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Class 10: Science ⬇️
Class 10: Optional Mathematics ⬇️
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Class 9 Optional Mathematics
Group A (1 mark questions)
1. (a) Evaluate the following limit:
lim x → 4 ( 10 x 2 ) .
Solution
Here,
lim x → 4 ( 10 x 2 )
= 10 ⨯ 42
= 160.
= 10 ⨯ 42
= 160.
(b) Find the value of sec 210°. [1]
Solution
Here,
sec (210)° = sec (180 + 30)°
= - sec 30°;
sec (210)° = sec (180 + 30)°
= - sec 30°;
since we know sec (180° + θ) = - sec θ
= -2 .
= -
2. (a) Calculate:
lim x → 1 (5 - x 2 ).
Solution
Here,
lim x → 1 (5 - x 2 )
= 5 - 12
= 5 - 1
= 4.
= 5 - 12
= 5 - 1
= 4.
(b) Calculate:
lim x → 2 ( x 2 - 4)
Solution
Here,
lim x → 2 ( x 2 - 4)
= 22 - 4
= 4 - 4
= 0.
= 22 - 4
= 4 - 4
= 0.
Group B (2 marks each)
3. (a) Prove that:
1 - cos 4 B sin 4 B = 1 + 2 cot 2 A
Proof
To prove , 1 - cos 4 B sin 4 B = 1 + 2 cot 2 cot 2 A.
LS
= 1 - ( cos 2 A) 2 sin 4 A
= ( 1 - cos 2 A ) ( 1 + cos 2 A ) sin 4 A
= ( 1 - cos 2 A ) ( 1 + cos 2 A ) sin 4 A
= sin 2 A ( 1 + cos 2 A ) sin 2 A. sin 2 A
= 1 sin 2 A + cos 2 A sin 2 A
= cose c 2 A + cot 2 A
= cot 2 A + 1 + cot 2 A
= 1 + 2 cot 2 A
= RS.
Proved.
4. (a) Evaluate :
Solution
Here, lim x → k x 3 + k3 x + k
= k3+ k3 k + k
= 2k3 2k
= k2.
=
=
= k2.
5. (a) Find the sum and the degree of the sum: p (y) = y 6 - 3 y 4 and g (y) = y 4 + y 3 + 2 y 2 - 6
Solution
Given that,
p (y) = y 6 - 3 y 4
and g (y) = y 4 + y 3 + 2 y 2 - 6.
Now, the sum of the polynomials is given by,
p( y) + g (y) = ( y 6 - 3 y 4 ) + ( y 4 + y 3 + 2 y 2 - 6 )
= y 6 - 2 y 4 + y 3 + 2 y 2 - 6.
Here, the highest power of y in p(y)+g(y)= 6.
Hence, the degree of the sum is 6.
Now, the sum of the polynomials is given by,
Here, the highest power of y in p(y)+g(y)= 6.
Hence, the degree of the sum is 6.
6. (a) Find the sum and the degree of the sum: p( x) = 3 x 2 + 5 x - 2 and g( x) = - 3 x 2 - 5 x + 6
Given that,
p (x) = 3 x 2 + 5 x - 2 and g (x) = - 3 x 2 - 5 x + 6
Now, the sum of the polynomials is given by,
p (x) + g (x) = 3 x 2 + 5 x - 2 + - 3 x 2 - 5 x + 6
= 4.
Here, the sum has no x terms.
Hence, the degree of the sum is 0.
Now, the sum of the polynomials is given by,
= 4.
Here, the sum has no x terms.
Hence, the degree of the sum is 0.
7. (a) What matrix should be added to 3 - 5 4 6 1 - 1 2 - 2
Here, the given matrices are:
A =
Let, matrix B should be added to A to get X. Then,
A+B = X
Hence, the required matrix is
8. (a) Define a unit matrix. If A =
Solution
The square matrix in which diagonal elements are equal to 1 and non-diagonal elements are equal to 0 is known as unit matrix.
For example:
U =
Here, A =
Let, X be the additive inverse of the matrix A.
Then,
X + A = O where, O is the null matrix.
⇒ X +
⇒ X =
∴ X =
=
9. (a) Evaluate : sin 112 ° + cos 80 ° - sin 68 ° + cos 100 °
Solution
We have,
10. (a) Write the range and the inverse relation of the given relation f = {(2,1),(3,3),(4,5)}. [2]
Solution
Here,
f = {(2,1),(3,3),(4,5)}.
Hence, the range of f = {set of second elements of order pairs of f}
= {1, 3, 5}.
Now, interchanging the elements of order pairs of f,
We get, inverse of f = f -1 = {(1, 2), (3, 3), (5, 4)}.
11. (a) Find the sum and the degree of the sum:
p( y) = 3 y 5 - 4 y 3 + y 2 + y - 7 and g(y) = - 3 y 5 + 4 y 3 + 6 y 2 + 7 y + 2
[2]
[2]
Solution
Given that,
Given that,
Now, the sum of the polynomials is given by,
Here, the highest power of y in p(y)+g(y)= 2.
Hence, the degree of the sum is 2.
Group C (4 marks each)
12 Prove that:
[4]
Proof
LHS = 1 cosec A - cot A 1 s in A
= cosec 2 A - c ot 2 A c osec A - cot A cosec A
= cosec A + cotA ( cosec A - cotA ) cosec A - cot cot A c osec A
= cosecA + cotA cosecA
= cot A
RHS = 1 sin A 1 cosec A + cot cot A
= cosec A - cosecs 2 A - c ot 2 A cosec A - cot A
= cosecA - ( c osec A - cot A )
osec A + cot A
= LHS .
Proved .
13 If I is the unit matrix of order 2×2, find the matrix A such that 3A + 4I = 7 6 9 16
[4]
[4]
Solution
Given, I = 1 0 0 1
and 3A + 4I = 7 6 9 16
⇒ 3A + 4 1 0 0 1 7 6 9 16
⇒ 3A + 4 0 0 4 7 6 9 16
⇒ 3A = 7 6 9 16 4 0 0 4
⇒ 3A = 7 - 4 6 - 0 9 - 0 16 - 4
⇒ 3A = 7 - 4 6 - 0 9 - 0 16 - 4
⇒ 3A = 3 6 9 12
⇒ A =
∴ A = 1 2 3 4
14 If p(x) = x2 – 3x + 2 and q(x) = x3 - 6x2 + x + 1, find the product of p(x) and q(x) and degree of product. [4]
Solution
Given p(x) = x2 - 3x + 2
q(x) = x3 - 6x2 + x + 1
Now,
Product p (x) and q (x) is
p(x). q (x) = (x2 - 3x + 2) (x3 - 6x2 + x + 1 )
= x2 (x3 - 6x2 + x +1) - 3x (x3 - 6x2 + x + 1) + 2(x3 - 6x2 + x + 1)
= x5 - 6x4 + x3 + x2 - 3x4 + 18x3 - 3x2 - 3x + 2x3- 12x2 + 2x + 2
= x5 - 10x4 - 21x3 - 14 x2 – x + 2 .
Now degree of polynomial is 5.
15 If A (a, b) and B (b, a) lies in 6x - y = 1 and 2x - 5y = 5 respectively. Find the equation of AB. [4]
Solution
Here,
A(a, b) lies on line, 6x - y = 1
∴ 6 × a - b = 1
⇒ 6a - 1 = b............. (i)
And B (b, a) lies on 2x - 5y = 5 i.e,
2 × b - 5a = 5
⇒ 2(6a - 1) - 5a = 5
⇒ 12a – 2 - 5a = 5
⇒ 7a = 7
a = 1
Putting value of b from (i)
Put a = 1 in (i)
6 × 1 - 1 = b
⇒ b = 5
∴ The point A (a, b) = (1, 5) & B (b, a) = (5, 1).
Hence equation of line passing through A & B is
y - y1 = y 2 - y 1 x 2 - x 1 x 1
⇒ y - 5 = 1 (x - 1)
⇒ y - 5 = -1 (x - 1).
⇒ x + y = 6 is the required equation.
