) 25
(3) 5
(4) –5

8) What is the y-intercept of the line represented by y + 8x + 22 = 0 ?

(1)
(2) –22
(3) 22

(4)

9) The area of a circle is given by A = πr2, where r is the radius. Calculate the rate of increase of
area w.r.t. radius.

(1) π r
(2) r2
(3) 2 πr
(4) 2r

10) The maximum and minimum value of the function f(x) = x3 – 3x is respectively.

(1) (–2, 2)
(2) (2, 0)
(3) (2, –2)
(4) (3, 0)

11) If and . Find the magnitude of .

(1)
(2)
(3)
(4)

12) What is the magnitude of resultant force, if the two forces of magnitude 6N and 8N respectively
are acting on a body and the angle between them is 90º.

(1) 100N
(2) 10N
(3)
(4) 20N

13) If is a unit vector along X-axis and , then what is the value of ?

(1)
(2)
(3)
(4)

14) Evaluate : , if and :

(1)
(2)
(3)
(4)

15) What happens for a negative vector ?

(1) Direction reverses and unit remains unchanged

(2) Direction reverses and unit changes.
(3) Direction remains unchanged but unit changes.
(4) Neither direction reverses nor unit changes.

16) If a unit vector is represented by , then the value of 'c' is :-

(1) 1
(2)
(3)
(4)

17) An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube
increasing when the edge is 10 cm long?

(1) 900 cm3/s

(2) 920 cm3/s
(3) 850 cm3/s
(4) 950 cm3/s

18) If equation of a curve is y = 2x2 + 4, then find area of the curve with x axis from x = 1 to x = 2.

(1)
unit

(2)
unit

(3)
unit

(4)
unit

19) Average value of y = 2x + 3 in the interval 0 ≤ x ≤ 1 :-

(1) 1
(2) 5
(3) 3
(4) 4

20) If three vectors A, B and C are 12, 5 and 13 in magnitude such that , then the angle
between A and B is

(1) 60°
(2) 90°
(3) 120°
(4) None

CHEMISTRY

1)

How many grams of concentrated nitric acid should be used to prepare 250 mL solution of 2.0 M
HNO3?

(1) 31.5 g
(2) 54.0
(3) 38.5 g
(4) 90.0 g

2) 1M HCl and 2M HCl are mixed in volume ratio of 4 : 1. What is the final molarity of HCl solution ?

(1) 1.5 M
(2) 1 M
(3) 1.2 M
(4) 1.8 M

3) Which property is temperature independent ?

(1) Strength of solution

(2) Molality
(3) Molarity
(4) Normality

4) 46 gm of C2H5OH and 54 gm of water are mixed find mole fraction of C2H5OH ?

(1) 0.25
(2) 0.75
(3) 1.00
(4) 0.30

5) The unit of mole fraction of a compound in solution are

(1) mol kg–1

(2) mol liter–1
(3) g litre–1
(4) Unitless

6)

If 25g of solute is present in 50g of solution. Then calculate % w/W.

(1) 50%
(2) 25%
(3) 75%
(4) 33.3%

7) A solution of sulphuric acid having 0.49 gram of it dissolved in 250 cm3 of solution will have its
normality -
(Equivalent mass of sulphuric acid = 49)

(1) 0.01 N
(2) 0.02 N
(3) 0.03 N
(4) 0.04 N

8) Calculated the formality of NaCl solution, 5.85 g of which have been dissolved to form 250 ml of
the given solution.

(1) 0.2 F
(2) 0.4 F
(3) 0.8 F
(4) 0.1 F

9) If cathode ray experiment is performed using hydrogen, Helium and Oxygen gases in seperate

experiments. For which gas the ratio of cathode rays will be maximum.

(1) Hydrogen
(2) Helium
(3) Oxygen
(4) All have same value

10) Which of the following is not the possible path of cathode rays ejecting from the surface of
cathode?

(1)
(2)

(3)

(4)

11) Statement type question:

Statement-I: Thomson proposed that an atom consists of a uniform sphere of positive charge in
which the electrons are distributed more or less uniformly.
Statement-II: Thomson model is a static modal. It does not reflect the movement of electron.

