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## Diploma 1st year Maths Important Question

Course | BSC (Bachelor of Science) |

Subject | Maths |

Content | Important Questions |

Provide by | Hindijankaripur |

Official Site | Hindijankaripur.com |

Telegram | https://t.me/studentcafeindia |

**CHAPTER 1: PARTIAL FRACTIONS (3M)**

Resolve 3xx-2(x+1) into partial fractions.

Resolve 2x+1x-1(2x+3) into partial fractions

Resolve 1x-4(x+9) into partial fractions

Resolve 1x-8(x+1) into partial fractions

Resolve 4x-2(x+5) into partial fractions

Resolve x-4x-2(x-3) into partial fractions

Resolve x-4x-2(x-7) into partial fractions

Resolve 1x-5(x-7) into partial fractions

Resolve 1x-3(x+1) into partial fractions

Resolve 2x+3x-1(x-2) into partial fractions

**CHAPTER 2: MATRICES (3M)**

If A=1 2 -2 3 and B=4 -1 3 2 Then find the value of 3A+5B=?

If A=1 2 3 4 and B=3 8 7 2 and 2X+B=A ,then find X ?

If A=cos sin -sin cos then show that A.AT=AT.A=I Where I is the identity matrix of order 2.

If ω is a cube root of unity, prove that1 2 2 1 2 1 =0.

If 2 -3 5 1 Then find A+AT=?

If A=1 3 1 0 and B=3 4 2 3 Then find the value of AB and BA.

If A=1 0 0 2 3 4 5 -6 x and *det*(*A*) = 45 then find the value of *x?*

If A=1 -3 2 2 1 -3 4 3 3 and B=2 -2 -4 -1 3 4 1 2 3 then find 2*A *+ 3*B.*

Using Laplace expansion, evaluate the determinant 8 2 5 2 -1 9 7 4 12 .

If A=1 2 3 4 and B=3 8 7 2 show that (*A* + *B* )* ^{T}* =

*A*+

^{T}*B*

*.*

^{T}If A=2 1 2 1 4 1 1 3 2 find the matrix *A*^{2}.

**CHAPTER 3:****ESSAY QUESTION Each question (5M).**

If A=2 0 1 2 1 3 1 -1 0 * Compute ,A*^{2}* *−* *5*A *+* *6*I *, where* I *is the unit Matrix of order 3*3.

Solve the system of equation x+y+z=3,x+2y+3z=4,x+4y+9z=6 using Gauss-Jordan method.

*Show that *a+b+2c a b c b+c+2a b c a c+a+2b =2(a+b+c)3.

Solve the following system of equations by using Cramer’s rule x+ 2*y* − *z* = −3, 3*x* + *y* + *z* = 4, x−* y *+* *2*z *=* *6.

Solve the equation *x *+ *y *+ 4*z *= 6, 3*x *+ 2*y − *2*z *= 9 and 5*x *+ *y *+ 2*z *= 13 by Crammer’s Rule.

Find the adjoint of the matrix A=2 1 2 1 4 1 1 3 2 .

Solve the equations *x *− *y *+ *z *= 2, 2*x *− 3*y *+ 4*z *= −4 and 3*x *+ *y *+ *z *= 8 by Cramer’s rule.

Find the inverse of the matrix 1 -3 2 2 1 -3 4 3 3 .

Solve the following equations by using Cramer’s rule :* x *− *y *+ *z *= 2, 2*x *+ 3*y *− 4*z *= − 4, and 3*x *+ *y *+ *z *= 8.

If A=2 3 4 5 7 9 -2 1 21 and B=4 0 5 1 2 0 0 3 1 then verify that (*AB *)* ^{T }*=

*B*.

^{T }A^{T }Solve the following equations by using Cramer’s rule : 2*x *− 3*y *+ *z *= − 1, *x *+ 4*y *− 2*z *= 3 and 4*x *− *y *+ 3*z *= 11.