16 Using trigonometrical ratios of complementary angles prove that tan 1° tan 2° tan 3° ......... tan 89° = 1. [4]
Solution
Here,
LHS= tan 1° tan 2° tan 3° ...... tan 89°
= tan 1° tan 2° ...... tan 44° tan 45° tan 46° ...... tan 88° tan 89°
= (tan 1° ∙ tan 89°) (tan 2° ∙ tan 88°) ...... (tan 44° ∙ tan 46°) ∙ tan 45°
= {tan 1° ∙ tan (90° - 1°)} ∙ {tan 2° ∙ (tan 90° - 2°)} ...... {tan 44° ∙ tan (90° - 44°)} ∙ tan 45°
= (tan 1° ∙ cot 1°)(tan 2° ∙ cot 2°) ...... (tan 44° ∙ cot 44°) ∙ tan 45°,
[Since tan (90° - θ) = cot θ]
= (1)(1) ...... (1) ∙ 1, [Since tan θ ∙ cot θ = 1 and tan 45° = 1]
= 1
Therefore, tan 1° tan 2° tan 3° ...... tan 89° = 1.`
17 If cos4A + cos2A = 1, then prove that: tan4A + tan2A = 1 [4]
proof
Given,
cos 4 A + cos 2 A = 1
⇒ cos 4 A = 1 - cos 2 A
⇒ cos 4 A = sin 2 A.
Now,
LHS = tan 4 A + tan 2 A
= cos4 A + cos 2 A (Given cos 4 A + cos 2 A = 1)
= 1.
∴ tan 4A + tan 2A = 1.
Proved.
18 If A - B = 5 6 7 8 2 - 1 3 - 4
Solution
Given that,
A - B = 5 6 7 8
B = 2 - 1 3 - 4
Put value of B in eqn (i)
A - 2 - 1 3 - 4 5 6 7 8
A = 5 6 7 8 2 - 1 3 - 4
= 5 + 2 6 - 1 7 + 3 8 - 4
= 7 5 10 4
∴ A = 7 5 10 4
19 Find the coordinates of two points which trisect the line segment joining points (5, 3) and (8, 6). [4]
Solution
Here,
Let the given point be P(5, 3) & Q (8, 6)
For the trisection,
PB: AQ = 1:2. = m1: m2
∴ Co- ordinate of A is = m 1 x 2 + m 2 x 1 m 1 + m 2 m 1 y 2 + m 2 y 2 m 1 + m 2
= 1 × 8 + 2 × 5 1 + 2 1 × 6 + 2 × 3 1 + 2
= (6, 4)
But B is midpoint of A & Q.
So, co- ordinate of B is = (6 , 6 )
= (7, 5).
20 Prove that:
1 = 1 + ( cot A sin B ) 2 1 + ( cot C sin B ) 2
Answer
RHS = 1 + ( cotA sin B ) 2 1 + ( c otC sin B ) 2
= 1 + ( c os A sin A . sin B ) 2 1 + ( cos c sin c × s in B ) 2
= 1 + cosec 2 A c os 2 A sin 2 B 1 + cosec 2 A c os 2 c sin 2 B
= LHS.
Proved.
21 If 15 sin2θ + 2 cos2 θ = 7, find the value of tanθ. [4]
Answer
Given,
15 sin2θ + 2 cos2θ = 7
⇒ 15 sin2θ + 2 (1 - sin2θ) = 7
⇒ 15 sin2θ + 2 - 2sin2θ) = 7
⇒ 13 sin2θ = 5
Finally,
22 Find the equation of a straight line passing through the point (5, 6) and making Y- intercept is three times the X- intercept. [4]
Solution
Let a be the x- intercept and be the y- intercept then,
b = 3a
Now equation of straight line is
x
⇒ 3x + y = 3a .........(i)
Since this line is passes through point (5, 6).
So, 3 × 5 + 6 = 3a
⇒ a = 7.
Put a = 7 in equation (i) we get,
3x + y = 3 × 7.
∴ 3x + y = 21 is the required equation.
23 Study the following patterns:
a. Add two more patterns in the sequence.
b. Find the formula for the nth dots in the term of the sequence.
c. Find the 10th term of the sequence.
[4]
a. Add two more patterns in the sequence.
b. Find the formula for the nth dots in the term of the sequence.
c. Find the 10th term of the sequence.
[4]
25 Evaluate : cos π c 8 + cos 3 π c 8 + cos 5 π c 8 + cos 7 π c 8
Solution
26 Find the equation of the straight line which makes intercepts twice the intercepts made by the line 4x + 5y = 20 on the axes. [4]
Solution
27 If A = {1, 2, 3} and B = {3, 4, 5} , then find A × B and B × A and show in the arrow diagram. [4]
Solution
Given,
A = {1, 2, 3}
B = {3, 4, 5}
Now,
A × B = {1, 2, 3} × {3, 4, 5}
= {(1, 3) , (1,4), (1,5), (2,3), (2,4), (2,5), (3,3), (3,4), (3,5), }
Arrow Diagram
Again,
B × A = {3, 4, 5} × {1, 2, 3}
= {(3, 1) , (3, 2) , (3, 3), (4, 1) (4, 2) (4, 3) (5, 1), (5, 2), (5, 3)}
Arrow Diagram
Class 10 Optional Mathematics
Group A (2 marks questions)
1. (a) If x3 – 21x – 20 = (x + 1). Q(x), find Q(x) by using synthetic division method.
[2]
Solution
Here,
x3 – 19x – 30 = (x + 2). Q(x)
Comparing (x - a) with (x + 1), then a = -1.
Using synthetic division,
∴ Q(x) = x2 – x – 20.
2. (a) If f (x)=2x-3 and g(x) =x2 +1, find the value of fog(3). [2]
Solution
Here,
f (x)=2x-3 and g(x) =x2 +1.
We have, fog (x) (3) = f(g (3))
= f(32+1)
=f(10)
= 2×10-3
=17.
Thus, the value of f go (x) (3) is 17.
3. (a) Define sequence and series.
[2]
[2]
Solution
A list of numbers is definite order is called sequence.
eg. 3,5,7..............
When the terms of sequence are connected by addition and subtraction signs :
e.g. 2+4+6+8...............which is a series.
4. (a) If A = 1 - 1 3 - 3 2 - 5 2 - 5
Solution
Here,
A = 1 - 1 3 - 3 2 - 5 2 - 5
Now,
AB = (1 - 1 3 - 3 2 - 5 2 - 5
= 2 - 2 - 5 + 5 6 - 6 - 15 + 15
∴ AB = 0 0 0 0
which show that AB is a null matrix
5. (a) If the inverse of matrix A is A-1 = 3 2 5 4
Solution
Here,
A-1 = 3 2 5 4
We have, |(A-1)| = 3 2 5 4
= 12 – 10
= 2
We know that,
A = (A-1)-1 = ¹/|A⁻¹| Adj. (A-1)
= 1 2 4 - 2 - 5 3
∴ A = 2 - 1 - 5 2 3 2
Thus, the required matrix A = 2 - 1 - 5 2 3 2
6. (a) State factor theorem. Use factor theorem to determine whether x + 3 is a factor of the polynomial x3 – 8x + 3.
[2]
Solution
i)
Statement of factor theorem:
"If a polynomial p(x) is divided by (x – a) and f(a) = R = 0 then (x – a) is a factor of p(x).
ii)
Here, p(x) = x3 – 8x + 3 and d(x) = x + 3
Comparing (x + 3) with (x – a) then, a = -3
Now,
P(a) = P(3)
= (-3)3 – 8 × (-3) + 3
= -27 + 24 + 3
= 0 .
Since P(3) = 0 shows that (x + 3) is a factor of P(x).
7. (a) Prove that: cot 22°. cot 23° - cot 22° - cot 23° = 1 [2]
Solution
Here,
We know,
22°+ 23° = 45
Taking cot on both the sides;
cot(22°+ 23°) = cot 45°
⇒ cot22°.cot 23°-1 = cot 23°+cot 22°
∴ cot 22°cot 23° - cot 23°-cot 22° =1.
Proved.
8. (a) If two straight lines px+3y-12=0 and 4y-3x+7=0 are perpendicular to each other,find the value of p. [2]
Solution
Here,
Given equation line;
px+3y-12=0..................... (i )
and -3x+7=0 ........................ (ii).
Slope of line (i ); m1 = - coefficent of x coefficent of y p 3
Slope of line (ii); m2- coefficent of x coefficent of y - 3 4 = 3 4
Since the lines are perpendicular to each other.
m1 . m2 = -1
⇒ - p 3 × 3 4
⇒ p =4.
Thus, the value of p is 4.
Group B (4 marks questions)
9 Solve:
x3 -21x – 20 = 0 [4]
x3 -21x – 20 = 0 [4]
Solution
Here,
x3 -21x – 20 = 0 = P(x).
Factors of 20 are ±1, ±2, ±4, ±5, ±10, ±20
For, x = -1,
Since R = 0 so (x + 1) is a factor,
And Quotient Q(x) = x2 – x – 20.
