Kathmandu BernHardt HS School
b) If C=30°, B=45°, c= 6√2, solve the triangle.
Balkhu, Kathmandu
First Term Examination- 2070
Set: I
7. a) Prove that: (b+c-a)(cot 2 + cot 2) = 2a cot 2
Faculty: Science
Subject: Mathematics
Grade: XI
Time: 3hrs
Group A[5×3×2=30]
B
FM: 100
PM: 35
c) In any ABC, prove that cos
=
b−c cosA
s(s−a)
√
.
bc
sinA
2. a) In any ABC, show that a−c cos B = sin B .
b) Show that the quadratic equations 3x²-8x+4=0 and 4x²-7x-2=0 have one root
in common.
c) For what value of a will the equation x²-(3a-1)x+2(a²-1)=0 have equal roots?
1+i
in
1−i
3. a) Express the complex number √
a−ib
then
a+ib
b) If x+iy=
prove that x²+y²=1.
c) Find the distance between the parallel lines 5x+12y+65=0 and 5x+12y+39=0.
y
b
1
1
1
= 1, prove that a² + b² = p² .
b) Evaluate:
c)
5. a)
b)
c)
6. a)
lim
x→∞
x²−5x+6
x²−2x+1
lim 1−cos7x
Evaluate:
x → 0 x²
Find the derivative of y=(x²+a²) (x²-a²)
Find the derivative of y=sec²(sin(ax+b)).
Find the acute angle between the lines x-3y-6=0 and y =2x+5.
Group B [5×2×4=40]
For any two real numbers x and y, prove that
(i) |x+y|≤|x|+|y|
(ii) |x-y|≥|x|-|y|.
b) If α and  are the roots of the equation x²-px+q=0, find the equation whose
roots are α²−1 and ²α−1 .
p
q
n
√p + √ l = 0 .
b) If p and p' be the length of perpendiculars from origin upon the straight line
whose equations are xsec+ycosec=a and xcos-ysin=acos2, prove that
4p²+p′2 =a².
9. a) Find the angle between the two straight lines y=m1x+c1 and y=m2x+c2. Also,
find the condition if the lines are (i) parallel and (ii) perpendicular
b)
Evaluate:
lim xcosθ−θcosx
x−θ
x→θ
10. a) Find from first principles, the derivative of √2𝑥 + 3.
b) Find from first principles, the derivative of cos²x.
Group C [5×6=30]
the form of a+ib.
4. a) If p be the length of perpendicular dropped from the origin on the line
A
8. a) If the roots of the equation ln²+nx+n=0 be in the ratio p:q prove that √ +
q
1. a) Let A=[-2, 1] and B=[0, 3]. Perform the operation (i) AB (ii) A-B.
b) If A and B are any two subsets of a universal set U, then prove that A −
(BC)=(A-B) (A-C) .
A
2
C
x
+
a
11. State and prove sine law. Also, in any ABC, prove that b²sin2C+c²sin2B=2absinC.
12. If the quadratic equations ax²+bx+c=0 and bx²+cx+a=0 have a common root,
then either a+b+c=0 or a=b=c. Also, if the roots of the equation ax²+bx+c=0 be
in the ratio of 3:4, prove that 12b²=49ac.
13. Define continuity of a function at a point. A function f(x) is defined as follows:
2𝑥 − 3, 𝑓𝑜𝑟 𝑥 < 2
𝑓𝑜𝑟 𝑥 = 2
𝑓(𝑥) = { 2
3𝑥 − 5 𝑓𝑜𝑟 𝑥 > 2
Is the function f(x) continuous at x=2? If not, how can the function be made
continuous at x=2?
14. What are the cube roots of unity? Find the square roots of 5+12i.
15. Find the equations of the bisectors of the angles between the pair of lines
represented by ax²+2hxy+by²=0.
The End
Kathmandu BernHardt HS School
A
Balkhu, Kathmandu
First Term Examination- 2070
Set: II
Faculty: Science
Subject: Mathematics
Grade: XI
Time: 3hrs
Group A[5×3×2=30]
FM: 100
PM: 35
1. a) Let A=[-3, 0] and B=(-1, 3), then perform the operation (i) B-A (ii) AB.
̅̅̅̅̅̅=A
̅ B
̅.
b) If A and B are subsets of a universal set U, then prove that AB
(s−b)(s−c)
A
c) In any ABC, prove that sin 2 = √
c−b cosA
bc
.
cosB
2. a) In any ABC, show that b−c Cos A = cosC .
b) Find the value of p if the quadratic equations 4x²+px-12=0 and 4x²+3px-4=0
may have one root in common.
c) If the equation x²+2(k+2)x+9k=0 has equal roots, find k.
1−i
3. a) Express the complex number (1+i)² in the form of a+ib.
b) If (3-4i)(x+iy)=3√5, show that 5x²+5y²=9.
c) Find the distance between the parallel lines 3x-4y+100=0 and 6x+8y+25=0.
4. a) If p be the length of perpendicular drawn from the point (a,b) on the line
x
y
1
1
1
+ = 1, prove that + = .
a
b
a²
b²
p²
lim 9x²+2x+7
b) Evaluate:
x → ∞ 3x²+4x+5
lim sinx−siny
c) Evaluate:
x → y x−y
5. a) Find the derivative of y=sin5(cos(ax²+b))
b) Find
dy
dx
when y =
1
x−√a²+x²
.
c) Find the acute angle between the lines x-√3y = a and √3x − y = b.
Group B [5×2×4=40]
6. a) For any two real numbers x and y, prove that
(i) |x+y|≤|x|+|y|
(ii) |x-y|≥|x|-|y|.
b) If a=2, b=√2, c=√3 + 1, solve the triangle.
B
C
7. a) Prove that: (a+b+c)(tan 2 + tan 2 ) = 2c cot 2
b) If α and  are the roots of the equation 2x²-3x-5=0, form a equation whose
1
1
roots are 2α +  and 2 + α .
α

q
8. a) If α and  are the roots of px²+qx+q=0, prove that √ + √ + √ = 0 .

α
p
b) Show that the product of the perpendiculars drawn from the two points
x
y
(±√a² − b², 0)upon the line cosθ + sinθ = 1 is b².
a
b
9. a) Find the length of perpendicular from the point (x1, y1) on the line
Ax+By+C=0.
lim xsinθ−θsinx
b) Evaluate:
x−θ
x→θ
10. a) Find from first principles, the derivative of √2 − 3𝑥 .
b) Find from first principles, the derivative of sin²x.
Group C [5×6=30]
11. State and establish the cosine law. Also, in any ABC, prove that
a(bcosC-ccosB)= b²-c².
12.If the equations x²+px+q=0 and x²+qx+p=0 have a common root, prove that
either p=q or p+q+1=0. Also, if one root of the equation x²-px+q=0 be twice the
other, show that 2p²=9q.
13. What do you understand by the limit of a function? Let a function f(x) be
defined by
2 − 𝑥² 𝑓𝑜𝑟 𝑥 < 2
𝑓(𝑥) = { 3 𝑓𝑜𝑟 𝑥 = 2
2𝑥 − 6 𝑓𝑜𝑟 𝑥 > 2
Verify that the limit of the function f(x) exists at x=2. Is the function f(x)
continuous at x=2? If not, why? State how can you make it continuous?
14. What do you mean by complex number? If a and b are real numbers, then prove
that the complex number (a, b) can be written as a+ib. And find the square
roots of 3-4i.
15. Prove that the homogeneous equation of second degree ax²+2hxy+by²=0,
always represents a pair of straight lines passing through origin. Also, find the
angle between them.
The End