(1) Statement-I is true only

(2) Statement-II is true only
(3) Both Statements are true
(4) Both Statements are false

12) Rutherford’s experiment on scattering of alpha particles showed for the first time that atom has
:-

(1) Electrons
(2) Protons
(3) Nucleus
(4) Neutrons

13) Number of protons, neutrons and electrons in the element 89Y231 is :-

(1) 89, 231, 89

(2) 89, 89, 242
(3) 89, 142, 89
(4) 89, 71, 89

14) In 7N14 if mass of electron were doubled and the mass of protons were halved, the atomic mass
would become approximately :-
(1) Halved
(2) Doubled
(3) Reduced by 25%
(4) Remain same

15) Atoms and are related to each other as :-

(1) Isotones
(2) Isoelectronic
(3) Isodiaphers
(4) Isosters

16) Two elements A and B are isotones. Their mass numbers are respectively 32 and 31 if atomic
number of A is 16, what will be the atomic number of B ?

(1) 15
(2) 30
(3) 32
(4) 31

17)

Which of the following set is not isosters?

(1)
(2)
(3)
(4)

18) Which of the following properties of Electromagnetic wave is incorrect according to

Electromagnetic wave theory?

(1) Electromagnetic wave do not require medium and can move in vacuum.
(2) Speed of the light is in vacuum.
Electric and magnetic fields are perpendicular to each other and parallel to the direction of
(3)
propagation of wave.
(4) None of the above.

19)

Radiocity broadcasts on a frequency of 5000 kHz. What is the wavelength of electromagnetic

radiation emitted by the transmitter?

(1) 50 m
(2) 60 m
(3) 70 m
(4) 80 m

20) The wavelength of a photon having energy ‘E’ is 6000 Å. Find wavelength of a photon having
energy ‘3E’.

(1) 18000 Å
(2) 6000 Å
(3) 2000 Å
(4) 3000 Å

MATHEMATICS

1) If x2 + 6x + 4 = 0 and 2x2 + px + q = 0 have both roots common, then find the value of p + q.

(1) 12
(2) 8
(3) 20
(4) 16

2) The graph of the quadratic polynomial y = ax2 + bx + c is as shown in the figure, then which of

the following is true?

(1) b2 – 4ac < 0

(2) b < 0
(3) a > 0
(4) c > 0

3) For what value of K, equation (K2 – 7K + 10)x2 + (K2 – 8K + 12)x + (K2 – 3K + 2) = 0 is an identity
?

(1) 1
(2) 2
(3) 5
(4) 6

4) If α, β are the roots of the equation 2x2 – 35x + 2 = 0, then find the value of .

(1) 35

(2)
(3)

(4) 70

5) If the roots of the equation x2 – 8x + a2 – 6a = 0 are real then the value of a will be

(1) –2 < a < 8

(2) –2 ≤ a ≤ 8
(3) 2 < a < 8
(4) 2 ≤ a ≤ 8

6) What will be the value of K, so that the roots of the equation x2 + 2Kx + 9 = 0 are imaginary ?

(1) –5 < K < 5

(2) –3 < K < 3
(3) 3 < K < 5
(4) – 5 < K < 3

7) Which of the following graph represents the expression f(x)=ax2+bx+c(a≠0)when a > 0, b < 0 & c
<0?

(1)

(2)

(3)

(4)

8) The value of p for which the equation 8x2 + 9px + 15 = 0 has equal roots is

(1)

(2)

(3) Both (1) and (2)

(4) None of these

9) If the equations x2 + x + 1 = 0 and x2 – ax + b = 0 have one root common, then 2a + 3b is equal to

(where a,b ∈ R)

(1) 1
(2) 2
(3) 3
(4) 4

10) If 3α2 – 6α + 5 = 0 & 3β2 – 6β + 5 = 0, α ≠ β, then value of is

(1)

(2)

(3)

(4)

11) If x is real, then the maximum value of is

(1) 5
(2) 6
(3) 9
(4) 1

12) If roots of equation are in ratio 3 : 2 then positive value of m is

(1)
(2)
(3)
(4) None of these

13) The minimum value of the expression is 'm' then the value of m2 is

(1)

(2)

(3)

(4)

14) If and are the roots of the equation x2 – 3x – 1 = 0 then the value of is

(1) 9
(2) 18
(3) 27
(4) 36

15) The roots of the quadratic equation , are

(1)
(2)

(3)

(4)

16) If α, β are the roots of the equation x2 + px + q = 0, then are the roots of the equation

(1) x2 – px + q = 0
(2) x2 + px + q =0
(3) qx2 + px + 1 =0
(4) qx2 – px + 1 =0

17) Find two natural numbers which differ by 3 and the sum of whose squares is 117.