Now, Factor × Q(x) = P(x)
⇒ (x + 1) (x2 – x – 20) = 0
⇒ (x + 1) (x2 – 5x + 4x – 20) = 0
⇒ (x + 1) {x(x – 5) + 4(x – 5)} = 0
⇒ (x + 1) (x – 5) (x + 4) = 0
Either, x + 1 = 0
∴ x = - 1
i.e. x – 5 = 0
∴ x = 5
i.e. x + 4 = 0
∴ x = - 4.
Thus, x = -4, -1, 5.
10 If the angle between the lines represented by 2x2 + kxy + 3y2 = 0 is 45°, find the value of k. [4]
Solution
Here,
Given equation of the line is 2x2 + kxy + 3y2 = 0
Angle between the lines is 45°,
Comparing given equation with ax2 + 2xhy + by2 = 0 then,
a = 2, 2h = k, i.e. h = 2 and b = 3.
Let θ
tan θ ± 2√ h 2 - ab a + b
i.e. tan 45 ° ± 2 √( k 2 ) 2 - 2 × 3 2 + 3
on solving we get
k2 = 49
Thus, the value of k is ±7.
11 If f = {x, 5x – 13}, g = {x, 2 } and and g-1(x) = f of(x), find the value of x. [4]
Solution
12 Find the sum of the first 8 terms of a geometrical series whose fourth term is 54 and common ratio is 3. [4]
Solution
Here,
Common ratio (r) = 3
Fourth term (t4) = 54
i.e. ar3= 54
get a = 2
Again,
Sum of 8 terms (S8) = 6560. (find yourself)
Thus, the sum of 8 terms is 6560.
13 The equation of a diagonal AC of given square ABCD is 3x - 4 y + 10 = 0 and the coordinates of vertex B are (4,-5), find the equation of diagonal BD.
[4]
[4]
solution
Here,
Equation of AC is
3x - 4y + 10 = 0 ....................(i)
Co- ordinates of B is (4,-5).
AC and BD are the perpendicular bisector to each other.
So,equstion of BD is
4x +3y + k = 0
It passes through B(4,-5)
So,
4 × 4 + 3 × (-5) = k
i.e. 16 -15 = k
∴ k = 1.
Putting k= 1 in equation (i) then,
4x + 3y = 1
which is the required equation of the line BD.
14 Find the sum of the first 9 terms of a geometric series whose third term and seventh term are 20 and 320 respectively. [4]
Solution
15 Maximize the objective function P = 9x + 7y under the given conditions:
x + 2y≤ 7, x - y ≥ 4, x ≥ 0 and y ≥ 0. [4]
x + 2y≤ 7, x - y ≥ 4, x ≥ 0 and y ≥ 0. [4]
Solution
Here,
Given constrains;
x + 2y≤ 7, x - y ≥ 4, x ≥ 0 and y ≥ 0.
The corresponding equation of x + 2y≤ 7 is x + 2y = 7.
x
|
1
|
3
|
y
|
3
|
2
|
Let (0, 0) be the test point in x + 2y≤ 7
0 + 2⨯0 ≤ 4
∴ 0 ≤ 4 (True)
Again, the corresponding equation of
x - y ≥ 4 is x - y = 4
x
|
0
|
4
|
y
|
-4
|
0
|
Let (0, 0) be a test point then,
0 -0 ≥ 4
0 ≥ 4 (False)
(x ≥ 0) ∩ (y ≥ 0) represents the first quadrant.
From the graph vertices of the feasible region are; A(4, 0), B(7, 0) and C(5,1).
Table:
Vertices
|
P = 9x + 7y
|
Remarks
|
A(4,0)
|
36
| |
B(7,0)
|
63
|
Maximum.
|
C(5,1)
|
52
|
Thus, the maximum value of P = 9x + 7y is 63 at (7, 0).
16 The 3rd and 7th terms of an arithmetic series are 18 and 30 respectively, find the sum of the first 20 terms. [4]
Solution
Here,
Third term (t3) = 18
Seventh term (t7) = 30
Find d=3 and a=12
Now,
Sn = 2 [2a + (n – 1)d]
S20 = 20 [2a + 19d]
= 20 (2 × 12 + 19 × 3)
= 10 × 81
= 810.
Thus, the sum of first 20th term is 810.
17 Solve by matrix method:
5x + 3y = 9 and 7x + 4y = 13 [4]
5x + 3y = 9 and 7x + 4y = 13 [4]
Solution
Here,
Given equation of lines are,
5x + 3y = 9 and
x + 4y = 13.
The matrix from of given equations are:
AX = B (say) ..............................i)
We have, |A| 5 3 7 4
We know that, A-1 = 1 |A| 4 - 3 - 7 5
= 1 - 1 4 - 3 - 7 5
We have, from(i),
X = A-1B
⇒ x y - 1 4 - 3 - 7 5 9 13
= - 1 36 - 39 - 63 + 65
= - 1 - 3 2 x y
= 3 - 2
Thus, x = 3 and y = -2 are the solutions.
18 Solve:
y3 – 6y2 + 11y – 6 = 0 [4]
y3 – 6y2 + 11y – 6 = 0 [4]
Solution
Here,
y3 – 6y2 + 11y – 6 = 0
The factors of 6 are: ±1, ±2, ±3 and ±6
Now, testing the factors by using the synthetic division method:
For, y = 1 using synthetic division,
∴ R = 0 shows that (y – 1) is a factor.
Q(y) = y2 – 5y + 6
Given polynomials = factor × quotient
So, y3 – 6y2 + 11y – 6 = 0
⇒ (y – 1) (y2 – 5y + 6) = 0
⇒ (y – 1) (y2 – 3y – 2y + 6) = 0
⇒ (y – 1) {y(y – 3) - 2(y – 3)} = 0
⇒ (y – 1) (y – 3) (y - 2) = 0
Either, y – 1 = 0………..(i)
Or, y – 3 = 0 …………(ii)
Or, y - 2 = 0 …………. (iii)
Now, from equation (i), (ii) and (iii);
y = 1, y = 3 or y = 2.
Thus, y = 1 or 3 or 2 is the solutions of the given equation.
19 Find the equation of the perpendicular bisector of the line segment joining the given points A (3, -7) and B (-5, 3). [4]
20 Solve:
6x3 + x2 – 19x + 6 = 0
[4]
6x3 + x2 – 19x + 6 = 0
[4]
Solution
Here,
Let,
p(x) = 6x3 + x2 – 19x + 6 = 0
The factors of 6 are: ±1, ±2, ±3 and ±6.
Now, testing the factors by using the synthetic division method:
At x = -2 using synthetic division,
Since remainder is 0. So (x + 2) is a factor.
Quotient Q(x) = 6x2 – 11x + 3 = 0
We have,
factor × Q(x) =P(x)
So, (x + 2) (6x2 – 11x + 3) = 0
⇒ (x + 2) (6x – 9x – 2x + 3) = 0
⇒ (x + 2) {3x(2x – 3) - 1(2x – 3)} = 0
⇒ (x + 2) (2x – 3) (3x - 1) = 0 .
Either, x + 2 = 0 ==> x = - 2
Or, 2x - 3 = 0 ==> x = 3/2
Or, 3x - 1 = 0 ==> x = 1/3.
Thus, x = -1 or 3/2 or 1/3 .
21 Solve:
2x3 + 6 - 3x2 – 11x = 0
[4]
2x3 + 6 - 3x2 – 11x = 0
[4]
Solution
Here,
2x3 + 6 - 3x2 – 11x = 0
⇒ 2x3 - 3x2 – 11x + 6 = 0.
The factors of 6 are: ±1, ±2, ±3 and ±6
At x = -2 using synthetic division,
R = 0, so (x + 2) is a factor and 2x2 - 7x + 3 is a quotient.
Now, 2x3 - 3x2 – 11x + 6 = 0
⇒ (x + 2) (2x2 - 7x + 3) = 0
⇒ (x + 2) (2x2 - 6x - x + 3) = 0
⇒ (x + 2) {2x(x – 3) – 1(x – 3)} = 0
⇒ (x + 2) (x – 3) (2x – 1) = 0.
Either, x + 2 = 0 Þ x = -2
⇒ x – 3 = 0 Þ x = 3
⇒ 2x – 1 = 0 Þ x = 1/2 .
Thus, the value of x are -2, 3 or 1/2.
22 Solve by matrix method:
2x + 3y = 3, x + 2y = 1 [4]
2x + 3y = 3, x + 2y = 1 [4]
Solution
Here,
given equations are
2x + 3y = 3 and
x + 2y = 1.