Solve the following equations by using Cramer’s rule :.* x *+ 2*y *− *z *= −1, 3*x *− *y *− 2*z *= 5 and *x *− *y *− 3*z *= 0.

Solve the following equations by using Matrix inversion method * x *+* *2*y *−* z *= −* *3, 3*x *+* y *+* z *=* *4 and *x *−* y *+2*z *=* *6

Show that 1 1 1 a b c a2 b2 c2 =a-bb-cc-a.

**CHAPTER **3: COMPOUND ANGLE (3M)

Show that cos 37°+sin 37° cos 37°-sin 37° =cot8°.

Show that cos 11°+sin 11° cos 11°-sin 11° =tan 56° .

If *A* + *B* = 45°, prove that (1 + tan *A*)(1 + tan *B* ) = 2

Prove that tan 75°+ cot 75° = 4.

If tan A=56 ,tan B=111 then show that A+B=4

If A+B+C=90° Then show that Tan A Tan B+Tan B Tan C+TanC TanA=1.

If *A *is acute and cos *A *= 35 then find sin 2*A*, cos 3*A *and tan 2*A*.

Prove that sin^{2 }45°-sin^{2 }15°=34.

if A+B+C=180° then show show that Tan A+Tan B+Tan C=Tan A TanB Tan C.

**CHAPTER 4: MULTIPLE & SUB MULTIPLE ANGLES(3M)**

Prove that sin 2θ 1-cos 2θ =cot θ.

Prove that sin 2θ 12θ =tan θ.

Prove that Tan 2θ1+sec 2θ =tan θ.

Prove that sin 10^{°} · sin 50^{°}· sin 70^{°} =38.

Prove that 1-cos sin =tan2.

Prove that 12θ sin 2θ =cot θ.

Show that sin A .Sin 60°-A.Sin 60°+A=14 sin 3A.

Show that tan (4 +θ)+ tan (4 -θ)= 2 tan 2.

**CHAPTER 5: TRANSFORAMTIONS (5M)**

In* ABC*, prove that sin 2*A* − sin 2*B* + sin 2*C* = 4 cos *A* sin *B* cos*C.*

Show that cos10° cos 30° cos 50° cos 70° = ^{316}.

Show that sin *A *+ sin (120°+ *A*) + sin (120°−*A*) = 0.

Show thatsin 5A-sin 3A cos 3A -cos 5A =cot 4A.

If sin (α+β) sin (α-β) =a+ba-b then show that *b tanα *= *a tanβ.*

If sin *x *+ sin *y *= 3 / 4 and sin *x *− sin *y *= 2 / 5, then prove that 8 cot ((*x *− *y*) / 2) = 15 cot ( *x *+ *y*) / 2).

If* A *+* B *+* C *=* *90°, then prove that sin 2*A* + sin 2*B* + sin 2*C* = 4 cos *A* cos *B* cos*C.*

*If A *+* B *+* C *=* *180°, then prove that cos 2A+cos 2B+cos 2C=-1-4 cos A cos B cos C.

If cos x+cos y=35 and cos x-cos y=27 then show that 21tan x-y2+10cot x+y2=0.

** CHAPTER 6: INVERSE TRIGONOMETRY FUNCTION (5M)**

Solve 1 + x + 1 – x = (12) .

If x+y +z=2 then show that x2+y2+z2+2xyz=1.

If +y +z=π then show that x2+y2+z2+2xyz=1.

If x +y +z =2 then show that xy+yz+zx=1.

If x +y +z =π then show that x+y+z=xyz.

Prove that 3365-513=(35) .

*Show that *(15) +(17) =(617) .

**CHAPTER 7: TRIGONOMETRIC EQUESTIONS (5M)**

Solve 2 cos^{2} θ + 3 sin θ = 0.

Solve cos θ+3 sin θ=1.

Solve the equation 4*sin*^{2}*θ *+ 2 *sin θ − *1 = 0.