(1) 6, 3
(2) 6, 9
(3) 8, 5
(4) 11, 8

18) Number of solutions of log4(x–1) = log2(x – 3) is

(1) 3
(2) 1
(3) 2
(4) 0

19) The sum of all the solutions to the equation 2log10x – log10(2x – 75) = 2

(1) 30
(2) 350
(3) 75
(4) 200

20) If , then x is equal to

(1) 125
(2) a2
(3) 25
(4) None of these
 ANSWER KEYS

PHYSICS

Q. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A. 4 2 4 3 3 4 3 2 3 3 3 2 2 4 1 2 1 4 4 2

CHEMISTRY

Q. 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
A. 1 3 2 1 4 1 4 2 4 4 3 3 3 3 3 1 3 3 2 3

MATHEMATICS

Q. 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
A. 3 2 2 3 2 2 2 3 1 2 3 1 2 4 3 4 2 2 4 3
 SOLUTIONS

PHYSICS

1)

2)

3)
θ > 90° (Negative slope)

4)
Parabola y = – kx2

5)

Centre (0, 0) and radius r = 10

(a2 = 100; )
x2 + y2 = a2
x2 + y2 = 100

6)
Length of semi-minor axis = 12

7)

Radius =
x2 + y2 – P = 0 ⇒ x2 + y2 = P
P = 5 as, x2 + y2 = (radius)2

8)
y + 8x + 22 = 0 ⇒ y = – 8x – 22
y-intercept = – 22 (∵ y = mx + C)

9)

Rate of increase of area w.r.t.

10)
∴ x = +1, –1
At x = 1, f(x) = (1)3 – 3(1) = –2 (minimum)
At x = –1, f(x) = (–1)3 – 3(–1) = 2 (maximum)

11) – =

12) R =
= 10N

13)

= –

14)
=

15) Direction reverses and unit remains unchanged.

16)

⇒ c2 = 1– [0.25 + 0.64]
⇒

17) = 3 cm/s

V = a3
 = 3 × (10)2 × 3
= 900 cm3/sec

18) Area =

19)

20) C2 = A2 + B2 + 2 AB cos θ
(13)2 = (12)2 + (5)2 + 2 . 12 . 5 . cos θ
cos θ = 0 ⇒ θ = 90°

CHEMISTRY

21)

Molarity =

w=

22)

[HCl] = = 1.2 M

23)

Molality is defined in terms of the mass of the solvent not its volume.
So molality of a solution does not change with temperature.
24)

Mole fraction of C2H5OH =

= = 1 mole

= = 3 mole

= = 0.25

25)

Mole fraction is a ratio.

So It is unitless expression.

26)

Mass of solute = 25 g
Mass of solution = 50 g

%
= 50%

27) Normality =

= N

28) Formality =

=
= 0.4 F

29)

In cathode ray experiment, the of particle (electron) forming cathode rays does
not depend on nature of gas.

30) Cathode rays have property that, they are emitted normal to surface of cathode.
31) Thomson proposed that an atom consists of a uniform sphere of positive charge in which
the electrons are distributed more or less uniformly.
Second statement is the drawback of Thomson model. It says it is a static model and does not
reflect the movement of electron.

32) Rutherford's experiment led to the discovery of nucleus.

33)

EA = 89Y231
Z

Number of protons = atomic number

= number of electrons
= 89
Number of neutrons = A – Z
= 231 – 89 = 142

34)

p n atomic mass
Old 7 7 7+7 = 14

New 7 3.5+7 = 10.5

% Change =

=– = 25% decrease

35) Isodiaphers n-p = 1 in both cases.

→ n = 7, p = 6 (7 – 6 = 1)
→ n = 9, p = 8 (9 – 8 = 1)

36)

Neutrons same
nA = nB
(A – Z)A = (A – Z)B
32 – 16 = 31 – ZB
 ZB = 15

37)

Isosters : Same number of atoms and electrons.

and CNO–
e– = 7 + 2 × 8 + 1 6+7+8+1
–
e = 24 e– = 22
Electrons are not same.

38) Electric and magnetic fields are perpendicular to each other and perpendicular to the
direction of propagation of wave.