Writing in matrix form,
AX = B...........................................i)
We have |A| 2 3 1 2
Adj.(A) = 2 - 3 - 1 2
We know that, A-1 = ¹/|A|Adj.(A)
= 1 2 - 3 - 1 2
= 2 - 3 - 1 2
Now,from(i),
X = A-1B = 2 - 3 - 1 2 3 1 6 - 3 - 3 + 2
∴ x y 3 1
Thus, x = 3 and y = -1 is the solution.
23 Maximize the finction Z = 3x + 4y subjected to the constraints
x + y ≤ 4, x ≥ 0 & y ≥ 0. [4]
x + y ≤ 4, x ≥ 0 & y ≥ 0. [4]
Solution
24 A(3,2) , B (1,-1) and C ( 5,-5) are the vertices of a triangle ABC. Find the equation of a straight the passing through the centroid of ∆ABC and parallel to the side BC. [4]
Solution
Here,
Vertices of ∆ABC are; A(3,2) , B (1,-1) and C ( 5,-5)
Centroid of ∆ABC (x,y) = 3 ,3
= (3, -4 3
Slope of line BC is; m1 = y 2 - y 1 x 2 - x 1
= 5
= -1
Now, Equation of line passing through (3,- 4 3
y - y 1 m ( x - x 1 )
⇒ y + 4/3 = -1 (x - 3)
⇒ 3 y + 4 3
⇒ 3y + 4 = -3x + 9
⇒ 3x + 3y = 5.
Thus, the equation of line is 3x + 3y = 5.
25 Find the values of x and y that make z(x , y) = 2x + 4y maximum subject to the conditions shown below and find the value of z at these values of x and y:
x ≥ 0
y ≥ 0
y ≤ x + 1
4y+x ≤ 10
y - x ≥ - 3. [4]
x ≥ 0
y ≥ 0
y ≤ x + 1
4y+x ≤ 10
y - x ≥ - 3. [4]
Solution
Point A is the intersection of lines x = 0 and y = 0. Solution A(0 , 0).
Point B is is the intersection of lines x = 0 and y = x + 1. Solution B(0 , 1)
Point C is is the intersection of lines y = x + 1 and 4 y + x = 10. Solution C(6/5 , 11/5)
Point D is is the intersection of lines 4y + x = 10 and y - x = - 3. Solution D(22/5 , 7/5)
Point E is is the intersection of lines y - x = -3 and y = 0. Solution E( 3 , 0).
Plotting in graph:
The vertices were also found (in the same example) to be:
A(0 , 0), B(0 , 1), C(6/5 , 11/5), D(22/5 , 7/5), E( 3 , 0)
We now evaluate function z(x , y) = 2 x + 4 y at all 5 vertices of the feasible set.
- at A: the x and y coordinates of A are x = 0 and y = 0.
Substitute x and y by 0 and 0 respectively in the linear function z = 2 x + 4 y to obtain z(A) = 2 (0) + 4 (0) = 0
- at B: the x and y coordinates of B are x = 0 and y = 1.
Substitute x and y by 0 and 1 respectively in the linear function z = 2 x + 4 y to obtain z(B) = 2 (0) + 4 (1) = 4
- at C: the x and y coordinates of C are x = 6/5 and y = 11/5.
Substitute x and y by 6/5 and 11/5 respectively in the linear function z = 2 x + 4 y to obtain z(B) = 2 (6/5) + 4 (11/5) = 11.2
- at D: the x and y coordinates of D are x = 22/5 and y = 7/5.
Substitute x and y by 22/5 and 7/5 respectively in the linear function z = 2 x + 4 y to obtain z(B) = 2 (22/5) + 4 (7/5) = 14.4
- at E: the x and y coordinates of D are x = 3 and y = 0.
Substitute x and y by 3 and 0 respectively in the linear function z = 2 x + 4 y to obtain z(B) = 2 (3) + 4 (0) = 6
The maximum value of z occur at vertex D with coordinates x = 22/5 and y = 7/5. Hence the solution of the given problem is Z has a maximum value at x = 22/5 and y = 7/5 and this value is
Group 4
26 Maximize the objective function F = 4x + 3y under the given conditions:
x + y ≤ 6, 2x - y ≤ 3, x ≥ 0 and y ≥ 0. [4]
x + y ≤ 6, 2x - y ≤ 3, x ≥ 0 and y ≥ 0. [4]
Solution
Here,
Given constrains;
x + y ≤ 6, 2x - y ≥ 3, x ≥ 0 and y ≥ 0.
The corresponding equation of x + y ≤ 6 is x + y = 6.
x
|
0
|
6
|
y
|
6
|
0
|
Let (0, 0) be the test point in x + y ≤ 6
or, 0 + 0 ≤ 6
∴ 0 < 6 (True)
Again, the corresponding equation of
2x - y ≤ 3 is 2x - y = 3
x
|
3
|
0
|
y
|
3
|
-3
|
Let (0, 0) be a test point then,
2x - y ≤ 3
or, 2 × 0 - 0≤ 3
or, 0≤ 3 (True).
(x ≥ 0) ∩ (y ≥ 0) represents the first quadrant.
Graph:
Table:
Vertices
|
F = 3x+4y
|
Remarks
|
A(0,1.5)
|
6
| |
B(3, 3)
|
21
| |
C(0,6)
|
24
|
Maximum.
|
O(0,0)
|
0
|
Therefore, function F = 4x + 3y has maximum value 24 at point C(0, 6).
mark questions)
1 What is rusting? [1]
Answer
The formation of red or brown scales or powder on the surface of iron when it comes in contact with moist air is called rusting.
2 Hydrogen is generally not found in air in free state. Why? [1]
Answer
Since hydrogen is more reactive gas; it reacts with other elements very easily, so it is generally not found in free state.
3 What is hydrogen? [1]
Answer
Hydrogen is the lightest gaseous element situated on first period and group IA of the periodic table.
4 Define 1 Watt power? [1]
Answer
1 Watt power is defined as the power of doing 1Joule work in 1 second.
5 How is hydrogen manufactured from methane ? [1]
Answer
In the industrial sector, hydrogen is produced by the reaction between hydrocarbon and water vapor.
For industrial purpose, hydrogen gas is prepared by reacting methane with water vapour at very high temperature in the presence of nickel powder, which acts as catalyst in a reaction.
CH₄ + H₂O →
For industrial purpose, hydrogen gas is prepared by reacting methane with water vapour at very high temperature in the presence of nickel powder, which acts as catalyst in a reaction.
CH₄ + H₂O
6 What is meant by mechanical energy? [1]
Answer
The energy possessed by a body by virtue of its position, configuration or motion is called mechanical energy.
7 Why are sodium and potassium metals stored in kerosene? [1]
Answer
Sodium and potassium highly reactive metals so they are stored in kerosene as they can easily catch fire even when they come in contact with air.
8 What is the refraction of a sound wave? [1]
Answer
The phenomenon of bending of sound waves when it passes from one medium to another medium is called refraction.
9 What is meant by acceleration due to gravity? [1]
Answer
The uniform acceleration produced in a freely falling body due to the pull of gravity of the earth or the other planet is known as acceleration due to gravity.
10 What do you understand by the term velocity ratio in a pulley? [1]
Answer
Velocity ratio in a pulley is defined as the number of pulleys used in the pulley system.
11 What is meant by refractive index of glass is 1.5? [1]
Answer
It means that the speed of light in glass is 1.5 times slower than the speed of light in vacuum.
12 What is transformation of energy? [1]
Answer
Transformation of energy is the process in which one form of energy is converted into another form.
13 What is Newton’s 2nd law of motion? [1]
Answer
Newton’s 2nd law of motion states that “the rate of change of momentum of a body is directly proportional to the resultant force on the body”.
14 What is static electricity? [1]
Answer
The electricity due to the collection of charge on a body is called static electricity.
15 What is duplet state? Give one example. [1]
Answer
The state of having two electrons in the K shell of an atom is duplet state.
E.g.Helium.
E.g.Helium.
Group 2 (2 marks questions)
16 A car starting from rest maintains an acceleration of 0.2 m/s² up to two kilometers. What would be the final velocity of the car and how much time would it take to cover 1.6 km? [2]
Answer
Here,
Initial Velocity (u) = 0 m/s
Distance Covered (s) = 1.6 km = 1.6 × 1000 = 1600 m
Acceleration (a) = 0.5 m/s²
Final velocity (v) = ?
Time (t) = ?
