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Published on: 29/10/2025
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1.
ABCD is a cyclic quadrilateral in which AB II CD. If ㄥD=70, find all the remaining angles.
2.
The probability of winning a game is \(\frac{1}{3}\) less than the twice of losing the game. Find probability of winning the game.
3.
In the figure, AB||CD, find the value if z, \(\angle DNM\) and \(\angle CNM.\)
4.
If a transversal intersects two parallel lines, then the bisectors of any pair of alternate angles are parallel. prove it.
5.
In figure, \(\angle DOB=87^o\) and \(\angle COA=82^o\). If \(\angle BOA=35^o\), then find \(\angle COB\) and \(\angle COD\)
6.
Simplify: \(\left( x+\frac { 1 }{ x } \right) \left( x-\frac { 1 }{ x } \right) \left( { x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } } \right) \left( { x }^{ 4 }+\frac { 1 }{ { x }^{ 4 } } \right) \)
7.
How many planes can be made to pass through
(i) Three collinear points.
(ii) Three non-collinear points.
8.
Find the value of the polynomial p(x) = x3 - 3x2 - 2x + 6 at x = \(\sqrt { 2 } \)
9.
In the given figure:
(a) name the two triangles on the same base AB and between the same parallels.
(b) name a triangle and a parallelogram on the same base BC and between the same parallels.
10.
In figure given below l || m Before of RQB and DRQ intersect at P find the measure of RPQ
11.
Plot the following points A(5,0), B(-1,2), C(2,-2), D(0,4), E(-3,-3), F(0,-1)
12.
The lengths of perpendiculars PM and PN drawn from a point P, on x-axis and y-axis are of 3 and 2 units respectively.Find the coordinates of points P,M and N.
13.
Evaluate 105 x106 without direct-multiplying.
14.
Divide p(x) by g(x), where p(x) = x + 3x2 – 1 and g(x) = 1 + x.
15.
If x = 5 and y = 2, find the value of \((i)\quad \left( { x }^{ y }+{ y }^{ y } \right) \ (ii)\ { \left( { x }^{ y }+{ y }^{ y } \right) }^{ -1 }\)
16.
Simplify \(\sqrt { 2 } (\sqrt { 6 } -\sqrt { 8 } )+\sqrt { 3 } (\sqrt { 27 } -\sqrt { 6 } )\)
17.
Express \(18.\overline{48} \) in the form of p/q, where p and q are integers, \(q\neq 0\) .
18.
Write three rational numbers between -2/5 and -1/5.
19.
Find two rational numbers between 3/4 and 5/9.
1.
Since sum of the opposite pairs of angles in a cyclic quadrilateral is 180°.
Hence ㄥB+ㄥD=180°
ㄥB=180°-70°
=110°
Again, AB II CD and AD is its transversal, so
ㄥA+ㄥD=180°
ㄥA=180°-70°
=110°
and ㄥA+ㄥC=180°
⇒ 110°+ㄥC=180°
ㄥC=180°-110°
=70°
2.
Let the probability of winning a game = p
and probability of losing a game = q
we know that p + q = 1...(i)
According to questions,
\(p=2q-\frac{1}{3}\)
\(\Rightarrow \) 6q - 3p = 1...(ii)
On solving (i) and (ii), we get
\(q=\frac{4}{9}\ and\ p=\frac{5}{9}\)
\(\therefore \) Probability of winning game \(=\frac{5}{9}\)
3.
3z-42o=2z+13o (Alternate interior angles)
z=42o+13o
z=55o
\(\angle DNM=2z+13^o=110^o+13^o\)
=123o
\(\angle CNM=180^o-\angle DNM\)(Linear pair)
=180o-123o=57o
4.
\(\angle BMN=\angle CNM(alternate \angle 's)\)
\(\frac{1}{2}\angle BMN=\frac{1}{2}\angle CNM\)
x=y
(But x and y are alternate angles)
MP || NQ
5.
Given, \(\angle COA=82^o\)
\(\Rightarrow \angle COB+\angle BOA=82^o\)
\(\Rightarrow \angle COB+35^o=82^o(\because \angle BOA=35^o)\)
\(\Rightarrow \angle COB=82^o-35^o=47^o\)
Similarly, \(\angle DOB=\angle DOC+\angle COB\)
\(\Rightarrow 87^o=\angle DOC+47^o\)
\(\Rightarrow \angle DOC=87^o-47^o=40^o\)
6.
\(\left( x+\frac { 1 }{ x } \right) \left( x-\frac { 1 }{ x } \right) \left( { x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } } \right) \left( { x }^{ 4 }+\frac { 1 }{ { x }^{ 4 } } \right) \)
\(=\left( { x }^{ 2 }-\frac { 1 }{ { x }^{ 2 } } \right) \left( { x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } } \right) \left( { x }^{ 4 }+\frac { 1 }{ { x }^{ 4 } } \right) \)
as (a+b)(a-b)=a2-b2
\(=\left( { x }^{ 4 }-\frac { 1 }{ { x }^{ 4 } } \right) \left( { x }^{ 4 }+\frac { 1 }{ { x }^{ 4 } } \right) \)
\(=\left( { x }^{ 8 }-\frac { 1 }{ { x }^{ 8 } } \right) \)
7.
(i) Infinite, if they are collinear.
(ii) Only one, if they are non-collinear points.
8.
p(x) = x3 - 3x2 - 2x + 6
Then, p(\(\sqrt { 2 } \)) = (\(\sqrt { 2 } \))3 -3(\(\sqrt { 2 } \))2 - 2(\(\sqrt { 2 } \)) + 6
=2\(\sqrt { 2 } \) - 6 - 2\(\sqrt { 2 } \) + 6
= 0
9.
\((a)\Delta DAB,DCAB\)
\(\\ (b)\Delta DAB,||gm\ ABCD\)
10.
\(90^{ 0 }\)
11.
12.
(2,3), (2,0), (0,3)
13.
11130
14.
Quotient = 3x - 2
Remainder = 1
15.
(i) 57
(ii) \(\frac { 1 }{ 3129 } \)
16.
\(5+2\sqrt { 3 } -3\sqrt { 2 } \)
17.
610/33
18.
-7/20,-3/10,-1/4
19.
101/144, 47/72
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