Solve sin 7θ+ sin 4θ+ sin θ+ 0.

Solve 4 sin^{2 }0 − 8 cos 0 +1 = 0.

Solve sin θ+cos θ=2 .

Solve 4 sin^{2 }0 − 8 cos 0 +1 = 0.

Solve 3 cos *x *+ sin *x *=2.

Solve 2sin2θ+3cos θ-3=0.

Solve sin θ-cos θ=2 .

Solve 2sin2θ+sin θ-1=0.

**CHAPTER 8: PROPERTIES AND SOLUTIONS OF A TRIANGLE**

Solve ∆ABC if b=1,c=3 ,A=30°.

In any ΔABC prove that sin A +sin B +sin C =sR.

Solve the ΔABC If a=2,c=3+1, B=60°

Solve the ∆ABC if a=1,b=3 ,c=2.

If cos A =b cos B, prove that ∆ABC is either isosceles or Right angled.

In Δ*ABC*, if *A* = 60°, show that bc+a+ca+b=1.

**CHAPTER 9:****Straight line**

Find the equation of the straight line passing through the points ( *−*4*,*6) and (6*, *8)

Find the perpendicular distance of the point (7 *,**−*2)from the line9*x*+17*y **− *13 =0

Find the p^{*}erpendicular distance of the point (−3, 5) from the line 3*x *− 4*y *− 26 = 0.

Find the equation of straight line passing through (3, parallel to 5*x *+ 7*y *− 3 = 0.

Find the equation of the straight line passing through the point (0,1) and perpendicular to 2*x *− 3*y *+ 5 = 0.

Find the distance between two parallel lines 3*x *+ 4*y *+ 5 = 0 and 3*x *− 4*y *− 2 = 0.

Find the distance between the parallel lines 2*x *+ 3*y *+ 5 = 0 and 2*x *+ 3*y *+ 9 =0

Find the equation of the line passing through the points (1, 2) and (−3, 5).

Find the va^{*}lue of *x *if the slope of the line joining two points (2, 5) and (*x*, 3) is 2.

Find the perpendicular distance from the point (2, − 1) to the line 3*x *+ 4*y *+ 5 = 0.

Find the equation of the straight line passing through the point (3 *, **−*4) and perpendicular to the line 5*x *+ 3*y **− *1 = 0

Find the angle between the lines 2*x *− *y *+ 3 = 0 and *x *+ *y *− 2 = 0.

Find the equation of the line passing through the points (1, – 2) and (– 2, 3).

Find the intercepts of the line 3*y *− 5*x *+ 7 = 0 made with coordinate axes.

Find the equation of the line passing through the points (1, − 2), (−2, 3).

Find the point of intersection of the straight lines 3*x* + 4*y* + 6 = 0 and 6*x *+5*y* − 9 = 0.

Find the intercepts made by the straight line 2*x* + 3*y* − 6 = 0 on the coordinate axes.

Find the equation of a straight line parallel to *x* − 2*y* + 1 = 0 and passing through the point (−1, 2).

Find the equation of the straight line passing through the points (–5, 2) and (3, –2).

**CHAPTER 10:****Conic sections (5M)**

Find the equation to the circle whose centre is (–1, 2) and radius is 3.

Find the equation of the point circle with centre (2, − 3).

Find the vertex, focus, directrix, axis and length of latus rectum* *of the parabola 7*x*^{2} + 4*y* = 0.

Find the equation of the ellipse whose axes are coordinate axes and passing through points (1, − 3) and (−2, 2).

Find the equation of the point circle with centre (7, − 9).

Find the equation of the ellipse whose axes are the coordinate axes and whose foci are (± 5, 0) and *e* = .-1/5

Find the equation of the rectangular hyperbola whose focus is (3, 4) and directrix is 4*x* − 3*y* + 1 = 0.