According to the formula:
v² = u² + 2as
= 0² + 2(0.5)1600
= 1600 v
= √1600
= 40 m/s²
Again,
v - u = at
i.e. t =v - u a = 40 - 0 0.5 = 80 s
Time taken =80 s
Initial Velocity (u) = 0 m/s
Distance Covered (s) = 1.6 km = 1.6 × 1000 = 1600 m
Acceleration (a) = 0.5 m/s²
Final velocity (v) = ?
Time (t) = ?
According to the formula:
v² = u² + 2as
= 0² + 2(0.5)1600
= 1600 v
= √1600
= 40 m/s²
Again,
v - u = at
i.e. t =
Time taken =80 s
17 Write the name of the SI unit used to measure the moment in SI system. Why it is called derived unit? [2]
Answer
The SI unit is used to measure the moment in SI system is Nm.
We have,
Moment = Force ⨉ Displacement (d)
= Force (f) ⨉Displacement (d)
= Mass(m) х acceleration(a) ⨉ dispalcement (d) [ Force = mass х acceleration]
= kg ⨉ m/s2 ⨉ m ⨉ m
= kgm3s-2
It is a derived unit because it is derived from fundamental units second, kilogram and metre.
We have,
Moment = Force ⨉ Displacement (d)
= Force (f) ⨉Displacement (d)
= Mass(m) х acceleration(a) ⨉ dispalcement (d) [ Force = mass х acceleration]
= kg ⨉ m/s2 ⨉ m ⨉ m
= kgm3s-2
It is a derived unit because it is derived from fundamental units second, kilogram and metre.
18 Write the electronic configuration of magnesium and calcium based on sub-shell. [2]
Answer
Electronic configuration of magnesium:
Mg (12) =1s2,2s2,2p6, 3s2
Electronic configuration of calcium:
Ca (20) = 1s2, 2s2, 2p6, 3s2, 3p6, 4s²
Mg (12) =1s2,2s2,2p6, 3s2
Electronic configuration of calcium:
Ca (20) = 1s2, 2s2, 2p6, 3s2, 3p6, 4s²
19 Mention an four things that can be known from the molecular formula of a compound. [2]
Answer
The things that we can know from molecular formula are as followings:
I. A molecular formula represents the percentage of each element present in the given compound.
II. It represents one molecule of the substance.
III. It tells about the number of atoms of each element present in one molecule of the compound.
IV. It helps to know about the molecular weight of the substance.
I. A molecular formula represents the percentage of each element present in the given compound.
II. It represents one molecule of the substance.
III. It tells about the number of atoms of each element present in one molecule of the compound.
IV. It helps to know about the molecular weight of the substance.
20 suggest any four methods of reducing noise.
Answer
Methods for reducing noise are as followings:
Cinema halls, saw mills and other noise producing factories should be constructed outside residential area.
Public transport should be used and use of private vehicles should be reduced as far as possible.
Plantation of vegetation should be done in living and working areas.
Sound absorbing materials such as glass, wool, carpet, etc. reduce noise. Hence, they should be used properly.
Cinema halls, saw mills and other noise producing factories should be constructed outside residential area.
Public transport should be used and use of private vehicles should be reduced as far as possible.
Plantation of vegetation should be done in living and working areas.
Sound absorbing materials such as glass, wool, carpet, etc. reduce noise. Hence, they should be used properly.
21 What is an unbalanced force? [2]
Answer
The sum of the total number of forces that can change the state or position of anybody is called an unbalanced force.
22 Express the SI unit of force in terms of fundamental units. [1]
Answer
The SI unit of force is 'Newton'.
As force is the product of mass and acceleration, its SI unit 'Newton' is obtained by multiplying the SI units of mass and acceleration.
The SI unit of mass is kilogram (kg) and the SI unit of acceleration is metre per squared second (m.s-2).
Thus, the unit of force; Newton in terms of a fundamental unit is kg.m.s-2.
As force is the product of mass and acceleration, its SI unit 'Newton' is obtained by multiplying the SI units of mass and acceleration.
The SI unit of mass is kilogram (kg) and the SI unit of acceleration is metre per squared second (m.s-2).
Thus, the unit of force; Newton in terms of a fundamental unit is kg.m.s-2.
23 What are exothermic and endothermic reactions? [2]
Answer
In endothermic reactions, a substance absorbs energy in the form of heat and undergoes a chemical reaction. An example is the heat energy absorbed during a chemical change involved in cooking, like baking a cake.
In exothermic reactions, heat energy is released. Explosion of fireworks is an example of exothermic chemical change. The explosion releases energy in the form of light.
In exothermic reactions, heat energy is released. Explosion of fireworks is an example of exothermic chemical change. The explosion releases energy in the form of light.
24 Define chemical reaction. What are the types of chemical reaction. [2]
Answer
The process by which a chemical change takes place by addition, decomposition or displacement of the atoms or molecules of the matter into new substances is called chemical reaction.
The types of chemical reactions are:
i) Addition/ Combination/ Synthesis reaction
ii) Substraction/ Decompostion reaction
iii) Displacement Reaction
iv) Acid-base reaction/ Neutralization reaction
The types of chemical reactions are:
i) Addition/ Combination/ Synthesis reaction
ii) Substraction/ Decompostion reaction
iii) Displacement Reaction
iv) Acid-base reaction/ Neutralization reaction
25 Write the name of devices that convert that the energies below:
I. Electrical energy to heat energy.
II. Light energy into electrical energy.
III. Sound energy into electrical energy.
IV. Electrical energy into light energy.
[2]
I. Electrical energy to heat energy.
II. Light energy into electrical energy.
III. Sound energy into electrical energy.
IV. Electrical energy into light energy.
[2]
Answer
| Energy conversion | Device |
| Electrical energy to heat energy. | Electric heater. |
| Light energy into electrical energy. | Photocell (the part of a solar panel that converts light into electrical energy). |
| Sound energy into electrical energy. | Microphone. |
| Electrical energy into light energy. | Electric bulb. |
26 Differentiate between MKS and SI system. [2]
Answer
The difference between MKS and SI system are as follows:
| MKS | SI |
| 1. It is a metric system of measurement. | 1. It is a revised and redefined extended version of the MKS system. |
| 2. It involves three fundamental quantities - length, mass and time. | 2. It involves seven fundamental quantities. |
27 Write two differences between energy and power. [2]
Answer
Differences between energy and power are as followings:
| Energy | Power |
| 1. It is the capacity of doing work. | 1. It is the rate of doing work or the rate of conversion of energy. |
| 2. Its SI unit is joule. | 2. Its SI unit is watt. |
28 What are the factors that bring out chemical change? [2]
Answer
The main factors that bring out a chemical change are:
• Heat
• Electricity
• light
• Simple contact
• Contact by solution
• Pressure
• Catalyst
• Heat
• Electricity
• light
• Simple contact
• Contact by solution
• Pressure
• Catalyst
Group 3 (3 marks questions)
29 Write the energy changes that take place when the light is produced by a torchlight using dry cells. Show numerical that how many times the kinetic energy of a body is changed if the mass is increased by two times and velocity is increased by four times. [3]
Answer
When light is produced by a torchlight using dry cells, chemical energy is converted into light energy.
Here,
Given,
Initially,
Mass = m
Velocity = v
Initial kinetic energy (K.E1) =1 mv2 .......... (i)
Again,
The mass is increased by two times and velocity is increased by four times.
So,
Mass = 2m
Velocity = 4v
Final kinetic energy (K.E2) =1 2m × (4v)2
=1 2m × 16v2
= 32 (1 mv2)
Putting the value from equation (i)
K.E2 = 32K.E1
The kinetic energy of a body is increased by 32 times if the mass is increased by two times and velocity is increased by four times.
Here,
Given,
Initially,
Mass = m
Velocity = v
Initial kinetic energy (K.E1) =
Again,
The mass is increased by two times and velocity is increased by four times.
So,
Mass = 2m
Velocity = 4v
Final kinetic energy (K.E2) =
=
= 32 (
Putting the value from equation (i)
K.E2 = 32K.E1
The kinetic energy of a body is increased by 32 times if the mass is increased by two times and velocity is increased by four times.
30 Riya is standing on the road by carrying a load of 100 N, what is work done by her? Why? Milan has a weight of 50 kg. If he can climb the ladder of 4 m height in 10 seconds, calculate his power. [3]
Answer
Riya is standing on the road by carrying a load of 100 N and she has not covered any displacement, i.e. displacement (d) = 0 m. Work done by a force acting on a body is defined as the product of force and displacement of the body in direction of the force. As displacement is 0 m, the product of force and displacement is also 0. So, she has not done any work or the work done by her is 0 J.