39) =
= 60 m

40)

⇒ Å

MATHEMATICS

41) x2 + 6x + 4 = 0 and 2x2 + px + q = 0 have both roots common.

⇒ p = 12, q = 8
⇒ p + q = 12 + 8 = 20

42)

∵ roots are real & distinct

∴ D > 0 ⇒ b2 – 4ac > 0
And f(0) < 0 ⇒ c < 0
by graph a < 0

and <0⇒b<0 {∵ a < 0}

43) K2 – 7K + 10 = 0 ⇒ (K – 2) (K – 5) = 0
⇒ K = 2, 5
(K2 – 8K + 12) = 0 ⇒(K – 2) (K – 6) = 0
⇒ K = 2, 6
K2 – 3K + 2 = 0 ⇒ (K – 1) (K – 2) = 0
⇒ K = 1, 2
equation is an identity for K = 2

44) 2x2 – 35x + 2 =0

⇒α+β= , αβ = 1

45) x2 – 8x + a2 – 6a = 0
Roots are real,
⇒D≥0
⇒ (–8)2 – 4(1) (a2 – 6a) ≥ 0
⇒ 64 – 4 (a2 – 6a) ≥ 0
⇒ a2 – 6a – 16 ≤ 0
⇒ (a – 8) (a + 2) ≤ 0
⇒ –2 ≤ a ≤ 8

46) Roots of x2 + 2Kx + 9 = 0 are imaginary

⇒D<0
⇒ (2K)2 –4(9) < 0
⇒ K2 – 9 < 0 ⇒ (K – 3) (K + 3) < 0
⇒ –3 < K < 3

47)

a > 0 & c < 0 is satisfied by (2) only [∵ f(0) < 0 & a > 0]
Further in (2)

>0⇒b<0 [∵ a > 0].

48) 8x2 + 9px + 15 = 0 has equal roots.

⇒D=0
⇒ (9p)2 – 4(8)(15) = 0
⇒ 81p2 = 480

⇒ p2 =

⇒p=

49) As the roots are imaginary, so both the roots are common

⇒
a = –1, b = 1
2a + 3b = –2 + 3 = 1
50)

Given: 3α2 – 6α + 5 = 0 and 3β2 – 6β + 5 = 0 with α ≠ β

So, α and β are the roots of quadratic equation
3x2 – 6x + 5 = 0

Sum of roots = (for f(x) = ax2 + bx + c)

Product of roots =

α+β= =2

Now,

= ​

51)

method (1)
let y = 5 + 4x – x2
= –(x2 – 4x – 5)
y = –(x2– 4x + 4 – 4 – 5)
y = –(x – 2)2 + 9
y = 9 – (x – 2)2
ymax = 9 at x = 2

method (2)
if coff. (x2) < 0
then

=9

52) Let the roots be 3α, 2α.

Sum of roots = 5α = ....(1)

Product of roots = 6α2 = ....(2)

From equation (1)

Substituting in equation (2) we have,

∴ positive value of m is .

53) a = 4 > 0

minimum value is

then

54) x2 – 3x – 1 = 0

= 36.

55) ⇒ x2 – 3x + 1 = 0

⇒ ,

56) Given equation: x2 + px+q = 0

Also, α and β are the roots of the given equation.
Then, sum of the roots = α + β = –p
Product of the roots = αβ = q

Now, for roots , we have:

Sum of the roots =

Product of roots =

Hence, the equation involving the roots is as follows:

⇒ x2– + =0

⇒ x2 –
⇒ qx2 – px + 1 = 0

57) Let the natural numbers be x and x + 3.

∴ x2 + (x + 3)2 = 117
x2 + x2 + 6x + 9 = 117
⇒ 2x2 + 6x – 108 = 0
⇒ x2 + 3x – 54 = 0
⇒ (x + 9) (x – 6) = 0
⇒ x + 9 = 0, or x – 6 = 0
⇒ x = – 9, or x = 6
∴ One number = 6
and other number = 6 + 3 = 9.
∴ Numbers are 6 and 9.

58)
=
⇒ x – 1 = (x – 3)2
⇒ x2 – 7x + 10 = 0
⇒ x = 2, 5
but x = 2 is rejected
∴x=5
hence only 1 solution.

59) log10x2 – log10(2x – 75) = log10100

⇒ x2 – 200x + 7500 = 0
⇒ (x – 150) (x – 50) = 0
⇒ x = 50, 150
∴ sum = 200

60)