Find the equation of the Circle with center at the point (2*, −*2) and passing through the point (*−*1*, *2)

Find the vertex, focus equation of axis, latus rectum, directrix and length of latus rectum of the Parabola *x*^{2} = *−*8*y*

Find the equation of the parabola whose focus is (−1, 2) and whose directrix is the line 3*x *− 4*y *− 5 = 0.

Find the equation of the circle passing through the points (0, 0), (1, 0) and (0, 1).

Find th^{*}e equation of the circle with (− 5, 1) and (3, − 7) as the end points of a diameter. Also find the radius and centre of the circle.

Find the equation of the parabola with focus (− 2, 3) and the directrix is the line 2*x *− 3*y *+ 4 = 0.

Find the equation of the circle with (1, 2) and (4, 5) as end points of a diameter.

Find the equation of the rectangular hyperbola whose focus is the point (3, − 4) and directrix is the line *x *− *y *+ 5 = 0.

Find the centre and radius of the circle 3*x*^{2 }+ 3*y *^{2 }−12*x *+ 6*y *+ 11 = 0.

Find the centre, vertices, lengths of axes, length of Latera recta, eccentricity, foci and the equations of Latera recta and directrices of the ellipse 4*x*^{2 }+ 9*y *^{2 }= 36.

Find the equation of the Circle whose center is at the point ( *−*3*, *2) and radius is 4 units

Find the vertex, focus equation of axis, latus rectum, directrix and length of latus rectum of the Parabola *x*^{2} = 4*y*

**CHAPTER 11:** LIMITS AND CONTINUITY (3M)

Evaluate sin 3θ sin 4θ .

Evaluate 1+2+3⋯+∞n2 .

Evaluate sin 37x tan 31x .

Find 2×2-x-6×3-2×2+x-2 .

Evaluate sin 11x tan 9x

Find 2×3-3×2+19×2+8x+7 .

Evaluate sin 8xtan 5x .

Evaluate 12+22+…+n2n3 .

**CHAPTER 12:** **DERIVATIVES**

Find the derivative of y=a-bcos xa+bcos x with respect to x. (3M)

Find dydx .if y=sin x log x. (3M)

If y=xxx……………………….∞then finddydx. (5M)

Find dydx, if x^{3}+y^{3}-24xy=0 (5M)

Find dydx if x=a(θ-sinθ) ,y=a(1-cosθ) (5M)

Find dydx. If u=(x3+y3x-y ) (5M)

Find the derivative of e8xsec x. (3M)

If y=tan x+tan x+tan x+…+∞ then find dydx. (5M)

Find dydx, if x^{3}+y^{3}-3axy=0 (5M)

Find dydx if x=at2,y=2at . (3M)

Find dydx. If y=tan^{-1}(1-cosx1+cosx). (5M)

If z=logx2+y2 then show that 2 z∂x∂y=2 z∂y∂x. (5M)

verify EULER’S theorem for fx,y=ax2+2hxy+by2 (5M)

Find dydx .if y= log x+x2+1 (5M)

If y=sinxsinxsinx……………………….∞then finddydx. (5M)

Find dydx .if y= t2+2t (3M)

Find dydx .if y=x^{2}^{e}^{x}^{ }(3M)

If y=sin xtan x then finddydx. (5M)

Find dydx. If u=x2+y2x+y show that x∂u∂x+y∂u∂y=tan u (5M)

if u=x^{2}y+y^{2}z+z^{2}x show that∂u∂x+∂u∂y+∂u∂z=x+y+z2. (5M)

find 2u∂x∂y,2u∂y∂x if u= x^{3}+y^{3}-3xy (5M)

Find the derivative of y=3 cos x+2 log x+21 x2+5 with respect to x. (3M)

Find dydx .if y=sin x log x. (3M)

Find dydx. If u=x2+y2x+y show that x∂u∂x+y∂u∂y=tan u (5M)

verify eulers theorem for u=x4+y4x+y . (5M)

if y=ax+bx find dydx ,d2 ydx2 . (5M)

Find dydx. If u=(x3+y3x-y ) show that x∂u∂x+y∂u∂y+z∂u∂y=sin 2u (5M)^{ 3}127.