Here,
Given,
Mass (m) = 50 kg
Height (h) = 4 m
Time (t) = 10 s
Acceleration due to gravity (g) = 9.8 m/s2
Power (P) =?
Here, the work is done against the gravity. Thus the work is done (W) is given by,
W = mgh
Again,
We have,
P =W t mgh t 50 = 196 watt
Hence, his power is 196 watt.
Here,
Given,
Mass (m) = 50 kg
Height (h) = 4 m
Time (t) = 10 s
Acceleration due to gravity (g) = 9.8 m/s2
Power (P) =?
Here, the work is done against the gravity. Thus the work is done (W) is given by,
W = mgh
Again,
We have,
P =
Hence, his power is 196 watt.
31 SI system is extended form of MKS system. Why? [3]
Answer
In MKS system, only three quantities, i.e. length, mass and time is used as fundamental quantities. But in SI system, four more quantities temperature, current, luminous intensity ad amount of substance are also included as fundamental quantities. It shows that, the units of length, mass and time are same in both systems. Thus, SI system is the extended form of MKS system.
32 Write three factors affecting the resistance of conductor wire. Mention the relation of the resistance and these factors. [3]
Answer
Three factors affecting the resistance of conductor wire are as followings:
Length of the conducting wire.
A cross-sectional area of the conducting wire.
The resistivity of the conducting wire.
We know that the resistance of a conductor is directly proportional to its length and inversely proportional to its cross-sectional area. If 'l' is the length of a wire, 'A' is the cross-sectional area of the wire and 'R' is its resistance, then
R α l............. (i)
R αl A
From equation (i) and (ii) we have
R αl A
i.e. R = ρl A
Where ρ (Rho) is the resistivity or specific resistance of the material. Equation (iii) is the required expression.
Length of the conducting wire.
A cross-sectional area of the conducting wire.
The resistivity of the conducting wire.
We know that the resistance of a conductor is directly proportional to its length and inversely proportional to its cross-sectional area. If 'l' is the length of a wire, 'A' is the cross-sectional area of the wire and 'R' is its resistance, then
R α l............. (i)
R α
From equation (i) and (ii) we have
R α
i.e. R = ρ
Where ρ (Rho) is the resistivity or specific resistance of the material. Equation (iii) is the required expression.
33 Study the given diagram and answer the following questions.
What is the mistake made in the diagram?
Identify the north and South Pole of the magnet.
The magnetic lines of forces are closer near the poles, why? [3]
What is the mistake made in the diagram?
Identify the north and South Pole of the magnet.
The magnetic lines of forces are closer near the poles, why? [3]
Answer
Magnetic lines of force will never intersect. But in figure, two magnetic lines of force are intersecting.
Magnetic lines of force start from the North Pole and reach the South Pole of a magnet externally. So A is the North Pole and B is the South Pole of given magnet.
The magnetic lines of forces are closer near the poles, because magnetic field is strong in the poles.
Magnetic lines of force start from the North Pole and reach the South Pole of a magnet externally. So A is the North Pole and B is the South Pole of given magnet.
The magnetic lines of forces are closer near the poles, because magnetic field is strong in the poles.
34 What happens when:-
a) Hydrogen is passed over hot ferric oxide.
b) Hydrogen combines with chlorine.
c) Iron reacts with hydrochloric acid.
d) Hydrogen burns with oxygen.
[3]
a) Hydrogen is passed over hot ferric oxide.
b) Hydrogen combines with chlorine.
c) Iron reacts with hydrochloric acid.
d) Hydrogen burns with oxygen.
[3]
Answer
a. Fe2O3 + 3H2 → 2Fe
b. H2 + Cl2 → 2HCl
c. Fe + 2HCl → FeCl2 + H2
d. When hydrogen is mixed in air, it burns with a pop sound and forms water. But pure hydrogen gas burns with light flame when mixed with oxygen to form water.
2H2 + O2 → 2H2O
b. H2 + Cl2 → 2HCl
c. Fe + 2HCl → FeCl2 + H2
d. When hydrogen is mixed in air, it burns with a pop sound and forms water. But pure hydrogen gas burns with light flame when mixed with oxygen to form water.
2H2 + O2 → 2H2O
Group 4 (4 marks questions)
35 State and prove second law of motion.
[4]
[4]
Answer
Newton's Second Law states that: " The acceleration produced in a body is directly proportional to the force applied and inversely proportional to the mass of the body":
Proof:
Let us consider a body of mass 'm' produces an acceleration of 'a' when the force 'f' is applied to it.
Then acceleration produced in a body is directly proportional to the force applied.
i.e. a∝ F ....................(i)
Again acceleration produced in a body is inversely proportional to the mass of the body.
i.e. a ∝ 1/m ........................(ii)
Combining equation (i) and (ii),
we get
a ∝ F/m
F ∝ ma
F = kma .....................(iii)
Where k is a proportionality constant, it's value depends upon the value of f, m and a. According to the definition of 1N force, we have
F = 1N
m = 1kg
a = 1 m/s2
Then,
∴ k = 1
Replacing the value of k in equation (iii), we get
F = 1. Ma
∴ F = ma
Proof:
Let us consider a body of mass 'm' produces an acceleration of 'a' when the force 'f' is applied to it.
Then acceleration produced in a body is directly proportional to the force applied.
i.e. a∝ F ....................(i)
Again acceleration produced in a body is inversely proportional to the mass of the body.
i.e. a ∝ 1/m ........................(ii)
Combining equation (i) and (ii),
we get
a ∝ F/m
F ∝ ma
F = kma .....................(iii)
Where k is a proportionality constant, it's value depends upon the value of f, m and a. According to the definition of 1N force, we have
F = 1N
m = 1kg
a = 1 m/s2
Then,
∴ k = 1
Replacing the value of k in equation (iii), we get
F = 1. Ma
∴ F = ma
36 A book has a dimension of 24cm x 15cm x 4cm and weight 1.5kg. It consists of 500 sheets of papers. Find:
i. the volume of the book
ii. thickness of each sheet
iii. the area of each sheet [4]
i. the volume of the book
ii. thickness of each sheet
iii. the area of each sheet [4]
Answer
Given,
Length of book (l) = 24cm
Breadth of book (b) = 15cm
Height of the book (h) = 4cm
Now,
Volume of the book = l x b x h = 24cm x 15cm x 4cm = 1440 cm3
Again,
Thickness of book of 500 pages = height of book = 4cm
Thickness of 1 page of the book = 4 / 500 cm = 0.008 cm= 8 x 10-3 cm
Area of each sheet = l x b = 24cm x 15cm = 360cm2
Length of book (l) = 24cm
Breadth of book (b) = 15cm
Height of the book (h) = 4cm
Now,
Volume of the book = l x b x h = 24cm x 15cm x 4cm = 1440 cm3
Again,
Thickness of book of 500 pages = height of book = 4cm
Thickness of 1 page of the book = 4 / 500 cm = 0.008 cm= 8 x 10-3 cm
Area of each sheet = l x b = 24cm x 15cm = 360cm2
37 Differentiate fundamental and derived units. [4]
Answer
The differences between the fundamental and derived units are as follows:
| Fundamental units | Derived units |
| 1. Fundamental units are all those units which are independent of any other unit (including themselves). | 1. Derived units are all those units which are obtained by using one or more fundamental units. |
| 2. Fundamental units cannot be broken down into elementary level; in fact, these are elementary units. | 2. Derived units can be broken down to its elementary level (composed of fundamental units). |
| 3. There are only seven fundamental units. | 3. There are a large number of derived units. |
| For example : • Mass (Kilogram, kg) • Time (Second, s) • Temperature (Kelvin, K) | For example: • Velocity (m/s) • Acceleration (m2/s) • Momentum (kg-m/s) |
38 Answer the following question after observing the figure .
a) Which gas is going to be prepared in the figure?
b) What are the mistakes in the arrangement? Correct the figure.
c) Write the formula equation for the preparation of the gas?
d) What happens when a burning match stick is held near the gas jar?
e) Give any three uses of the gas?
f) What happens when concentrated sulphuric acid is used instead of dilute sulphuric acid in it? [4]
a) Which gas is going to be prepared in the figure?
b) What are the mistakes in the arrangement? Correct the figure.
c) Write the formula equation for the preparation of the gas?
d) What happens when a burning match stick is held near the gas jar?
e) Give any three uses of the gas?
f) What happens when concentrated sulphuric acid is used instead of dilute sulphuric acid in it? [4]
Answer
a. Hydrogen gas
b. The end of thistle funnel must be under the acid in Woulfe’s bottle and delivery tube should not be dipped under the acid in Woulfe’s bottle.
c. Granulated Zn + H2SO4 → ZnSO4 + H2
d. When a burning matchstick is brought near a test tube full of hydrogen gas then the fire goes out producing pop sound forming blue flume.
e. The main uses of hydrogen are as follows:
• It is used as a reducing agent in industries and laboratory.
• It is used to fill in the balloons as it is the lightest gas but it is mixed with helium as hydrogen is highly inflammable.
• It is used to prepare chemical fertilizers.
f. If conc. sulphuric acid is used instead of dilute sulphuric acid, suphur dioxide is produced instead of hydrogen gas.
Zn + conc.H2SO4 → ZnO + H2O + SO2
Instead when the dilute H2SO4 is used, the following reaction occurs to form the hydrogen gas.
Zn + dil. H2SO4 → ZnSO4 + H2
b. The end of thistle funnel must be under the acid in Woulfe’s bottle and delivery tube should not be dipped under the acid in Woulfe’s bottle.
c. Granulated Zn + H2SO4 → ZnSO4 + H2
d. When a burning matchstick is brought near a test tube full of hydrogen gas then the fire goes out producing pop sound forming blue flume.
e. The main uses of hydrogen are as follows:
• It is used as a reducing agent in industries and laboratory.
• It is used to fill in the balloons as it is the lightest gas but it is mixed with helium as hydrogen is highly inflammable.
• It is used to prepare chemical fertilizers.
f. If conc. sulphuric acid is used instead of dilute sulphuric acid, suphur dioxide is produced instead of hydrogen gas.
Zn + conc.H2SO4 → ZnO + H2O + SO2
Instead when the dilute H2SO4 is used, the following reaction occurs to form the hydrogen gas.
Zn + dil. H2SO4 → ZnSO4 + H2
Class 10 Science
Group A (1 mark questions)
1 What is long-sightedness? [1]
Answer
The defect of vision in which a person can see the distant object clearly but cannot see distinctly the near objects is called long sightedness.
2What is a fuse? [1]
Answer
A piece of wire made of a material of high resistance and a very low melting point is called Fuse.
3 Define humus. [1]
Answer
Humus is a large group of natural organic compound found in the soil formed from the chemical and biological decomposition of plants and animals residues and from the synthetic activities of microorganisms.
4 What is kaolin? [1]
Answer
The pure white soil which is used to make ceramics is called kaolin.
5 What is eddy current? [1]
Answer
The circulating current produced within any metal when magnetic flux linked with it is changed is called eddy current.
6 What is far point of the eyes? [1]
Answer
The farthest point from where an object can be seen clearly is called a far point of the eyes.
7 Define synthetic fiber. [1]
Answer
Artificial fibre like nylon, polyester, arcylene and olefin that are prepared by a chemical process are called synthetic fibres.
8 What is saponification? [1]
Answer
The process of making soap by the hydrolysis of fats and oils with alkali is called saponification.
9 What are fertilizers? [1]
Answer
The substances which are added in the soil to increase the fertility of soil are called fertilizers.
10 What is electromagnetic induction? [1]
Answer
The method of production of the current by changing the magnetic flux linked with coil or conductor is called electromagnetic induction.
11 What is homologous series? [1]
Answer
A series of compounds which have general formula, general methods of preparation and similar chemical properties are called homologous series.
12 What is an alkane? [1]
Answer
The hydrocarbon in which all the four valencies of a carbon atom are used as a single covalent bonds is called an alkane.
13 What is a functional group? [1]
Answer
A functional group is an atom or group of atoms which defines the structure of a particular family of organic compounds.
14 Define mortar. [1]
Answer
Mortar is the mixture of cement, sand and water which is used as plastering material and to join bricks, stone etc.
15 What is the principle of caloriemetry? [1]
Answer
The amount of heat gained is equal to the amount of heat lost, this relation is called principle of caloriemetry.
Group B (2 marks questions)
16
Calculate the resistance of a device if 0.3 A of current flows through it when the potential difference across it is 9 V.
[2]
Solution
Here,
Current (I) = 0.3 A
Potential difference (V) = 9 V
Now,
V = I
On solving we get:
R = 30
17 If three bulbs of 40 W each are used 5 hours daily and a heater of 1500 W is used at the rate of 2 hours daily. How much electricity is consumed in one month? [2]
Solution
Here,
Given,
For bulb,
Power (P1) = 40 watt = (40 / 1000) kW = 0.04 kW
Number (N1) = 3
Time (t1) = 5 х 30 hrs. = 150 hrs.
Energy consumptions of 30 days = P1 х N1 х t1= 0.04 х 3 х 150= 18 kWh
For heater,
Power (P1) = 1500 watt = (1500 / 1000) kW = 1.5 kW
Number (N2) = 1
Time (t2) = 2 х 30 hrs. = 60 hrs.
Energy consumptions of 30 days = P2 х N2 х t2 = 1.5 х 1 х 60 = 90 kWh
Again,
Total energy consumption for 1 month = (18 + 90) kWh = 108 unit
Hence, units of electricity consumed in 30 days are 108 units.
Given,
For bulb,
Power (P1) = 40 watt = (40 / 1000) kW = 0.04 kW
Number (N1) = 3
Time (t1) = 5 х 30 hrs. = 150 hrs.
Energy consumptions of 30 days = P1 х N1 х t1= 0.04 х 3 х 150= 18 kWh
For heater,
Power (P1) = 1500 watt = (1500 / 1000) kW = 1.5 kW
Number (N2) = 1
Time (t2) = 2 х 30 hrs. = 60 hrs.
Energy consumptions of 30 days = P2 х N2 х t2 = 1.5 х 1 х 60 = 90 kWh
Again,
Total energy consumption for 1 month = (18 + 90) kWh = 108 unit
Hence, units of electricity consumed in 30 days are 108 units.
18 Differentiate between hard glass and soft glass. [2]
Answer
The difference between the hard and soft glass is tabulated below:
| Soft Glass | Hard Glass |
| 1. It is produced by heating sodium carbonate and calcium carbonate with silica. | 1. It is obtained by heating potassium carbonate and calcium carbonate with silica. |
| 2. It is used for making window panes, mirror etc | 2. It is used for making heat-resistant equipment. |
19 When electric current is passed through a nichrome wire, it becomes hot, but if same current is passed through copper wire it does not become hot, why? [2]
Answer
Heat produced is directly proportional to the resistance generated in the wire in an electrical circuit, to the time and to the square of the current.
Nichrome wire has very high resistance as compared to copper wire. Due to the high resistance present in nichrome wire, large amount of current is converted into heat energy. Hence, when electric current is passed through a nichrome wire, it becomes hot, but if same current is passed through copper wire it does not become hot.
Nichrome wire has very high resistance as compared to copper wire. Due to the high resistance present in nichrome wire, large amount of current is converted into heat energy. Hence, when electric current is passed through a nichrome wire, it becomes hot, but if same current is passed through copper wire it does not become hot.
20 Write two-two examples of nitrogen fertilizers and phosphorus fertilizer. [2]
Answer
The two examples of nitrogen fertilizers are:
a. Ammonium Nitrate (NH4NO3)
b. Ammonium Sulphate (NH4)2SO4
The two examples of phosphorus fertilizers are:
a. Ammonium Phosphate (NH4)3PO4
b.Calcium super phosphate Ca(H2PO4)2.2CaSO4
a. Ammonium Nitrate (NH4NO3)
b. Ammonium Sulphate (NH4)2SO4
The two examples of phosphorus fertilizers are:
a. Ammonium Phosphate (NH4)3PO4
b.Calcium super phosphate Ca(H2PO4)2.2CaSO4
21 Why is detergent called soapless soap? [2]
Answer
Detergents are called the soap less soap because they have cleansing property in water like soap but their chemical nature is different from soap.
22 Write the disadvantages of plastics. [2]
Answer
The disadvantages of plastics are:
1. It non-biodegradable
2. It causes soil water and water pollution.
3. They are harmful to aquatic creatures.
1. It non-biodegradable
2. It causes soil water and water pollution.
3. They are harmful to aquatic creatures.
23 "The use of alternating current would be limited, if transformer was not invented." Prove the statement with two clues. [2]
Answer
The use of alternating current would be limited, if transformer was not invented due to the following reasons:
I. If transformer were not invented, then sending current over long distance while keeping constant voltage would cause loss of power in form of heat.