* *If *y *= sin (log *x*), then prove that *x*^{2}*y*_{2 }+ *xy*_{1 }+ *y *= 0.

**CHAPTER 13:** **GEOMETRICAL APPLICATIONS & PHYSICAL APPLICATIONS (5M)**

find the length of the tangent, normal, sub-tangent, sub-normal to the curve 2x^{3}+3xy-2y^{2}-8=0 at (2,3).

Find the length of tangent, normal, sub-tangent and sub-normal to the curve *x*^{2} + *y*^{2} − 6*x* − 2*y* + 5 =0, at (2, −1).

Find the angle between the curves *y*^{2} = 8*x *and *x*^{2} = 8*y *at the point (8*, *8)

Find the lengths of tangent, normal, sub-tangent and subnormal of the curve *y *= *x *^{3 }− 2*x*^{2 }+ 4 at the point (3, 13).

Find the equation of the tangent and normal to the curve *y *= *x*^{2 }+ 2*x *−1 at (1, 2).

Find the lengths of tangent, normal subtangent and subnormal for the curve *y *= *x *^{3 }− 3*x *+ 2 at the point (0, 2).

the volume of a sphere is increasing at the rate of 400cm^{3}/sec. find the rate of increase if its radius and its surface area is at the instant when the radius of the sphere is 40 cm.

A particle is moving along a straight line according to the law S=2t^{3}-3 t^{2}+15 t+18. Find its velocity when its acceleration is zero.

A man of 2 m tall is approaching a lamp post at the rate of 0·5 m/sec. If the lamp is situated at a height of 8 m, then find the rate at which the length of the shadow of the man is decreasing

A light is hung 8 m, directly above a straight horizontal floor. A man of 2 m tall is walking away from the lamp at the rate of 5·4 m/min. Find the rate at which his shadow is lengthening.

The edge of a cube is decreasing at the rate of 0 *.*03 *cm/sec*. Find the rate at which the volume is decreasing when the edge is 12 *cm*. Also find the rate of decrease in surface area

A circular patch of oil spreads out on water and the area is growing at the rate of 3 sq.cm/sec. How fast does the radius increase, when the radius is 4 cm?

A sphere of radius 10 cm shrinks to 9·8 cm. Find the approximate decrease in volume of the sphere.

The volume of a sphere is increasing at the rate of 0 *.*3 *cc/sec*. Find the rate of increase of its surface area and radius at the instant when the radius of the sphere is 20 *cm.*

**CHAPTER 14:** **ERRORS AND APPROXIMATIONS &MAXIMA AND MINIMA**

Find the dimensions of a rectangle of maximum area having a* *perimeter of 36 ft.

An error of 0·05 cm is committed in measuring a length of 10 cm. If so, find the absolute error, relative error and percentage error.

Find the dimensions of a rectangle of maximum area having a perimeter of 26 ft.

The radius of spherical balloon is increased by 2%. Find the approximate the percentage increase in its surface area.

Show that the area of a rectangle of given fixed perimeter is maximum when the rectangle is a square.

If the radius of a spherical balloon is increased by 0·2%, find the approximate percentage in its volume.

The sum of two numbers is 24. Find the numbers when the sum of their squares is minimum.

Find the maximum and minimum values of 4*x *^{3 }−18*x*^{2 }+ 24*x *− 7.

The radius of a spherical balloon is increased by 1%. Find the approximate percentage increase in its volume.

Find the maximum and minimum values of *f *(*x*) = *x*^{3} *− *4*x*^{2} + 5*x.*

Each side of a cube is increased by 0 *.*2%. Find the approximate percentage increase in its volume. Also find the approximate percentage increase in its surface area

Find the approximate value o f3127.

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