II. To use alternating current, there would be need of generation/power plant everywhere. But due to the invention of transformer, there is no need of establishing power plant everywhere.
I. If transformer were not invented, then sending current over long distance while keeping constant voltage would cause loss of power in form of heat.
II. To use alternating current, there would be need of generation/power plant everywhere. But due to the invention of transformer, there is no need of establishing power plant everywhere.
24 How is plaster of paris prepared? Write its uses. [2]
Answer
Plaster of Paris (CaSO4. ½H2O) is prepared by heating gypsum (CaSO4.2H2O). It is used for making casts and patterns and for plastering the walls, pillars and ceilings and to make ornamental patterns on them.
25 Write any two differences between polythene and Bakelite. [2]
Answer
| Polythene | Bakelite |
| It becomes soft on heating. | It becomes hard on heating. |
| It can be remolded. | It cannot be remolded. |
26 What is polyethene? [2]
Answer
Polythene is a polymer made from ethene (CH2=CH2). It is one of the most commonly used materials. It is a soft plastic, which softens on heating. It is used for making bottles, buckets, and pipes, as covering for electrical wires and cables and as film for making bags.
27 What is Bakelite? [2]
Answer
Bakelite (Phenol-formaldehyde resin) is made by reacting phenol and formaldehyde. It is hard and quite a strong material. It is used for making combs, electrical switches, and plugs and for making handles of many kitchen utensils and electrical appliances like pans, pressure cookers, electric irons, kettles, and toasters.
28 Define plastics and enlist their uses. [2]
Answer
Plastics are the artificial materials made by the polymerization of carbon containing monomers.
The uses of plastics are:
1. They are used as shopping bags.
2. They can be used as insulators in wires.
3. They are used as waterproof.
4. It is used in surgical cases.
The uses of plastics are:
1. They are used as shopping bags.
2. They can be used as insulators in wires.
3. They are used as waterproof.
4. It is used in surgical cases.
Group C (3 marks questions)
29 What is NPK fertilizer? [1]
Answer
The chemical fertilizer containing all plants nutrients ie. nitrogen, phosphorous and potassium is called NPK fertilizer. NPK fertilizer is also called the complete fertilizer.
30 Differentiate between soap and detergent. [3]
Answer
The difference between soap and detergent is tabulated below:
| Soap | Detergent |
| 1. It is a sodium or potassium salt of fatty acid. | 1. It is a synthetic petrochemical obtained from hydrocarbon. |
| 2. It is not suitable for washing purpose when the water is hard. | 2. Synthetic detergent can be used for washing ever when the water is hard. |
| 3. Soaps are biodegradable. | 3. It is not biodegradable. |
31 Describe the processes involved during the manufacture of cement. [3]
Answer
Limestone and clay are mixed in definite proportion and ground to a fine powdery state. This dry powder is used as such or mixed with water to form a paste and heated in a rotary kiln (a type of furnace). It is slowly made to pass through the kiln wherein limestone and clay combine chemically and form a mixture of calcium silicate, CaSiO3 and calcium aluminate, CaAl2O3. This mixture is in the form of small greenish black or grey-coloured hard balls known as clinkers. These clinkers are allowed to cool down and then ground to very fine powder. To this powder, 2-3% gypsum is added and the mixture is again ground to obtain a grayish coloured powder, which is cement. It is then packed in airtight bags to exclude the moisture. Gypsum is added to decrease the setting time of cement.
32 What are organic fertilizers? How do the chemical fertilizers trigger pollution? [3]
Answer
Organic fertilizers are made by decaying plants and waste products of animals. They do not cause chemical pollution. Chemical fertilizers are made by using different chemicals. They cause chemical pollution.
33 Write short notes on:
Borosilicate glass [2]
Borosilicate glass [2]
Answer
It is sodium aluminium borosilicate. It can withstand rapid heating and cooling without breaking. It is used for making kitchenware and laboratory apparatus. It is sold under the trade names Borosil and Pyrex.
34 Derive Principle of caloriemetry [3]
Answer
“When two bodies of different temperatures are brought together, the body at higher temperature looses heat energy whereas the body at lower temperature gains heat energy. The process will continue until they reach at thermal equilibrium.”
i.e Heat lost = Heat gained
This process continues until both bodies are at same temperature.
Let the mass of a body at higher temperature is m1, its specific heat capacity is s1 and the temperature of body is t1.
Similarly, let the mass of a body at lower temperature is m2, its specific heat capacity is s2 and the temperature of body is t2.
If both bodies are brought together, then let the final temperature of the mixture be t.
If dt and dt’ are difference in temperature of bodies at higher and lower temperature respectively, then their values will be
dt = (t1 – t2)
and
dt’ = (t2 – t1)
So, the amount of heat energy lost by a body at higher temperature will be
Heat lost = m1s1dt
Similarly, the amount of heat energy gained by a body at lower temperature will be
Heat gained = m2s2dt’
According to the principle of caloriemetry,
Heat lost = Heat gained
i.e. m1s1dt = m2s2dt’
i.e. m1s1 (t1 – t2) = m2s2 (t2 – t1)
Group D (4 marks questions)
35 Three liquids A, B, C of equal mass are kept in the same type of container and placed in the sun for 30 minutes. The increase in temperature is given in the table. Study the table and answer the following questions:
I. Which liquid has the highest specific heat capacity? Why?
II. If equal mass of all the liquid at same temperature cooled, which one will cool faster? Why?
[4]
| Liquid | Increase in temperature |
| A | 17°C |
| B | 30°C |
| C | 23°C |
II. If equal mass of all the liquid at same temperature cooled, which one will cool faster? Why?
[4]
Answer
The specific heat capacity of a body is defined as the amount of heat required to raise the temperature of unit mass of that body by 1°C or 1K.
Different substances have different specific heat capacities. The bodies, which have more specific heat capacity, change their temperature slowly and those, which have less specific heat capacity, change their temperature faster.
I. The same amount of heat is applied to all three liquids for the equal time interval and liquid A has the lowest increase in temperature. As we know that, the substance which has more specific heat capacity, change their temperature slowly. So, the liquid A has more specific heat capacity.
II. If the equal mass of all three liquids at the same temperature is cooled, liquid C will cool faster, because the same amount of heat is applied to all three liquids for the equal time interval and liquid C has the highest increase in the temperature. That means liquid C has the lowest specific heat capacity and we know that substance which has less heat capacity, change their temperature faster.
Different substances have different specific heat capacities. The bodies, which have more specific heat capacity, change their temperature slowly and those, which have less specific heat capacity, change their temperature faster.
I. The same amount of heat is applied to all three liquids for the equal time interval and liquid A has the lowest increase in temperature. As we know that, the substance which has more specific heat capacity, change their temperature slowly. So, the liquid A has more specific heat capacity.
II. If the equal mass of all three liquids at the same temperature is cooled, liquid C will cool faster, because the same amount of heat is applied to all three liquids for the equal time interval and liquid C has the highest increase in the temperature. That means liquid C has the lowest specific heat capacity and we know that substance which has less heat capacity, change their temperature faster.
36 Study the diagram and answer the following questions.
I. Complete the following ray diagram.
II. Calculate the power of this lens.
III. Write any two nature of image formed by it.
[4]
I. Complete the following ray diagram.
II. Calculate the power of this lens.
III. Write any two nature of image formed by it.
[4]
Answer
I.
II. Here,
Given,
Focal length (f) = 2 cm
Power (p) =?
We have,
p =1
=1 2 100 cm
= 50 dioptre (D)
III. The nature of image thus formed is,
real
inverted
II. Here,
Given,
Focal length (f) = 2 cm
Power (p) =?
We have,
p =
=
= 50 dioptre (D)
III. The nature of image thus formed is,
real
inverted
37 Calculate the specific heat capacity of the alloy of which a pressure cooker of mass 1.5 kg is made of if the quantity of heat necessary to raise its temperature by 60°C is 81kJ. [4]
Answer
s = 900 Jkg-1 °C-1. ( Find yourself )
38 What is chemical pollution? What are the causes of environmental pollution? [4]
Answer
The environment pollution caused by various chemicals like insecticides, chemical fertilizers, synthetic cleanser and other industrial chemicals is called chemical pollution.
The causes of environmental pollution are:
1. The use of insecticides and fertilizers.
2. Household waste and plastics.
3. Smoke from vehicles and factories.
4. Synthetic cleanser.
The causes of environmental pollution are:
1. The use of insecticides and fertilizers.
2. Household waste and plastics.
3. Smoke from vehicles and factories.
4. Synthetic cleanser.
....